QT5201U wiki - User contributions [en]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T07:40:01Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

Since we already have the expression of transmission (T), we can also derive some important parameters:

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_{max}</math>, <math>I_{min}</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_{min}</math>, the maximum transmission is <math>T_{max}</math>, we can rewrite the visibility

<math>V=I\frac{T_{max}^2-T_{min}^2}{T_{max}^2+T_{min}^2}</math>

<math>T_{min}=\frac{1}{1+0}</math>, <math>T_{max}=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V_{max}=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V_{min}=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity, in general, is given by

<math>|\Delta \lambda_{FSR}|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

<math>|\frac{\partial \beta}{\partial \lambda}|=\frac{2\pi}{\lambda^2}[n(\lambda)-\lambda\frac{\partial n}{\partial \lambda}]=\frac{2\pi}{\lambda^2}n_g</math>

The FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda^2}{n_gL}</math>, <math>n_g</math> is the group index of the media within the cavity.

In etalon, the FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda_0^2}{2nl\cos\theta}</math>

Where <math>\lambda_0</math> is the central wavelength of the nearest transmission peak, <math>n</math> is the index of refraction of the cavity, <math>l</math> is the thickness of the cavity, <math>\theta</math> is the angle of incidence.

3. Full width at half maximum

The full width at half maximum (FWHM) is a parameter commonly used to describe the width of a "bump" on a curve or function. It is given by the distance between points on the curve at which the function reaches half its maximum value.

4.Central wavelength

Central Wavelength, used in defining bandpass filters, describes the midpoint of spectral bandwidth over which the filter transmits.

[[File:FWHM_CENTRAL WAVELENGTH.jpg]]

5.The relationship between FSR and FWHM

The FSR is related to the full-width half-maximum <math>\delta\lambda</math> of any one transmission band by a quantity known as the finesse

<math>\mathcal{F}=\frac{\Delta \lambda}{\delta \lambda}=\frac{\pi}{2\arcsin\frac{1}{\sqrt{F}}}</math>

In a word, If we want to observe more clear interference pattern, we should make <math>F</math> large, or make <math>\Delta\lambda(FSR)</math> large and <math>\delta\lambda (FWHM)</math> small.

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T07:36:54Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

Since we already have the expression of transmission (T), we can also derive some important parameters:

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_{max}</math>, <math>I_{min}</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_{min}</math>, the maximum transmission is <math>T_{max}</math>, we can rewrite the visibility

<math>V=I\frac{T_{max}^2-T_{min}^2}{T_{max}^2+T_{min}^2}</math>

<math>T_{min}=\frac{1}{1+0}</math>, <math>T_{max}=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V_{max}=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V_{min}=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity, in general, is given by

<math>|\Delta \lambda_{FSR}|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

<math>|\frac{\partial \beta}{\partial \lambda}|=\frac{2\pi}{\lambda^2}[n(\lambda)-\lambda\frac{\partial n}{\partial \lambda}]=\frac{2\pi}{\lambda^2}n_g</math>

The FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda^2}{n_gL}</math>, <math>n_g</math> is the group index of the media within the cavity.

In etalon, the FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda_0^2}{2nl\cos\theta}</math>

Where <math>\lambda_0</math> is the central wavelength of the nearest transmission peak, <math>n</math> is the index of refraction of the cavity, <math>l</math> is the thickness of the cavity, <math>\theta</math> is the angle of incidence.

3. Full width at half maximum

The full width at half maximum (FWHM) is a parameter commonly used to describe the width of a "bump" on a curve or function. It is given by the distance between points on the curve at which the function reaches half its maximum value.

4.Central wavelength

Central Wavelength, used in defining bandpass filters, describes the midpoint of spectral bandwidth over which the filter transmits.

[[File:FWHM_CENTRAL WAVELENGTH.jpg]]

5.The relationship between FSR and FWHM

The FSR is related to the full-width half-maximum <math>\delta\lambda</math> of any one transmission band by a quantity known as the finesse

<math>\mathcal{F}=\frac{\Delta \lambda}{\delta \lambda}=\frac{pi}{2\arctan\frac{1}{\sqrt{F}}}</math>

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T07:32:47Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

Since we already have the expression of transmission (T), we can also derive some important parameters:

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_{max}</math>, <math>I_{min}</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_{min}</math>, the maximum transmission is <math>T_{max}</math>, we can rewrite the visibility

<math>V=I\frac{T_{max}^2-T_{min}^2}{T_{max}^2+T_{min}^2}</math>

<math>T_{min}=\frac{1}{1+0}</math>, <math>T_{max}=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V_{max}=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V_{min}=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity, in general, is given by

<math>|\Delta \lambda_{FSR}|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

<math>|\frac{\partial \beta}{\partial \lambda}|=\frac{2\pi}{\lambda^2}[n(\lambda)-\lambda\frac{\partial n}{\partial \lambda}]=\frac{2\pi}{\lambda^2}n_g</math>

The FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda^2}{n_gL}</math>, <math>n_g</math> is the group index of the media within the cavity.

In etalon, the FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda_0^2}{2nl\cos\theta}</math>

Where <math>\lambda_0</math> is the central wavelength of the nearest transmission peak, <math>n</math> is the index of refraction of the cavity, <math>l</math> is the thickness of the cavity, <math>\theta</math> is the angle of incidence.

3. Full width at half maximum

The full width at half maximum (FWHM) is a parameter commonly used to describe the width of a "bump" on a curve or function. It is given by the distance between points on the curve at which the function reaches half its maximum value.

4.Central wavelength

Central Wavelength, used in defining bandpass filters, describes the midpoint of spectral bandwidth over which the filter transmits.

[[File:FWHM_CENTRAL WAVELENGTH.jpg]]

5.The relationship between FSR and FWHM

The FSR is related to the full-width half-maximum <math>\delta\lambda</math> of any one transmission band by a quantity known as the finesse

<math>\mathcal{F}=\frac{\Delta \lambda}{\delta \lambda}</math>

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T07:24:52Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

Since we already have the expression of transmission (T), we can also derive some important parameters:

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_{max}</math>, <math>I_{min}</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_{min}</math>, the maximum transmission is <math>T_{max}</math>, we can rewrite the visibility

<math>V=I\frac{T_{max}^2-T_{min}^2}{T_{max}^2+T_{min}^2}</math>

<math>T_{min}=\frac{1}{1+0}</math>, <math>T_{max}=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V_{max}=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V_{min}=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity, in general, is given by

<math>|\Delta \lambda_{FSR}|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

<math>|\frac{\partial \beta}{\partial \lambda}|=\frac{2\pi}{\lambda^2}[n(\lambda)-\lambda\frac{\partial n}{\partial \lambda}]=\frac{2\pi}{\lambda^2}n_g</math>

The FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda^2}{n_gL}</math>, <math>n_g</math> is the group index of the media within the cavity.

In etalon, the FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda_0^2}{2nl\cos\theta}</math>

Where <math>\lambda_0</math> is the central wavelength of the nearest transmission peak, <math>n</math> is the index of refraction of the cavity, <math>l</math> is the thickness of the cavity, <math>\theta</math> is the angle of incidence.

3. Full width at half maximum

The full width at half maximum (FWHM) is a parameter commonly used to describe the width of a "bump" on a curve or function. It is given by the distance between points on the curve at which the function reaches half its maximum value.

4.Central wavelength

Central Wavelength, used in defining bandpass filters, describes the midpoint of spectral bandwidth over which the filter transmits.

[[File:FWHM_CENTRAL WAVELENGTH.jpg]]

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

File:FWHM CENTRAL WAVELENGTH.jpg

2021-03-10T07:23:46Z

Zhang qian:

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T07:17:36Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

Since we already have the expression of transmission (T), we can also derive some important parameters:

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_{max}</math>, <math>I_{min}</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_{min}</math>, the maximum transmission is <math>T_{max}</math>, we can rewrite the visibility

<math>V=I\frac{T_{max}^2-T_{min}^2}{T_{max}^2+T_{min}^2}</math>

<math>T_{min}=\frac{1}{1+0}</math>, <math>T_{max}=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V_{max}=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V_{min}=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity, in general, is given by

<math>|\Delta \lambda_{FSR}|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

<math>|\frac{\partial \beta}{\partial \lambda}|=\frac{2\pi}{\lambda^2}[n(\lambda)-\lambda\frac{\partial n}{\partial \lambda}]=\frac{2\pi}{\lambda^2}n_g</math>

The FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda^2}{n_gL}</math>, <math>n_g</math> is the group index of the media within the cavity.

In etalon, the FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda_0^2}{2nl\cos\theta}</math>

Where <math>\lambda_0</math> is the central wavelength of the nearest transmission peak, <math>n</math> is the index of refraction of the cavity, <math>l</math> is the thickness of the cavity, <math>\theta</math> is the angle of incidence.

3. Full width at half maximum

The full width at half maximum (FWHM) is a parameter commonly used to describe the width of a "bump" on a curve or function. It is given by the distance between points on the curve at which the function reaches half its maximum value.

4.Central wavelength

Central Wavelength, used in defining bandpass filters, describes the midpoint of spectral bandwidth over which the filter transmits.

[[Pictures:FWHM_CENTRAL WAVELENGTH.jpg]]

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T07:14:28Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

Since we already have the expression of transmission (T), we can also derive some important parameters:

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_{max}</math>, <math>I_{min}</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_{min}</math>, the maximum transmission is <math>T_{max}</math>, we can rewrite the visibility

<math>V=I\frac{T_{max}^2-T_{min}^2}{T_{max}^2+T_{min}^2}</math>

<math>T_{min}=\frac{1}{1+0}</math>, <math>T_{max}=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V_{max}=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V_{min}=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity, in general, is given by

<math>|\Delta \lambda_{FSR}|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

<math>|\frac{\partial \beta}{\partial \lambda}|=\frac{2\pi}{\lambda^2}[n(\lambda)-\lambda\frac{\partial n}{\partial \lambda}]=\frac{2\pi}{\lambda^2}n_g</math>

The FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda^2}{n_gL}</math>, <math>n_g</math> is the group index of the media within the cavity.

In etalon, the FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda_0^2}{2nl\cos\theta}</math>

Where <math>\lambda_0</math> is the central wavelength of the nearest transmission peak, <math>n</math> is the index of refraction of the cavity, <math>l</math> is the thickness of the cavity, <math>\theta</math> is the angle of incidence.

3. Full width at half maximum

The full width at half maximum (FWHM) is a parameter commonly used to describe the width of a "bump" on a curve or function. It is given by the distance between points on the curve at which the function reaches half its maximum value.

4.Central wavelength

Central Wavelength, used in defining bandpass filters, describes the midpoint of spectral bandwidth over which the filter transmits.

[[File:FWHM_CENTRAL WAVELENGTH.jpg]]

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T07:09:08Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

Since we already have the expression of transmission (T), we can also derive some important parameters:

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_{max}</math>, <math>I_{min}</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_{min}</math>, the maximum transmission is <math>T_{max}</math>, we can rewrite the visibility

<math>V=I\frac{T_{max}^2-T_{min}^2}{T_{max}^2+T_{min}^2}</math>

<math>T_{min}=\frac{1}{1+0}</math>, <math>T_{max}=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V_{max}=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V_{min}=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity, in general, is given by

<math>|\Delta \lambda_{FSR}|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

<math>|\frac{\partial \beta}{\partial \lambda}|=\frac{2\pi}{\lambda^2}[n(\lambda)-\lambda\frac{\partial n}{\partial \lambda}]=\frac{2\pi}{\lambda^2}n_g</math>

The FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda^2}{n_gL}</math>, <math>n_g</math> is the group index of the media within the cavity.

In etalon, the FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda_0^2}{2nl\cos\theta}</math>

Where <math>\lambda_0</math> is the central wavelength of the nearest transmission peak, <math>n</math> is the index of refraction of the cavity, <math>l</math> is the thickness of the cavity, <math>\theta</math> is the angle of incidence.

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T07:04:20Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

Since we already have the expression of transmission (T), we can also derive some important parameters:

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_{max}</math>, <math>I_{min}</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_{min}</math>, the maximum transmission is <math>T_{max}</math>, we can rewrite the visibility

<math>V=I\frac{T_{max}^2-T_{min}^2}{T_{max}^2+T_{min}^2}</math>

<math>T_{min}=\frac{1}{1+0}</math>, <math>T_{max}=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V_{max}=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V_{min}=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity, in general, is given by

<math>|\Delta \lambda_{FSR}|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

<math>|\frac{\partial \beta}{\partial \lambda}|=\frac{2\pi}{\lambda^2}[n(\lambda)-\lambda\frac{\partial n}{\partial \lambda}]=\frac{2\pi}{\lambda^2}n_g</math>

The FSR is <math>\Delta \lambda_{FSR}=\frac{\lambda^2}{n_gL}</math>, <math>n_g</math> is the group index of the media within the cavity.

In etalon, the FSR is <math>\Delta \lambda=\frac{\lambda_0^2}{2nl\cos\theta}</math>

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T06:59:06Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

Since we already have the expression of transmission (T), we can also derive some important parameters:

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_{max}</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity, in general, is given by

<math>|\Delta \lambda|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

<math>|\frac{\partial \beta}{\partial \lambda}|=\frac{2\pi}{\lambda^2}[n(\lambda)-\lambda\frac{\partial n}{\partial \lambda}]=\frac{2\pi}{\lambda^2}n_g</math>

The FSR is <math>\Delta \lambda=\frac{\lambda^2}{n_gL}</math>, <math>n_g</math> is the group index of the media within the cavity.

In etalon, the FSR is <math>\Delta \lambda=\frac{\lambda_0^2}{2nl\cos\theta}</math>

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T04:08:02Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|\Delta \lambda|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

<math>|\frac{\partial \beta}{\partial \lambda}|=\frac{2\pi}{\lambda^2}[n(\lambda)-\lambda\frac{\partial n}{\partial \lambda}]=\frac{2\pi}{\lambda^2}n_g</math>

The FSR is <math>\Delta \lambda=\frac{\lambda^2}{n_gL}</math>, <math>n_g</math> is the group index of the media within the cavity.

In etalon, the FSR is <math>\delta \lambda=\frac{\lambda_0^2}{2nl\cos\theta}</math>

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T03:27:03Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|\Delta \lambda|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

<math>|\frac{\partial \beta}{\partial \lambda}|=\frac{2\pi}{\lambda^2}[n(\lambda)-\lambda\frac{\partial n}{\partial \lambda}]=\frac{2\pi}{\lambda^2}n_g</math>

The FSR is <math>\delta \lambda=\frac{\lambda^2}{n_gL}</math>, <math>n_g</math> is the group index of the media within the cavity.

In etalon, the FSR is \delta \lambda=\frac{\lambda_0^2}{2nl\cos\theta}</math>

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T03:21:19Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|\Delta \lambda|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

<math>|\frac{\partial \beta}{\partial \lambda}|=\frac{2\pi}{\lambda^2}[n(\lambda)-\lambda\frac{\partial n}{\partial \lambda}]=\frac{2\pi}{\lambda^2}n_g</math>

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T03:20:40Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|\Delta \lambda|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

<math>|\frac{\partial \beta}{\partial \lambda}|=\frac{2\pi}{\lambda^2}[n(\lambda)-\lambda\frac{\partial n}{\partial \lambda}]=\frac{2\pi}{\lambda^2}n_g

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T03:16:09Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|\Delta \lambda|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,
<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa_0</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T03:13:57Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|\Delta \lambda|=\frac{2\pi}{L}|\frac{1}{\frac{\partial \beta}{\partial \lambda}}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T03:10:56Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|\Delta \lambda|=\frac{2\pi}{L}|(\frac{\partial \beta}{\partial \lambda})^-1|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T03:10:04Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|\Delta \lambda|=\frac{2\pi}{L}|\frac{\partial \beta}{\partial \lambda}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T03:08:48Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|\Delta \lambda|=\frac{2\pi}{L}|frac{\partial \beta}{\partial \lambda}|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T03:07:12Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|\Delta \lambda|=\frac{2\pi}{L}</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T03:05:40Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|\Delta \lambda|=</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T02:31:51Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|\Delta \lambda|=\frac{2\pi}{L}|(\frac{\partial \beta}{\partial \lambda}）^-1|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T02:31:00Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|&delta \lambda|=\frac{2\pi}{L}|(\frac{\partial \beta}{\partial \lambda}）^-1|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T02:28:24Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|&delta \lambda|=\frac{2\pi}{L}|(\frac{\partial \beta}{\partial \lambda}^-1)|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T02:24:54Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|&delta \lambda|=\frac{2\pi}{\lambda^2}|(\frac{\partial \beta}^-1)|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T02:22:44Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>|&triangle \lambda|=\frac{2\phi}{\lambda^2}|(\frac{\partial \beta}^-1)|</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T02:17:57Z

Zhang qian: /* Characteristic Parameters of an Etalon */

====Members====
Shi Yicheng (A0054800R), Du Jinyi (A0227185B), Zhang Qian(A0228752Y)

=Rationale=
A Fabry-Perot interferometer (or an Etalon), being probably the simplest form of all interferometers, is found useful in a variety of optical applications such as spectral filtering or frequency referencing.

An etalon is typically constructed out of the two parallel reflecting surfaces of a transparent plate. The plate needs to have low absorption loss for the desired working wavelengths to ensure a relatively high finesse of the etalon. The material choice for visible wavelengths is usually fused silica with an absorption coefficient of [bla] and a thermal expansion coefficient of [bla].

For optical telecommunication wavelengths, which range from about 1260nm to 1625nm, pure silicon becomes a more practical choice with an absorption coefficient of [bla] and thermal expansion coefficient of [bla].

==Characteristic Parameters of an Etalon==
The performance of etalon is characterized by several main parameters: including visibility (V), free spectral range （FSR), full width half maximum (FWHM), and central wavelength.

Suppose that the <math>a_1</math>, <math>a_2</math>, <math>a_3</math> are the electrical field intensity of input, oscillating, output light.
[[File:Etalon_principle.jpg]]

The relationship is as follows

<math>a_2=ta_1+r^2a_2e^{-i\phi}</math>

<math>a_3=ta_2</math>

Where \phi means that phase delay in the cavity and the gap between two reflective surfaces is d.

<math>\phi=2kd=2\frac{2\pi}{\lambda}d</math>

From the upper two equations, we have

<math>a_2=a_1\frac{t}{1-r^2e^{-i\phi}}</math>

<math>a_3=a_1\frac{t^2}{1-r^2e^{i\phi}}</math>

Then, we can calculate the transmission ratio:

<math>T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2e^{i\phi}}|^2=\frac{t^4}{(1-r^2e^{i\phi})^2+(r^2sin\phi)^2}=\frac{t^4}{1-r^2-2rcos\phi}</math>

Here, we have <math>cos\phi=1-2sin^2\frac{\phi}{2}</math>

So, <math>T_R</math> can be simplified as follows

<math>T_R=\frac{t^4}{(1-r^2)^2+4r^2sin^2\frac{\phi}{2}}=\frac{1}{1+\frac{4r^2}{t^4}sin^2\frac{\phi}{2}}</math>

Here, we define a new parameter: <math>F=\frac{4r^2}{t^4}</math>, which is called coefficient finesse.

So, <math>T_R=\frac{1}{1+Fsin^2\frac{\phi}{2}}</math>

1.Visibility

The interferometric visibility quantifies the contrast of interference in an optical system. The ratio of the amplitude of oscillations to the sum of the powers of the individual waves is defined as the visibility.

Assume <math>I_2</math>, <math>I_1</math> are the maximum intensity of the oscillations and the minimum intensity of the oscillations, <math>V</math> is the visibility of the interference pattern.

<math>V=\frac{I_2-I_1}{I_2+I_1}</math>

Suppose the intensity of incident light of etalon is <math>I</math>,the minimum transmission is <math>T_1</math>, the maximum transmission is <math>T_2</math>, we can rewrite the visibility

<math>V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}</math>

<math>T_1=\frac{1}{1+0}</math>, <math>T_2=\frac{1}{1+F}</math>

<math>V=\frac{(1+F)^2-1}{(1+F)^2+1}</math>

This indicate the visibility of interference pattern is associated with coefficient finesse. When <math>V=1</math>, <math>F</math> is approximately equal to infinite, get the best interference pattern; when <math>V=0</math>, <math>F=0</math>, can’t observe the interference pattern.

2.Free spectral range(FSR)

The free spectral range(FSR) of a cavity in general is given by

<math>\triangle \lambda=\frac{2\phi}{\lambda^2}(\frac{\partial \beta}^-1)</math>

Where <math>\beta</math> is the wavevector of the light inside the cavity,<math>\beta=\kappa_0n(\lambda)=\frac{2\pi}{\lambda}n(\lambda)</math>. <math>\kappa</math> and <math>\lambda</math> are the wavevector and wavelength in vacuum, <math>n</math> is the refractive index of the cavity, <math>L</math> is the length of the cavity(for a standing-wave cavity, <math>L</math> is equal to twice the physical length of the cavity)

==Building an Etalon Out of Silicon Wafer==

=Design=
[[A few photos for now.]]

=Performance=
[[File:Drawing.png|options|caption]]

==Bare Silicon Wafer==
[[File:100um-whole.png|options|Transmission spectrum of a bare silicon wafer of 100μm]]
[[File:100um-zoom.png|options|Zoom in of the spectrum, showing a free spectral range of ~2.3nm (~400GHz)]]

==HR Coated Silicon Wafer==

==Temperature Tuning of silicon etalon==
A small piece of the silicon wafer is mounted on a copper block with a 4mm diameter hole through hole. The copper serves as a thermally conductive mount for the silicon etalon and is placed on top of a Peltier stage. The temperature of this entire stack is adjusted and stabilized with a TEC controller. A few different temperatures were tried. Just to quickly note that the thermal expansion coefficient of silicon is <math>2.6\times 10^{-6} {^{\circ}C}^{-1}</math> ([http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html])

[[File:temperature_shift.png|options|]]
[[File:wavelength_v_temp.png|options|]]

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T02:08:12Z

Zhang qian: /* Characteristic Parameters of an Etalon */

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T02:07:04Z

Zhang qian: /* Characteristic Parameters of an Etalon */

A temperature-tunable etalon for optical telecommunication wavelength

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Zhang qian: /* Characteristic Parameters of an Etalon */

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T01:51:22Z

Zhang qian: /* Characteristic Parameters of an Etalon */

A temperature-tunable etalon for optical telecommunication wavelength

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Zhang qian: /* Rationale */

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T01:36:37Z

Zhang qian: /* Rationale */

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T01:34:52Z

Zhang qian: /* Characteristic Parameters of an Etalon */

A temperature-tunable etalon for optical telecommunication wavelength

2021-03-10T01:33:20Z

Zhang qian: /* Characteristic Parameters of an Etalon */

A temperature-tunable etalon for optical telecommunication wavelength

2021-02-09T01:09:43Z

Zhang qian: /* Members */