A temperature-tunable etalon for optical telecommunication wavelength

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 $a_1$, $a_2$, $a_3$ are the electrical field intensity of input, oscillating, output light.

The relationship is as follows

$a_2=t{a}_1+r^2{a}_2{e}^{-i\varphi }$

$a_3=t{a}_2$

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

$\varphi =2kd=2\frac{2\pi }{\lambda }d$

From the upper two equations, we have

$a_2=a_1\frac{t}{1-r^2{e}^{-i\varphi }}$

$a_3=a_1\frac{t^2}{1-r^2{e}^{i\varphi }}$

Then, we can calculate the transmission ratio:

$T_R=|\frac{a_3}{a_1}|=|\frac{t^2}{1-r^2{e}^{i\varphi }}{|}^2=\frac{t^4}{(1-r^2{e}^{i\varphi }{)}^2+(r^{2}sin\varphi {)}^2}=\frac{t^4}{1-r^2-2rcos\varphi }$

Here, we have $cos\varphi =1-2si{n}^2\frac{\varphi }{2}$

So, $T_R$ can be simplified as follows

$T_R=\frac{t^4}{(1-r^2{)}^2+4r^{2}si{n}^2\frac{\varphi }{2}}=\frac{1}{1+\frac{4r^2}{t^4}si{n}^2\frac{\varphi }{2}}$

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

So, $T_R=\frac{1}{1+Fsi{n}^2\frac{\varphi }{2}}$

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 $I_2$, $I_1$ are the maximum intensity of the oscillations and the minimum intensity of the oscillations, $V$ is the visibility of the interference pattern.

$V=\frac{I_2-I_1}{I_2+I_1}$

Suppose the intensity of incident light of etalon is $I$,the minimum transmission is $T_1$, the maximum transmission is $T_2$, we can rewrite the visibility

$V=I\frac{T_2^2-T_1^2}{T_2^2+T_1^2}$

$T_1=\frac{1}{1+0}$, $T_2=\frac{1}{1+F}$

$V=\frac{(1+F{)}^2-1}{(1+F{)}^2+1}$

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

2.Free spectral range(FSR)

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

$|\Delta \lambda |=\frac{2\pi }{L}|frac\partial \beta \partial \lambda |$

Where $\beta$ is the wavevector of the light inside the cavity,$\beta ={\kappa }_{0}n(\lambda )=\frac{2\pi }{\lambda }n(\lambda )$. $\kappa$ and $\lambda$ are the wavevector and wavelength in vacuum, $n$ is the refractive index of the cavity, $L$ is the length of the cavity(for a standing-wave cavity, $L$ is equal to twice the physical length of the cavity)

Design

A few photos for now.

Performance

Bare 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 $2.6\times 10^{-6}{{}^{\circ }C}^{-1}$ ([1])