APhO 2017, Експеримент — Задача 1. Photonic Crystal

Part A. Sample X. Spectral measurements (3,5 т.)

Sample X has a simple structure (fig. 10). Its period consists of two layers with the same thickness $D_X/2$ and with close refractive indices $n_1$ and $n_2$:

1. Assemble the experimental setup for the spectral measurements. Place the diffraction arm into the left hole of the main bench;

2. Turn on the lamp. Look at the diffraction grating from the other end of the optical bench. Rotate the diffraction arm until you see the first diffraction order (a rainbow strip).

3. Install the sample holder into the right hole of the main bench (set the triangular mark on zero). Place Sample X in the holder (the light should fall onto the red frontface). Look at the rainbow strip of the diffraction grating through the sample.

4. If you rotate the sample clockwise at the right angle, you can see a dark region in the transmitted spectrum (fig. 12). This is the transmittance minimum corresponding to $m = 1$ in the Bragg-Snell law. When you rotate the sample, the wavelength of the transmittance minimum changes.

5. By moving the lens, you can see wider or narrower part of the spectrum. For precise measurements place the lens in such a manner, that the lamp will be located in the focal point.

A.1 Light falls on the diffraction grating perpendicular to its surface. Write down the equation, connecting the diffraction angle $\varphi$ and the wavelength $\lambda$. Grating period is $h = 1000$ nm. You can write down the answer without deriving it. [0,1 т.]

A.2 Measure the angle of incidence $\theta$, at which you see the transmittance minimum, as a function of the diffraction angle $\varphi$. Take reading at such $\theta$ that transmittance minimum (dark region) can be seen in the middle of the grating window. [1 т.]

A.3 Find new coordinates instead of $\theta$ on $\varphi$ to test that graph would be linear. Plot the graph in new coordinates. [1,5 т.]

A.4 Determine the sample period $D_X$ and the mean refractive index $n_X$. [0,9 т.]

Part B. Sample X. Laser measurements (5 т.)

The porosity of the crystal $p$ is a fraction of the crystal volume filled with air (channels). Crystal effective refractive index can be expressed in terms of air refractive index $n_a = 1$ and aluminum oxide refractive index $n_{AAO}$:

If channels are filled with water, the refractive index of crystal changes:

where $n_w = 1.33$ is the refractive index of water.

In this part we will determine the porosity $p$ of the photonic crystal by comparing the experimental data for dry and wet samples. We will use the setup for laser measurements.

B.1 Choose the laser such that the transmittance minimum may be observed at laser wavelength $\lambda$ at some sample rotation angle $\theta$. Write down the wavelength of the laser you have chosen. [0,1 т.]

Use the chosen laser in all tasks of part B.

Assemble the experimental setup for laser measurements (fig. 13).

Do not stare into the laser beam directly! Keep track of reflected beams, they must not get into your eyes. Even short exposure may seriously damage your eyes. Do not hold your head on beam level when you conduct any laser experiments.

Before collecting any data in laser experiment, make sure that laser intensity is constant. If it is not, wait for 3-5 minutes.

B.2 Measure the intensity of laser light $I_t$ (in μA) transmitted through the photonic crystal as a function of $\theta$. [1 т.]

B.3 Plot the graph $I_t(\theta)$. [1 т.]

B.4 Determine the angle of incidence $\theta_1$, corresponding to the transmittance minimum at the laser wavelength. Determine the width of the valley $\Delta\theta_1$ on plot $I_t(\theta)$ at the level of half-depth (fig. 14a). [0,2 т.]

B.5 Using the Bragg-Snell law (2), determine the transmittance minimum normal wavelength $\lambda_X$ for sample X. Use the value $n_X$ from part A, $\theta_1$ from B.4 and laser wavelength $\lambda$. [0,2 т.]

When light falls perpendicular to the crystal ($\theta = 0$), spectral width $\Delta\lambda$ (fig. 14b) of transmittance minimum, corresponding to $m = 1$, can be expressed in terms of $n$ and $\Delta n$ as follows:

To estimate $\Delta n_X$ for the sample X, assume that relational spectral width of transmittance minimum $\frac{\Delta\lambda}{\lambda}$ does not change with rotation.

B.6 Estimate refractive indices difference $\Delta n_X$ of sample X. [0,6 т.]

Place some water on the face side of the sample X. Carefully place a coverslip over the sample to prevent water evaporation during the experiment. There is a solid oxide layer 15 nm thick on the back side of the sample which prevents water from pouring out.

Be careful! Samples are very fragile. Their width is a quarter of a hair diameter. Operate samples only by the frame. The coverslip is very fragile too! Hold it only with tweezers.

B.7 Determine the angle of incidence $\theta_2$ corresponding to the transmittance minimum of the wet sample at the laser wavelength. [0,3 т.]

B.8 Determine the mean porosity $p_X$ of the sample X and aluminum oxide refractive index $n_{AOO}$. [1 т.]

B.9 Determine the mean porosities $p_1$ and $p_2$ of the layers in sample X. [0,6 т.]

Part C. Sample Y. Several transmittance minimums (4,5 т.)

Sample Y structure is more complicated than the one of sample X. Sample Y has 4 transmittance minimums in the visible spectral range (400–800 nm) corresponding to 4 consecutive integers $m$ in the Bragg-Snell law (2). The refractive indices of samples Y and X are the same $n_Y = n_X$.

C.1 Although there are 4 transmittance minimums for sample Y, you can observe only three of them in spectral measurements. Using spectral measurements (see part A), determine the normal wavelengths of these three transmittance minimums $\lambda_1^{sp}$, $\lambda_2^{sp}$, $\lambda_3^{sp}$. [0,6 т.]

Now conduct laser measurements for the sample Y (see part B). Laser measurements let you determine normal wavelengths of transmittance minimums with greater accuracy. Moreover, you will be able to determine transmittance values which you will need in part E.

C.2 For red laser, measure transmitted light intensity $I_{red}$ as a function of the sample rotation angle $\theta$. [0,5 т.]

C.3 For green laser, measure transmitted light intensity $I_{green}$ as a function of the sample rotation angle $\theta$. [0,5 т.]

C.4 For blue laser, measure transmitted light intensity $I_{blue}$ as a function of the sample rotation angle $\theta$. [0,5 т.]

C.5 Using data obtained in tasks C.2–C.4, determine normal wavelengths of 4 transmittance minimums. [0,6 т.]

C.6 Determine integers $m$ corresponding to 4 normal wavelengths obtained in C.5. You may plot a graph if you need. [1 т.]

C.7 Determine the period $D_Y$ of the sample Y in nanometers. [0,2 т.]

C.8 Determine the transmittance values $t$ of 4 transmittance minimums of the sample Y (fig. 15).

These values will be used in part E to determine the structure of the sample Y. You may plot graphs if you need it, though it is not required. [0,6 т.]

Part D. Sample Z. Missed transmittance minimums (4,5 т.)

The Bragg-Snell law (2) determines angles and wavelengths at which transmittance minimum may be observed. Transmittance values $t$ depend on the internal structure of the period. A sample may have a specific structure such that some transmittance minimums predicted by the Bragg-Snell law have $t = 1$ and therefore can not be seen. We will call these transmittance minimums missed transmittance minimums. We will call minimums that are not missed visible.

Regard transmittance minimum as missed if it can not be seen in spectral or laser experiments at the position predicted by the Bragg-Snell law. The sample Y has no missed transmittance minimums. Conversely, the sample Z has 2 missed minimums in the visible spectrum. This means that not all visible transmittance minimums correspond to consecutive $m$.

Mean refractive indices of the samples X and Z are the same: $n_Z = n_X$.

D.1 Determine normal wavelengths $\lambda_Z$ of visible transmittance minimums of the sample Z. Use spectral and laser experiments. Describe your experiments with sketches and equations. [1,2 т.]

D.2 Determine integers $m$ corresponding to visible transmittance minimums. You may plot a graph if you need. [2 т.]

D.3 Determine period $D_Z$ of the sample Z in nanometers. [0,3 т.]

D.4 Determine, which transmittance minimums are missing. Derive their $m'$ in Bragg-Snell law and normal wavelengths $\lambda'_Z$. [1 т.]

Part E. Samples Y and Z. Internal structure of the period. (2,5 т.)

The Bragg-Snell law (2) describes angles and wavelengths, where reflectance maximums and transmittance minimums may be observed (fig. 9). Transmittance values depend on the internal structure of the period.

Simple theory for the reflectance values is discussed in this part. Normal incidence of light ($\theta = 0$) is considered. Such being the case, the Bragg-Snell law simplifies:

where $D$ is the period of the crystal.

In the model we assume that light reflects only from boundaries where it travels from high to low refractive index layer. Reflective boundaries considered in the model are marked in fig. 16 with arrows.

The samples X, Y, and Z consist of the layers with the same thickness and 2 different refractive indices, so we will limit ourselves only to this particular case.

Let $d_l$ be the thickness of bilayer, $D$ crystal period, $\delta_j$ the distance between the beginning of the period and the reflective boundary $j$.

For example, structure on Fig. 16 has crystal period $D = 3.5d_l$, the distances to reflective boundaries are $\delta_1 = 0$, $\delta_2 = d_l$, $\delta_3 = 2.5d_l$.

For the beam reflected from layer $j$ phase advance is

Considering interference between beams reflected from one period, we deduce that reflective intensity for reflectivity maximum $m$ of the Bragg-Snell law is:

where $i$ – imaginary unit.

The model considered above is not quantitatively correct. But it keeps the order of reflectance maximums: you can compare the magnitudes of maximums and say, which would be more intensive. If reflectance maximum is intense, transmittance value $t$ of transmittance minimum is low. If reflectance maximum is low enough, we consider it as missed transmittance minimum.

In the appendix there is a table with the values $I_m/I_0$ calculated according to equation (9) for the integer $m$ from 1 to 20. The intensity $I_0$ is calculated according to (9) for $m = 0$. In that case all the reflected beams interfere without phase shift to output the maximum intensity possible. Values $I_m/I_0$ are expressed in $\%$.

The structures of the samples Y and Z are in the table among others.

E.1 Compare transmittance values for the sample Y obtained in C.8 with the table in the appendix. Determine the structure of the sample Y. Write the name of the structure in the answer sheet. [1,2 т.]

E.2 Determine the structure of the sample Z. Write the name of the structure in your answer sheet. [1,3 т.]

Appendix

Possible variants of structures of samples Y and Z. Names of the structures are given under the pictures. Numbers in the table are $I_m/I_0$.