Calculate the thickness of a quartz quarter- wave plate for sodium light `(lambda=5893Å)` given that the index of refraction of quartz for ordinary and extraordinary rays are 1.5442 and 1.5533 respectively.

Answer

To calculate the thickness of a quartz quarter-wave plate for sodium light, we can follow these steps:

### Step 1: Convert the wavelength from Angstroms to meters
The wavelength of sodium light is given as \( \lambda = 5893 \, \text{Å} \).

To convert Angstroms to meters:
\[ \lambda = 5893 \, \text{Å} = 5893 \times 10^{-10} \, \text{m} \]

### Step 2: Identify the refractive indices
We are given the refractive indices for ordinary and extraordinary rays in quartz:
- Refractive index for ordinary ray \( \mu_o = 1.5442 \)
- Refractive index for extraordinary ray \( \mu_e = 1.5533 \)

### Step 3: Use the formula for the thickness of a quarter-wave plate
The thickness \( T \) of a quarter-wave plate is given by the formula:
\[ T = \frac{\lambda}{4(\mu_e - \mu_o)} \]

### Step 4: Substitute the values into the formula
First, calculate \( \mu_e - \mu_o \):
\[ \mu_e - \mu_o = 1.5533 - 1.5442 = 0.0091 \]

Now substitute \( \lambda \) and \( \mu_e - \mu_o \) into the thickness formula:
\[ T = \frac{5893 \times 10^{-10}}{4 \times 0.0091} \]

### Step 5: Calculate the thickness
Calculating the denominator:
\[ 4 \times 0.0091 = 0.0364 \]

Now calculate \( T \):
\[ T = \frac{5893 \times 10^{-10}}{0.0364} \approx 1.620 \times 10^{-5} \, \text{m} \]

### Step 6: Convert the thickness to micrometers
To express the thickness in micrometers:
\[ T \approx 1.620 \times 10^{-5} \, \text{m} = 16.20 \, \mu m \]

Thus, the thickness of the quartz quarter-wave plate is approximately \( 16.20 \, \mu m \).

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