To produce a minimum reflection of wavelength near the middle of visible spectrum (550 nm), how thick should a coating of `MgF_(2) (mu = 1.38)` be vaccum-coated on a glass surface?

Options

A. `10^(-7)` m
B. `10^(-10)` m
C. `10^(-9)` m
D. `10^(-8)` m

Explanation

To solve the problem of determining the thickness of the MgF₂ coating needed to produce minimum reflection for a wavelength of 550 nm, we can follow these steps:

### Step 1: Understand the condition for minimum reflection
For a thin film to produce minimum reflection, the path difference between the light waves reflected from the top and bottom surfaces of the film must be an odd multiple of half the wavelength. This condition can be expressed as:
\[ \Delta x = (2n - 1) \frac{\lambda}{2} \]
where \( n \) is an integer (1, 2, 3, ...) and \( \lambda \) is the wavelength of light in vacuum.

### Step 2: Calculate the total path difference
The total path difference for light reflecting off the two surfaces of the film is given by:
\[ \Delta x = 2 \mu t \]
where \( \mu \) is the refractive index of the film (MgF₂ in this case), and \( t \) is the thickness of the film.

### Step 3: Set the two expressions for path difference equal
To find the thickness \( t \) that results in minimum reflection, we set the two expressions for path difference equal:
\[ 2 \mu t = (2n - 1) \frac{\lambda}{2} \]

### Step 4: Solve for thickness \( t \)
Rearranging the equation to solve for \( t \):
\[ t = \frac{(2n - 1) \lambda}{4 \mu} \]

### Step 5: Substitute known values
We are given:
- Wavelength \( \lambda = 550 \, \text{nm} = 550 \times 10^{-9} \, \text{m} \)
- Refractive index \( \mu = 1.38 \)

For minimum thickness, we can take \( n = 1 \):
\[ t = \frac{(2 \cdot 1 - 1) \cdot 550 \times 10^{-9}}{4 \cdot 1.38} \]
\[ t = \frac{1 \cdot 550 \times 10^{-9}}{4 \cdot 1.38} \]
\[ t = \frac{550 \times 10^{-9}}{5.52} \]
\[ t \approx 99.64 \times 10^{-9} \, \text{m} = 0.09964 \, \text{μm} \]

### Step 6: Final answer
Thus, the thickness of the MgF₂ coating should be approximately:
\[ t \approx 100 \, \text{nm} \]