What is wedge-shaped thin film? Obtain an expression for Fringe width of the interference due to reflected rays in wedge-shaped thin film.
2 Answers
A **wedge-shaped thin film** is a thin, transparent film (usually made of glass or other dielectric material) with a thickness that varies linearly across its surface. It is typically formed by placing two flat glass plates together at a small angle, leaving a wedge-shaped air gap between them.
When light is incident on such a film, part of it is reflected at the top surface (air-to-film interface) and part at the bottom surface (film-to-glass interface). The two reflected rays interfere, producing a pattern of bright and dark fringes, known as interference fringes. These fringes arise because the optical path difference between the two rays changes across the wedge due to the varying film thickness.
---
### Derivation of Fringe Width in Wedge-Shaped Thin Film
**Assumptions and Setup:**
1. The film thickness at a point is \( t(x) = x \tan\theta \), where \( \theta \) is the wedge angle and \( x \) is the distance from the thin end of the wedge.
2. The light is monochromatic with wavelength \( \lambda \).
3. The interference occurs due to the reflected rays.
---
**Path Difference Calculation:**
When light reflects from the upper and lower surfaces of the film, the optical path difference between the two rays is influenced by:
1. The **additional path** traveled by the second ray through the film. This is \( 2t \), as the light traverses the film twice.
2. A **phase change of \(\pi\)** (equivalent to \(\lambda/2\)) upon reflection at the lower surface because the reflection occurs at a denser medium (glass).
Thus, the net optical path difference is:
\[
\Delta = 2t - \frac{\lambda}{2}.
\]
For constructive interference (bright fringes), the path difference must satisfy:
\[
\Delta = m\lambda \quad \text{(where \( m \) is an integer, \( m = 0, 1, 2, \ldots \))}.
\]
Substituting \(\Delta\) into the condition for constructive interference:
\[
2t - \frac{\lambda}{2} = m\lambda.
\]
Rearranging for \( t \):
\[
t = \frac{(m + \frac{1}{2})\lambda}{2}.
\]
For destructive interference (dark fringes), the path difference must satisfy:
\[
\Delta = (m + \frac{1}{2})\lambda \quad \text{(where \( m \) is an integer)}.
\]
Substituting \(\Delta\) into this condition gives:
\[
t = \frac{m\lambda}{2}.
\]
---
**Fringe Spacing:**
The thickness of the wedge at any point is \( t(x) = x \tan\theta \). Substituting this into the fringe condition:
For bright fringes:
\[
x \tan\theta = \frac{(m + \frac{1}{2})\lambda}{2}.
\]
For dark fringes:
\[
x \tan\theta = \frac{m\lambda}{2}.
\]
The distance between two successive bright or dark fringes (fringe width) is:
\[
\Delta x = x_{m+1} - x_m.
\]
Substituting for \( x_m \) and \( x_{m+1} \) from the fringe condition:
\[
\Delta x = \frac{(m+1)\lambda}{2\tan\theta} - \frac{m\lambda}{2\tan\theta}.
\]
Simplifying:
\[
\Delta x = \frac{\lambda}{2\tan\theta}.
\]
---
### Final Expression for Fringe Width
The fringe width (distance between successive bright or dark fringes) in a wedge-shaped thin film is:
\[
\Delta x = \frac{\lambda}{2\tan\theta},
\]
where:
- \( \lambda \) is the wavelength of light,
- \( \theta \) is the wedge angle.
---
**Explanation of Variables:**
- The fringe width depends on \( \lambda \), so using light of a different color will alter the spacing.
- Smaller wedge angles (\( \theta \)) increase the fringe spacing, as the thickness variation becomes more gradual.
- This expression holds true for monochromatic light and for small angles of incidence, ensuring the approximation \( \tan\theta \approx \theta \) is valid.
### Conclusion
Wedge-shaped thin films create an interference pattern due to the varying film thickness, and the fringe width is inversely proportional to the wedge angle. This principle is widely used in precision measurements and optical testing setups, such as Newton's rings.
When light is incident on such a film, part of it is reflected at the top surface (air-to-film interface) and part at the bottom surface (film-to-glass interface). The two reflected rays interfere, producing a pattern of bright and dark fringes, known as interference fringes. These fringes arise because the optical path difference between the two rays changes across the wedge due to the varying film thickness.
---
### Derivation of Fringe Width in Wedge-Shaped Thin Film
**Assumptions and Setup:**
1. The film thickness at a point is \( t(x) = x \tan\theta \), where \( \theta \) is the wedge angle and \( x \) is the distance from the thin end of the wedge.
2. The light is monochromatic with wavelength \( \lambda \).
3. The interference occurs due to the reflected rays.
---
**Path Difference Calculation:**
When light reflects from the upper and lower surfaces of the film, the optical path difference between the two rays is influenced by:
1. The **additional path** traveled by the second ray through the film. This is \( 2t \), as the light traverses the film twice.
2. A **phase change of \(\pi\)** (equivalent to \(\lambda/2\)) upon reflection at the lower surface because the reflection occurs at a denser medium (glass).
Thus, the net optical path difference is:
\[
\Delta = 2t - \frac{\lambda}{2}.
\]
For constructive interference (bright fringes), the path difference must satisfy:
\[
\Delta = m\lambda \quad \text{(where \( m \) is an integer, \( m = 0, 1, 2, \ldots \))}.
\]
Substituting \(\Delta\) into the condition for constructive interference:
\[
2t - \frac{\lambda}{2} = m\lambda.
\]
Rearranging for \( t \):
\[
t = \frac{(m + \frac{1}{2})\lambda}{2}.
\]
For destructive interference (dark fringes), the path difference must satisfy:
\[
\Delta = (m + \frac{1}{2})\lambda \quad \text{(where \( m \) is an integer)}.
\]
Substituting \(\Delta\) into this condition gives:
\[
t = \frac{m\lambda}{2}.
\]
---
**Fringe Spacing:**
The thickness of the wedge at any point is \( t(x) = x \tan\theta \). Substituting this into the fringe condition:
For bright fringes:
\[
x \tan\theta = \frac{(m + \frac{1}{2})\lambda}{2}.
\]
For dark fringes:
\[
x \tan\theta = \frac{m\lambda}{2}.
\]
The distance between two successive bright or dark fringes (fringe width) is:
\[
\Delta x = x_{m+1} - x_m.
\]
Substituting for \( x_m \) and \( x_{m+1} \) from the fringe condition:
\[
\Delta x = \frac{(m+1)\lambda}{2\tan\theta} - \frac{m\lambda}{2\tan\theta}.
\]
Simplifying:
\[
\Delta x = \frac{\lambda}{2\tan\theta}.
\]
---
### Final Expression for Fringe Width
The fringe width (distance between successive bright or dark fringes) in a wedge-shaped thin film is:
\[
\Delta x = \frac{\lambda}{2\tan\theta},
\]
where:
- \( \lambda \) is the wavelength of light,
- \( \theta \) is the wedge angle.
---
**Explanation of Variables:**
- The fringe width depends on \( \lambda \), so using light of a different color will alter the spacing.
- Smaller wedge angles (\( \theta \)) increase the fringe spacing, as the thickness variation becomes more gradual.
- This expression holds true for monochromatic light and for small angles of incidence, ensuring the approximation \( \tan\theta \approx \theta \) is valid.
### Conclusion
Wedge-shaped thin films create an interference pattern due to the varying film thickness, and the fringe width is inversely proportional to the wedge angle. This principle is widely used in precision measurements and optical testing setups, such as Newton's rings.
### **Wedge-Shaped Thin Film:**
A wedge-shaped thin film is a thin, transparent layer of material (like glass or air) that has a slight thickness difference across its surface, forming a wedge shape. It is thicker on one end and gradually becomes thinner towards the other end. This shape creates an angle between the two surfaces of the film.
When light falls on this film, some light is reflected from the top surface, and some from the bottom surface. These two reflected light rays interfere with each other, leading to an interference pattern of bright and dark fringes (lines) due to the variation in thickness of the film.
---
### **Expression for Fringe Width:**
Fringe width (\( \beta \)) is the distance between two consecutive bright or dark fringes in the interference pattern.
1. **Path Difference**:
The extra distance traveled by the light ray reflected from the bottom surface is given by:
\[
\Delta = 2 \, \mu \, t \, \cos r
\]
Here:
- \( \mu \) = refractive index of the thin film
- \( t \) = thickness of the film at a given point
- \( r \) = angle of refraction
2. For constructive interference (bright fringe):
\[
2 \, \mu \, t \, \cos r = (2n + 1) \frac{\lambda}{2}
\]
Where \( n = 0, 1, 2, ... \) (order of the fringe)
For destructive interference (dark fringe):
\[
2 \, \mu \, t \, \cos r = n \lambda
\]
3. **Fringe Width Calculation**:
The fringe width is the distance between two consecutive bright or dark fringes and is given by:
\[
\beta = \frac{\lambda}{2 \, \alpha}
\]
Where:
- \( \lambda \) = wavelength of light
- \( \alpha \) = angle of the wedge
---
### **Key Points to Memorize:**
1. A wedge-shaped thin film has gradually changing thickness.
2. Interference occurs due to light reflecting off the top and bottom surfaces of the film.
3. Bright and dark fringes are formed due to constructive and destructive interference.
4. Fringe width (\( \beta \)) depends on the wavelength of light and the wedge angle (\( \alpha \)).
This explanation with the formula can help you score full marks!
A wedge-shaped thin film is a thin, transparent layer of material (like glass or air) that has a slight thickness difference across its surface, forming a wedge shape. It is thicker on one end and gradually becomes thinner towards the other end. This shape creates an angle between the two surfaces of the film.
When light falls on this film, some light is reflected from the top surface, and some from the bottom surface. These two reflected light rays interfere with each other, leading to an interference pattern of bright and dark fringes (lines) due to the variation in thickness of the film.
---
### **Expression for Fringe Width:**
Fringe width (\( \beta \)) is the distance between two consecutive bright or dark fringes in the interference pattern.
1. **Path Difference**:
The extra distance traveled by the light ray reflected from the bottom surface is given by:
\[
\Delta = 2 \, \mu \, t \, \cos r
\]
Here:
- \( \mu \) = refractive index of the thin film
- \( t \) = thickness of the film at a given point
- \( r \) = angle of refraction
2. For constructive interference (bright fringe):
\[
2 \, \mu \, t \, \cos r = (2n + 1) \frac{\lambda}{2}
\]
Where \( n = 0, 1, 2, ... \) (order of the fringe)
For destructive interference (dark fringe):
\[
2 \, \mu \, t \, \cos r = n \lambda
\]
3. **Fringe Width Calculation**:
The fringe width is the distance between two consecutive bright or dark fringes and is given by:
\[
\beta = \frac{\lambda}{2 \, \alpha}
\]
Where:
- \( \lambda \) = wavelength of light
- \( \alpha \) = angle of the wedge
---
### **Key Points to Memorize:**
1. A wedge-shaped thin film has gradually changing thickness.
2. Interference occurs due to light reflecting off the top and bottom surfaces of the film.
3. Bright and dark fringes are formed due to constructive and destructive interference.
4. Fringe width (\( \beta \)) depends on the wavelength of light and the wedge angle (\( \alpha \)).
This explanation with the formula can help you score full marks!