What is the formula for diffraction of a double-slit?
2 Answers
The diffraction pattern of a double-slit experiment can be understood using both interference and diffraction principles. Here's the basic breakdown:
### Key Variables
- \( \lambda \): Wavelength of the light
- \( d \): Distance between the two slits
- \( L \): Distance from the slits to the screen
- \( \theta \): Angle from the central maximum (bright spot in the middle)
- \( m \): Order of the maximum (interference maxima)
### Double-Slit Interference Pattern Formula
For a simple double-slit interference pattern, the angle \( \theta_m \) to the \( m \)-th bright fringe (maximum) is given by the formula:
\[
d \sin(\theta_m) = m \lambda, \quad m = 0, \pm 1, \pm 2, \pm 3, \dots
\]
Where:
- \( m \) is the order of the bright fringe (where \( m = 0 \) corresponds to the central maximum, and \( m = \pm 1 \) corresponds to the first order maxima, etc.).
- \( d \) is the distance between the two slits.
- \( \lambda \) is the wavelength of the light.
This gives the positions for the bright fringes (constructive interference) in the diffraction pattern.
### Double-Slit Diffraction (Envelope) Formula
Additionally, the diffraction envelope that modulates the interference pattern is due to the diffraction of light through a single slit. For each individual slit, the angle \( \theta_{\text{dark}} \) for the dark fringes in a single-slit diffraction pattern is given by:
\[
a \sin(\theta_{\text{dark}}) = m \lambda, \quad m = \pm 1, \pm 2, \dots
\]
Where:
- \( a \) is the width of each individual slit.
This diffraction envelope essentially defines the overall intensity pattern for the double-slit interference fringes, and it can cause the interference fringes to become less sharp or dimmer at certain angles.
### Final Result
- The double-slit diffraction pattern is a series of bright and dark fringes due to interference, with the intensity modulated by the envelope of single-slit diffraction.
The combined pattern is formed by two effects:
1. **Interference fringes** from the double slit (bright maxima spaced by \( \lambda d / L \)).
2. **Single-slit diffraction envelope** modulating the intensity of the interference fringes.
Both the position and the intensity depend on the wavelength of light (\( \lambda \)), the slit separation (\( d \)), and the screen distance (\( L \)).
### Key Variables
- \( \lambda \): Wavelength of the light
- \( d \): Distance between the two slits
- \( L \): Distance from the slits to the screen
- \( \theta \): Angle from the central maximum (bright spot in the middle)
- \( m \): Order of the maximum (interference maxima)
### Double-Slit Interference Pattern Formula
For a simple double-slit interference pattern, the angle \( \theta_m \) to the \( m \)-th bright fringe (maximum) is given by the formula:
\[
d \sin(\theta_m) = m \lambda, \quad m = 0, \pm 1, \pm 2, \pm 3, \dots
\]
Where:
- \( m \) is the order of the bright fringe (where \( m = 0 \) corresponds to the central maximum, and \( m = \pm 1 \) corresponds to the first order maxima, etc.).
- \( d \) is the distance between the two slits.
- \( \lambda \) is the wavelength of the light.
This gives the positions for the bright fringes (constructive interference) in the diffraction pattern.
### Double-Slit Diffraction (Envelope) Formula
Additionally, the diffraction envelope that modulates the interference pattern is due to the diffraction of light through a single slit. For each individual slit, the angle \( \theta_{\text{dark}} \) for the dark fringes in a single-slit diffraction pattern is given by:
\[
a \sin(\theta_{\text{dark}}) = m \lambda, \quad m = \pm 1, \pm 2, \dots
\]
Where:
- \( a \) is the width of each individual slit.
This diffraction envelope essentially defines the overall intensity pattern for the double-slit interference fringes, and it can cause the interference fringes to become less sharp or dimmer at certain angles.
### Final Result
- The double-slit diffraction pattern is a series of bright and dark fringes due to interference, with the intensity modulated by the envelope of single-slit diffraction.
The combined pattern is formed by two effects:
1. **Interference fringes** from the double slit (bright maxima spaced by \( \lambda d / L \)).
2. **Single-slit diffraction envelope** modulating the intensity of the interference fringes.
Both the position and the intensity depend on the wavelength of light (\( \lambda \)), the slit separation (\( d \)), and the screen distance (\( L \)).
The diffraction pattern produced by a double-slit is a result of the interference between light waves passing through the two slits. The formula for the diffraction pattern observed on a screen can be derived based on principles of wave optics. Here's the main formula for the diffraction angles:
### Condition for Constructive Interference (Bright Fringes):
For the points where constructive interference occurs, the condition is given by:
\[
d \sin(\theta) = m \lambda
\]
Where:
- \(d\) is the distance between the two slits (slit separation),
- \(\theta\) is the angle of the diffraction maxima (bright fringe) relative to the central maximum (zeroth-order),
- \(m\) is an integer representing the order of the maxima (\(m = 0, \pm 1, \pm 2, \dots\)),
- \(\lambda\) is the wavelength of the light passing through the slits.
### Condition for Destructive Interference (Dark Fringes):
For the points where destructive interference occurs (dark fringes), the condition is given by:
\[
d \sin(\theta) = \left(m + \frac{1}{2}\right) \lambda
\]
Where:
- The terms are the same as for constructive interference, but \(m\) now represents integer values for dark fringes.
### Derivation Notes:
1. **Wave Interference**: The double-slit experiment relies on the wave nature of light. Each slit acts as a coherent source of light waves that interfere with each other.
2. **Path Difference**: The conditions above are based on the principle that the path difference between the two waves must either correspond to a whole number (constructive) or half-integer (destructive) multiple of the wavelength for maxima or minima to occur, respectively.
### Additional Considerations:
- The diffraction fringes get closer as the wavelength of the light increases, or as the slit separation decreases.
- If the screen is at a distance \(L\) from the slits, the position \(y_m\) of the \(m\)-th order maximum on the screen can be found by the relation:
\[
y_m = L \tan(\theta_m)
\]
For small angles, \(\tan(\theta) \approx \sin(\theta)\), which simplifies the calculation of fringe positions.
These formulas describe the diffraction and interference pattern for a double-slit experiment, a classic demonstration of wave behavior in physics.
### Condition for Constructive Interference (Bright Fringes):
For the points where constructive interference occurs, the condition is given by:
\[
d \sin(\theta) = m \lambda
\]
Where:
- \(d\) is the distance between the two slits (slit separation),
- \(\theta\) is the angle of the diffraction maxima (bright fringe) relative to the central maximum (zeroth-order),
- \(m\) is an integer representing the order of the maxima (\(m = 0, \pm 1, \pm 2, \dots\)),
- \(\lambda\) is the wavelength of the light passing through the slits.
### Condition for Destructive Interference (Dark Fringes):
For the points where destructive interference occurs (dark fringes), the condition is given by:
\[
d \sin(\theta) = \left(m + \frac{1}{2}\right) \lambda
\]
Where:
- The terms are the same as for constructive interference, but \(m\) now represents integer values for dark fringes.
### Derivation Notes:
1. **Wave Interference**: The double-slit experiment relies on the wave nature of light. Each slit acts as a coherent source of light waves that interfere with each other.
2. **Path Difference**: The conditions above are based on the principle that the path difference between the two waves must either correspond to a whole number (constructive) or half-integer (destructive) multiple of the wavelength for maxima or minima to occur, respectively.
### Additional Considerations:
- The diffraction fringes get closer as the wavelength of the light increases, or as the slit separation decreases.
- If the screen is at a distance \(L\) from the slits, the position \(y_m\) of the \(m\)-th order maximum on the screen can be found by the relation:
\[
y_m = L \tan(\theta_m)
\]
For small angles, \(\tan(\theta) \approx \sin(\theta)\), which simplifies the calculation of fringe positions.
These formulas describe the diffraction and interference pattern for a double-slit experiment, a classic demonstration of wave behavior in physics.