Diffraction at Multiple Slits and Gratings

Experimental Objectives

• Investigation of diffraction at double slits with varying slit separations.
• Investigation of diffraction at double slits with varying slit widths.
• Investigation of diffraction at multiple slits with varying numbers of slits.
• Investigation of diffraction at a transmission grating and a cross grating.

Product Code: UE4030200

General Principles

The diffraction of light at multiple slits and gratings can be described by the superposition of coherent elementary waves originating from each illuminated point within a multiple slit, following Huygens' principle. This superposition leads to constructive or destructive interference in specific directions, explaining the observed pattern of bright and dark fringes behind the multiple slit.

Behind a double slit, the intensity is maximal at an observation angle α when for every elementary wave from the first slit, there is a corresponding elementary wave from the second slit that constructively interferes. This condition is met when the path difference Δs between the elementary waves from the slit centers is an integer multiple of the light's wavelength λ (see Fig. 2).

The condition for constructive interference is given by:

Δs = d sin α = n ⋅ λ

where n = 0, ±1, ±2,... represents the diffraction order.

For small observation angles, the relationship between the path difference Δs and the coordinate x of an intensity maximum is:

d ⋅ (x / L) ≈ d ⋅ α = n ⋅ λ

This leads to maxima occurring at regular intervals:

x_n+1 - x_n = (λ ⋅ L) / d

This principle also applies to diffraction at multiple slits with more than two equidistant slits. Equation (1) defines the condition for constructive interference of elementary waves from all N slits. Thus, equations (2) and (3) can also be applied to multiple slits.

Mathematically, determining the intensity minima is more complex. While in double-slit diffraction, an intensity minimum occurs exactly between two intensity maxima, in multiple-slit diffraction, a minimum is found between the n-th and (n+1)-th maximum when the elementary waves from the N slits interfere such that the total intensity becomes zero. This occurs when the path difference between elementary waves from the slit centers satisfies:

Δs = n ⋅ λ + m ⋅ (λ / N)

where n = 0, ±1, ±2,... and m = 1, ..., N-1. This results in N-1 minima and N-2 so-called secondary maxima between them, whose intensity is less than that of the principal maxima.

As the number of slits N increases, the contribution of the secondary maxima diminishes. The phenomenon is then referred to as a transmission grating. A cross grating can be considered an arrangement of two transmission gratings rotated by 90° relative to each other. The diffraction maxima form points on a rectangular grid, with the mesh size determined by equation (3).

The brightness of the principal maxima is modulated by the diffraction pattern of a single slit. The brighter the principal maxima, the more concentrated they are at smaller angles α, which is dependent on the slit width b. For precise calculation, the amplitudes of all elementary waves are summed, considering the path differences, to obtain the total amplitude A. At any position x on the screen, the intensity is:

I = A² ⋅ [sin(N ⋅ π ⋅ a / (λ ⋅ L)) / sin(π ⋅ a / (λ ⋅ L))]² ⋅ [sin(π ⋅ b ⋅ α / (λ ⋅ L)) / (π ⋅ b ⋅ α / (λ ⋅ L))]²

The function f(x) on the right side of equation (5) is given by the following limit at x = 0 (center of the intensity distribution):

lim f(x) = N²

The first factor of f(x) describes the diffraction at a single slit, and the second factor describes the interference between N slits.

Equipment List

Setup and Procedure

• Place an optical rider at 0 cm and another at 70 cm on the optical bench and fix them (Fig. 1).
• Insert the diode laser into the holder for the diode laser and fix it. Place the holder with the diode laser onto the optical rider at the 70 cm position.
• Place the clamping holder onto the optical rider at the 0 cm position. Clamp a slit mask with diffraction objects into the clamping holder such that the diffraction object to be measured is centered on the optical axis.
• Position a projection screen at a distance L ≈ 7 m from the diffraction object (Fig. 2) perpendicular to the optical axis. Measure and record the distance between the diffraction object and the projection screen accurately. Do not change the position of the optical bench anymore; if necessary, re-measure the distance L for each change of slit mask or diffraction object.
• Connect the power supply of the diode laser to the mains and switch on the diode laser.
• Clamp all slit masks with diffraction objects into the clamping holder one by one and observe, or photograph if necessary, the intensity distribution for all diffraction objects on the screen.
• For the 4 double slits with different slit separations, measure the distance x_n between a maximum of order n ≠ 0 and the maximum of order 0 in the center of the intensity distribution (example for n = 2 shown in Fig. 2) and record it in Table 1 (see Evaluation).

Measurement Example

Evaluation

For a detailed evaluation of the intensity distributions observed on the screen, the intensity distributions or the functions f(x) according to equation (5) are calculated using the given slit widths, slit separations, and slit numbers, and compared with the observed intensity distributions on the screen (Fig. 3, 4, and 5).

For slits with different separations (Fig. 3), it is observed that the number of interference maxima increases with increasing slit separation, and their width becomes narrower, as the width of the diffraction maxima (envelope of the interference maxima) remains the same. With increasing slit separation, more diffracted waves can interfere.

For slits with different widths (Fig. 4), it is observed that the number of interference maxima remains the same with increasing slit width, and their intensity for orders n ≠ 0 decreases because the width of the diffraction maxima (envelope of the interference maxima) becomes narrower. Due to the increasingly weaker diffraction with increasing slit width, the interference between the diffracted waves is correspondingly weaker.

For multiple slits (Fig. 5), as expected, N-2 secondary maxima are observed: none for a double slit, one for a triple slit, two for a quadruple slit, and three for a quintuple slit.

Generally, no interference maxima can be observed at positions where diffraction minima occur. This is the case when the first factor of f(x) in equation (5) becomes zero, which happens at integer multiples of x = (λ / b) ⋅ L. For b = 0.15 mm, this results in:

x = (λ / 0.15 mm) ⋅ 7 m = 30.3 mm

The wavelength of the diffracted light can be determined for double slits with different slit separations using equation (3) from the regular distances a of the maxima.

• For the 4 double slits with different slit separations, calculate the quotients L / d (Table 1).
• To determine the distance a of the maxima, divide the measured distances x_n by the diffraction order n (Table 1).
• Plot the values of a determined from the measurements against L / d in a diagram and fit a straight line to the data points (Fig. 8).

According to equation (3), the slope of the fitted line corresponds exactly to the wavelength λ:

a = (λ / d) ⋅ L

This yields a value of λ = 630 nm, which agrees with the specified value of λ = 650 nm within 3%.