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Multiple-Slit Diffraction and Diffraction Gratings

Multiple-slit diffraction involves the simultaneous interference of light waves diffracted from a large number of parallel, equally spaced slits. It produces sharp principal maxima interspersed with weaker secondary maxima and diffraction minima.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Qualitative understanding of grating spectra, condition for principal maxima, number of orders, and relation between slit width and spacing.

PGT extension

Quantitative derivation of intensity distribution, missing orders, dispersive power, resolving power of optical diffraction gratings, and angular half-width of principal maxima.

Prerequisites

Huygens-Fresnel principle and Fraunhofer diffraction
Superposition of coherent waves and interference
Double-slit interference and single-slit diffraction envelope

Intensity Distribution in Multiple-Slit Diffraction

The resultant intensity is a combination of interference due to N slits and diffraction due to an individual slit.
The intensity equation is given by the product of the interference term and the diffraction term: $I = I_0 (\frac{\sin^2 \beta}{\beta^2}) (\frac{\sin^2 Nalpha}{\sin^2 \alpha})$.
The term (\frac{sin^2 \beta}{\beta^2}) represents the single-slit diffraction envelope, while (\frac{sin^2 Nalpha}{sin^2 alpha}) represents the N-slit interference pattern.

Principal Maxima and Minima

Principal maxima occur when the path difference between light from adjacent slits is an integral multiple of the wavelength: (a + b) sin \theta = n lambda.
There are (N - 1) secondary minima between two adjacent principal maxima due to phase cancellation.
There are (N - 2) subsidiary maxima situated between adjacent secondary minima, giving a total of (N - 2) minor peaks between principal peaks.

Missing Orders in a Diffraction Grating

A missing order occurs when a principal maximum coincides with a diffraction minimum.
The condition for principal maxima is (a + b) sin \theta = n lambda, and the condition for single-slit diffraction minima is a sin \theta = m lambda.
Dividing the two conditions yields \frac{a + b}{a} = \frac{n}{m}. If this ratio is an integer, that particular order n of the principal maximum will be completely missing.

Resolving Power of a Diffraction Grating

Resolving power measures the ability of a grating to form separate, distinguishable diffraction maxima of two close wavelengths.
According to Rayleigh's criterion, two wavelengths lambda and lambda + Delta lambda are resolved if the principal maximum of one coincides with the first minimum of the other.
The resolving power is expressed mathematically as \frac{lambda}{Delta lambda} = n N, where n is the spectral order and N is the total number of ruled lines on the grating.

Derivations

Derivation of Intensity Distribution for N Slits

Conditions: Fraunhofer diffraction through N parallel, identical, equidistant slits of width 'a' and separation 'b'.

Let each slit contribute an amplitude E_0 \frac{sin \beta}{\beta} where \beta = \frac{pi a sin \theta}{lambda}.
The phase difference between disturbances from corresponding points of two successive slits is 2alpha, where alph$a = \frac{\pi (a + b) \sin \theta}{\lambda}$.
The resultant amplitude R of N harmonic waves of equal amplitude $A = E_0 \frac{\sin \beta}{\beta}$$ and common phase difference 2alpha is given by R = A \frac{sin(Nalpha)}{sin alpha}.
Squaring the amplitude to find intensity $I = R^2 yields I = I_0 (\frac{\sin^2 \beta}{\beta^2}) (\frac{\sin^2 Nalpha}{\sin^2 \alpha})$.

Result: $I = I_0 (\frac{\sin^2 \beta}{\beta^2}) (\frac{\sin^2 Nalpha}{\sin^2 \alpha})$

Resolving Power of a Diffraction Grating

Conditions: Limit of resolution based on Rayleigh's criterion for adjacent spectral lines.

The nth principal maximum for wavelength lambda occurs at angle \theta_n given by (a + b) sin \thet$a_n = n \lambda$.
The first minimum adjacent to this maximum occurs when the total phase difference across all N slits changes by 2pi, giving the condition for the adjacent minimum as N(2alpha) = 2pi + n \text{ (or corresponding differential change)}.
Differentiating the grating equation with respect to lambda and \theta gives (a + b) cos \theta d\theta = n dlambda.
Equating the angular separation to the position of the first minimum d$\theta = \frac{\lambda}{N (a + b) \cos \theta}$, we obtain \frac{lambda}{Delta lambda} = n N.

Result: \frac{\lambda}{\Delta \lambda} = n N

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Grating Element	$$e = a + b$$	a = slit width, b = opaque spacing metres (m) or cm	Constant spatial periodicity
Grating Equation (Principal Maxima)	$$(a + b) \sin \theta = n \lambda$$	n = order of spectrum, lambda = wavelength, \theta = diffraction angle dimensionless (n), m (lambda)	Normal incidence
Intensity of N-Slit Pattern	$$I = I_0 \left(\frac{\sin \beta}{\beta}\right)^2 \left(/frac{\sin N\alpha}{\sin \alpha}\right)^2$$	\beta = \frac{pi a sin \theta}{lambda}, alph$a = \frac{\pi (a+b) \sin \theta}{\lambda}$ W/m^2	Fraunhofer diffraction
Direction of Secondary Minima	$$N\alpha = m\pi \quad (m \neq 0, \pm N, \pm 2N...)$$	m = integer, N = number of slits dimensionless	Between principal maxima
Missing Order Condition	$$\frac{a + b}{a} = \frac{n}{m}$$	n = order of principal maximum, m = order of diffraction minimum dimensionless	Coincidence of principal maximum and diffraction minimum
Dispersive Power of Grating	$$\frac{d\theta}{d\lambda} = \frac{n}{(a + b) \cos \theta}$$	\frac{d\theta}{dlambda} = change in angle per unit wavelength radians per metre (rad/m)	For a given spectral order n
Resolving Power of Grating	$$\frac{\lambda}{\Delta \lambda} = n N$$	N = total number of lines, n = order dimensionless	Rayleigh resolution criterion
Maximum Possible Order	$$n_{max} = \frac{a + b}{\lambda}$$	a + b = grating element, lambda = wavelength dimensionless	Since sin \theta le 1

Graphs and experiments

Intensity vs. Diffraction Angle (\theta)

Draw: A rapid-oscillating interference fringe pattern modulated by a smooth, broad single-slit diffraction envelope curve, showing sharp principal peaks and minor secondary peaks.

Interpret: Demonstrates how the single-slit envelope limits the height of the N-slit interference principal maxima.

Exam use: Frequently asked in identifying missing orders and counting secondary maxima between principal peaks.

Plane Transmission Grating Spectrometer Setup

Draw: A collimator tube providing parallel light, a central telescope mounted on a rotating table, and a transmission diffraction grating placed normally on the table.

Interpret: Used experimentally to measure wavelengths of spectral lines and calculate dispersive and resolving powers.

Exam use: Core practical physics setup questions in PGT interviews and written concept verification.

Grating Spectra (First and Second Orders)

Draw: A central white band (n=0) flanked symmetrically by distinct, dispersed color spectra (violet deviated least, red deviated most) for n=1 and n=2.

Interpret: Highlights the angular dispersion of polychromatic light due to wavelength dependence in the grating equation.

Exam use: Explains why spectra overlap at higher orders.

Solved numericals

Tgt core

A diffraction grating has 5000 lines per centimetre. Find the angle of diffraction for the second-order principal maximum for monochromatic light of wavelength 5890 Å.

Calculate the grating element (a + b): \frac{1 \text{ cm}}{5000} = 2 \times 10^{-4} \text{ cm} = 2 \times 10^{-6} \text{ m}.
Use the grating equation: (a + b) sin \theta = n lambda.
Substitute the given values: (2 \times 10^{-6}) sin $\theta = 2 \times (5.89 \times 10^{-7})$.
Simplify: sin $\theta = \frac{11.78 \times 10^{-7}}{2 \times 10^{-6}} = 0.589$.
Find angle: $\theta = arcsin(0.589) \approx 36.09^\circ$.

Answer: $\theta = 36.09^\circ$

Pgt extension

Light containing two wavelengths lambd$a_1 = 5890 Å$ and lambd$a_2 = 5896 Å$ is incident normally on a grating. Calculate the minimum number of lines in the grating required to resolve these two wavelengths in the second order.

Identify given parameters: lambd$a = 5.89 \times 10^{-7} \text{ m}$, Delta lambd$a = \lambda_2 - \lambda_1 = 6 \text{ Å} = 6 \times 10^{-10} \text{ m}$, n = 2.
Use the formula for resolving power of a grating: \frac{lambda}{Delta lambda} = n N.
Rearrange to solve for N: N = \frac{lambda}{n Delta lambda}.
Substitute the values: $N = \frac{5.89 \times 10^{-7}}{2 \times (6 \times 10^{-10})}$.
Calculate result: $N = \frac{5890 \times 10^{-10}}{12 \times 10^{-10}} = \frac{5890}{12} \approx 490.8$.

Answer: N = 491 lines

Pgt extension

In a transmission grating, the width of each slit is equal to the width of the opaque space between them. Which orders of principal maxima will be missing from the diffraction pattern?

Given condition: slit width a equals opaque space b, so b = a, meaning the grating element is (a + b) = 2a.
Apply the missing order condition: \frac{a + b}{a} = \frac{n}{m}.
Substitute (a + b) = 2a into the equation: \frac{2a}{a} = \frac{n}{m} implies \frac{n}{m} = 2 implies n = 2m.
Since m can take any integer values (1, 2, 3, ...), the missing orders correspond to n = 2, 4, 6, ..., which are all even orders.

Answer: All even orders (n = 2, 4, 6, ...)

Assumptions and limitations

Assumes normal incidence of plane wavefronts unless explicitly stated otherwise for oblique incidence.
Assumes the grating surface is perfectly flat and rulings are precisely equidistant and parallel, which is an idealization not fully realized in commercial gratings.
Edge effects at individual slits are neglected in the Fraunhofer approximation.

Common misconceptions

Common error: The number of secondary maxima between two principal maxima is equal to N.

Correct: The number of secondary maxima is (N - 2), while the number of secondary minima is (N - 1).

Between two principal maxima, there are (N-1) valleys (minima), which bound (N-2) peaks (subsidiary maxima).

Common error: Increasing the number of slits N changes the angular positions of the principal maxima.

Correct: The positions of principal maxima depend solely on the grating element (a + b) and wavelength lambda, not on N.

N only controls the sharpness and intensity (width) of the principal maxima, making them narrower and brighter.

Common error: A missing order means light does not undergo diffraction at all for that angle.

Correct: Light does diffract, but the direction corresponds to a zero (minimum) of the single-slit diffraction envelope, making the net intensity zero.

The interference maximum coincides with a single-slit diffraction null.

Quick revision

Grating formula is (a + b) sin \theta = n lambda.
Resolving power of grating is given by \frac{lambda}{Delta lambda} = n N.
Dispersive power is \frac{d\theta}{dlambda} = \frac{n}{(a+b) cos \theta}.
Missing orders occur when \frac{a+b}{a} = \frac{n}{m} is an integer.
For N slits, there are N-1 secondary minima and N-2 subsidiary maxima between principal peaks.

Must remember

The resultant intensity in a multiple-slit arrangement is the product of the single-slit diffraction term and the N-slit interference term: $I = I_0 \left(\frac{\sin^2 \beta}{\beta^2}\right) \left(\frac{\sin^2 N\alpha}{\sin^2 \alpha}\right)$.
Principal maxima occur when $(a + b) \sin \theta = n \lambda$.
Between two adjacent principal maxima, there are $(N - 1)$ secondary minima and $(N - 2)$ subsidiary (subsidiary/minor) maxima.
A missing order occurs when a principal maximum coincides with a single-slit diffraction minimum, given by $\frac{a + b}{a} = \frac{n}{m}$.
The resolving power of a diffraction grating is given by $\frac{\lambda}{\Delta \lambda} = n N$, depending on the spectral order $n$ and total lines $N$.

Glossary

Grating ElementThe sum of the slit width $a$ and opaque spacing $b$, denoted as $e = a + b$, representing the spatial period of the grating.

Principal MaximaBright peaks in a diffraction grating spectrum formed by constructive interference, governed by $(a + b) \sin \theta = n \lambda$.

Missing OrdersSpecific spectral orders $n$ that fail to appear because the principal maximum coincides with a single-slit diffraction minimum.

Resolving PowerThe measure of a grating's ability to distinguish two closely spaced spectral lines, expressed as $\frac{\lambda}{\Delta \lambda} = n N$.

Dispersive PowerThe rate of change of diffraction angle with respect to wavelength, $\frac{d\theta}{d\lambda} = \frac{n}{(a + b) \cos \theta}$.

Interference vs Diffraction Terms in Multiple Slit

Feature	Single-Slit Diffraction Term	N-Slit Interference Term
Mathematical Expression	$\left(\frac{\sin \beta}{\beta}\right)^2$	$\left(\frac{\sin N\alpha}{\sin \alpha}\right)^2$
Governing Parameter	Slit width $a$	Number of slits $N$ and grating element $a+b$
Role in Pattern	Provides the overall intensity envelope	Determines the sharpness and position of principal peaks

Practice Quiz

Choose an option to reveal the explanation.

1. What is the intensity distribution equation for multiple-slit Fraunhofer diffraction?

Explanation: The resultant intensity is the product of the single-slit diffraction factor $\left(\frac{\sin \beta}{\beta}\right)^2$ and the N-slit interference factor $\left(\frac{\sin N\alpha}{\sin \alpha}\right)^2$.

2. Condition for principal maxima in a diffraction grating with normal incidence is:

Explanation: Principal maxima occur when the path difference between light from adjacent slits is an integral multiple of wavelength, i.e., $(a + b) \sin \theta = n \lambda$.

3. How many secondary minima lie between two adjacent principal maxima for a grating with $N$ slits?

Explanation: There are always $(N - 1)$ secondary minima between any two adjacent principal maxima due to phase cancellation.

4. How many subsidiary (minor) maxima lie between two adjacent principal maxima for a grating with $N$ slits?

Explanation: Between two adjacent principal maxima, there are $(N - 1)$ secondary minima which bound $(N - 2)$ subsidiary maxima.

5. Under what condition does a missing order occur in a diffraction grating spectrum?

Explanation: A missing order occurs when the condition for a principal maximum coincides with the condition for a single-slit diffraction minimum.

6. If the ratio of the grating element to the slit width is $(a + b)/a = 3$, which order of the principal maximum will be missing?

Explanation: Using $\frac{a + b}{a} = \frac{n}{m}$, if $(a+b)/a = 3$, then for $m = 1$, $n = 3$. Thus, the 3rd order principal maximum is missing.

7. The resolving power of a diffraction grating is expressed as:

Explanation: According to Rayleigh's criterion, the resolving power is given by $\frac{\lambda}{\Delta \lambda} = n N$, where $n$ is the spectral order and $N$ is the total number of lines.

8. What is the physical significance of the grating element $(a + b)$?

Explanation: The grating element $e = a + b$ is the spatial period, representing the distance between corresponding points of two adjacent slits.

9. Dispersive power of a diffraction grating is defined as:

Explanation: Dispersive power measures the change in diffraction angle per unit change in wavelength and is given by $\frac{d\theta}{d\lambda} = \frac{n}{(a + b) \cos \theta}$.

10. What happens to the sharpness of principal maxima when the number of slits $N$ is increased?

Explanation: Increasing $N$ makes the principal maxima sharper and more intense, as more interfering beams contribute to the peak.

11. The maximum possible order $n_{max}$ for a diffraction grating is limited by:

Explanation: Since $\sin \theta \le 1$ in the grating equation $(a + b) \sin \theta = n \lambda$, the maximum possible order is $n_{max} = \frac{a + b}{\lambda}$.

12. In the intensity equation, what does the variable $\beta$ represent?

Explanation: The single-slit phase term is $\beta = \frac{\pi a \sin \theta}{\lambda}$.

13. Which term in the multiple-slit intensity expression dictates the envelope of the diffraction pattern?

Explanation: The single-slit diffraction envelope is determined by the term $\left(\frac{\sin \beta}{\beta}\right)^2$.

14. Does increasing the number of slits $N$ change the angular positions of the principal maxima?

Explanation: The positions of principal maxima are governed strictly by $(a + b) \sin \theta = n \lambda$, which is independent of $N$.

15. According to Rayleigh's criterion, two spectral lines are just resolved if:

Explanation: Rayleigh's criterion states that two lines are resolved when the principal maximum of one wavelength coincides with the first diffraction minimum of the second wavelength.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. A diffraction grating has $500$ lines per cm. The angular position of the 2nd order principal maximum for a wavelength of $5000\text{ \AA}$ is:

Explanation: Grating element $a + b = \frac{1}{500} \text{ cm} = 2 \times 10^{-5} \text{ m}$. Using $(a + b) \sin \theta = n \lambda$, $2 \times 10^{-5} \sin \theta = 2 \times (5000 \times 10^{-10})$, giving $\sin \theta = 0.5$, so $\theta = 30^\circ$.

2. If the width of the transparent slits is equal to the opaque spacing in a diffraction grating, which spectral orders will be missing?

Explanation: If $a = b$, then $\frac{a + b}{a} = \frac{2a}{a} = 2$. Using $\frac{a+b}{a} = \frac{n}{m} = 2$, for $m = 1, 2, 3...$, $n = 2, 4, 6...$. Thus, even orders ($2, 4, 6...$) are missing.

3. A grating has $N$ total lines and is used in the $n$-th order. Its resolving power can be increased by:

Explanation: Resolving power is given by $\frac{\lambda}{\Delta \lambda} = n N$. Thus, increasing either the order $n$ or total lines $N$ increases resolving power.

4. Between two adjacent principal maxima in a grating spectrum with $10$ slits, the number of subsidiary maxima is:

Explanation: The number of subsidiary maxima is $N - 2$. For $N = 10$, $10 - 2 = 8$.

5. What is the dispersive power of a grating having $e = 2 \times 10^{-6} \text{ m}$ in the 1st order spectrum at normal incidence for $\theta = 30^\circ$?

Explanation: Dispersive power $\frac{d\theta}{d\lambda} = \frac{n}{(a + b) \cos \theta} = \frac{1}{(2 \times 10^{-6}) \cos 30^\circ} = \frac{1}{2 \times 10^{-6} \times 0.866} \approx 5.77 \times 10^5 \text{ rad/m}$.

6. If the slit width $a$ is halved while keeping the grating element $(a + b)$ constant, how does the diffraction envelope change?

Explanation: Halving slit width $a$ doubles the single-slit diffraction envelope width ($\beta$ terms scale with $a$), changing which principal maxima coincide with diffraction minima.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. The combination of interference due to multiple slits and diffraction from a single slit results in:

Explanation: The intensity pattern is characterized by sharp principal maxima whose heights are modulated (constrained) by the single-slit diffraction envelope.

2. For a grating with $N$ slits, what is the total number of secondary minima between the central maximum and the first principal maximum?

Explanation: There are always $(N - 1)$ secondary minima between any two adjacent principal maxima, including the region between the central and first principal maximum.

3. If $\frac{a + b}{a} = 4$, which orders of principal maxima will be missing?

Explanation: Using $\frac{a+b}{a} = \frac{n}{m} = 4$, for integer values of $m = 1, 2, 3...$, the missing orders are $n = 4, 8, 12...$

4. What happens to the angular separation of principal maxima when light of a longer wavelength is used?

Explanation: Since $(a + b) \sin \theta = n \lambda$, a larger wavelength $\lambda$ requires a larger $\sin \theta$, hence increasing the angular separation.

5. The angular half-width of a principal maximum in a diffraction grating is inversely proportional to:

Explanation: The angular width of principal maxima is inversely proportional to $N(a + b)$, which is the total aperture width of the grating.

6. In a diffraction grating, the intensity of secondary maxima is:

Explanation: Secondary maxima are much dimmer than principal maxima; their relative intensity decreases as $N$ increases.

7. If a grating has $4000$ lines per cm and is illuminated normally by light of wavelength $5000\text{ \AA}$, what is the maximum order visible?

Explanation: $a + b = \frac{1}{4000} \text{ cm} = 2.5 \times 10^{-6} \text{ m}$. $n_{max} = \frac{a+b}{\lambda} = \frac{2.5 \times 10^{-6}}{5 \times 10^{-7}} = 5$.

8. Which of the following parameters does NOT affect the resolving power of a diffraction grating?

Explanation: Resolving Power = $n N$. It depends on $n$ and $N$, and inversely on $\lambda$ (via resolving limit), but the individual grating element $(a+b)$ does not appear in the resolving power formula directly unless expressed via total width $W = N(a+b)$, where $RP = \frac{W \sin \theta}{\lambda}$.

9. The phase difference $\alpha$ between light waves from adjacent slits in normal incidence is given by:

Explanation: The phase difference between adjacent slits is $\alpha = \frac{\pi (a + b) \sin \theta}{\lambda}$.

10. Why are diffraction gratings preferred over prisms for high-resolution spectroscopy?

Explanation: Gratings offer linear dispersion properties and high resolving power governed by the total number of lines, making them superior for separating close spectral lines.