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Working Principles of Diffraction Gratings
A diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams travelling in different directions. The directions of these beams depend on the spacing of the grating elements and the wavelength of the incident light.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Qualitative understanding of grating spectra, principal maxima conditions, grating element, and basic numericals on diffraction grating formula.
PGT extension
Detailed wave-front analysis, derivation of principal maxima and minima conditions, angular dispersion, resolving power of diffraction grating, and overlapping orders.
Prerequisites
Huygens' Principle and Wave Theory of Light
Fraunhofer Diffraction at a Single Slit and Double Slit
Superposition and Interference of Light Waves
Structure of a Transmission Grating
A plane transmission grating consists of an optically flat glass plate upon which a large number of equidistant, parallel lines are ruled using a fine diamond point.
The opaque ruled lines act as obstacles that block light, while the transparent spaces between them act as slits.
The combined width of a ruling and a transparent space is termed the grating element ($d = a + b)$, where 'a' is the width of the slit and 'b' is the width of the opaque space.
Principle of Action
When monochromatic light is normally incident on the grating, light is diffracted by each transparent slit and interference occurs between secondary wavelets originating from corresponding points of all slits.
The resultant intensity distribution is a combination of single-slit diffraction (which acts as an envelope) and multiple-slit interference (which produces sharp principal maxima).
The number of diffraction orders observed depends on the ratio of the grating element to the wavelength of light, given by N $\langle = d/\lambda$.
Principal Maxima and Secondary Minima
Principal maxima are bright lines formed when the path difference between light waves from adjacent slits is an integral multiple of the wavelength.
Between any two adjacent principal maxima, there are (N-2) secondary minima and (N-1) secondary maxima, where N is the total number of parallel slits.
As N becomes very large, the secondary maxima become extremely weak and negligible, leaving only sharp, well-defined principal maxima.
Absent Orders in a Grating Spectrum
An absent order occurs when the condition for a principal maximum coincides with the condition for a minimum due to single-slit diffraction.
If (a+b)sin\theta = n\lambda and asin\theta = m\lambda simultaneously, the n-th principal maximum will be missing if (a+b)/$a = n/m$ is an integer.
This phenomenon depends strictly on the relative widths of the transparent slit 'a' and the opaque space 'b'.
Derivations
Grating Equation for Principal Maxima
Conditions: Normal incidence of monochromatic plane wavefront on a transmission grating with N slits
Consider a plane wavefront incident normally on a transmission grating with grating element $d = a + b$.
The path difference between diffracted rays from corresponding points of two successive slits separated by distance d at an angle of diffraction \theta is given by \Delta = d \sin\theta.
For constructive interference to produce a principal maximum, this path difference must be an integral multiple of the wavelength \lambda.
Applying the condition: d \sin\theta = n\lambda, where n = 0, 1, 2, 3, ... represents the order of the spectrum.
Result: (a + b) \sin\theta = n\lambda
Resolving Power of a Diffraction Grating
Conditions: Two nearby spectral wavelengths \lambda and \lambda + \Delta\lambda resolved at the n-th order diffraction maximum according to Rayleigh criterion
According to Rayleigh's criterion, two spectral lines are just resolved if the principal maximum of one falls on the first secondary minimum of the other.
The n-th principal maximum for wavelength \lambda occurs at angle \theta_n given by d \sin\thet$a_n = n\lambda$.
The first secondary minimum adjacent to the n-th principal maximum occurs when the total path difference across the entire grating of width W (= N d) is (Nn + 1)\lambda, yielding d \sin(\theta_n + \Delta\theta) = \frac{(Nn + 1)\lambda}{N} ... wait, using N d \sin(\theta_n + \Delta\theta) = n\lambda + \lambda / N is more direct.
Differentiating the grating equation d \sin\theta = n\lambda gives d \cos\theta \Delta\theta = n \Delta\lambda.
Equating the angular half-width to the separation yields the resolving power limit.
Result: \frac{\lambda}{\Delta\lambda} = n N
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Grating Equation
$$(a + b) \sin\theta = n\lambda$$
a = slit width, b = opaque space width, $d = a+b = grating element$, \theta = angle of diffraction, n = order, \lambda = wavelength Distance in metres (m), angles in radians or degrees
Normal incidence
Number of Lines per Unit Length
$$N_{0} = \frac{1}{a + b} = \frac{1}{d}$$
$N_{0} = number of lines per unit length$, d = grating element Lines per metre (m^{-1}) or lines per cm
Uniform ruling across the grating surface
Condition for Absent Orders
$$\frac{a + b}{a} = \frac{n}{m}$$
a = slit width, b = opaque space width, n = missing order, m = integer representing single slit minimum Dimensionless ratio
Occurs when a principal maximum coincides with a single-slit minimum
Resolving Power of Grating
$$R = \frac{\lambda}{\Delta\lambda} = n N$$
\lambda = mean wavelength, \Delta\lambda = wavelength difference, n = spectral order, N = total number of lines on grating Dimensionless
$N_{0} = lines per unit length$, n = order, \theta = diffraction angle rad/m
Constant ruling density
Graphs and experiments
Intensity Distribution Curve of a Grating
Draw: Intensity (I) on the y-axis plotted against path difference or diffraction angle on the x-axis, showing sharp principal maxima bounded by a single-slit diffraction envelope.
Interpret: The sharp peaks represent multiple-slit interference maxima whose heights are modulated by the single-slit diffraction envelope.
Exam use: Frequently asked in identifying secondary minima count between principal maxima (N-2).
Determination of Wavelength using Spectrometer
Draw: A schematic layout showing a spectrometer source, collimator, prism table with diffraction grating, and telescope.
Interpret: Allows measurement of angles of diffraction for different spectral lines in various orders.
Exam use: Practical optics experiment questions regarding least count and angle measurement.
Angular Dispersion vs. Order Graph
Draw: Straight line or progressive curve showing the variation of diffraction angle \theta with wavelength \lambda for different orders n.
Interpret: Demonstrates that the grating spectrum is more spread out in higher orders and violet light deviates less than red light.
Exam use: Conceptual interpretation of spectral dispersion.
Solved numericals
TGT core
A diffraction grating has 5000 lines per cm. Calculate the grating element.
Identify given data: Number of lines per unit length N_0 = 5000 lines/cm = 500,000 lines/m.
Use the relation for grating element: $d = 1 / N_0$.
Light of wavelength 589 nm is normally incident on a diffraction grating. If the third-order principal maximum is observed at an angle of 30 degrees, find the total number of lines per cm on the grating.
Write down the grating equation: d \sin\theta = n\lambda.
Substitute given values: $\lambda = 589 \times 10^{-9} \text{ m}$, n = 3, $\theta = 30^{\circ}$.
Calculate lines per metre: $N_0 = 1 / d = \frac{1}{3.534 \times 10^{-6}} = 2.829 \times 10^5 \text{ m}^{-1}$.
Convert to lines per cm: N_0(\text{per cm}) = \frac{2.829 \times 10^5}{100} = 2829 \text{ cm}^{-1}.
Answer: 2829 \text{ lines/cm}
PGT extension
A grating has 4000 lines per cm and is 5 cm wide. Find the resolving power in the second order and the minimum resolvable wavelength difference for sodium doublet lines around 589 nm.
Total number of lines N = N_0 \times W = 4000 \text{ lines/cm} \times 5 \text{ cm} = 20,000.
Order n = 2.
Calculate resolving power R = n N = 2 \times 20,000 = 40,000.
Use $R = \lambda / \Delta\lambda$ to find \Delta\lambda: \Delta\lambda = \lambda / R = \frac{589 \text{ nm}}{40,000} = 0.014725 \text{ nm}.
Answer: Resolving power = 40,000; \Delta\lambda = 0.0147 \text{ nm}
Assumes infinite width of incident plane wavefront and identical slit dimensions across the entire grating surface.
Limitation: Imperfections in ruling (such as periodic errors in groove spacing) introduce ghost lines and false spectra.
Common misconceptions
Common error: The grating spectrum is identical to a prism spectrum in terms of order distribution.
Correct: A grating produces multiple orders of spectra where red deviates more than violet, whereas a prism produces a single normal dispersive spectrum where violet deviates more than red.
Prism dispersion relies on refractive index variation with wavelength (Cauchy's equation), whereas grating dispersion relies on diffraction geometry (d sin\theta = n\lambda).
Common error: Increasing the number of lines per cm increases the wavelength of diffracted light.
Correct: The wavelength of light is an intrinsic property; increasing line density decreases the grating element d, which increases the angle of diffraction for a given wavelength.
The grating equation couples d, \theta, and \lambda; changing d alters \theta, not \lambda.
Quick revision
Grating equation is (a+b)sin\theta = n\lambda.
Resolving power of a grating is given by R = nN.
Absent spectra condition occurs when (a+b)/$a = n/m$.
Angular dispersion is given by d\theta/d$\lambda = n / (d \cos\theta)$.
Must remember
The grating element is defined as $d = a + b$, where '$a$' is the width of the transparent slit and '$b$' is the width of the opaque space.
The principal maxima condition for a transmission grating is given by $(a + b) \sin\theta = n\lambda$.
An absent order occurs when the principal maximum condition coincides with a single-slit diffraction minimum, satisfying $\frac{a + b}{a} = \frac{n}{m}$.
The resolving power of a diffraction grating is given by $R = \frac{\lambda}{\Delta\lambda} = n N$, depending on the spectral order '$n$' and total lines '$N$.
Angular dispersion of a grating is given by $\frac{d\theta}{d\lambda} = \frac{n}{d \cos\theta}$ or $\frac{N_{0} n}{\cos\theta}$.
Glossary
Grating ElementThe combined width of a transparent slit and an opaque ruled space, represented as $d = a + b$.
Principal MaximaSharp bright spectral lines formed when path differences from adjacent slits are integral multiples of the wavelength $\lambda$.
Absent OrderA missing spectral order in a grating spectrum caused by the coincidence of a principal maximum with a single-slit diffraction minimum.
Angular DispersionThe rate of change of angle of diffraction with respect to the wavelength, given by $\frac{d\theta}{d\lambda}$.
Resolving PowerThe ability of a grating to form separate diffraction maxima of two close wavelengths, expressed as $R = \frac{\lambda}{\Delta\lambda} = n N$.
Diffraction Grating vs Prism Spectrum
Parameter
Diffraction Grating
Prism
Dispersive Principle
Diffraction and interference of light
Refraction and variation of refractive index with wavelength
Order of Spectra
Produces multiple orders of spectra
Produces only a single spectrum
Deviation of Colors
Red deviates more than violet ($\theta_r > \theta_v$)
Violet deviates more than red ($\delta_v > \delta_r$)
Dispersive Power
Uniform and depends on grating element and order
Non-uniform and depends on material properties
Practice Quiz
Choose an option to reveal the explanation.
1. What does the grating element '$d$' represent in a plane transmission grating?
Explanation: The grating element $d = a + b$ is defined as the combined width of one transparent slit '$a$' and one opaque ruled space '$b$.
2. The condition for principal maxima in a plane transmission grating is given by:
Explanation: Principal maxima are formed when the path difference between light waves from adjacent slits is an integral multiple of wavelength, expressed as $(a + b) \sin\theta = n\lambda$.
3. If the total number of lines on a grating is '$N$' and the order of the spectrum is '$n$', the resolving power of the grating is:
Explanation: According to Rayleigh's criterion, the resolving power of a diffraction grating is given by $R = \frac{\lambda}{\Delta\lambda} = n N$.
4. What is the angular dispersion of a diffraction grating having grating element '$d$' for wavelength $\lambda$ at diffraction angle $\theta$?
Explanation: Differentiating the grating equation $(a+b)\sin\theta = n\lambda$ with respect to $\lambda$ yields the angular dispersion $\frac{d\theta}{d\lambda} = \frac{n}{d \cos\theta}$.
5. An absent order occurs in a grating spectrum when the condition for a principal maximum coincides with:
Explanation: An absent order occurs when a principal maximum condition $(a+b)\sin\theta = n\lambda$ coincides with a single-slit diffraction minimum $a\sin\theta = m\lambda$.
6. Between any two adjacent principal maxima, the number of secondary minima for '$N$' parallel slits is:
Explanation: For $N$ slits, the intensity distribution contains $N-1$ secondary maxima and $N-2$ secondary minima between any two adjacent principal maxima.
7. What is the maximum possible order '$n_{\text{max}}$' observable with a grating of element '$d$' for normal incidence?
Explanation: Since $\sin\theta \le 1$, the maximum possible order is given by $n_{\text{max}} = \frac{d}{\lambda}$ (taking $\sin\theta = 1$).
8. In a grating spectrum, which color deviates the most from the direct beam?
Explanation: Since $\sin\theta = \frac{n\lambda}{d}$, the angle of diffraction $\theta$ is directly proportional to the wavelength $\lambda$. Red light has the longest wavelength, hence it deviates the most.
9. The number of lines per unit length '$N_{0}$' of a grating is related to the grating element '$d$' by:
Explanation: The number of lines per unit length is $N_{0} = \frac{1}{a + b} = \frac{1}{d}$.
10. If the ratio of the grating element to the slit width is $\frac{a+b}{a} = 3$, which principal maximum will be missing?
Explanation: An order is absent when $\frac{a+b}{a} = \frac{n}{m}$ is an integer. Here $\frac{a+b}{a} = 3$, so for $m=1$, $n = 3$, meaning the $3^{\text{rd}}$ order is absent.
11. What happens to the sharpness of principal maxima as the total number of lines '$N$' on the grating increases?
Explanation: As $N$ increases, the principal maxima become extremely sharp, intense, and well-defined because destructive interference between secondary wavelets becomes more effective.
12. What is the oblique incidence grating equation when light is incident at angle '$i$' to the normal?
Explanation: For oblique incidence at angle '$i$', the total path difference is $d(\sin i + \sin\theta)$, leading to the equation $d(\sin i + \sin\theta) = n\lambda$.
13. Why does a diffraction grating produce multiple orders of spectra whereas a prism produces only one?
Explanation: Multiple orders arise because the grating equation $(a+b)\sin\theta = n\lambda$ can be satisfied for multiple integer values of '$n$ ($n = 1, 2, 3, \dots$).
14. How does increasing the line density '$N_{0}$' of a grating affect the angle of diffraction for a given wavelength?
Explanation: Increasing line density decreases $d$ ($d = \frac{1}{N_{0}}$). Since $\sin\theta = \frac{n\lambda}{d}$, a smaller '$d$' increases $\sin\theta$ and hence the angle of diffraction $\theta$.
15. What is the dimensionless ratio used to determine missing orders when a principal maximum coincides with a single-slit minimum?
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A diffraction grating has 5000 lines per cm. The angle of diffraction for the second-order principal maximum for a monochromatic light of wavelength $5000\text{ \AA}$ is:
2. In a diffraction grating, if the width of the transparent slit '$a$' is equal to the width of the opaque space '$b$', which orders of spectra will be absent?
Explanation: If $a = b$, then $\frac{a+b}{a} = \frac{2a}{a} = 2$. Therefore, $\frac{n}{m} = 2 \implies n = 2m$. For $m = 1, 2, 3, \dots$, the missing orders are $n = 2, 4, 6, \dots$ (even orders).
3. Calculate the maximum possible order of diffraction '$n_{\text{max}}$' for a grating with $4000\text{ lines/cm}$ illuminated by light of wavelength $6000\text{ \AA}$.
Explanation: Grating element $d = \frac{1}{4000\text{ cm}^{-1}} = 2.5 \times 10^{-6}\text{ m}$. Wavelength $\lambda = 6000 \times 10^{-10}\text{ m} = 6 \times 10^{-7}\text{ m}$. Maximum order $n_{\text{max}} = \frac{d}{\lambda} = \frac{2.5 \times 10^{-6}}{6 \times 10^{-7}} = \frac{25}{6} \approx 4.16$. Since '$n$' must be an integer, $n_{\text{max}} = 4$.
4. The resolving power of a diffraction grating of width $5\text{ cm}$ having $600\text{ lines/cm}$ in the second-order spectrum is:
Explanation: Total number of lines $N = \text{width} \times \text{line density} = 5\text{ cm} \times 600\text{ lines/cm} = 3000$. Resolving power $R = n N = 2 \times 3000 = 6000$.
5. What is the angular separation $\Delta\theta$ between two sodium yellow lines of wavelengths $5890\text{ \AA}$ and $5896\text{ \AA}$ in the first order spectrum of a grating having $500\text{ lines/mm}$?
6. A plane transmission grating is normally incident with light. If the grating element is $d$ and wavelength is $\lambda$, what happens to the angular dispersion when the order '$n$' is doubled?
Explanation: Angular dispersion $\frac{d\theta}{d\lambda} = \frac{n}{d \cos\theta}$ is directly proportional to order '$n$'. Doubling '$n$' doubles the angular dispersion.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. In a transmission grating experiment, the diffraction pattern is a combination of:
Explanation: The resultant intensity distribution is a combination of single-slit diffraction (acting as an envelope) and multiple-slit interference (producing sharp principal maxima).
2. If the total number of slits '$N$' in a grating is doubled while keeping the grating element constant, how does the intensity of principal maxima change?
Explanation: The intensity of principal maxima is proportional to $N^{2}$. Doubling $N$ increases the intensity by a factor of $2^{2} = 4$.
3. The condition for absent spectra in a diffraction grating depends strictly on:
Explanation: Absent orders occur when $\frac{a+b}{a} = \frac{n}{m}$ is an integer, which depends directly on the relative widths '$a$' and '$b$.
4. What is the dispersive power '$D$' of a grating defined as?
Explanation: Dispersive power is defined as the rate of change of angle of diffraction with wavelength, $D = \frac{d\theta}{d\lambda}$.
5. Calculate the resolving power of a grating having $4000\text{ lines/cm}$ across a $2.5\text{ cm}$ width in the $3^{\text{rd}}$ order.
Explanation: Total lines $N = 4000 \times 2.5 = 10000$. Resolving power $R = n N = 3 \times 10000 = 30000$.
6. Which of the following statements is correct regarding the spectrum produced by a diffraction grating?
Explanation: Since $\sin\theta = \frac{n\lambda}{d}$, longer wavelengths (red) have larger diffraction angles than shorter wavelengths (violet).
7. If the grating element '$d$' is smaller than the wavelength $\lambda$ of incident light:
Explanation: Since $n_{\text{max}} = \frac{d}{\lambda}$, if $d < \lambda$, then $n_{\text{max}} < 1$, meaning only the zero-order maximum ($n = 0$) can be observed.
8. Between two adjacent principal maxima of a grating with '$N$' slits, how many secondary maxima are present?
Explanation: There are $N-1$ secondary maxima between any two adjacent principal maxima.
9. If light is incident obliquely at angle '$i$' on a grating, the path difference between waves from two consecutive slits is:
Explanation: For oblique incidence, the path difference contributed by the incident and diffracted wavefronts is $d(\sin i + \sin\theta)$.
10. Which parameter does NOT directly affect the resolving power of a diffraction grating?
Explanation: Resolving power is given by $R = n N$, which depends on order '$n$' and total lines '$N$', but not directly on the individual opaque space width '$b$.