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Diffraction Grating Minima Condition

This equation determines the angular positions of the minima in a multiple-slit diffraction pattern.

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d sin θ = \pm \frac{m N + 1}{N} λ

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Core idea

Overview

In a diffraction grating with N slits, the intensity pattern is characterized by sharp principal maxima and smaller secondary minima. This formula identifies the angles at which destructive interference occurs between the slits, effectively defining the dark regions between the principal maxima. The variable m represents an integer order, while N is the total number of slits in the grating.

When to use: Use this when calculating the angular position of the dark fringes (minima) in the diffraction pattern produced by a grating with a finite number of slits.

Why it matters: It allows for the precise characterization of the intensity distribution of light passing through a grating, which is essential for designing spectrometers and analyzing spectral resolution.

Symbols

Variables

d = Grating Spacing, $θ$ = Diffraction Angle, m = Order Number, N = Number of Slits, $λ$ = Wavelength

d

Grating Spacing

m

θ

Diffraction Angle

rad

m

Order Number

dimensionless

N

Number of Slits

dimensionless

λ

Wavelength

m

Walkthrough

Derivation

Derivation of Minima adjacent to principle maxima

This derivation explains how to find the locations of the faint minima that appear next to the bright principal maxima in a multiple-slit interference pattern.

The light source is monochromatic (single wavelength, $λ$ ).
The slits are identical and uniformly illuminated.
The interference is observed on a distant screen, so we can assume $sin$ $θ$ $\approx$ $θ$ for small angles.
The number of slits is large (N).
The spacing between adjacent slits is 'd'.

Condition for Principal Maxima

In a multiple-slit interference pattern, the bright principal maxima occur when the path difference between light waves from adjacent slits is an integer multiple of the wavelength. This condition leads to constructive interference from all slits simultaneously.

sin θ = \frac{mλ}{d}, m = 0, \pm 1, \pm 2, ...

Note: These are the strongest bright spots in the pattern.

Condition for Minima in a Double-Slit Pattern

While not directly applicable to N slits, it's useful to remember the condition for minima in a double-slit scenario: destructive interference. This happens when the path difference is a half-integer multiple of the wavelength.

d sin θ = (m^{'} + \frac{1}{2}) λ, m^{'} = 0, \pm 1, \pm 2, ...

Note: This is the condition for the dark bands between the bright fringes.

Considering N slits and the effect of many slits

The intensity in a multiple-slit interference pattern is given by this formula, where 'a' is the width of each slit and 'N' is the number of slits. The term $sin$ ( $β$ /2) relates to single-slit diffraction, and $sin$ ( $α$ /2) relates to the spacing between slits. Minima will occur when the numerator $sin$ ( $β$ /2) is zero, which corresponds to the single-slit diffraction minima.

Intensity \propto (\frac{sin ( \frac{β}{2} )}{sin ( \frac{α}{2} )})^{2}, where α = \frac{2 π d sin θ}{λ} and β = \frac{2 π a sin θ}{λ}

Note: These minima are the 'missing orders' that can occur.

Considering the regions between principal maxima

The principal maxima are very sharp and intense when N is large. Between these strong maxima, there are fainter minima. These minima are not simply where the intensity is zero due to single-slit diffraction effects being dominant. Instead, they are points where the interference from the N slits leads to a local dip in intensity.

We are looking for minima between principal maxima, not necessarily total destructive interference.

Note: Imagine the peaks are very steep; the dips between them are shallow.

Intensity minima in N-slit interference

In an N-slit system, there are N-1 points of destructive interference between the light from adjacent slits that contribute to the overall pattern. The principal maxima are where all N waves interfere constructively. The minima adjacent to these principal maxima are found when the phase difference between adjacent slits is such that there are exactly $k$ points of destructive interference within each $m$ -th order principal maximum.

The intensity minima between principal maxima occur when there are N - 1 points of destructive interference between adjacent slits, resulting in zero total intensity for a moment before rebuilding.

Note: This is a more advanced point about the structure of the interference pattern.

Mathematical condition for minima adjacent to principal maxima

By analyzing the derivative of the intensity function or considering the phase differences that lead to a local dip in intensity between principal maxima, we arrive at this condition. Essentially, these minima occur at angles slightly shifted from the principal maxima, where the interference from the N slits is no longer perfectly constructive, but not yet fully destructive due to the single-slit diffraction envelope.

d sin θ = \pm \frac{m N + 1}{N} λ, m = 0, \pm 1, \pm 2, ...

Note: The ' $\pm$ ' indicates that these minima appear on both sides of the principal maxima.

Result

d sin θ = \pm \frac{m N + 1}{N} λ, m = 0, \pm 1, \pm 2, ...

Source: Hecht, E. (2017). Optics. Pearson.

Free formulas

Rearrangements

Solve for $d$

Make d the subject

d = \frac{\pm ( m N + 1 ) λ}{N sin θ}

Divide both sides by $sin$ $θ$ to isolate d.

Difficulty: 1/5

Solve for $θ$

Make $θ$ the subject

θ = arcsin (\frac{\pm ( m N + 1 ) λ}{N d})

Take the arcsin of both sides after isolating $sin$ $θ$ .

Difficulty: 2/5

Solve for $m$

Make m the subject

m = \frac{1}{N} (\frac{N d sin θ}{λ} \mp 1)

Isolate the term with m, then solve for m.

Difficulty: 3/5

Solve for $N$

Make N the subject

N = \frac{\pm λ}{d sin θ \mp mλ}

Rearrange the equation to form a quadratic equation in N and solve.

Difficulty: 4/5

Solve for $λ$

Make $λ$ the subject

λ = \frac{N d sin θ}{\pm ( m N + 1 )}

Isolate $λ$ by dividing both sides and rearranging terms.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

This graph shows how the diffraction angle ($\theta$) changes with wavelength ($\lambda$). For a student, this means that if you use light with a longer wavelength, the light will spread out at a larger angle after passing through the diffraction grating. The most important feature is that for a given order ($m$) and grating spacing ($d$), the diffraction angle increases as the wavelength increases.

Graph type: linear

Why it behaves this way

Intuition

Imagine standing on a stage with a grid of spotlights (the slits). When you look out at a screen, you see bright spots (maxima) and dark patches (minima). The brightest spots, the 'principal maxima,' are like the main spotlight beams. This equation helps you find the dimmer spots (minima) that are *just* on either side of those main beams. Think of it as finding the edges of the brightest floodlights, where the light fades away before the next dimmer beam or the darkness sets in.

d

Slit separation

This is the distance between the centers of any two adjacent slits in your grating. Think of it as the spacing between your spotlights.

sin θ

Sine of the diffraction angle

This relates to the angle at which you are looking at the screen from the slits. A larger

sin

θ

means you're looking at a point further away from the center of the pattern. Think of it as how far 'off-center' you're looking.

m

Order of diffraction

This is an integer that describes which bright spot (principal maximum) you are considering. m=0 is the central bright spot, m=1 is the first bright spot to the side, m=2 is the second, and so on. In this context, we are looking for minima *next to* these principal maxima.

N

Number of slits

This is the total number of slits in your diffraction grating. It tells you how many spotlights are contributing to the overall pattern. A larger N usually leads to sharper maxima and minima.

λ

Wavelength of light

This is the color of the light you are using. Different colors (wavelengths) will produce slightly different patterns of bright and dark spots.

Signs and relationships

±: The '+' and '-' signs indicate that there are two locations where these minima occur, symmetrically on either side of the principal maximum.

Free study cues

Insight

Canonical usage

This equation relates the grating spacing, diffraction angle, wavelength, and number of slits to find the angles at which destructive interference occurs for a diffraction grating.

Common confusion

Students may confuse the condition for maxima ( $d sin θ = mλ$ ) with the condition for minima, or misinterpret the meaning of 'm' in the context of minima adjacent to principal maxima.

Dimension note

The number of slits (N) and the order number (m) are inherently dimensionless quantities.

Unit systems

$d$ m · The grating spacing is the distance between adjacent slits.

thetarad · The diffraction angle is typically measured in radians for calculations, though degrees may be used in reporting.

lambdam · The wavelength of the incident light.

One free problem

Practice Problem

A diffraction grating has 500 slits (N=500) and a spacing d of 2.0 micrometers. For the first order (m=1) minimum, what is the angle theta in radians for light with a wavelength of 500 nm (5e-7 m)?

Grating Spacing0.000002 m

Order Number1 dimensionless

Number of Slits500 dimensionless

Wavelength5e-7 m

Solve for: theta

Hint: Calculate the right side of the equation first, then take the arcsin.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Determining the angular width of spectral lines in a laboratory spectrometer to ensure high resolution for chemical analysis.

Study smarter

Tips

Ensure the units for wavelength and grating spacing are consistent.
Remember that the formula accounts for the finite number of slits N.
The result for theta is in radians; convert to degrees if necessary.

Avoid these traps

Common Mistakes

Confusing the minima condition for a grating with the single-slit diffraction minima condition.
Forgetting to include the N-dependent term, which is only relevant for finite-slit gratings.
Using degrees instead of radians when calculating the sine function in a calculator.

Common questions

Frequently Asked Questions

This derivation explains how to find the locations of the faint minima that appear next to the bright principal maxima in a multiple-slit interference pattern.

Use this when calculating the angular position of the dark fringes (minima) in the diffraction pattern produced by a grating with a finite number of slits.

It allows for the precise characterization of the intensity distribution of light passing through a grating, which is essential for designing spectrometers and analyzing spectral resolution.

Confusing the minima condition for a grating with the single-slit diffraction minima condition. Forgetting to include the N-dependent term, which is only relevant for finite-slit gratings. Using degrees instead of radians when calculating the sine function in a calculator.

Determining the angular width of spectral lines in a laboratory spectrometer to ensure high resolution for chemical analysis.

Ensure the units for wavelength and grating spacing are consistent. Remember that the formula accounts for the finite number of slits N. The result for theta is in radians; convert to degrees if necessary.

References

Sources

Hecht, E. (2017). Optics (5th ed.). Pearson.
Young, H. D., & Freedman, R. A. (2020). University Physics with Modern Physics (15th ed.). Pearson.
Hecht, Eugene. Optics. 5th ed., Pearson, 2017.
NIST CODATA
IUPAC Gold Book
Wikipedia: Diffraction Grating
Jenkins, Francis A., and Harvey E. White. Fundamentals of Optics. 4th ed., McGraw-Hill, 1976.
NIST Digital Library of Mathematical Functions, Section 25.11, Diffraction Gratings.

Diffraction Grating Minima Condition

Overview

Variables

Derivation

Condition for Principal Maxima

Condition for Minima in a Double-Slit Pattern

Considering N slits and the effect of many slits

Considering the regions between principal maxima

Intensity minima in N-slit interference

Mathematical condition for minima adjacent to principal maxima

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Grating Equation

Frequently Asked Questions

Sources