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Conditions for Sustained Interference of Light

Sustained interference refers to a steady, time-independent spatial distribution of light intensity where distinct maxima and minima are permanently maintained. It requires coherent light sources that emit waves of identical frequency with a constant phase difference.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Definition of interference, coherent sources, phase difference, path difference, and basic conditions like constant wavelength and same frequency.

PGT extension

Mathematical formulation of intensity distribution, visibility of fringes, broad/narrow source requirements, and effects of polarization on interference.

Prerequisites

Superposition principle of waves
Wavefront and Huygens' principle
Phase and path difference relations

Requirement of Coherent Sources

Two independent sources of light cannot emit waves with a constant phase relationship because light emission is a molecular/atomic phenomenon occurring in independent, random bursts (each lasting ~10^-8 s).
For sustained interference, the phase difference delta(phi) between the two interfering waves must remain invariant with time at any given point in space.
Coherent sources are practically achieved by dividing a single wavefront or amplitude from a single primary source into two secondary sources.

Same Frequency and Wavelength

The interfering waves must have identical frequencies (nu) and wavelengths (lambda) in the medium.
If frequencies differ, the phase difference delta(phi) = 2π(nu_1 - nu_2)t will fluctuate rapidly with time, averaging out the intensity at every point to a uniform background (I_max + I_min)/2.
This rapid fluctuation prevents the formation of observable stationary fringe patterns.

Constant Phase Difference

The initial phase difference between the two sources must remain constant over time.
If delta(phi) changes randomly, the positions of maxima and minima shift continuously, resulting in uniform illumination.
Derived from a single parent source, division of wavefront (Young's double slit) or division of amplitude (thin films) inherently guarantees this condition.

Narrow Sources and Amplitude Condition

Sources must be sufficiently narrow; a broad source is equivalent to a large number of independent adjacent sources, resulting in overlapping fringe patterns that wash out the contrast.
For maximum contrast (visibility) of fringes, the amplitudes (a_1 and a_2) of the interfering waves should be equal or nearly equal, yielding complete darkness at destructive interference minima.
The sources must be close to each other to produce fringe widths large enough to be clearly observed.

Derivations

Resultant Intensity and Condition for Sustained Interference

Conditions: Two harmonic light waves of identical frequency meeting at a point with a constant phase difference.

Let the displacement of the two waves be given by $y_1 = a_1 \sin(\omega t)$ and $y_2 = a_2 \sin(\omega t + \phi)$, where \phi is the constant phase difference.
By the superposition principle, the resultant displacement is $y = y_1 + y_2 = a_1 \sin(\omega t) + a_2 \sin(\omega t + \phi)$.
Expanding and rearranging terms: $y = (a_1 + a_2 \cos\phi)\sin(\omega t) + (a_2 \sin\phi)\cos(\omega t)$.
Let a_1 + a_2 \cos\phi = A \cos\theta and a_2 \sin\phi = A \sin\theta, where A is the resultant amplitude.
Squaring and adding gives the resultant amplitude squared: $A^2 = a_1^2 + a_2^2 + 2a_1a_2 \cos\phi$.
Since intensity I is proportional to amplitude squared ($I = kA^2)$, the resultant intensity is $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$.

Result: $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$, where intensity distribution is stationary if \phi is time-independent.

Relation Between Path Difference and Phase Difference

Conditions: Monochromatic plane or spherical wave propagating in a homogeneous medium.

A path difference of \lambda corresponds to a phase difference of 2\pi radians.
Therefore, a unit path difference corresponds to a phase difference of 2\pi / \lambda.
For a general path difference \Delta x, the corresponding phase difference \phi is obtained by multiplying by the path difference.
\phi = \frac{2\pi}{\lambda} \Delta x.

Result: \phi = \frac{2\pi}{\lambda} \Delta x

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Resultant Intensity in Interference	$$I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$$	I_1, $I_2 = individual intensities$, \phi = phase difference W/m^2	Coherent sources with constant phase difference
Phase and Path Difference Relation	$$\phi = \frac{2\pi}{\lambda} \Delta x$$	\phi = phase difference, \lambda = wavelength, \Delta x = path difference radians (phase), meters (path)	Homogeneous isotropic medium
Condition for Constructive Interference (Maxima)	$$\Delta x = n\lambda \quad \text{or} \quad \phi = 2n\pi$$	n = 0, 1, 2, 3... meters, radians	Maximum intensity
Condition for Destructive Interference (Minima)	$$\Delta x = \left(2n + 1\right)\frac{\lambda}{2} \quad \text{or} \quad \phi = \left(2n + 1\right)\pi$$	n = 0, 1, 2, 3... meters, radians	Minimum intensity
Fringe Width Expression	$$\beta = \frac{\lambda D}{d}$$	\lambda = wavelength, D = distance to screen, d = slit separation meters	D >> d
Maximum Intensity (Equal Amplitudes)	$$I_{\max} = 4I_0$$	$I_0 = intensity from each individual source (I_1 = I_2 = I_0)$ W/m^2	$a_1 = a_2$
Minimum Intensity (Equal Amplitudes)	$$I_{\min} = 0$$	$I_0 = intensity from each individual source$ W/m^2	$a_1 = a_2$, complete destructive interference
Fringe Visibility (Contrast)	$$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}}$$	I_{\max}, $I_{\min} = maximum$ and minimum intensities dimensionless	Measure of fringe sharpness

Graphs and experiments

Intensity vs Phase Difference / Position

Draw: A sinusoidal-like or squared-cosine intensity distribution curve showing sharp, equally spaced peaks of height 4I_0 and zero minima.

Interpret: Demonstrates the conservation of energy where constructive interference peaks are four times the single-source intensity, balanced by zero-intensity destructive minima.

Exam use: Tested in identifying maximum/minimum positions and calculating visibility ratio.

Young's Double-Slit Experiment (YDSE Setup)

Draw: A monochromatic source S illuminating a single slit S_0, which in turn illuminates two closely spaced narrow slits S_1 and S_2, forming alternate bright and dark fringes on a screen at distance D.

Interpret: Shows wavefront division to generate two coherent secondary sources required for sustained interference.

Exam use: Conceptual questions on why a single source and narrow slits are mandatory.

Effect of Broad Source on Interference Fringes

Draw: Overlapping fringe patterns resulting in overall uniform illumination on the screen.

Interpret: A broad source acts as multiple independent point sources, creating independent fringe patterns that wash out the contrast and destroy sustained interference.

Exam use: Frequently asked conceptual question in TGT/PGT regarding source constraints.

Solved numericals

TGT core

In a Young's double-slit experiment, the ratio of maximum to minimum intensity in the interference pattern is 9:1. What is the ratio of the amplitudes of the two interfering waves?

Given the ratio of maximum to minimum intensity: I_max / $I_min = 9 / 1$.
We know that I_max proportional to (a_1 + a_2)^2 and I_min proportional to (a_1 - a_2)^2.
Therefore, (a_1 + a_2)^2 / (a_1 - $a_2)^2 = 9 / 1$.
Taking the square root on both sides: (a_1 + a_2) / (a_1 - $a_2) = 3 / 1$.
Cross-multiplying: a_1 + $a_2 = 3a_1 - 3a_2 => 2a_1 = 4a_2 => a_1 / a_2 = 2 / 1$.

Answer: 2:1

PGT extension

Two coherent sources of light of intensity ratio 4:1 interfere. Calculate the ratio of the intensity at the maxima to that at the minima in the interference pattern, and also find the fringe visibility.

Let I_1 / $I_2 = 4 / 1$. Let $I_1 = 4I_0$ and $I_2 = I_0$.
The ratio of amplitudes a_1 / $a_2 = \sqrt{I_1 / I_2} = \sqrt{4} / 1 = 2 / 1$.
Maximum intensity $I_max = (a_1 + a_2)^2 = (2 + 1)^2 = 9 units (in terms of a_2^2)$.
Minimum intensity $I_min = (a_1 - a_2)^2 = (2 - 1)^2 = 1 unit$.
Ratio I_max / $I_min = 9 / 1$.
Visibility $V = (I_max - I_min) / (I_max + I_min) = (9 - 1) / (9 + 1) = 8 / 10 = 0.8$.

Answer: Intensity ratio = 9:1, Visibility = 0.8

PGT extension

Two independent sodium lamps emitting light of wavelength 589 nm are used in an interference setup. State with reason whether a sustained interference pattern will be observed.

Identify that two independent sodium lamps are separate light sources.
Independent sources undergo independent atomic transitions with random phase jumps every ~10^-8 seconds.
Thus, the phase difference between the waves emitted by the two lamps changes randomly with time.
A sustained or stationary interference pattern cannot be formed; instead, rapid temporal averaging results in uniform illumination.

Answer: No sustained interference pattern will be observed because independent sources are incoherent.

Assumptions and limitations

Assumes monochromatic light sources emitting purely sinusoidal wave trains.
Limited by the finite coherence length and coherence time of real light sources, meaning interference is lost if path differences exceed coherence length.

Common misconceptions

Common error: Two independent identical bulbs of 100W each can produce a sustained interference pattern.

Correct: Independent bulbs cannot produce interference.

Light emission is independent and random per atom; independent sources lack a constant phase relationship (incoherence).

Common error: Interference violates the law of conservation of energy.

Correct: Energy is conserved.

Dark fringes have zero intensity, but bright fringes have four times the single-source intensity (for equal amplitudes), redistributing energy rather than creating or destroying it.

Common error: Two light waves polarized in mutually perpendicular planes can interfere.

Correct: Perpendicularly polarized light waves do not interfere.

Interference requires oscillations along the same line or parallel components of electric vectors; perpendicular vectors result in a constant resultant intensity equal to the sum of individual intensities.

Quick revision

Sustained interference strictly requires two coherent sources with identical frequency and a constant phase difference.
Resultant Intensity in Interference: $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$
Phase and Path Difference Relation: \phi = \frac{2\pi}{\lambda} \Delta x

Must remember

Two independent sources of light cannot emit waves with a constant phase relationship because light emission is a random molecular or atomic phenomenon.
For sustained interference, the interfering waves must have the exact same frequency, wavelength, and a constant phase difference.
Division of wavefront or division of amplitude from a single parent source are the primary practical methods to achieve coherent sources.
Maximum intensity for equal individual intensities $I_0$ is given by $I_{\max} = 4I_0$, and minimum intensity is $I_{\min} = 0$.
Fringe visibility or contrast is defined as $V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}}$.

Glossary

Coherent SourcesSources of light that emit waves of the same frequency, wavelength, and maintain a constant phase difference with time.

Phase Difference ($\phi$)The difference in phase between two waves at a given point in space, related to path difference by $\phi = \frac{2\pi}{\lambda} \Delta x$.

Fringe Visibility ($V$)A dimensionless measure of the sharpness and contrast of an interference fringe pattern, defined as $(I_{\max} - I_{\min}) / (I_{\max} + I_{\min})$.

Division of AmplitudeA method of producing coherent sources by splitting the amplitude of a single incident wave into reflected and refracted parts (e.g., thin films).

Constructive vs Destructive Interference

Feature	Constructive Interference	Destructive Interference
Path Difference ($\Delta x$)	$n\lambda$ (where $n = 0, 1, 2, \dots$)	$(2n + 1)\frac{\lambda}{2}$ (where $n = 0, 1, 2, \dots$)
Phase Difference ($\phi$)	$2n\pi$	$(2n + 1)\pi$
Resultant Intensity (Equal amplitudes)	Maximum ($4I_0$)	Minimum ($0$)

Practice Quiz

Choose an option to reveal the explanation.

1. Why can two independent sodium lamps not produce a sustained interference pattern?

Explanation: Two independent sources cannot emit waves with a constant phase relationship because light emission occurs in random atomic bursts lasting about $10^{-8}$ s.

2. What is the primary condition regarding the frequency of two light waves to observe sustained interference?

Explanation: The interfering waves must have identical frequencies and wavelengths; differing frequencies cause rapid phase fluctuation and wash out the pattern.

3. If two interfering waves have individual intensities $I_1$ and $I_2$, what is the expression for resultant intensity $I$ at a point where the phase difference is $\phi$?

Explanation: The general resultant intensity formula derived from superposition is $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$.

4. What is the relationship between path difference $\Delta x$ and phase difference $\phi$ for a homogeneous isotropic medium?

Explanation: The phase difference is proportional to path difference via the relation $\phi = \frac{2\pi}{\lambda} \Delta x$.

5. What is the path difference condition for constructive interference (maximum intensity)?

Explanation: Constructive interference occurs when path difference is an integral multiple of wavelength, i.e., $\Delta x = n\lambda$.

6. What is the minimum intensity for two interfering waves of equal individual intensity $I_0$ undergoing destructive interference?

Explanation: For equal amplitudes, complete destructive interference yields $I_{\min} = 0$.

7. How is fringe visibility ($V$) defined in terms of maximum and minimum intensities?

Explanation: Fringe visibility or contrast is given by $V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}}$.

8. Which of the following methods guarantees a constant phase difference between two secondary sources?

Explanation: Deriving two secondary sources from a single parent source via division of wavefront or amplitude inherently maintains a constant initial phase difference.

9. Why must the sources be narrow for sustained and clear interference?

Explanation: A broad source is equivalent to a large number of independent adjacent sources, causing overlapping fringe patterns that wash out contrast.

10. What happens to the fringe pattern if two light waves are polarized in mutually perpendicular planes?

Explanation: Perpendicularly polarized light waves do not interfere because interference requires oscillations along the same line or parallel components of electric vectors.

11. If the amplitudes of two interfering beams are $a_1 = 3a$ and $a_2 = a$, what is the ratio of maximum to minimum intensity ($I_{\max} / I_{\min}$)?

Explanation: $I_{\max} \propto (3a + a)^2 = 16a^2$ and $I_{\min} \propto (3a - a)^2 = 4a^2$. Thus, the ratio is $16a^2 / 4a^2 = 4:1$ wait, let's recalculate: $(3+1)^2 / (3-1)^2 = 16 / 4 = 4:1$. Let's check option indices: option B is $4:1$.

12. Does interference violate the law of conservation of energy?

Explanation: Energy is conserved; dark fringes have zero intensity while bright fringes have enhanced intensity, representing a spatial redistribution of energy.

13. What is the condition for destructive interference in terms of phase difference $\phi$?

Explanation: Destructive interference occurs when phase difference is an odd multiple of $\pi$, i.e., $\phi = (2n + 1)\pi$.

14. What is the effect of unequal amplitudes ($a_1 \neq a_2$) on the visibility of interference fringes?

Explanation: When amplitudes are unequal, complete destructive interference is not achieved ($I_{\min} \neq 0$), which reduces fringe visibility ($V < 1$).

15. What is the expression for fringe width $\beta$ in Young's double slit experiment?

Explanation: Fringe width is given by $\beta = \frac{\lambda D}{d}$, where $D$ is screen distance and $d$ is slit separation.

PYQ Patterns

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

1. For sustained interference of light, the essential condition is that the two sources must:

Explanation: Sustained interference strictly requires sources to emit waves of the same frequency/wavelength with a constant phase difference.

2. Two coherent sources of intensity ratio $4:1$ produce interference fringes. The ratio of maximum to minimum intensity in the fringe system is:

Explanation: Given $I_1/I_2 = 4/1$, amplitudes ratio $a_1/a_2 = 2/1$. $I_{\max}/I_{\min} = (a_1+a_2)^2 / (a_1-a_2)^2 = (2+1)^2 / (2-1)^2 = 9/1$.

3. The phase difference corresponding to a path difference of $\frac{\lambda}{4}$ is:

Explanation: Using $\phi = \frac{2\pi}{\lambda} \Delta x$, substituting $\Delta x = \frac{\lambda}{4}$ gives $\phi = \frac{2\pi}{\lambda} \times \frac{\lambda}{4} = \frac{\pi}{2}$ radians.

4. Why is it difficult to observe interference with two independent light bulbs?

Explanation: Independent sources lack a constant phase relationship due to random atomic emission bursts, causing the interference pattern to average out into uniform illumination.

5. In an interference pattern, if the amplitudes of two interfering waves are equal ($a_1 = a_2 = a$), the ratio of maximum intensity to minimum intensity is:

Explanation: $I_{\max} \propto (2a)^2 = 4a^2$ and $I_{\min} \propto 0$. Thus, $I_{\max}/I_{\min} = 4a^2 / 0 = \infty$.

6. Which of the following phenomena proves the transverse nature of light waves?

Explanation: Polarization uniquely demonstrates the transverse nature of light waves, as longitudinal waves cannot be polarized.

Timed Topic Test

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

1. Two waves having intensities in the ratio $9:1$ produce interference. What is the ratio of maximum to minimum intensity?

Explanation: Amplitudes ratio $a_1/a_2 = \sqrt{9}/\sqrt{1} = 3/1$. $I_{\max}/I_{\min} = (3+1)^2 / (3-1)^2 = 16/4 = 4:1$.

2. What path difference corresponds to a phase difference of $5\pi$ radians?

Explanation: Using $\Delta x = \frac{\lambda}{2\pi} \phi$, substituting $\phi = 5\pi$ yields $\Delta x = \frac{\lambda}{2\pi} \times 5\pi = \frac{5\lambda}{2}$.

3. If the source in Young's double slit experiment is white light, what change occurs in the fringe pattern?

Explanation: With white light, the central maximum is white (all wavelengths constructively interfere at $\Delta x = 0$), surrounded by colored fringes because different wavelengths have different fringe widths.

4. What is the visibility of fringes when the two interfering beams have equal intensities ($I_1 = I_2$)?

Explanation: When $I_1 = I_2$, $I_{\min} = 0$, so visibility $V = (I_{\max} - 0) / (I_{\max} + 0) = 1$.

5. Which condition is essential for obtaining distinct and broad interference fringes?

Explanation: Fringe width $\beta = \frac{\lambda D}{d}$ increases when $d$ (slit separation) is small and $D$ (screen distance) is large.

6. In an interference experiment, third bright fringe is obtained at a point with path difference:

Explanation: For constructive interference (bright fringes), $\Delta x = n\lambda$. For the third bright fringe ($n = 3$), $\Delta x = 3\lambda$ (assuming $n=0$ is central maximum, or $n=1,2,3$... depending on convention; standard $n\lambda$ gives $3\lambda$).

7. Two monochromatic light beams of intensity $I$ superimpose with a phase difference of $\pi/3$. What is the resultant intensity?

Explanation: $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi = I + I + 2I\cos(\pi/3) = 2I + 2I(1/2) = 3I$.

8. Why is division of amplitude preferred in thin film interference experiments like Newton's rings?

Explanation: Division of amplitude (such as reflection from thin films) allows the use of broad extended sources because each point on the film creates coherent counterparts.

9. If the distance between slits $d$ in Young's double slit experiment is halved and the distance to the screen $D$ is doubled, the fringe width becomes:

Explanation: New fringe width $\beta' = \frac{\lambda (2D)}{(d/2)} = 4 \left(\frac{\lambda D}{d}\right) = 4\beta$.

10. What happens to the interference pattern if the two interfering light waves have random polarization vectors perpendicular to each other?

Explanation: Mutually perpendicular polarization states cannot interfere; the resultant intensity is simply the sum of individual intensities, resulting in uniform background illumination ($V = 0$).