Interference intensity

Blocking one beam can make a detector brighter. To understand why, follow the two fields before adding up what the detector measures.

We develop a method that handles unequal beams, phase ambiguity, path changes and real observation conditions.

What you will learn to do

Convert optical path changes into phase without counting a replaced medium twice.
Calculate intensity for unequal coherent beams and diagnose incomplete darkness.
Infer amplitude ratios from measured extrema.
Distinguish coherence, polarization and detector averaging.

Useful before this chapterInterference, diffraction and the structure of a fringe pattern →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Interference: phase, brightness and contrast
Learning goals:
- Convert optical path changes into phase without counting a replaced medium twice.
- Calculate intensity for unequal coherent beams and diagnose incomplete darkness.
- Infer amplitude ratios from measured extrema.
- Distinguish coherence, polarization and detector averaging.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

If intensity quadruples, does electric-field amplitude quadruple too?

Compare your reasoning

No. Intensity is proportional to amplitude squared in a fixed observation medium, so a fourfold intensity means a doubled field amplitude.

Revisit the explanation →

01 / Interference: phase, brightness and contrast

Keep track of where each oscillation is in its cycle

Imagine two people pushing a swing. The number of pushes is not the whole story: their timing matters. A wave phase records that timing as an angle. Advancing through one complete oscillation changes phase by 2π radians, or 360°, and returns the displacement or electric field to the same part of its cycle.

For a travelling sinusoid, phase depends on both position and time. At a fixed detector, two waves of the same frequency can retain a fixed phase difference even while both fields oscillate rapidly. We compare their fields at the same place and time. Comparing one crest here with a later crest elsewhere answers a different question.

Choose a reference wave and declare which phase is subtracted from which. Interchanging that choice reverses the sign of phase difference; the two-wave intensity stays the same because cosine is even. A signed phase is still useful when following how a pattern moves or reconstructing the source timing.

y(x,t)=A\cos(kx-\omega t+\phi_0),\qquad k=\frac{2\pi}{\lambda},\quad\omega=2\pi f

The argument of cosine is dimensionless. For this sign convention, a constant-phase point moves toward increasing x.

WORKED EXAMPLE

One quarter of a wavelength is not half a cycle

Two points on a sinusoidal wave are separated by λ/4 along its direction at the same instant. What is the magnitude of their phase difference?

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Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Interference: phase, brightness and contrast
Focus: Keep track of where each oscillation is in its cycle
Learning goals:
- Convert optical path changes into phase without counting a replaced medium twice.
- Calculate intensity for unequal coherent beams and diagnose incomplete darkness.
- Infer amplitude ratios from measured extrema.
- Distinguish coherence, polarization and detector averaging.
Equations in this section (LaTeX):
y(x,t)=A\cos(kx-\omega t+\phi_0),\qquad k=\frac{2\pi}{\lambda},\quad\omega=2\pi f
Meaning: The argument of cosine is dimensionless. For this sign convention, a constant-phase point moves toward increasing x.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Interference: phase, brightness and contrast

Convert a journey into phase, medium by medium

Light spends more oscillation cycles crossing a given distance in a medium with a larger refractive index. Its frequency is unchanged at a stationary boundary, while its wavelength becomes λ₀/n. Optical path length packages this information: a physical segment L contributes nL. Add those contributions along the route before comparing routes.

An inserted plate replaces a segment of the original medium; it does not add its full optical path on top of the old one. If an air segment of length t is replaced by glass, the extra optical path is (n−1)t. For a varying medium, the corresponding quantity is the integral of n along the path.

Propagation is only one part of phase. The source may already have a phase offset, and reflection may introduce another. Keep these terms in a small ledger rather than hiding all of them inside a memorized bright-fringe rule. The next chapter will use precisely this ledger for thin films.

\mathrm{OPL}=\int n\,ds,\qquad \delta=\frac{2\pi\Delta\mathrm{OPL}}{\lambda_0}+\delta_{\rm source}+\delta_{\rm reflection}

Here ΔOPL is route 2 minus route 1; δ is a consistent accumulated phase lag, used modulo 2π for intensity. λ₀ is the vacuum wavelength.

WORKED EXAMPLE

A plate changes phase even when the route looks unchanged

A 1.20 μm plate of index 1.50 replaces air in one normally incident beam. The vacuum wavelength is 600 nm. Find the added optical path and phase lag.

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STOP & THINK

A plate of index 1.4 replaces 1.5 μm of air at normal incidence. What extra optical path does it add, in nm?

Answer in nm.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Interference: phase, brightness and contrast
Focus: Convert a journey into phase, medium by medium
Learning goals:
- Convert optical path changes into phase without counting a replaced medium twice.
- Calculate intensity for unequal coherent beams and diagnose incomplete darkness.
- Infer amplitude ratios from measured extrema.
- Distinguish coherence, polarization and detector averaging.
Equations in this section (LaTeX):
\mathrm{OPL}=\int n\,ds,\qquad \delta=\frac{2\pi\Delta\mathrm{OPL}}{\lambda_0}+\delta_{\rm source}+\delta_{\rm reflection}
Meaning: Here ΔOPL is route 2 minus route 1; δ is a consistent accumulated phase lag, used modulo 2π for intensity. λ₀ is the vacuum wavelength.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

03 / Interference: phase, brightness and contrast

Add fields first; square afterwards

A detector measures a quantity proportional to the time average of the squared electric field. The fields themselves obey superposition. These are two different operations, and their order matters. If E₁ and E₂ arrive together, the detector responds to the square of E₁+E₂, including a cross term.

For two same-frequency, equally polarized beams in the same observation medium, each intensity is proportional to its amplitude squared. Their amplitude ratio is therefore the square root of their intensity ratio. Draw two rotating arrows, or phasors, with those amplitude lengths; the length of their vector sum determines the resultant intensity.

This construction does not require the beams to be equally strong. Equal beams make particularly dark minima, but unequal beams still interfere. Ordinary independent lamps usually fail the stable-phase condition; their cross term averages away during observation. That is a different reason for losing visible fringes.

I=I_1+I_2+2\sqrt{I_1I_2}\cos\delta

I₁ and I₂ are the intensities measured at the same detector with the other beam blocked. The formula assumes stable relative phase and parallel polarization.

Explain this equation

Read it in words

Two coherent waves contribute their separate intensities plus a phase-dependent cross term. Fields add before the detector squares and averages them.

Symbols and units

$I$: Combined time-averaged intensity at the detector.W/m², or a common consistent intensity scale.
$I_1,\ I_2$: Intensities measured from the individual beams in the same observation medium.Same units as I.
$\delta$: Phase difference between the waves at the detector.Radians; dimensionless.
$\sqrt{I_1I_2}$: Geometric mean of the two intensities, arising from the product of field amplitudes.Same intensity units as I.

Signs and operations

The square root is essential: field amplitude is proportional to the square root of intensity. cosδ ranges from −1 to +1, so the cross term can reduce or increase the intensity.

When it applies

Use overlapping coherent waves of the same frequency with parallel polarization in the same medium. A detector averaging a rapidly and uniformly varying relative phase measures a zero average cross term.

Make it concrete

For separate intensities 9 and 4 W/m², in-phase waves give 25 W/m² and opposite-phase waves give 1 W/m². Phase opposition does not make unequal amplitudes cancel completely.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Equation (LaTeX): I=I_1+I_2+2\sqrt{I_1I_2}\cos\delta
Read it as: Two coherent waves contribute their separate intensities plus a phase-dependent cross term. Fields add before the detector squares and averages them.
Symbols:
I: Combined time-averaged intensity at the detector.; W/m², or a common consistent intensity scale.
I_1,\ I_2: Intensities measured from the individual beams in the same observation medium.; Same units as I.
\delta: Phase difference between the waves at the detector.; Radians; dimensionless.
\sqrt{I_1I_2}: Geometric mean of the two intensities, arising from the product of field amplitudes.; Same intensity units as I.
Signs and operations: The square root is essential: field amplitude is proportional to the square root of intensity. cosδ ranges from −1 to +1, so the cross term can reduce or increase the intensity.
Conditions: Use overlapping coherent waves of the same frequency with parallel polarization in the same medium. A detector averaging a rapidly and uniformly varying relative phase measures a zero average cross term.

Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.

WORKED EXAMPLE

A bright point can exceed the sum of the separate intensities

Two coherent, parallel-polarized beams separately give 9 W/m² and 4 W/m² at a detector. Their phase difference is 60°. Find the combined intensity.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Interference: phase, brightness and contrast
Focus: Add fields first; square afterwards
Learning goals:
- Convert optical path changes into phase without counting a replaced medium twice.
- Calculate intensity for unequal coherent beams and diagnose incomplete darkness.
- Infer amplitude ratios from measured extrema.
- Distinguish coherence, polarization and detector averaging.
Equations in this section (LaTeX):
I=I_1+I_2+2\sqrt{I_1I_2}\cos\delta
Meaning: I₁ and I₂ are the intensities measured at the same detector with the other beam blocked. The formula assumes stable relative phase and parallel polarization.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

04 / Interference: phase, brightness and contrast

A minimum is not necessarily darkness

At a maximum the field amplitudes add; at a minimum they oppose. A smaller amplitude cannot cancel a larger one completely. This is why making one slit much dimmer leaves a visible background even when the relative phase reaches exactly π.

Fringe visibility measures contrast by comparing the brightest and dimmest readings. It is a ratio, not a brightness. Multiplying both beam intensities by the same factor brightens the entire pattern without improving its ideal visibility. Balancing the two beams improves contrast even if it reduces their total power.

Measured extrema can reveal the separate intensities when the coherent, parallel-polarized model applies. First take square roots of the extrema to reconstruct the sum and difference of amplitudes. Without those model conditions, a shallow fringe could instead arise from imperfect coherence or polarization mismatch.

I_{\max}=(\sqrt{I_1}+\sqrt{I_2})^2,\quad I_{\min}=(\sqrt{I_1}-\sqrt{I_2})^2,\quad\mathcal V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}

Visibility lies between zero and one for nonnegative readings with a nonzero sum. A perfectly dark minimum requires equal amplitudes in this model.

WORKED EXAMPLE

Recover the hidden beams from a fringe scan

A coherent, parallel-polarized two-beam scan has maximum 36 W/m² and minimum 4 W/m². Find the separate beam intensities, up to interchange.

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STOP & THINK

Two unequal coherent beams are exactly out of phase. Must blocking the weaker beam make the detector darker?

Yes; removing light always lowers the reading.
No; removing the opposing field increases the reading.
Only if their frequencies differ.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Interference: phase, brightness and contrast
Focus: A minimum is not necessarily darkness
Learning goals:
- Convert optical path changes into phase without counting a replaced medium twice.
- Calculate intensity for unequal coherent beams and diagnose incomplete darkness.
- Infer amplitude ratios from measured extrema.
- Distinguish coherence, polarization and detector averaging.
Equations in this section (LaTeX):
I_{\max}=(\sqrt{I_1}+\sqrt{I_2})^2,\quad I_{\min}=(\sqrt{I_1}-\sqrt{I_2})^2,\quad\mathcal V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}
Meaning: Visibility lies between zero and one for nonnegative readings with a nonzero sum. A perfectly dark minimum requires equal amplitudes in this model.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

05 / Interference: phase, brightness and contrast

Change one thing and predict the contrast

Before moving a control, predict the maximum, minimum and mean for the chosen intensity ratio. Then move only the phase. The phasor lengths should remain fixed while their resultant changes; the graph tells you where the current reading sits in a complete phase scan.

Next change the weaker beam into the stronger beam. Watch the minimum rise once the amplitudes become unequal. Finally switch to a uniformly drifting phase during the detector exposure. The averaged reading becomes flat even though the individual fields continue to exist.

INTERACTIVE LAB

Phase and contrast explorer

Beam 1 has unit intensity. Vary beam 2 and the phase, then compare coherent addition with a uniformly averaged relative phase. All curves and phasors use the displayed model.

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Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Interference: phase, brightness and contrast
Focus: Change one thing and predict the contrast
Learning goals:
- Convert optical path changes into phase without counting a replaced medium twice.
- Calculate intensity for unequal coherent beams and diagnose incomplete darkness.
- Infer amplitude ratios from measured extrema.
- Distinguish coherence, polarization and detector averaging.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

06 / Interference: phase, brightness and contrast

The detector can average away a real oscillating cross term

Coherence asks how reliably a phase relationship survives the comparison being made. It depends on the source, path delay and observation time. Equal nominal frequency is useful but does not by itself guarantee a stable interference pattern. Two independent sources can wander in relative phase.

If the relative phase samples a complete cycle uniformly during one detector reading, positive and negative values of the cosine cross term cancel. The detector reports I₁+I₂. If the phase remains nearly fixed during that reading, the cross term survives. Partial stability gives intermediate contrast, so a missing fringe is not proof that one beam is absent.

Polarization supplies another condition. Perpendicular electric-field directions have zero dot-product cross term at a polarization-insensitive detector. An analyzer can project both fields onto one direction, changing the individual intensities and allowing a cross term if the projected waves also have a stable relative phase. State what the apparatus measures before diagnosing coherence.

WORKED EXAMPLE

The same beams, a different observation

Two parallel-polarized beams separately give 5 and 20 W/m². Their phase samples 0°, 90°, 180° and 270° for equal times within one exposure. Find the average reading.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Interference: phase, brightness and contrast
Focus: The detector can average away a real oscillating cross term
Learning goals:
- Convert optical path changes into phase without counting a replaced medium twice.
- Calculate intensity for unequal coherent beams and diagnose incomplete darkness.
- Infer amplitude ratios from measured extrema.
- Distinguish coherence, polarization and detector averaging.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

07 / Interference: phase, brightness and contrast

Use the pattern to check the model

A bright interference fringe is a local redistribution of energy, not an extra source of energy. For a phase scan with fixed beam intensities, averaging uniformly over phase removes the cross term. For a full physical interference pattern, the complete field and optical arrangement determine how energy is redistributed; a bright point alone cannot be compared with the total source power.

An intensity measurement also does not uniquely determine signed phase. Solving for cosine generally leaves two phase angles in a full cycle. Extra information about motion, a known path change or a second measurement is needed to choose between them.

A reliable method is to identify the detector, establish polarization and coherence, find the separate intensities, calculate the total phase ledger, and only then evaluate intensity. Finish by checking that the answer lies between the correct extrema. A value below zero or above the amplitude-sum maximum signals a model or arithmetic error.

WORKED EXAMPLE

One intensity can correspond to two phases

Two equal coherent beams each give 7 W/m². Their combined reading is 7 W/m². Find the possible phase differences between 0° and 360°.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Interference: phase, brightness and contrast
Focus: Use the pattern to check the model
Learning goals:
- Convert optical path changes into phase without counting a replaced medium twice.
- Calculate intensity for unequal coherent beams and diagnose incomplete darkness.
- Infer amplitude ratios from measured extrema.
- Distinguish coherence, polarization and detector averaging.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

Add electric fields before forming intensity.
Optical path, source phase and reflection phase belong in one ledger.
Unequal amplitudes leave a nonzero minimum even at opposite phase.
Intensity alone usually cannot determine the sign of phase.
Observation conditions determine whether the interference term survives.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

Parallel-polarized coherent beams separately give 16 and 9 W/m². Their phase difference is 90°. What is the combined intensity?

Answer in W/m².

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SELF CHECK 2

Two stable beams have perpendicular polarizations at a detector with no analyzer. What is the correct first conclusion?

Their electric fields cancel.
Their intensities add without a polarization cross term.
They must have different wavelengths.

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SELF CHECK 3

A coherent pair separately gives 25 and 9 W/m². What is the minimum combined intensity?

Answer in W/m².

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

A stable pattern never becomes completely dark. Does that prove that the beams are incoherent?

Compare your reasoning

No. Unequal amplitudes leave a nonzero minimum even with stable phase and parallel polarization. Compare extrema or separate beam readings before diagnosing coherence.

Revisit the explanation →

A few days later: change the situation

A phase switches between 0° and 180°, spending three times longer at 180°. Does the detector necessarily record I₁+I₂?

Compare your reasoning

No. Its average cosine is −1/2, not zero. Weight the intensity or cross term by actual exposure time; a changing phase alone does not establish uniform averaging.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

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