# Waves and optics: distinguish unequal beams from lost coherence — Study worksheet

[Study guides](https://unireadinglist.com/guides/waves-optics/)

[Course topic: Waves and Optics](https://unireadinglist.com/subjects/physics/#course-waves-optics)

Sources checked: 2026-11-04

Study route and exercise: University Reading List editorial project.

A focused route through superposition, interference and temporal averaging. The free Delft text expects Maxwell equations and late-undergraduate preparation. Hecht is a broader quantitative course; Walmsley offers conceptual orientation. The invented phase ensemble is not a measured noise process.

## Choose your starting point

- [ ] I know Maxwell equations and want free depth
  Use BSc Optics for field models, approximations and interference.

- [ ] I need a broader quantitative course
  Use Hecht with calculus and basic electromagnetic waves.

- [ ] I want to orient myself before the mathematics
  Use Walmsley for rays, waves and quantum light, then move to a problem-solving spine.

## Compare the three books

### BSc Optics

Sander Konijnenberg; Aurèle J.L. Adam; H. Paul Urbach · 2 · 2024
Free advanced undergraduate route

Choose it when: You understand Maxwell equations, complex fields and vector notation.
Use another route when: Free access is your reason for expecting a school-level introduction.
Selected reading: Wave-optical foundations, interference and coherence, with polarization prerequisites as needed.
Limits and trade-offs: Open access removes a price barrier, not the mathematical prerequisites; approximation choices are part of the argument.

#### Try before choosing

Set complex amplitudes sqrt(I0) and 2sqrt(I0)e^(iδ), then square their sum.
Check your work: I/I0=5+4cosδ; the cross term is essential.

Your notes:

________________________________

ISBN: 9789463668460

[Official free reading](https://books.open.tudelft.nl/home/catalog/book/174)

[Book record and access options](https://unireadinglist.com/subjects/physics/bsc-optics-konijnenberg-adam-urbach/)

### Optics

Eugene Hecht · 5 · 2021
Broad quantitative spine

Choose it when: You want examples spanning geometrical and physical optics.
Use another route when: You only need a short qualitative overview.
Selected reading: Fifth-edition interference and coherence material, reviewing wave superposition first.
Limits and trade-offs: The selected ISBN is a Pearson+ digital product; access terms differ from the older hardcover. Breadth requires choosing a bounded route.

#### Try before choosing

Find the visibility when the two intensities are I0 and 4I0.
Check your work: Max/min are 9I0 and I0, so visibility is 8/10, not 1.

Your notes:

________________________________

ISBN: 9780137526420

[Book record and access options](https://unireadinglist.com/subjects/physics/optics-hecht/)

### Light: A Very Short Introduction

Ian A. Walmsley · 2015
Conceptual orientation

Choose it when: You need to distinguish rays, waves, duality and light–matter questions.
Use another route when: You need a detailed derivation and a bank of quantitative interference problems.
Selected reading: Waves and neighboring discussions of light and matter; use the overview to frame questions.
Limits and trade-offs: The short narrative gives context, while the calculation must come from a mathematical text.

#### Try before choosing

Explain verbally why phase π need not make unequal beams dark.
Check your work: Opposite fields cancel only when their magnitudes also match.

Your notes:

________________________________

ISBN: 9780199682690

[Book record and access options](https://unireadinglist.com/subjects/physics/light-a-very-short-introduction/)

## Check your preparation

### Can I distinguish an amplitude from intensity?

Your notes:

________________________________

### What permits the interference term?

Your notes:

________________________________

### What does the detector average?

Your notes:

________________________________

## A six-step study plan

- [ ] Specify the two fields

Selected reading: Superposition.

What to do: Write field amplitudes, polarization and relative phase.

Check your work: Amplitudes have ratio 1:2, intensities 1:4.

Your notes:

________________________________

- [ ] Derive rather than add intensities

Selected reading: Interference.

What to do: Expand the squared magnitude including conjugate cross terms.

Check your work: The result is (5+4cosδ)I0.

Your notes:

________________________________

- [ ] Calculate contrast

Selected reading: Visibility.

What to do: Find extrema and visibility, naming the scanned variable.

Check your work: δ is scanned; Imax=9I0, Imin=I0.

Your notes:

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- [ ] Predict a dark fringe

Selected reading: Field cancellation.

What to do: State the extra condition needed to reach zero intensity.

Check your work: Equal field magnitudes as well as opposite phase.

Your notes:

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- [ ] Average a specified ensemble

Selected reading: Coherence and the follow-up.

What to do: Compute intensity in each phase state and then its weighted average.

Check your work: The modulation coefficient becomes 2, not 4.

Your notes:

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- [ ] Write an identifiable conclusion

Selected reading: Your two models.

What to do: Explain why one measured minimum alone cannot identify a coherence loss.

Check your work: Beam imbalance and phase statistics must be separately specified.

Your notes:

________________________________

## Worked exercise: Perfect coherence without a dark minimum

Two same-frequency, co-polarized beams overlap in one mode with intensities I1=I0 and I2=4I0. Their stable controlled phase difference is δ. Derive total intensity, extrema and visibility V=(Imax−Imin)/(Imax+Imin).

Assumptions: I0\>0; compatible field normalization, stable phase during each measurement, linear superposition and ideal intensity detection. No other background light.

Your notes:

________________________________

## Change an assumption: A prescribed phase ensemble reduces the modulation

Keep beam powers and polarization, but add a phase θ that is 0 with probability 3/4 and π with probability 1/4. At every scanned δ the detector averages many such switches. Find the mean fringe and visibility.

Your notes:

________________________________

## Before moving on

- Derive an unequal-beam fringe from fields.
- Calculate visibility with a specified averaging model.
- Separate amplitude imbalance from phase statistics.

## Answer and interpretation

### Can I distinguish an amplitude from intensity?

For compatible fields intensity is proportional to squared magnitude. Four times the intensity means twice the field amplitude.

### What permits the interference term?

Same frequency, overlapping spatial mode and the same polarization in this model. Orthogonal polarizations would be a different assumption.

### What does the detector average?

It averages intensity over many prescribed phase switches at each controlled phase δ; it does not simply square the mean field.

### Perfect coherence without a dark minimum

1. Normalize the fields as sqrt(I0) and 2sqrt(I0)e^(iδ). Squaring the summed magnitude gives I=(1+4+4cosδ)I0.
2. The maximum occurs at δ=0 modulo 2π and equals 9I0. The minimum at δ=π equals I0, since the opposite amplitudes have unequal magnitudes.
3. V=(9−1)/(9+1)=4/5. The fields remain perfectly coherent in this specified stable-phase model.

I=(5+4cosδ)I0, Imax=9I0, Imin=I0, V=0.8.

Test the result: At phase π the residual field is −sqrt(I0), giving intensity I0. Vanishing minimum would require matching amplitudes as well as opposing phase.

### Change an assumption: A prescribed phase ensemble reduces the modulation

1. Conditional intensities are I(δ,0)=(5+4cosδ)I0 and I(δ,π)=(5−4cosδ)I0.
2. Average the intensities: (3/4)I(δ,0)+(1/4)I(δ,π)=(5+2cosδ)I0. The single-beam powers have not changed.
3. The mean maximum is 7I0 and minimum 3I0, so V=(7−3)/(7+3)=2/5. Averaging the field first and squaring would also alter the second beam’s own intensity and is not this detector model.

Mean intensity is (5+2cosδ)I0, with visibility 0.4.

Test the result: The intensity averaged over a full δ cycle remains 5I0 in both cases. The changed modulation follows from the stated ensemble, not an inferred experimental noise distribution.

## Common mistakes

### Adding only intensities

Compatible coherent fields also have a cross term.

### Calling every bright minimum incoherent

Unequal amplitudes can cause it without phase fluctuations.

### Squaring a mean field

The mean of intensity generally differs from intensity of the mean field.

## Sources and edition checks

- [TU Delft: BScOptics second edition](https://books.open.tudelft.nl/home/catalog/book/174)
  The official second edition explicitly targets late BSc/early MSc readers familiar with Maxwell equations.

- [Pearson: selected fifth-edition digital product](https://www.pearson.com/en-us/subject-catalog/p/optics/P200000006793/9780137526420)
  The exact selected fifth-edition identifier is digital, distinct from the older hardcover.

- [Oxford: conceptual light overview](https://academic.oup.com/book/857)
  Oxford’s contents frame rays, waves and quantum light conceptually; it is not a quantitative course substitute.

These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.

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