Waves and optics: distinguish unequal beams from lost coherence
A nonzero interference minimum does not by itself show lost coherence. Unequal beam strengths can leave a bright minimum even with perfectly stable phase. Model the amplitudes and the averaging separately.
Course topic: Waves and Optics
Study route and exercise: University Reading List editorial project. How we curate · Sources checked:
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Scope and responsibility
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 · Edition: 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.
Optics
Eugene Hecht · Edition: 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.
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.
Check your preparation
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.
A six-step study plan
Steps completed: 0 / 6
These are your own study marks, not an assessment. They are not sent to this site.
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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.
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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.
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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.
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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.
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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.
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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.
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.
- Normalize the fields as sqrt(I0) and 2sqrt(I0)e^(iδ). Squaring the summed magnitude gives I=(1+4+4cosδ)I0.
- 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.
- V=(9−1)/(9+1)=4/5. The fields remain perfectly coherent in this specified stable-phase model.
Answer and interpretation
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
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.
- Conditional intensities are I(δ,0)=(5+4cosδ)I0 and I(δ,π)=(5−4cosδ)I0.
- Average the intensities: (3/4)I(δ,0)+(1/4)I(δ,π)=(5+2cosδ)I0. The single-beam powers have not changed.
- 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.
Answer and interpretation
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.
Before moving on
- Derive an unequal-beam fringe from fields.
- Calculate visibility with a specified averaging model.
- Separate amplitude imbalance from phase statistics.
Sources and edition checks
Sources checked:
- TU Delft: BScOptics second edition
The official second edition explicitly targets late BSc/early MSc readers familiar with Maxwell equations.
- Pearson: selected fifth-edition digital product
The exact selected fifth-edition identifier is digital, distinct from the older hardcover.
- Oxford: conceptual light overview
Oxford’s contents frame rays, waves and quantum light conceptually; it is not a quantitative course substitute.
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