Division of amplitude

Division of amplitude

1. Two Ways to Get Coherent Beams

• Division of wavefront: split the wavefront spatially (Young's double slit, Fresnel biprism, Lloyd's mirror) — two parts of same wavefront become separate sources.
• Division of amplitude: split the amplitude (intensity) at a partial reflector — one beam partly reflected, partly transmitted, then recombined after different paths.

This topic is the amplitude type — used when source is extended but still needs coherence via same wave.

2. How Amplitude Division Works

At a thin film / glass slab / beam splitter:

I → [BS] → Ir (reflected) + It (transmitted)
Ir reflects off M1, It off M2 → recombine at BS → interference at detector/eye

Amplitudes: $A_r = r A$, $A_t = t A$ where $r,t$ are reflection/transmission coefficients ($r²+R, t²+T$ with $R+T=1$ ignoring absorption). Both beams originate from same incident wave ⇒ coherent, even if source is not point-like.

Phase: reflection off denser medium adds π (half-wave) phase flip ($Δ=λ/2$). Transmission no flip.

3. Conditions

• Film thickness $t$ and refractive index $μ$: optical path $2μt cos r$ matters (see thin-film theory). But the coherence comes from strict amplitude split, not spatial separation.
• Needs path difference within coherence length $L_c = λ²/Δλ$ of source — otherwise beams decorrelate (white light gives only few orders).

4. Examples of Amplitude Division

System	Splitter	Paths
Thin film / Newton's rings	film surfaces	reflection from top vs bottom of film
Michelson interferometer	beam splitter	two arms M1, M2
Fabry-Perot	two highly reflecting plates	multiple reflections (many beams)
Mach-Zehnder	two beam splitters	two separate arms

Vs wavefront: Young uses two holes in screen; amplitude uses one beam split into two intensities.

5. Thin-Film Path Difference (preview for next topics)

For film thickness $t$, index $μ$, refraction angle $r$:

(± for half-wave at one surface). Bright when $Δ=nλ$.

6. Worked Example

Amp division vs wavefront: Young with extended source washes out (spatial coherence small), but Newton's rings with extended source still gives rings (amplitude division tolerates extended source because each point's split beams recombine locally) ✓

7. Exam Points

• Distinguish division of wavefront vs amplitude with diagram — 5 marks, very frequent (almost certain to ask one line difference).
• Explain how beam splitter gives coherent beams — 3-5 marks, frequent.
• Mention half-wave flip on reflection — 3 marks, very frequent (used in next derivations).
• Give examples: Young's (wavefront) vs Newton's/Michelson (amplitude) — 3 marks.