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What is the Maxima Condition in Holography?

Thread starter Debam1224
Start date Dec 25, 2023
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Holography Laser Physics

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Debam1224

Thread starter
Dec 25, 2023

#1

Homework Statement: I came across one problem from Holography which is a subtopic in my DSE Paper(Laser and Fibre Optics) in Bsc. The problem is as follows:

Suppose the reference wave in hologram is given by. Eᵣ = E₀ e^{-i(ωt- k𝓏 - kₙ sin θ)}and the spherical wave emanating from point source is given as
Eₛₚₕ= E₁ e^{-i(ωt- k𝓏 - (kₓ²/2z₀))}
Then find out the resultant intensity in the plate. Show that the condition for maxima in the hologram is given by ∆x = 4λ (Choose θ = 15°)

I've attached what I've done. The intensity part is correct. But I'm confused about the maxima part because the question has specifically asked for
4λ whereas I got nλ. So, should I just put n=4 or is it wrong? I would really appreciate your views and answers. P.S.: Entirely typed in latex, not copied from any book.

Relevant Equations: Resultant Electric Field,
Eᵣ= E₁ + E₂
Intensity, I = |Eᵣ|²

Typeset derivation: Given E_r = E0 e^(-i(wt - kz - kn sin θ)) and E_sph = E1 e^(-i(wt - kz - kx^2/(2z0))), resultant E_res = E_r + E_sph, giving E_res = E0 e^(-i(wt - kz - kn sin θ)) + E1 e^(-i(wt - kz - kx^2/(2z0))).

Handwritten-style derivation of intensity I = |E_rec|^2 expanded into E_0^2 + E_1^2 + 2E_0E_1 cos(k_n sin θ - k_x^2/(2z_0)), ending with the boxed-off final result.

Handwritten-style derivation of the maximum intensity condition: cos(k_n sin θ - k_x^2/(2z_0)) = 1, then k_n sin θ - k_x^2/(2z_0) = 2nπ, then k_n/4 - k_x^2/(2z_0) = 2nπ.

Handwritten-style typeset derivation of a phase difference: Δφ = φr − φsph expanded into terms with ωt, kz, kn sin θ and kx²/(2z0), reduced to Δφ = kn/4 − kx²/(2z0) = 2nπ, then Δx = (λ/2π)Δφ = nλ.

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