Synthesis of asymmetric profiles from a double grating interferometer

C. C. lemmi; Juan M. Simon; J. O. Ratto

doi:10.1364/AO.25.003171

Applied Optics
Vol. 25,
Issue 18,
pp. 3171-3178
(1986)
•https://doi.org/10.1364/AO.25.003171

Synthesis of asymmetric profiles from a double grating interferometer

C. C. lemmi, Juan M. Simon, and J. O. Ratto

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C. C. lemmi, Juan M. Simon, and J. O. Ratto, "Synthesis of asymmetric profiles from a double grating interferometer," Appl. Opt. 25, 3171-3178 (1986)

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Abstract

We study the possibility of obtaining holographic diffraction gratings with blaze. The analysis is performed with an interferometric system containing two gratings, where a substrate with an emulsion of photosensitive resin is exposed. By an appropriate choice of phase and amplitude of the light beams that converge onto the resist, it is possible to synthesize asymmetric profiles on it. The theory is developed to first order and applied to a specific example, where two different ways of carrying out the desired synthesis are presented. The stability of the proposed interferometric system, the dependence on the size of the light source employed, and its bandwidth tolerance are discussed; the influence of these factors is displayed graphically.

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sin(2πf²lΔλ)	Solution	a₀	a₁	a₂	β₁	β₂
	1	0.540	0.414	0.192	2.005	0.434
1	2	0.540	0.192	0.414	2.707	1.136
	3	0.260	0.646	0.123	1.759	0.188
	4	0.260	0.123	0.646	2.953	1.382
	1	0.486	0.486	0.163	1.895	0.324
0.8	2	0.486	0.163	0.486	2.817	1.246
	3	0.364	0.591	0.135	1.795	0.224
	4	0.364	0.135	0.591	2.917	1.347

Deviations from ideal disposition	Comments and tolerances	Examples for Δθ = 0.01, λ₀ = 4500 Å f = 600 lines/mm
f₁ = f	(i) Does not generate moire fringes	—
f₂ = f + Δf	(ii) f + Δf is registered
The plane of grating G₂ forms an angle φ with G₁	Does not generate moire fringes to first order in (φ ⇒ large tolerance	—
Positioning errors of grating G₂ and plane P on the z axis (or problems of mechanical instability)	Negligible effect if α ≡ 3πλ₀f²Δl < 0.5	Requires Δl < 0.33 μm
Extended light source of angular halfwidth Δθ	Negligible effect to first order in θ	Requires Δl < 5 μm
Light source of spectral halfwidth Δλ	(i) Negligible effect for sinc (2πlf²Δλ) ≥ 0.77
	(ii) If parameters a_j,β_j are not adjusted, the effect is negligible for sinc(2πlf²Δλ) > 0.70	—
Positioning errors Δ of the three gratings placed in P₂ (or construction errors if it is synthesized holographically)	Negligible if $Δ < \frac{1}{8 f}$	—

sin(2πf²lΔλ)	Solution	a₀	a₁	a₂	β₁	β₂
	1	0.540	0.414	0.192	2.005	0.434
1	2	0.540	0.192	0.414	2.707	1.136
	3	0.260	0.646	0.123	1.759	0.188
	4	0.260	0.123	0.646	2.953	1.382
	1	0.486	0.486	0.163	1.895	0.324
0.8	2	0.486	0.163	0.486	2.817	1.246
	3	0.364	0.591	0.135	1.795	0.224
	4	0.364	0.135	0.591	2.917	1.347

Deviations from ideal disposition	Comments and tolerances	Examples for Δθ = 0.01, λ₀ = 4500 Å f = 600 lines/mm
f₁ = f	(i) Does not generate moire fringes	—
f₂ = f + Δf	(ii) f + Δf is registered
The plane of grating G₂ forms an angle φ with G₁	Does not generate moire fringes to first order in (φ ⇒ large tolerance	—
Positioning errors of grating G₂ and plane P on the z axis (or problems of mechanical instability)	Negligible effect if α ≡ 3πλ₀f²Δl < 0.5	Requires Δl < 0.33 μm
Extended light source of angular halfwidth Δθ	Negligible effect to first order in θ	Requires Δl < 5 μm
Light source of spectral halfwidth Δλ	(i) Negligible effect for sinc (2πlf²Δλ) ≥ 0.77
	(ii) If parameters a_j,β_j are not adjusted, the effect is negligible for sinc(2πlf²Δλ) > 0.70	—
Positioning errors Δ of the three gratings placed in P₂ (or construction errors if it is synthesized holographically)	Negligible if $Δ < \frac{1}{8 f}$	—

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Applied Optics