Optimum holographic elements recorded with nonspherical wave fronts

K. A. Winick; J. R. Fienup

doi:10.1364/JOSA.73.000208

Journal of the Optical Society of America
Vol. 73,
Issue 2,
pp. 208-217
(1983)
•https://doi.org/10.1364/JOSA.73.000208

Optimum holographic elements recorded with nonspherical wave fronts

K. A. Winick and J. R. Fienup

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K. A. Winick and J. R. Fienup, "Optimum holographic elements recorded with nonspherical wave fronts," J. Opt. Soc. Am. 73, 208-217 (1983)

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Abstract

The problem of designing a flat, aspheric, holographic optical element that images a finite set of input wave fronts into a finite set of output wave fronts is rigorously analyzed. The optimum phase transfer function of the holographic optical element is analytically determined. The optimum phase transfer function is defined as the one for which the element has minimum mean-squared wave-front error averaged over the set of input wave fronts. It is also shown that in general it may not be possible to obtain a low value for this average mean-squared wave-front error by using a single holographic element. Furthermore, the performance trade-off of spherical aberration, coma, and astigmatism versus geometric distortion is clearly indicated by example.

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l	α_l	β_l	f_x′,l (lines/min)	f_y′,l (lines/mm)	(a) Ideal Spot Location		(b) Conventional HOE Spot Location		(c) Fairchild–Fienup Aspheric HOE Spot Location

					x″ (mm) y″ (mm)		x″ (mm) y″ (mm)		x″ (mm) y″ (mm)
1	−2.4°	0°	−81.39	0	−20.93783	0.00000	−19.8463	0.00000	−19.8287	0.00000
2	−1.2°	0°	−40.70	0	−10.47121	0.00000	−9.87993	0.00000	−9.8764	0.00000
3	0°	0°	0	0	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
4	1.2°	0°	40.70	0	10.47121	0.00000	9.80504	0.00000	9.8014	0.00000
5	2.4°	0°	81.39	0	20.93783	0.00000	19.5483	0.00000	19.5291	0.00000
6	0°	−2.4°	0	−81.39	0.00000	−20.93783	0.15001	−20.9562	0.1500	−20.9396
7	0°	−1.2°	0	−40.70	0.00000	−10.97121	0.0375	−10.4735	0.0375	−10.4686
8	0°	1.2°	0	40.70	0.00000	10.47121	0.0375	10.4735	0.0375	10.4686
9	0°	2.4°	0	81.39	0.00000	20.93783	0.15001	−20.9562	0.1500	20.9396

l	α_l	β_l	(a) Conventional HOE [∼E_l (in wavelengths)]	(b) Fairchild–Fienup Aspheric HOE [∼E_l (in wavelengths)]	(c) Optimum Aspheric HOE [∼E_l (in wavelengths)]
1	−2.4°	0°	2.75 × 10⁻¹	3.72 × 10⁻²	3.95 × 10⁻²
2	−1.2°	0°	0.84 × 10⁻¹	2.55 × 10⁻²	2.75 × 10⁻²
3	0°	0°	0	3.85 × 10⁻²	2.52 × 10⁻²
4	1.2°	0°	0.89 × 10⁻¹	2.48 × 10⁻²	2.76 × 10⁻²
5	2.4°	0°	3.07 × 10⁻¹	3.88 × 10⁻²	3.99 × 10⁻²
6	0°	−2.4°	2.33 × 10⁻¹	4.59 × 10⁻²	5.59 × 10⁻²
7	0°	−1.2°	0.71 × 10⁻¹	2.68 × 10⁻²	4.00 × 10⁻²
8	0°	1.2°	0.71 × 10⁻¹	2.68 × 10⁻²	4.05 × 10⁻²
9	0°	2.4°	2.33 × 10⁻¹	4.59 × 10⁻²	5.59 × 10⁻²
			E = 1.84 × 10⁻¹^a	E = 3.54 × 10⁻²^a	E = 4.06 × 10⁻²^a

l	α_l	β_l	f_x′,l (lines/min)	f_y′,l (lines/mm)	(a) Ideal Spot Location		(b) Conventional HOE Spot Location		(c) Fairchild–Fienup Aspheric HOE Spot Location

					x″ (mm) y″ (mm)		x″ (mm) y″ (mm)		x″ (mm) y″ (mm)
1	−2.4°	0°	−81.39	0	−20.93783	0.00000	−19.8463	0.00000	−19.8287	0.00000
2	−1.2°	0°	−40.70	0	−10.47121	0.00000	−9.87993	0.00000	−9.8764	0.00000
3	0°	0°	0	0	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
4	1.2°	0°	40.70	0	10.47121	0.00000	9.80504	0.00000	9.8014	0.00000
5	2.4°	0°	81.39	0	20.93783	0.00000	19.5483	0.00000	19.5291	0.00000
6	0°	−2.4°	0	−81.39	0.00000	−20.93783	0.15001	−20.9562	0.1500	−20.9396
7	0°	−1.2°	0	−40.70	0.00000	−10.97121	0.0375	−10.4735	0.0375	−10.4686
8	0°	1.2°	0	40.70	0.00000	10.47121	0.0375	10.4735	0.0375	10.4686
9	0°	2.4°	0	81.39	0.00000	20.93783	0.15001	−20.9562	0.1500	20.9396

l	α_l	β_l	(a) Conventional HOE [∼E_l (in wavelengths)]	(b) Fairchild–Fienup Aspheric HOE [∼E_l (in wavelengths)]	(c) Optimum Aspheric HOE [∼E_l (in wavelengths)]
1	−2.4°	0°	2.75 × 10⁻¹	3.72 × 10⁻²	3.95 × 10⁻²
2	−1.2°	0°	0.84 × 10⁻¹	2.55 × 10⁻²	2.75 × 10⁻²
3	0°	0°	0	3.85 × 10⁻²	2.52 × 10⁻²
4	1.2°	0°	0.89 × 10⁻¹	2.48 × 10⁻²	2.76 × 10⁻²
5	2.4°	0°	3.07 × 10⁻¹	3.88 × 10⁻²	3.99 × 10⁻²
6	0°	−2.4°	2.33 × 10⁻¹	4.59 × 10⁻²	5.59 × 10⁻²
7	0°	−1.2°	0.71 × 10⁻¹	2.68 × 10⁻²	4.00 × 10⁻²
8	0°	1.2°	0.71 × 10⁻¹	2.68 × 10⁻²	4.05 × 10⁻²
9	0°	2.4°	2.33 × 10⁻¹	4.59 × 10⁻²	5.59 × 10⁻²
			E = 1.84 × 10⁻¹^a	E = 3.54 × 10⁻²^a	E = 4.06 × 10⁻²^a

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Tables (2)

Equations (53)

Journal of the Optical Society of America