Diffractive optical elements in hybrid lenses: modeling and design by zone decomposition

Hervé Sauer; Pierre Chavel; Gábor Erdei

doi:10.1364/AO.38.006482

Applied Optics
Vol. 38,
Issue 31,
pp. 6482-6486
(1999)
•https://doi.org/10.1364/AO.38.006482

Diffractive optical elements in hybrid lenses: modeling and design by zone decomposition

Hervé Sauer, Pierre Chavel, and Gábor Erdei

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Hervé Sauer, Pierre Chavel, and Gábor Erdei, "Diffractive optical elements in hybrid lenses: modeling and design by zone decomposition," Appl. Opt. 38, 6482-6486 (1999)

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Abstract

We propose to model hybrid optical systems (i.e., lenses with conventional and diffractive optical elements) as multiaperture systems in which the images formed by each zone of the diffractive optical element should be summed up coherently. This new zone decomposition concept is explained and compared with the standard diffraction-order expansion with the help of a hybrid triplet example.

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Surface Number	Surface Type	Radius of Curvature (mm)	Thickness (mm)	Material
0 (Object)	Flat	Infinity	Infinity	—
1 (Stop)	Sphere	15.94	5.80	SK2 (Schott)
2	Sphere	-11.53	6.50	SF4 (Schott)
3	Sphere	-29.34	0.40	—
4	Flat + DOEa	Infinity	1.50	BK7 (Schott)
5	Flat	Infinity	14.70	—
6 (Image)	Flat	Infinity	—	—

Components and Curve Numbers	Definitions
MTF	Modulus of the complex optical transfer function (OTF), i.e., MTF = \|OTF\|, where the OTF is the Fourier transform of the real PSF.6 The MTF and the OTF are normalized to 1 at the spatial frequency of 0 cycle/mm.
OTF_{m, λ}	Monochromatic OTF of the diffraction order m at λ.
OTF_ZD,λ	Monochromatic OTF at λ for the zone-decomposition computation.
w _λ	Spectral weights with ∑_λ w _λ = 1.
η_m,λ	Efficiency of diffraction order m at the wavelength λ. We have $\sum_{m = - \infty}^{+ \infty}$ η_m,λ = 1 for all λ. In the case of the assumed perfect continuous DOE etching profile (see Fig. 2), we have η_m,λ = sinc²(m- λ_Nominal/λ).7
Curve 2	MTF_{+1 order(η₁} _{≡100%∀λ)} = \|∑_λ OTF_1,λ w _λ\|.
Curve 3	MTF_ZD = \|∑_λ OTF_ZD,λ w _λ\|.
Curve 4	MTF_{+1 order+uniform background} = [\|∑_λ OTF_1,λ w _λη_1,λ\|/∑_λ w _λη_1,λ] [∑_λ w _λη_1,λ], at all nonzero spatial frequencies (1 at 0 cycle/mm). Here, because of the nearly optimal choice of λ_Nominal, ∑_λ w _λη_1,λ reaches 0.927.
Curve 5	MTF_{5 orders} = \| $\sum_{m = - 1}^{+ 3}$ ∑_λ OTF_m,λ w _λη_m,λ\|/ $\sum_{m = - 1}^{+ 3}$ ∑_λ w _λη_m,λ, where $\sum_{m = - 1}^{+ 3}$ ∑_λ w _λη_m,λ = 0.984.
Curve 6	MTF_{19 orders} = \| $\sum_{m = - 8}^{+ 10}$ ∑_λ OTF_m,λ w _λη_m,λ\|/ $\sum_{m = - 8}^{+ 10}$ ∑_λ w _λ (η_m,λ), where $\sum_{m = - 8}^{+ 10}$ ∑_λ w _λη_m,λ = 0.996.

Surface Number	Surface Type	Radius of Curvature (mm)	Thickness (mm)	Material
0 (Object)	Flat	Infinity	Infinity	—
1 (Stop)	Sphere	15.94	5.80	SK2 (Schott)
2	Sphere	-11.53	6.50	SF4 (Schott)
3	Sphere	-29.34	0.40	—
4	Flat + DOEa	Infinity	1.50	BK7 (Schott)
5	Flat	Infinity	14.70	—
6 (Image)	Flat	Infinity	—	—

Components and Curve Numbers	Definitions
MTF	Modulus of the complex optical transfer function (OTF), i.e., MTF = \|OTF\|, where the OTF is the Fourier transform of the real PSF.6 The MTF and the OTF are normalized to 1 at the spatial frequency of 0 cycle/mm.
OTF_{m, λ}	Monochromatic OTF of the diffraction order m at λ.
OTF_ZD,λ	Monochromatic OTF at λ for the zone-decomposition computation.
w _λ	Spectral weights with ∑_λ w _λ = 1.
η_m,λ	Efficiency of diffraction order m at the wavelength λ. We have $\sum_{m = - \infty}^{+ \infty}$ η_m,λ = 1 for all λ. In the case of the assumed perfect continuous DOE etching profile (see Fig. 2), we have η_m,λ = sinc²(m- λ_Nominal/λ).7
Curve 2	MTF_{+1 order(η₁} _{≡100%∀λ)} = \|∑_λ OTF_1,λ w _λ\|.
Curve 3	MTF_ZD = \|∑_λ OTF_ZD,λ w _λ\|.
Curve 4	MTF_{+1 order+uniform background} = [\|∑_λ OTF_1,λ w _λη_1,λ\|/∑_λ w _λη_1,λ] [∑_λ w _λη_1,λ], at all nonzero spatial frequencies (1 at 0 cycle/mm). Here, because of the nearly optimal choice of λ_Nominal, ∑_λ w _λη_1,λ reaches 0.927.
Curve 5	MTF_{5 orders} = \| $\sum_{m = - 1}^{+ 3}$ ∑_λ OTF_m,λ w _λη_m,λ\|/ $\sum_{m = - 1}^{+ 3}$ ∑_λ w _λη_m,λ, where $\sum_{m = - 1}^{+ 3}$ ∑_λ w _λη_m,λ = 0.984.
Curve 6	MTF_{19 orders} = \| $\sum_{m = - 8}^{+ 10}$ ∑_λ OTF_m,λ w _λη_m,λ\|/ $\sum_{m = - 8}^{+ 10}$ ∑_λ w _λ (η_m,λ), where $\sum_{m = - 8}^{+ 10}$ ∑_λ w _λη_m,λ = 0.996.

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Applied Optics