## Diffraction efficiency improvement of diffractive cylinder lenses by Gaussian-beam illumination.

Optics Express, Vol. 1, Issue 8, pp. 234-239 (1997)

http://dx.doi.org/10.1364/OE.1.000234

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### Abstract

To maximize the diffraction efficiency of cylinder lenses with high numerical apertures (such as F/0.5 lenses) we use an iterative algorithm to determine the optimum field distribution in the lens plane. The algorithm simulates the free-space propagation between the lens and the focal plane applying the angular spectrum of plane waves. We show that the optimum field distribution in the lens plane is the phase distribution of a converging cylindrical wave-front and an amplitude distribution with Gaussian-profile. The computed results are verified by rigorous calculations, simulating a F/0.5 lens with subwavelength structures.

© Optical Society of America

## 1. Introduction

## 2. Optimization of the electrical field distribution in the lens plane

1. E. Noponen, J. Turunen, and A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A **10**,434–443(1993) [CrossRef]

*k*

_{0}= 2π/λ, and λ is the free-space wavelength. The corresponding transmitted field, i.e. the electrical field in the xy-plane (lens plane), for TE-polarization is

*F*form the lens plane. In practice, the aperture of the lens has finite boundaries, which leads to a significant loss in diffraction efficiency.

*F*with a maximum in energie density. In order to find such a complex field distribution, the angular spectrum of plane waves was used in an iterative algorithm (see Fig. 1) to simulate the free-space propagation between the lens and the focal plane.[4] To develop a field distribution that is inherent within the physical boundary conditions, we started with a random complex distribution

*E*

_{y}(

*x*,

*z*=0) in the lens plane.

*E*

_{y}(

*x*,

*z*=0) can be written as the inverse Fourier transform

*z*can be regarded as a change in the relative phase of the spatial Fourier components

*ψ*(

*k*

_{x}), because each component

*ψ*(

*k*

_{x}) propagates at a different angle. Across the focal plane at

*z*=-

*F*, the electric-field can be described by the function

*O*(

*x*,

*ν*) with

*x*

_{0}is hereby used to define the symmetrical window in the focal plane and the parameter

*ν*regulates the speed of the reduction process. Backward propagation of the angular spectrum of plane waves was used to determine the optimized transmitted field in the lens plane at z=0. The function

*A(x,a)*is used to set the electrical field to zero outside the aperture-radius of

*a*. This algorithm converges after a few iterations. The transmitted field that optimizes the diffraction efficiency of the central peak has a phase distribution according to Eq. (1) and an amplitude distribution with a Gaussian-profile. However, it is not possible to control the phase and amplitude modulation simultaniously by the use of a DOE. If, on the other hand, a conventionel diffractive cylinder lens is illuminated with light that has a Gaussian-beam profile and the Gaussian-profile can be preserved while the lens adds the cylindrical phase modulation, a significant rise in diffraction efficiency can be expected.

## 3. Rigorous calculations

3. M. Schmitz and O. Bryngdahl, “Rigorous concept for the design of diffractive microlenses with high numerical apertures,” J. Opt. Soc. Am. A **14**, 901–906(1997) [CrossRef]

*E*

_{y}(

*x*,

*z*=0), calculated with an updated version of the modal method,[5–9

5. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. **56**, 1502–1507(1966) [CrossRef]

*E*

_{y}(

*x*,

*z*=0) assuming a Gaussian-beam illumination with the amplitude distribution

*ξ*=20

*λ*. The ratio between the half-width and the aperture-width was determined using the procedures described in section 2. The desired Gaussian-profile is basically preserved, though modulated with a high frequency, and the phase-modulation is similar to that generated with an incident plane wave (see Figs. 3(b) and 4(b)).

*z*-component of the Poynting vector

*x*-component of the magnetic-field, which can be derived from the electrical field component

*E*

_{y}with the use of Maxwell’s equations. Further we divide <

*S*

_{z}> by the energy of the incident field to introduce the normalized

*z*-component of the Poynting vector (assuming an aperture width of 40

*λ*)

*x*

_{0}of the central peak in the focal region.

*x*

_{0}=0.675

*λ*. According to Eq. (10), this leads to a diffraction efficiency of 56%. Assuming an incident Gaussian-beam with

*ξ*=20

*λ*, the location of the first minimum becomes

*x*

_{0}=1.175

*λ*, which leads to

*η*=77%. In case of a Gaussian-beam

*x*

_{0}becomes a function of the half-width

*ξ*. The variation of the diffraction efficiency

*η*with the half-width

*ξ*of the incident Gaussian-beam is illustrated in Fig. 6. The dashed curve indicates that

*x*

_{0}is the location of the first minimum. The solid curve, on the other hand, indicates that

*x*

_{0}is fixed to 0.675λ, which provides a better comparison with the diffraction efficiency obtained with a normally incident plane wave. The half-width that maximizes the diffraction efficiency correlates with the half-width determined by the iterative algorithm.

## 4. Conclusion

## References and links

1. | E. Noponen, J. Turunen, and A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A |

2. | K. Hirayama, E. N. Glytsis, and T. K. Gaylord, “Riogorous electromagnetic analysis of diffractive cylinder lenses,” J. Opt. Soc. Am. A |

3. | M. Schmitz and O. Bryngdahl, “Rigorous concept for the design of diffractive microlenses with high numerical apertures,” J. Opt. Soc. Am. A |

4. | J. W. Goodman, |

5. | C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. |

6. | F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectriv constant,” J. Opt. Soc. Am. |

7. | K. Knop, “Rigorous diffraction theory ofor transmissionphase gratings with deep rectangular grooves,” J. Opt. Soc. Am. |

8. | M. Schmitz, R. Bräuer, and O. Bryngdahl, “Comment on numerical stability of rigorous differential methods of diffraction,” Opt. Commun. |

9. | Lifeng Li, “Use of Fourier series in the analysis of discontinuous periodic structures”, J. Opt. Soc. Am. A |

**OCIS Codes**

(050.1220) Diffraction and gratings : Apertures

(230.3990) Optical devices : Micro-optical devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 27, 1997

Revised Manuscript: July 24, 1997

Published: October 13, 1997

**Citation**

Felix Fuerer, Martin Schmidt, and Olof Bryngdahl, "Diffractive efficiency improvement of diffractive cylinder lenses by Gaussian-beam illumination.," Opt. Express **1**, 234-239 (1997)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-8-234

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### References

- E. Noponen, J. Turunen and A. Vasara, Electromagnetic theory and design of diffractive-lens arrays, J. Opt. Soc. Am. A 10,434-443(1993) [CrossRef]
- K. Hirayama, E. N. Glytsis and T. K. Gaylord, Riogorous electromagnetic analysis of diffractive cylinder lenses, J. Opt. Soc. Am. A 13, 2219-2231(1996) [CrossRef]
- M. Schmitz and O. Bryngdahl, Rigorous concept for the design of diffractive microlenses with high numerical apertures, J. Opt. Soc. Am. A 14, 901-906(1997) [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968)
- C. B. Burckhardt, Diffraction of a plane wave at a sinusoidally stratified dielectric grating, J. Opt. Soc. Am. 56, 1502-157(1966) [CrossRef]
- F. G. Kaspar, Diffraction by thick, periodically stratified gratings with complex dielectric constant, J. Opt. Soc. Am. 63, 37-45(1973) [CrossRef]
- K. Knop, Rigorous diffraction theory ofor transmissionphase gratings with deep rectangular grooves, J. Opt. Soc. Am. 68, 1206-1210(1978) [CrossRef]
- M. Schmitz, R. Bruer and O. Bryngdahl, Comment on numerical stability of rigorous differential methods of diffraction, Opt. Commun. 124, 1-8(1996) [CrossRef]
- Lifeng Li, Use of Fourier series in the analysis of discontinuous periodic structures, J. Opt. Soc. Am. A 13, 1870-1876(1996) [CrossRef]

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