## 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.

## 1. Introduction

Cylinder lenses are used to couple light into waveguides and other optical elements with small feature size. Further miniaturization of optical elements raise the need for cylinder lenses with small focus diameters, high diffraction efficiency and a high numerical aperture. Recent developments in the production procedures of diffractive optical elements (DOE) show that it is possible to produce cylinder lenses with high numerical aperture.

Usually an incident plane wave is assumed within the design of diffractive lenses. In this case, the simulation of the performance of these lenses can be done in two steps. First, one has to determine the field distribution in the lens plane, i.e. the field in a plane immediately behind the lens structure. In case of DOE with high numerical apertures this has to be done by the use of rigorous calculations. The second step is to calculate the field distribution in the focal plane. This can be done by simulating the free-space propagation between the lens and the focal plane applying the angular spectrum of plane waves. To encode a specific optical function in a DOE one simply has to follow the two steps backwards. This means that, starting from a desired field distribution in the focal plane, the field distribution in the lens plane (ideal transmitted field) can be found applying the angular spectrum of plane waves or by means of geometrical optics in general. Afterwards a DOE can be designed, by means of rigorous calculations, which transforms the incident plane wave into a field distribution, that is close to the ideal transmitted field. So far a lot of work has been done on optimizing the structure of the DOE with respect to the phase modulation it has to perform upon incidence of plane waves.

In the following section of this paper we optimze the field in the lens plane applying the angular spectrum of plane waves with respect to a finite aperture. The result is, that the electrical field distribution in the lens plane, which produces the focus with the highest diffraction efficiency, has a conventional sawtooth phase distribution, describing a converging cylindrical wave and an amplitude distribution with Gaussian-profile. Motivated by these results, we compare the diffraction efficiency of a cylinder lens illuminated with a plane wave and with a light beam that has a Gaussian-profile. The rigorous calculations on an example of a binary lens are presented in section 3.

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

As an example, for a cylinder lens, [1–3] the phase function of the electrical field in the lens plane is often choosen to be of the form of a converging cylindrical wave front

$φ(x)=k0n(F−F2+x2),$

where F is the desired focal length, n the refractive index of the focal region, 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

$Ey(x,z=0)=∣Ey(x,z=0)∣exp(iφ(x)).$

Assuming an infinite aperture, the electrical field according to Eqs. (1) and (2) produces a sharp peak on the optical axis at a distance F form the lens plane. In practice, the aperture of the lens has finite boundaries, which leads to a significant loss in diffraction efficiency.

Our idea to optimize the performance of a cylinder lens is to find an electrical field in the lens plane that is restricted to the extents of the lens aperture and is optimized to produce a single peak at the distance 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. To develop a field distribution that is inherent within the physical boundary conditions, we started with a random complex distribution Ey (x,z=0) in the lens plane. Ey (x,z=0) can be written as the inverse Fourier transform

$Ey(x,z=0)=12π∫ψ(kx)exp(ikxx)dkx.$ Figure 1: Iterative algorithm to evaluate the propagation of light between the lens and the focal plane.

The propagation over a distance z can be regarded as a change in the relative phase of the spatial Fourier components ψ(kx ), because each component ψ(kx ) propagates at a different angle. Across the focal plane at z=-F, the electric-field can be described by the function

$Ey(x,z=−F)=12π∫ψ(kx)exp(ikz(−F))exp(ikxx)dkx$

with

$kz=k02−kx2.$

The function O(x, ν) with

$O(x,ν)={∙ν∣x∣≥x00≤ν≤1∙1∣x∣

is used to reduce the amplitude of the electrical field outside a given window in the focal plane. x0 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. Figure 2: a) Diffractive binary lens and b) the ideal phase distribution of the transmitted electrical field.

## 3. Rigorous calculations

The result of our rigorous analysis of the complex field distribution transmitted by a diffractive lens illuminated with Gaussian-beams is presented in the following.

The diffractive cylinder lens shown in Fig. 2(a) was used as an example to demonstrate the effects of phase-modulation by diffractive lenses  on Gaussian-beams. A F/0.5 lens (aperture 40λ, focal length 20λ) with a minimum lateral feature-size of δ=0.1λ was chosen. The desired phase distribution of Eq. (1) is depicted in Fig. 2(b). Figure 3 illustrates the amplitude and the phase distribution of the transmitted field Ey (x,z=0), calculated with an updated version of the modal method,[5–9] where a plane wave under normal incidence is assumed. Some high frequency amplitude-modulation is present (see Fig. 3(a)) which is not desireable. Unfortunately, it can not be controlled or surpressed by the concept used to design the lens-structure.

The modal method enables us to calculate the transmitted field of an incident plane wave with a wave-vector located in a plane normal to the y-direction. Therefore, it is necessary to decompose the Gaussian-beam profile into its spectral components according to Eqs. (3)–(5). For our example, we figured that it is sufficient to use 100 plane waves with the angles of incident ranging from -4 to 4 degrees. Figure 3: a) Amplitude - and b) phase distribution of the transmitted electrical field assuming a normal incident plane wave. Solid curves : calculated field, dashed curves : ideal field. Figure 4: a) Amplitude- and b) phase distribution of the transmitted electrical field assuming an incident Gaussian-beam. Solid curves : calculated field, dashed curves : ideal field.

The field of each spectral component is then computed and combined to the total field transmitted by the lens. Figure 4 illustrates the transmitted field Ey (x,z=0) assuming a Gaussian-beam illumination with the amplitude distribution

$∣Ey(x,z=0)∣=exp(−4ln(2)x2ξ2),$

and the half-width ξ=20λ. 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)).

To compare the different incident beams we evaluated the z-component of the Poynting vector

$〈Sz〉=12RE(EyHx*),$

with RE(..) indicating ‘the real part’ and $Hx*$ the complex conjugate of the x-component of the magnetic-field, which can be derived from the electrical field component Ey with the use of Maxwell’s equations. Further we divide <Sz > by the energy of the incident field to introduce the normalized z-component of the Poynting vector (assuming an aperture width of 40λ)

$〈S¯z(x,z)〉=〈Sz(x,z)〉∫∣x∣≤20λdx〈Sz(x,z=λ)〉inc,$

and the diffraction efficiency

$η=∫∣x∣≤x0dx〈S¯z(x,z=−F)〉.$

The integration is performed over the extent 2x0 of the central peak in the focal region.

Figure 5 shows a comparison of the Poynting vector in the focal plane with incident plane wave (solid curve) and Gaussian-beam (dashed curve). The central part illustrated in Fig. 5(b) was chosen to compare the energy without the disturbances resulting from inefficient phase modulation in the outer regions of the diffractive element (see Fig 4(b)). The higher order maxima have been eliminated and the half-width of the central peak has been enlarged. This enlargement is caused by illuminating the aperture with a Gaussian-beam, which can be interpreted as a further reduction of the aperture. Figure 5: Energie distribution in the focal plane assuming an incident plane wave (solid curve) and Gaussian-beam illumination (dashed curve).

Usually one cannot choose the beam-profile. It makes more sense to design the diffractive element that suits the light source in terms of diffraction efficiency. In case of a normally incident plane wave, the first minimum in the focal plane is located at x0 =0.675λ. According to Eq. (10), this leads to a diffraction efficiency of 56%. Assuming an incident Gaussian-beam with ξ=20λ, the location of the first minimum becomes x0 =1.175λ, which leads to η=77%. In case of a Gaussian-beam x0 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 x0 is the location of the first minimum. The solid curve, on the other hand, indicates that x0 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. Figure 6: Diffraction efficiency η as a function of the half-width ξ of the incident Gaussian-beam. The dashed curve indicates the diffraction efficiency with x0 being the location of the first minimum. The solid curve indicates the diffraction efficiency with x0 = 0.675λ.

## 4. Conclusion

In conclusion, we have shown by the use of an iterative algorithm, evaluating the propagation of light between the lens and the focal plane, that the loss in diffraction efficiency of a diffractive cylinder lens caused by a finite aperture can be eliminated if the transmitted field in the lens plane has a Gaussian-beam profile and a phase distribution of a converging cylindrical wave front. Furthermore, the half-width of the Gaussian-profile was determined which maximizes the diffraction efficiency. Finally, we have shown by the use of rigorous calculations on an example of a binary diffractive cylinder lens, that the Gaussian-profile of an illuminating light source is preserved sufficiently in order to reduce the loss of diffraction efficiency caused by the finite aperture of the lens. The diffraction efficiency can be raised from 56% to 77% using a light source with Gaussian-beam profile.

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]

2. 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]

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]

4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968)

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

6. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectriv constant,” J. Opt. Soc. Am. 63, 37–45(1973) [CrossRef]

7. K. Knop, “Rigorous diffraction theory ofor transmissionphase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210(1978) [CrossRef]

8. M. Schmitz, R. Bräuer, and O. Bryngdahl, “Comment on numerical stability of rigorous differential methods of diffraction,” Opt. Commun. 124, 1–8(1996) [CrossRef]

9. Lifeng Li, “Use of Fourier series in the analysis of discontinuous periodic structures”, J. Opt. Soc. Am. A 13, 1870–1876(1996) [CrossRef]

### References

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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]
2. 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]
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]
4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968)
5. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1507(1966)
[Crossref]
6. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectriv constant,” J. Opt. Soc. Am. 63, 37–45(1973)
[Crossref]
7. K. Knop, “Rigorous diffraction theory ofor transmissionphase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210(1978)
[Crossref]
8. M. Schmitz, R. Bräuer, and O. Bryngdahl, “Comment on numerical stability of rigorous differential methods of diffraction,” Opt. Commun. 124, 1–8(1996)
[Crossref]
9. Lifeng Li, “Use of Fourier series in the analysis of discontinuous periodic structures”, J. Opt. Soc. Am. A 13, 1870–1876(1996)
[Crossref]

#### 1996 (3)

M. Schmitz, R. Bräuer, and O. Bryngdahl, “Comment on numerical stability of rigorous differential methods of diffraction,” Opt. Commun. 124, 1–8(1996)
[Crossref]

#### Bräuer, R.

M. Schmitz, R. Bräuer, and O. Bryngdahl, “Comment on numerical stability of rigorous differential methods of diffraction,” Opt. Commun. 124, 1–8(1996)
[Crossref]

#### Bryngdahl, O.

M. Schmitz, R. Bräuer, and O. Bryngdahl, “Comment on numerical stability of rigorous differential methods of diffraction,” Opt. Commun. 124, 1–8(1996)
[Crossref]

#### Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968)

#### Schmitz, M.

M. Schmitz, R. Bräuer, and O. Bryngdahl, “Comment on numerical stability of rigorous differential methods of diffraction,” Opt. Commun. 124, 1–8(1996)
[Crossref]

#### Opt. Commun. (1)

M. Schmitz, R. Bräuer, and O. Bryngdahl, “Comment on numerical stability of rigorous differential methods of diffraction,” Opt. Commun. 124, 1–8(1996)
[Crossref]

#### Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968)

### Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

### Figures (6)

Figure 1: Iterative algorithm to evaluate the propagation of light between the lens and the focal plane.
Figure 2: a) Diffractive binary lens and b) the ideal phase distribution of the transmitted electrical field.
Figure 3: a) Amplitude - and b) phase distribution of the transmitted electrical field assuming a normal incident plane wave. Solid curves : calculated field, dashed curves : ideal field.
Figure 4: a) Amplitude- and b) phase distribution of the transmitted electrical field assuming an incident Gaussian-beam. Solid curves : calculated field, dashed curves : ideal field.
Figure 5: Energie distribution in the focal plane assuming an incident plane wave (solid curve) and Gaussian-beam illumination (dashed curve).
Figure 6: Diffraction efficiency η as a function of the half-width ξ of the incident Gaussian-beam. The dashed curve indicates the diffraction efficiency with x0 being the location of the first minimum. The solid curve indicates the diffraction efficiency with x0 = 0.675λ.

### Equations (10)

$φ ( x ) = k 0 n ( F − F 2 + x 2 ) ,$
$E y ( x , z = 0 ) = ∣ E y ( x , z = 0 ) ∣ exp ( i φ ( x ) ) .$
$E y ( x , z = 0 ) = 1 2 π ∫ ψ ( k x ) exp ( i k x x ) d k x .$
$E y ( x , z = − F ) = 1 2 π ∫ ψ ( k x ) exp ( i k z ( − F ) ) exp ( i k x x ) d k x$
$k z = k 0 2 − k x 2 .$
$O ( x , ν ) = { ∙ ν ∣ x ∣ ≥ x 0 0 ≤ ν ≤ 1 ∙ 1 ∣ x ∣ < x 0 ,$
$∣ E y ( x , z = 0 ) ∣ = exp ( − 4 ln ( 2 ) x 2 ξ 2 ) ,$
$〈 S z 〉 = 1 2 RE ( E y H x * ) ,$
$〈 S ¯ z ( x , z ) 〉 = 〈 S z ( x , z ) 〉 ∫ ∣ x ∣ ≤ 20 λ dx 〈 S z ( x , z = λ ) 〉 inc ,$
$η = ∫ ∣ x ∣ ≤ x 0 dx 〈 S ¯ z ( x , z = − F ) 〉 .$