## Abstract

A theoretical model is established using Rayleigh-Sommerfeld diffraction theory to describe the diffraction focusing characteristics of low F-number diffractive optical elements with continuous relief fabricated by laser direct writing, and continuous-relief diffractive optical elements with a design wavelength of 441.6nm and a F-number of F/4 are fabricated and measured to verify the validity of the diffraction focusing model. The measurements made indicate that the spot size is 1.75µm and the diffraction efficiency is 70.7% at the design wavelength, which coincide well with the theoretical results: a spot size of 1.66µm and a diffraction efficiency of 71.2%.

© 2007 Optical Society of America

## Corrections

Mingguang Shan and Jiubin Tan, "Modeling focusing characteristics of low F-number diffractive optical elements with continuous relief fabricated by laser direct writing: errata," Opt. Express**16**, 11680-11681 (2008)

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-16-15-11680

## 1. Introduction

Diffractive optical elements (DOEs) with continuous relief are widely used for multichannel optical interconnect, fiber coupling, beam splitter [1–3] and especially parallel laser direct writing (PLDW) [4–6] because of their advantages over the widely used binary DOEs, including higher diffraction efficiency and one step fabrication. It is the fast development of such direct writing technology as laser direct writing (LDW) [7–9], single-point diamond turning [7], electron beam direct writing [7,10] and focused ion beam direct writing [2,7] that makes it possible to fabricate continuous-relief DOEs with high diffraction efficiency. Compared to electron beam direct writing and focused ion beam direct writing which cause high fabrication cost, LDW and single point diamond turning can both be used to produce DOEs with large aperture and deep continuous relief at low cost, but single-point diamond turning can be used on some material for fabrication of low NA Fresnel zone elements only. So, LDW is an advanced technology more suitable for fabrication of DOEs with large aperture and deep continuous relief. However, the convoluted-relief smoothing resulting from the finite size of the writing laser beam cause a significant reduction in diffraction efficiency of DOEs [7–9,11–14]. Much work has been done on the fabrication of continuous-relief DOEs [9,11–14]. However, to the best of our knowledge, most of the work on the focusing characteristics of continuous-relief DOEs fabricated using LDW focus on blazed gratings. The analyses of the focusing characteristics of the convoluted-relief DOEs are done with the elements taken as local blazed gratings or on the basis of paraxial approximation. However, in such application as PLDW, a lower F-number (*F*/#) array is usually utilized to enhance the writing resolution. It is therefore of great significance to accurately characterize the focusing characteristics of each DOE in the array to improve the pattern fidelity through exposure dose modulation [5]. So a theoretical model with nonparaxial approximation is established to describe the diffraction focusing characteristics of low F-number DOEs with continuous relief fabricated by LDW, and the experimental results are compared with the theoretical results to verify the validity of the diffraction focusing model.

## 2. Diffraction focusing model of DOEs with continuous relief

According to the theories of blazed grating and geometrical optics, a phase function of continuous relief DOEs with nonparaxial approximation [2,13] can be expressed as

where, *r _{m}* is the radius of the

*m*-th zone of DOEs, which can be given by

${r}_{m}=\sqrt{2mp{\lambda}_{0}{f}_{0}+{\left(mp{\lambda}_{0}\right)}^{2}},0\le m\le M$

where, *p* is the phase depth factor used to optimize the relief depth and the radius of zone to make the fabrication easy; *f*
_{0} is the focal length of the DOEs for design wavelength λ_{0}; *n*
_{0} is the refractive index of DOEs material for design wavelength λ_{0}; *c* and *K* are the factors used to determine the relief curvature which can be given by

$c=\frac{1}{{f}_{0}*\left(1-{n}_{0}\right)+mp{\lambda}_{0}},$
$K=-{{n}_{0}}^{2}$A continuous relief can be generated using a LDW system in a single-exposure step by scanning an intensity modulated focused laser beam across a photoresist coated substrate and produced through the development process later on. However, due to the finite extension of the focused laser beam, the continuous relief fabricated by laser direct writing is slightly different in profile from the relief designed, for example, the sharp edges of profile are all smoothed. Mathematically, the smoothing function can be described using the convolution of sampled profile *φ _{s}* (

*r*) with Gaussian intensity distribution

*I*(

*r*) in the writing spot [9,11–14] as shown below.

where, *φ _{c}*(

*r*) is the phase function of the DOEs with continuous relief fabricated;

*φ*(

_{s}*r*) is the phase function

*φ*

_{0}(

*r*) sampled by the interscan distance rs, which can be given by

Gaussian intensity distribution *I*(*r*) in the writing spot can be expressed as

where, *ω* is the radius of *I*(*r*) at maximum intensity level *e*
^{-2} of the writing spot.

When the characteristic period of the DOEs is longer than the diameter of the writing spot, the effect of convolution can be analyzed using scalar diffraction theory. According to Eqs. (1)–(4), the phase function of DOEs with continuous relief fabricated by LDW can be given by

It can be seen from Eq. (5) that the fidelity of the relief fabricated depends on the modulated laser intensity, the diameter of the writing spot and the interscan distance if other fabrication conditions are ideal.

The first Rayleigh-Sommerfeld diffraction theory in convolution form [15] is used in this paper to analyze complex amplitude distribution *U*
_{1}(*x*
_{1}, *y*
_{1}) in the back focal plane of DOEs. *U*
_{1}(*x*
_{1}, *y*
_{1}) can then be given by

$$H({v}_{x},{v}_{y})=\{\begin{array}{cc}\mathrm{exp}\left[i2\pi {z}_{12}\sqrt{\frac{1}{{\lambda}^{2}}-{{v}_{x}}^{2}-{{v}_{y}}^{2}}\right]& \frac{1}{{\lambda}^{2}}-{{v}_{x}}^{2}-{{v}_{y}}^{2}>0\\ 0& \mathrm{otherwise}\end{array}$$

where, *F* and *F*
^{-1} denote the forward and inverse Fourier transforms, respectively; *U*
_{0}(*x, y*) is the complex amplitude distribution of the incident plane wave; *H*(*v _{x}, v_{y}*) is the transform function of free-space propagation from DOEs to the focal plane;

*z*

_{12}is the distance of propagation from DOEs to the back focal plane;

*v*=sin

_{x}*θ*/λ and

_{x}*v*=sin

_{y}*θ*/λ are spatial frequencies.

_{y}As the intensity distribution is the squared modulus of complex amplitude distribution, the intensity distribution in the focal region of DOEs with continuous relief can be expressed as

In the way similar to that used in Ref.7, the diffraction focusing spot size is defined as the full width at half-maximum (FWHM), and the diffraction efficiency is the ratio of the energy inside the spot to the energy in the focal plane. The diffraction efficiency can then be expressed as

where rFWHM is the radius at the FWHM of the spot.

## 3. Analysis

In order to verify the validity of our model, a DOE is designed using Eq. (1) with a wavelength of 441.6nm, a radius of 64.0µm, a focal length of 512.0µm and a phase depth factor of 3. The DOE is made of fused quartz for its excellent optical properties with a refractive index of 1.466. The DOE was fabricated with LDW system CLWS300 [9] using He-Cd writing laser on a positive photoresist with a writing-spot radius of 1.4µm and an interscan distance of 0.4µm. After the development, the pattern written in photoresist is transferred by ion etching into fused quartz. As shown in Fig.1, the sharp edges are smoothed by the smooth function of LDW as we have discussed.

The refractive properties of DOEs deriving from the phase depth factor *p*=3 [16] cause the actual focal plane to depart from the designed focal plane. And so the actual focal plane is defined as the lateral plane at the position of the maximum intensity along the light axis. As shown in Fig.2, in order to evaluate the focusing characteristics of the DOE, a He-Cd laser with a wavelength of 441.6nm and a CCD with a pixel size of 4.65µm×4.65µm are used. The DOE is mounted on a microstage with a position resolution of 10nm. An attenuator is used to attenuate the point spread function (PSF) intensity and an objective is used to magnify the PSF within the CCD detection range. The microstage can be adjusted along the optical axis to moves the DOE forward or back until the optimal focus is formed on the PC monitor connected to the CCD as shown in Fig. 3(a), and then the 2D intensity profile can be obtained as is shown in Fig. 3(b).

The normalized intensity distributions are shown in Fig. 4 for comparison of experimental results with theoretical results and ideal design results. It can be seen from Fig.4 that the experimental results coincide well with the theoretical results although the experimental spot size (FWHM_{E}=1.75µm) is slightly larger than the theoretical spot size (FWHM_{T}=1.66µm).

The experimental diffraction efficiency is 70.7%, which coincides well with the theoretical diffraction efficiency of 71.2%, but it is far away from the ideal design diffraction efficiency of 79.1%, and all these prove the validity of our model.

However, as shown in Fig. 4, there is still a slight difference between experimental and theoretical results. So we analyzed the estimated precision of the technique used to measure efficiency and found such factors as quality of collimated beam, background noise and pixel size of CCD, and effect of external interference have their effect on the estimated precision of the technique used to measure efficiency, and it is therefore recommended to enhance measurement accuracy by using quality expanders & collimators with high magnification power, objective systems with a magnification power of 40× or 60×, and digital CCD with high sensitivity, low noise and small pixel size, and/or blocking parasitic light.

## 4. Conclusion

A theoretical model with nonparaxial approximation is established using Rayleigh-Sommerfeld diffraction theory for DOEs with continuous relief with the focusing characteristics of continuous relief fabricated by LDW taken into consideration. The good agreement between theoretical and experimental results further verifies the validity of the diffraction focus model established.

## Acknowledgments

We would like to thank the National Natural Science Foundation of China (50675052) for its financial support, Prof. H.P. Herzig of University of Neuchâtel and Markus Rossi of Heptagon Oy for assistance, and Victor P. Korolkov of the Russian Academy of Science for useful discussion and assistance provided.

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