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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 25 — Dec. 10, 2007
  • pp: 17032–17037
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Modeling focusing characteristics of low F-number diffractive optical elements with continuous relief fabricated by laser direct writing

Mingguang Shan and Jiubin Tan  »View Author Affiliations


Optics Express, Vol. 15, Issue 25, pp. 17032-17037 (2007)
http://dx.doi.org/10.1364/OE.15.017032


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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

1. Introduction

Diffractive optical elements (DOEs) with continuous relief are widely used for multichannel optical interconnect, fiber coupling, beam splitter [1

1. P. Ehbets, M. Rossi, and H. P. Herzig, “Continuous-relief fan-out elements with optimized fabrication tolerance,” Opt. Eng. 34, 3456–3464 (1995). [CrossRef]

3

3. D. Feng, Y. Yan, G. Jin, and S. Fan, “Design and fabrication of continuous-profile diffractive micro-optical elements as a beam splitter,” Appl. Opt. 43, 5476–5480 (2004). [CrossRef] [PubMed]

] and especially parallel laser direct writing (PLDW) [4

4. R. Yang, K.F. Chan, Z. Feng, I. Akihito, and W. Mei, “Design and fabrication of microlens and spatial filter array by self-alignment for maskless lithography systems,” J. Microlith. Microfab. Microsyst. 2, 210–219 (2003). [CrossRef]

6

6. J. B. Tan, M.G. Shan, J. Liu, H. Zhang, and C.G. Zhao. “Model analysis of effect of diffraction focus characteristics of microlens arrays on parallel laser direct writing quality,” Opt. Commun. 277, 237–240(2007). [CrossRef]

] 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

7. M. T. Gale , “Direct writing of continuous-relief elements,” in Micro-Optics-Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997).

9

9. V. P. Korolkov, R. K. Nasyrov, and R. V. Shimansky, “Zone-boundary optimization for direct laser writing of continuous-relief diffractive optical elements,” Appl. Opt. 45, 53–62 (2006). [CrossRef] [PubMed]

], single-point diamond turning [7

7. M. T. Gale , “Direct writing of continuous-relief elements,” in Micro-Optics-Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997).

], electron beam direct writing [7

7. M. T. Gale , “Direct writing of continuous-relief elements,” in Micro-Optics-Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997).

,10

10. M. Okano, H. Kikuta, Y. Hirai, K. Yamamoto, and T. Yotsuya, “Optimization of diffraction grating profiles in fabrication by electron-beam lithography,” Appl. Opt. 43, 5137–5142 (2004). [CrossRef] [PubMed]

] and focused ion beam direct writing [2

2. F. Yong-Qi, N. Kok Ann Bryan, and O. Shing, “Diffractive optical elements with continuous relief fabricated by focused ion beam for monomode fiber coupling,” Opt. Express 7, 141–147 (2000). [CrossRef] [PubMed]

,7

7. M. T. Gale , “Direct writing of continuous-relief elements,” in Micro-Optics-Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997).

] 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

7. M. T. Gale , “Direct writing of continuous-relief elements,” in Micro-Optics-Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997).

9

9. V. P. Korolkov, R. K. Nasyrov, and R. V. Shimansky, “Zone-boundary optimization for direct laser writing of continuous-relief diffractive optical elements,” Appl. Opt. 45, 53–62 (2006). [CrossRef] [PubMed]

,11

11. M. Kuittinen, H. P. Herzig, and P. Ehbets, “Improvements in diffraction efficiency of gratings and microlenses with continuous relief structures,” Opt. Commun. 120, 230–234 (1995). [CrossRef]

14

14. I. Kallioniemi, T. Ammer, and M. Rossi, “Optimization of continuous-profile blazed gratings using rigorous diffraction theory,” Opt. Commun. 177, 15–24 (2000). [CrossRef]

]. Much work has been done on the fabrication of continuous-relief DOEs [9

9. V. P. Korolkov, R. K. Nasyrov, and R. V. Shimansky, “Zone-boundary optimization for direct laser writing of continuous-relief diffractive optical elements,” Appl. Opt. 45, 53–62 (2006). [CrossRef] [PubMed]

,11

11. M. Kuittinen, H. P. Herzig, and P. Ehbets, “Improvements in diffraction efficiency of gratings and microlenses with continuous relief structures,” Opt. Commun. 120, 230–234 (1995). [CrossRef]

14

14. I. Kallioniemi, T. Ammer, and M. Rossi, “Optimization of continuous-profile blazed gratings using rigorous diffraction theory,” Opt. Commun. 177, 15–24 (2000). [CrossRef]

]. 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

5. R. Menon, D. Gil, and H. I. Smith, “Experimental characterization of focusing by high-numerical-aperture zone plates,” J. Opt. Soc. Am. A 23, 567–571 (2006). [CrossRef]

]. 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

2. F. Yong-Qi, N. Kok Ann Bryan, and O. Shing, “Diffractive optical elements with continuous relief fabricated by focused ion beam for monomode fiber coupling,” Opt. Express 7, 141–147 (2000). [CrossRef] [PubMed]

,13

13. T. Hessler, M. Rossi, R. E. Kunz, and M. T. Gale, “Analysis and optimization of fabrication of continuous-relief diffractive optical elements,” Appl. Opt. 37, 4069–4079 (1998). [CrossRef]

] can be expressed as

φ0(r)=2πpm+2π(n01)λ0cr21+1(K+1)c2r2,rmrrm+1
(1)

where, rm is the radius of the m-th zone of DOEs, which can be given by

rm=2mpλ0f0+(mpλ0)2,0mM

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 λ0; n 0 is the refractive index of DOEs material for design wavelength λ0; c and K are the factors used to determine the relief curvature which can be given by

c=1f0*(1n0)+mpλ0, K=n02A 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 φs (r) with Gaussian intensity distribution I(r) in the writing spot [9

9. V. P. Korolkov, R. K. Nasyrov, and R. V. Shimansky, “Zone-boundary optimization for direct laser writing of continuous-relief diffractive optical elements,” Appl. Opt. 45, 53–62 (2006). [CrossRef] [PubMed]

,11

11. M. Kuittinen, H. P. Herzig, and P. Ehbets, “Improvements in diffraction efficiency of gratings and microlenses with continuous relief structures,” Opt. Commun. 120, 230–234 (1995). [CrossRef]

14

14. I. Kallioniemi, T. Ammer, and M. Rossi, “Optimization of continuous-profile blazed gratings using rigorous diffraction theory,” Opt. Commun. 177, 15–24 (2000). [CrossRef]

] as shown below.

φc(r)=φs(r)I(r)
(2)

where, φc(r) is the phase function of the DOEs with continuous relief fabricated; φs (r) is the phase function φ 0(r) sampled by the interscan distance rs, which can be given by

φs(r)=φ0(r)n=δ(rnrs)=n=φ0(nrs)δ(rnrs)
(3)

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

I(r)=I0exp(2r2ω2)
(4)

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

φc(r)=I0n=[2πpm+2π(n01)λ0c(nrs)21+1(K+1)c2(nrs)2]exp[2(rnrs)2ω02]
(5)

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

15. N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A 15, 857–867 (1998). [CrossRef]

] 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

U1(x1,y1)=F1{F[U0(x,y)exp[iφc(x,y)]]H(vx,vy)}
H(vx,vy)={exp[i2πz121λ2vx2vy2]1λ2vx2vy2>00otherwise
(6)

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(vx, vy) 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; vx=sinθx/λ and vy=sinθy/λ are spatial frequencies.

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

Ic(x1,y1)=U1(x1,y1)2
(7)

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

η=0rFWHMIc(r1)r1dr10Ic(r1)r1dr1
(8)

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

9. V. P. Korolkov, R. K. Nasyrov, and R. V. Shimansky, “Zone-boundary optimization for direct laser writing of continuous-relief diffractive optical elements,” Appl. Opt. 45, 53–62 (2006). [CrossRef] [PubMed]

] 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.

Fig. 1. AFM image of DOE with continuous relief

The refractive properties of DOEs deriving from the phase depth factor p=3 [16

16. M. Rossi, R. E. Kunz, and H. P. Herzig, “Refractive and diffractive properties of planar micro-optical elements,” Appl. Opt. 34, 5996–6007 (1995). [CrossRef] [PubMed]

] 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).

Fig. 2. Schematic diagram of experimental set-up
Fig. 3. Experimental results: (a) CCD image of focused spot (b) Intensity profile

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 (FWHME=1.75µm) is slightly larger than the theoretical spot size (FWHMT=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.

Fig. 4. Comparison of experimental results with theoretical results and ideal design results

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.

References and links

1.

P. Ehbets, M. Rossi, and H. P. Herzig, “Continuous-relief fan-out elements with optimized fabrication tolerance,” Opt. Eng. 34, 3456–3464 (1995). [CrossRef]

2.

F. Yong-Qi, N. Kok Ann Bryan, and O. Shing, “Diffractive optical elements with continuous relief fabricated by focused ion beam for monomode fiber coupling,” Opt. Express 7, 141–147 (2000). [CrossRef] [PubMed]

3.

D. Feng, Y. Yan, G. Jin, and S. Fan, “Design and fabrication of continuous-profile diffractive micro-optical elements as a beam splitter,” Appl. Opt. 43, 5476–5480 (2004). [CrossRef] [PubMed]

4.

R. Yang, K.F. Chan, Z. Feng, I. Akihito, and W. Mei, “Design and fabrication of microlens and spatial filter array by self-alignment for maskless lithography systems,” J. Microlith. Microfab. Microsyst. 2, 210–219 (2003). [CrossRef]

5.

R. Menon, D. Gil, and H. I. Smith, “Experimental characterization of focusing by high-numerical-aperture zone plates,” J. Opt. Soc. Am. A 23, 567–571 (2006). [CrossRef]

6.

J. B. Tan, M.G. Shan, J. Liu, H. Zhang, and C.G. Zhao. “Model analysis of effect of diffraction focus characteristics of microlens arrays on parallel laser direct writing quality,” Opt. Commun. 277, 237–240(2007). [CrossRef]

7.

M. T. Gale , “Direct writing of continuous-relief elements,” in Micro-Optics-Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997).

8.

M.T. Gale, M. Rossi, J. Pedersen, and H. Schutz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994). [CrossRef]

9.

V. P. Korolkov, R. K. Nasyrov, and R. V. Shimansky, “Zone-boundary optimization for direct laser writing of continuous-relief diffractive optical elements,” Appl. Opt. 45, 53–62 (2006). [CrossRef] [PubMed]

10.

M. Okano, H. Kikuta, Y. Hirai, K. Yamamoto, and T. Yotsuya, “Optimization of diffraction grating profiles in fabrication by electron-beam lithography,” Appl. Opt. 43, 5137–5142 (2004). [CrossRef] [PubMed]

11.

M. Kuittinen, H. P. Herzig, and P. Ehbets, “Improvements in diffraction efficiency of gratings and microlenses with continuous relief structures,” Opt. Commun. 120, 230–234 (1995). [CrossRef]

12.

T. Hessler and R. E. Kunz, “Relaxed fabrication tolerances for low-Fresnel-number lenses,” J. Opt. Soc. Am. A 14, 1599–1606 (1997). [CrossRef]

13.

T. Hessler, M. Rossi, R. E. Kunz, and M. T. Gale, “Analysis and optimization of fabrication of continuous-relief diffractive optical elements,” Appl. Opt. 37, 4069–4079 (1998). [CrossRef]

14.

I. Kallioniemi, T. Ammer, and M. Rossi, “Optimization of continuous-profile blazed gratings using rigorous diffraction theory,” Opt. Commun. 177, 15–24 (2000). [CrossRef]

15.

N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A 15, 857–867 (1998). [CrossRef]

16.

M. Rossi, R. E. Kunz, and H. P. Herzig, “Refractive and diffractive properties of planar micro-optical elements,” Appl. Opt. 34, 5996–6007 (1995). [CrossRef] [PubMed]

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(220.4000) Optical design and fabrication : Microstructure fabrication
(260.1960) Physical optics : Diffraction theory

ToC Category:
Diffraction and Gratings

History
Original Manuscript: September 28, 2007
Revised Manuscript: November 13, 2007
Manuscript Accepted: November 23, 2007
Published: December 5, 2007

Citation
Mingguang Shan and Jiubin Tan, "Modeling focusing characteristics of low Fnumber diffractive optical elements with continuous relief fabricated by laser direct writing," Opt. Express 15, 17032-17037 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-17032


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References

  1. P. Ehbets, M. Rossi, and H. P. Herzig, "Continuous-relief fan-out elements with optimized fabrication tolerance," Opt. Eng. 34, 3456-3464 (1995). [CrossRef]
  2. F. Yong-Qi, N. Kok Ann Bryan, and O. Shing, "Diffractive optical elements with continuous relief fabricated by focused ion beam for monomode fiber coupling," Opt. Express 7, 141-147 (2000). [CrossRef] [PubMed]
  3. D. Feng, Y. Yan, G. Jin, and S. Fan, "Design and fabrication of continuous-profile diffractive micro-optical elements as a beam splitter," Appl. Opt. 43, 5476-5480 (2004). [CrossRef] [PubMed]
  4. R. Yang, K.F. Chan, Z. Feng, I. Akihito, and W. Mei, "Design and fabrication of microlens and spatial filter array by self-alignment for maskless lithography systems," J. Microlithogr. Microfabr. Microsyst. 2, 210-219 (2003). [CrossRef]
  5. R. Menon, D. Gil, and H. I. Smith, "Experimental characterization of focusing by high-numerical-aperture zone plates," J. Opt. Soc. Am. A 23, 567-571 (2006). [CrossRef]
  6. J. B. Tan, M.G. Shan, J. Liu, H. Zhang, and C.G. Zhao. "Model analysis of effect of diffraction focus characteristics of microlens arrays on parallel laser direct writing quality," Opt. Commun. 277, 237-240(2007). [CrossRef]
  7. M. T. Gale, "Direct writing of continuous-relief elements," in Micro-Optics-Elements, Systems, and Applications, H. P. Herzig, ed., (Taylor & Francis, London, 1997).
  8. M. T. Gale, M. Rossi, J. Pedersen, and H. Schutz, "Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists," Opt. Eng. 33, 3556-3566 (1994). [CrossRef]
  9. V. P. Korolkov, R. K. Nasyrov, and R. V. Shimansky, "Zone-boundary optimization for direct laser writing of continuous-relief diffractive optical elements," Appl. Opt. 45, 53-62 (2006). [CrossRef] [PubMed]
  10. M. Okano, H. Kikuta, Y. Hirai, K. Yamamoto, and T. Yotsuya, "Optimization of diffraction grating profiles in fabrication by electron-beam lithography," Appl. Opt. 43, 5137-5142 (2004). [CrossRef] [PubMed]
  11. M. Kuittinen, H. P. Herzig, and P. Ehbets, "Improvements in diffraction efficiency of gratings and microlenses with continuous relief structures," Opt. Commun. 120, 230-234 (1995). [CrossRef]
  12. T. Hessler and R. E. Kunz, "Relaxed fabrication tolerances for low-Fresnel-number lenses," J. Opt. Soc. Am. A 14, 1599- 1606 (1997). [CrossRef]
  13. T. Hessler, M. Rossi, R. E. Kunz, and M. T. Gale, "Analysis and optimization of fabrication of continuous-relief diffractive optical elements," Appl. Opt. 37, 4069-4079 (1998). [CrossRef]
  14. I. Kallioniemi, T. Ammer, and M. Rossi, "Optimization of continuous-profile blazed gratings using rigorous diffraction theory," Opt. Commun. 177, 15-24 (2000). [CrossRef]
  15. N. Delen and B. Hooker, "Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach," J. Opt. Soc. Am. A 15, 857-867 (1998). [CrossRef]
  16. M. Rossi, R. E. Kunz, and H. P. Herzig, "Refractive and diffractive properties of planar micro-optical elements," Appl. Opt. 34, 5996-6007 (1995). [CrossRef] [PubMed]

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