Abstract

The properties of planar-integrated imaging systems are studied on the basis of algebraic ray tracing. A third-order approximation yields analytical expressions for ray aberrations in the image plane. Ray aberrations are compared with the diffraction-limited spot size to estimate optimum system performance from the minimum of the second moment of the point-spread function. We use as a figure of merit the space–bandwidth product of the planar imaging setups. The investigation is restricted to imaging systems consisting of three optical elements.

© 1999 Optical Society of America

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References

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  1. J. Jahns, A. Huang, “Planar integration of free-space optical components,” Appl. Opt. 28, 1602–1605 (1989).
    [CrossRef] [PubMed]
  2. J. Jahns, “Integrated optical imaging system,” Appl. Opt. 29, 1998 (1990).
    [CrossRef] [PubMed]
  3. J. Jahns, S. J. Walker, “Imaging with planar optical systems,” Opt. Commun. 76, 313–317 (1990).
    [CrossRef]
  4. M. Testorf, J. Jahns, “Paraxial theory of planar integrated systems,” J. Opt. Soc. Am. A 14, 1569–1575 (1997).
    [CrossRef]
  5. R. Barakat, A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
    [CrossRef]
  6. M. Testorf, J. Jahns, “Imaging in planar optics: system design with respect to the angle of light propagation,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 364–367.
  7. Ch. Gimkiewicz, D. Hagedorn, J. Jahns, E. B. Kley, F. Thoma, “Fabrication of microprisms for planar-optical interconnections using analog gray-scale lithography with high-energy beam-sensitive glass,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 175–176.
  8. W. Eckert, K.-H. Brenner, C. Passon, “Planar integration of free-space micro-optical systems with refractive elements,” presented at the International Conference on Optical Computing, Edinburgh, August 22–25, 1994.
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    [CrossRef]
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  12. W. T. Welford, “Aberration theory of gratings and grating mountings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. 4, pp. 243–280.
  13. G. Schulz, “Primary aberration-free imaging by three refracting surfaces,” J. Opt. Soc. Am. 70, 1149–1152 (1980).
    [CrossRef]
  14. A. Offner, “New concepts in projection mask aligners,” Opt. Eng. 14, 130–132 (1975).
    [CrossRef]
  15. W. G. Fastie, “A small plane grating monochromator,” J. Opt. Soc. Am. 42, 641–647 (1952).
    [CrossRef]
  16. N. Bareket, “Second moment of the diffraction point spread function as an image quality criterion,” J. Opt. Soc. Am. 69, 1311–1312 (1979).
    [CrossRef]
  17. E. H. Linfoot, “Contrast transmission function at low spatial frequencies,” Opt. Acta 6, 387–403 (1959).
    [CrossRef]

1997

1996

1992

1990

J. Jahns, “Integrated optical imaging system,” Appl. Opt. 29, 1998 (1990).
[CrossRef] [PubMed]

J. Jahns, S. J. Walker, “Imaging with planar optical systems,” Opt. Commun. 76, 313–317 (1990).
[CrossRef]

1989

1980

1979

1975

A. Offner, “New concepts in projection mask aligners,” Opt. Eng. 14, 130–132 (1975).
[CrossRef]

1966

R. Barakat, A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

1959

E. H. Linfoot, “Contrast transmission function at low spatial frequencies,” Opt. Acta 6, 387–403 (1959).
[CrossRef]

1952

1950

Barakat, R.

R. Barakat, A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

Bareket, N.

Brenner, K.-H.

W. Eckert, K.-H. Brenner, C. Passon, “Planar integration of free-space micro-optical systems with refractive elements,” presented at the International Conference on Optical Computing, Edinburgh, August 22–25, 1994.

Davis, J. A.

Eckert, W.

W. Eckert, K.-H. Brenner, C. Passon, “Planar integration of free-space micro-optical systems with refractive elements,” presented at the International Conference on Optical Computing, Edinburgh, August 22–25, 1994.

Fastie, W. G.

Gimkiewicz, Ch.

Ch. Gimkiewicz, D. Hagedorn, J. Jahns, E. B. Kley, F. Thoma, “Fabrication of microprisms for planar-optical interconnections using analog gray-scale lithography with high-energy beam-sensitive glass,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 175–176.

Hagedorn, D.

Ch. Gimkiewicz, D. Hagedorn, J. Jahns, E. B. Kley, F. Thoma, “Fabrication of microprisms for planar-optical interconnections using analog gray-scale lithography with high-energy beam-sensitive glass,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 175–176.

Harris, C. W.

Houston, A.

R. Barakat, A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

Huang, A.

Jahns, J.

M. Testorf, J. Jahns, “Paraxial theory of planar integrated systems,” J. Opt. Soc. Am. A 14, 1569–1575 (1997).
[CrossRef]

J. Jahns, “Integrated optical imaging system,” Appl. Opt. 29, 1998 (1990).
[CrossRef] [PubMed]

J. Jahns, S. J. Walker, “Imaging with planar optical systems,” Opt. Commun. 76, 313–317 (1990).
[CrossRef]

J. Jahns, A. Huang, “Planar integration of free-space optical components,” Appl. Opt. 28, 1602–1605 (1989).
[CrossRef] [PubMed]

M. Testorf, J. Jahns, “Imaging in planar optics: system design with respect to the angle of light propagation,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 364–367.

Ch. Gimkiewicz, D. Hagedorn, J. Jahns, E. B. Kley, F. Thoma, “Fabrication of microprisms for planar-optical interconnections using analog gray-scale lithography with high-energy beam-sensitive glass,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 175–176.

Kley, E. B.

Ch. Gimkiewicz, D. Hagedorn, J. Jahns, E. B. Kley, F. Thoma, “Fabrication of microprisms for planar-optical interconnections using analog gray-scale lithography with high-energy beam-sensitive glass,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 175–176.

Lilly, R. A.

Linfoot, E. H.

E. H. Linfoot, “Contrast transmission function at low spatial frequencies,” Opt. Acta 6, 387–403 (1959).
[CrossRef]

Offner, A.

A. Offner, “New concepts in projection mask aligners,” Opt. Eng. 14, 130–132 (1975).
[CrossRef]

Passon, C.

W. Eckert, K.-H. Brenner, C. Passon, “Planar integration of free-space micro-optical systems with refractive elements,” presented at the International Conference on Optical Computing, Edinburgh, August 22–25, 1994.

Schulz, G.

Testorf, M.

M. Testorf, J. Jahns, “Paraxial theory of planar integrated systems,” J. Opt. Soc. Am. A 14, 1569–1575 (1997).
[CrossRef]

M. Testorf, J. Jahns, “Imaging in planar optics: system design with respect to the angle of light propagation,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 364–367.

Thoma, F.

Ch. Gimkiewicz, D. Hagedorn, J. Jahns, E. B. Kley, F. Thoma, “Fabrication of microprisms for planar-optical interconnections using analog gray-scale lithography with high-energy beam-sensitive glass,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 175–176.

Walker, S. J.

J. Jahns, S. J. Walker, “Imaging with planar optical systems,” Opt. Commun. 76, 313–317 (1990).
[CrossRef]

Walter, A.

Welford, W. T.

W. T. Welford, “Aberration theory of gratings and grating mountings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. 4, pp. 243–280.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

R. Barakat, A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

E. H. Linfoot, “Contrast transmission function at low spatial frequencies,” Opt. Acta 6, 387–403 (1959).
[CrossRef]

Opt. Commun.

J. Jahns, S. J. Walker, “Imaging with planar optical systems,” Opt. Commun. 76, 313–317 (1990).
[CrossRef]

Opt. Eng.

A. Offner, “New concepts in projection mask aligners,” Opt. Eng. 14, 130–132 (1975).
[CrossRef]

Other

W. T. Welford, “Aberration theory of gratings and grating mountings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. 4, pp. 243–280.

M. Testorf, J. Jahns, “Imaging in planar optics: system design with respect to the angle of light propagation,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 364–367.

Ch. Gimkiewicz, D. Hagedorn, J. Jahns, E. B. Kley, F. Thoma, “Fabrication of microprisms for planar-optical interconnections using analog gray-scale lithography with high-energy beam-sensitive glass,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 175–176.

W. Eckert, K.-H. Brenner, C. Passon, “Planar integration of free-space micro-optical systems with refractive elements,” presented at the International Conference on Optical Computing, Edinburgh, August 22–25, 1994.

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Figures (6)

Fig. 1
Fig. 1

Planar-integrated optical imaging systems with three diffractive optical elements.

Fig. 2
Fig. 2

Coordinate system used for ray tracing in planar-integrated systems: unfolded geometry.

Fig. 3
Fig. 3

Examples of imaging systems that can be derived from (a) the three-element setup: (b) single-lens system, (c) 4f system.

Fig. 4
Fig. 4

Spot array patterns of object points in the image plane of the single-lens system assuming a parabolic lens: (a) second-order aberrations, (b) third-order aberrations.

Fig. 5
Fig. 5

Imaging performance of the single-lens system: (a) optimum tilt angle of the optical axis, (b) maximum achievable SBP.

Fig. 6
Fig. 6

Imaging performance of the 4f system. Plots correspond to those in Fig. 5.

Equations (56)

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vout=vin,wout=win.
xout=xin+Δsp vin(1-vin2-win2)1/2,
yout=yin+Δsp win(1-vin2-win2)1/2.
Δsp=Δs1-v tan ϑ1-v2-w2-1.
uout=uin+kg(r),
ϕ(x, y)=2πmnamnxmyn.
k(r)=12πϕ(r),
vout=vin+mnmamnxinm-1yinn,
wout=win+mnnamnxinmyinn-1.
xout=xin,yout=yin.
a1,m n=-cos ϑ2zDcos ϑ+2a2,m nzDcos ϑ+a2,m nzD,
fx=-12a20 cos2 ϑ,fy=-12a02.
fx=zDcos3 ϑ,fy=zDcos ϑ.
Δx=p,qXpqpq+o,p,qXopqopq,
Δy=p,qYpqpq+o,p,qYopqopq,
Δx(2)=Xxxx2+Xxvxv,
Δy(2)=Yxyxy+Yxwxw+Yyvyv.
SBP=DxDyδxδy,
δx=2mx,δy=2my,
m=md+mr.
md=md,x+md,y=2δxdlπ2+2δydl2π2,
δxdl=λΔsAi,x,δydl=λΔsAi,y,
Ai,x2zD sin ϑ.
mr=mr,x+mr,y=vminvmaxwminwmax(Δx2+Δy2)dvdw(vmax-vmin)(wmax-wmin).
Xxx=2 sin ϑzD,Xxv=4 tan ϑ;
Yxy=2 sin ϑzD,Yxw=2 tan ϑ,Yvy=2 tan ϑ.
Xxxx=-1zD22-cos2 ϑ-8 zD3cos ϑa2,40,
Xxxv=-3zD cos ϑ2-cos2 ϑ-16 zD3cos ϑa2,40,
Xxyy=-cos2 ϑzD21-4 zD3cos3 ϑa2,22,
Xxyw=-2 cos ϑzD1-8 zD3cos3 ϑa2,22,
Xxvv=1cos2 ϑ2-5 cos2 ϑ+96 zD3cos ϑa2,40,
Xxww=-1+16 zD3cos3 ϑa2,22,
Xyyv=-cos ϑzD1-8 zD3cos3 ϑa2,22,
Xyvw=2-1+16 zD3cos3 ϑa2,22,
Xvvv=2zDcos3 ϑ-3+2 cos2 ϑ+32 zD4cos ϑa2,40,
Xvww=2zDcos ϑ-1+16 zD3cos3 ϑa2,22;
Yxxy=-1zD22-1 cos2 ϑ-4 zD3cos ϑa2,22,
Yxxw=-1zD cos ϑ2-cos2 ϑ-8 zD3cos ϑa2,22,
Yxyv=-2 1zD cos ϑ2-cos2 ϑ-8 zD3cos ϑa2,22,
Yxvw=-21-16 zD3cos3 ϑa2,22,
Yyyy=-cos2 ϑzD21-8 zD3cos3 ϑa2,04,
Yyyw=-3 cos ϑzD1-16 zD3cos3 ϑa2,04,
Yyvv=1cos2 ϑ2-3 cos2 ϑ+16 zD3cos ϑa2,22,
Yyww=-31-32 zD3cos3 ϑa2,04,
Yvvw=2 zDcos3 ϑ2-3 cos2 ϑ+16 zD3cos ϑa2,22,
Ywww=-2 zDcos ϑ1-32 zD3cos3 ϑa2,04.
Xxxv=4 1zD cos ϑ1-cos2 ϑ+6 zD3cos ϑa1,40,
Xxyw=8 zD2cos2 ϑa1,22,
Xyyv=4 zD2cos2 ϑa1,22,
Xvvv=-zDcos3 ϑ2-cos2 ϑ-4 zD3cos ϑ(2a1,40+a2,40),
Xvww=-zDcos ϑ1-2 zD3cos3 ϑ(2a1,22+a2,22);
Yxxw=2 1zD cos ϑsin2 ϑ+2 zD3cos ϑa1,22,
Yxyv=2 1zD cos ϑsin2 ϑ+4 zD3cos ϑa1,22,
Yyyw=24 zD2cos2 ϑa1,04,
Yvvw=-zDcos3 ϑ1+sin2 ϑ-2 zD3cos ϑ(2a1,22+a2,22),
Ywww=-zDcos ϑ1-4 zD3cos3 ϑ(2a1,04+a2,04).

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