Abstract

We discuss the paraxial approximation for optical systems with an oblique orientation of the optical axis. Our investigation provides modifications necessary to adapt the parabolic approximation of conventional optics to planar integrated free-space optics. Our results are applied to a planar imaging system. For a single-lens system we discuss a design for arbitrary magnification factors. In addition, we calculate the space–bandwidth product of the imaging system, considering geometrical constraints of the concept of planar integration. This provides an upper bound for the number of spatial data channels that can be realized with planar optics.

© 1997 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. S. J. Walker, J. Jahns, L. Li, W. M. Mansfield, P. Mulgrew, D. M. Tennant, C. W. Roberts, L. C. West, N. K. Ailawadi, “Design and fabrication of high-efficiency beam splitters and beam deflectors for integrated planar micro-optic systems,” Appl. Opt. 32, 2494–2501 (1993).
    [CrossRef] [PubMed]
  3. J. Jahns, B. A. Brumback, “Integrated-optical split and shift module based on planar optics,” Opt. Commun. 79, 318–320 (1989).
  4. J. Jahns, “Integrated optical imaging,” Appl. Opt. 29, 1998 (1990).
    [CrossRef]
  5. M. M. Downs, J. Jahns, “Integrated optical array generation,” Opt. Lett. 15, 769–770 (1990).
    [CrossRef] [PubMed]
  6. M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129, 167–172 (1996).
    [CrossRef]
  7. T. P. Kosoburd, N. S. Stepanov, “Talbot effect on inclined illumination,” Opt. Spektrosk. 67, 708–709 (1989).
  8. S. Reinhorn, S. Gorodeisky, A. A. Friesem, Y. Amitai, “Fourier transformation with a planar holographic doublet,” Opt. Lett. 20, 495–497 (1995).
    [CrossRef] [PubMed]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw- Hill, New York, 1968), Chap. 3, pp. 30–56.
  10. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 3, pp. 30–56.
  11. A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 16.2, pp. 630–637.
  12. Z. Zhou, T. Drabik, “Coplanar refractive-diffractive doublets for optoelectronic integrated systems,” Appl. Opt. 34, 3048–3054 (1995).
    [CrossRef] [PubMed]
  13. M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Design of Talbot array illuminators for planar optics,” Opt. Commun. 132, 205–211 (1996).
    [CrossRef]
  14. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 8.7, pp. 353–356.
  15. G. Einarsson, Principles of Lightwave Communications (Wiley, Chichester, UK, 1996), Chap. 3, pp. 31–51.
  16. M. Testorf, J. Jahns, “Imaging in planar optics: system design for oblique deflection angles,” 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.

1996 (2)

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129, 167–172 (1996).
[CrossRef]

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Design of Talbot array illuminators for planar optics,” Opt. Commun. 132, 205–211 (1996).
[CrossRef]

1995 (2)

1993 (1)

1990 (2)

1989 (3)

J. Jahns, B. A. Brumback, “Integrated-optical split and shift module based on planar optics,” Opt. Commun. 79, 318–320 (1989).

T. P. Kosoburd, N. S. Stepanov, “Talbot effect on inclined illumination,” Opt. Spektrosk. 67, 708–709 (1989).

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

Ailawadi, N. K.

Amitai, Y.

Brumback, B. A.

J. Jahns, B. A. Brumback, “Integrated-optical split and shift module based on planar optics,” Opt. Commun. 79, 318–320 (1989).

Downs, M. M.

Drabik, T.

Einarsson, G.

G. Einarsson, Principles of Lightwave Communications (Wiley, Chichester, UK, 1996), Chap. 3, pp. 31–51.

Friesem, A. A.

Goncharenko, A. M.

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Design of Talbot array illuminators for planar optics,” Opt. Commun. 132, 205–211 (1996).
[CrossRef]

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129, 167–172 (1996).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw- Hill, New York, 1968), Chap. 3, pp. 30–56.

Gorodeisky, S.

Huang, A.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 8.7, pp. 353–356.

Jahns, J.

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Design of Talbot array illuminators for planar optics,” Opt. Commun. 132, 205–211 (1996).
[CrossRef]

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129, 167–172 (1996).
[CrossRef]

S. J. Walker, J. Jahns, L. Li, W. M. Mansfield, P. Mulgrew, D. M. Tennant, C. W. Roberts, L. C. West, N. K. Ailawadi, “Design and fabrication of high-efficiency beam splitters and beam deflectors for integrated planar micro-optic systems,” Appl. Opt. 32, 2494–2501 (1993).
[CrossRef] [PubMed]

M. M. Downs, J. Jahns, “Integrated optical array generation,” Opt. Lett. 15, 769–770 (1990).
[CrossRef] [PubMed]

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

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

J. Jahns, B. A. Brumback, “Integrated-optical split and shift module based on planar optics,” Opt. Commun. 79, 318–320 (1989).

M. Testorf, J. Jahns, “Imaging in planar optics: system design for oblique deflection angles,” 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.

Khilo, N. A.

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Design of Talbot array illuminators for planar optics,” Opt. Commun. 132, 205–211 (1996).
[CrossRef]

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129, 167–172 (1996).
[CrossRef]

Kosoburd, T. P.

T. P. Kosoburd, N. S. Stepanov, “Talbot effect on inclined illumination,” Opt. Spektrosk. 67, 708–709 (1989).

Li, L.

Mansfield, W. M.

Mulgrew, P.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 3, pp. 30–56.

Reinhorn, S.

Roberts, C. W.

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 16.2, pp. 630–637.

Stepanov, N. S.

T. P. Kosoburd, N. S. Stepanov, “Talbot effect on inclined illumination,” Opt. Spektrosk. 67, 708–709 (1989).

Tennant, D. M.

Testorf, M.

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Design of Talbot array illuminators for planar optics,” Opt. Commun. 132, 205–211 (1996).
[CrossRef]

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129, 167–172 (1996).
[CrossRef]

M. Testorf, J. Jahns, “Imaging in planar optics: system design for oblique deflection angles,” 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.

Walker, S. J.

West, L. C.

Zhou, Z.

Appl. Opt. (4)

Opt. Commun. (3)

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Design of Talbot array illuminators for planar optics,” Opt. Commun. 132, 205–211 (1996).
[CrossRef]

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129, 167–172 (1996).
[CrossRef]

J. Jahns, B. A. Brumback, “Integrated-optical split and shift module based on planar optics,” Opt. Commun. 79, 318–320 (1989).

Opt. Lett. (2)

Opt. Spektrosk. (1)

T. P. Kosoburd, N. S. Stepanov, “Talbot effect on inclined illumination,” Opt. Spektrosk. 67, 708–709 (1989).

Other (6)

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 8.7, pp. 353–356.

G. Einarsson, Principles of Lightwave Communications (Wiley, Chichester, UK, 1996), Chap. 3, pp. 31–51.

M. Testorf, J. Jahns, “Imaging in planar optics: system design for oblique deflection angles,” 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.

J. W. Goodman, Introduction to Fourier Optics (McGraw- Hill, New York, 1968), Chap. 3, pp. 30–56.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 3, pp. 30–56.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 16.2, pp. 630–637.

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Figures (7)

Fig. 1
Fig. 1

Planar integrated optical imaging system.

Fig. 2
Fig. 2

Unfolded configuration of the planar system shown in Fig. 1.

Fig. 3
Fig. 3

Illustration of the coordinate systems.

Fig. 4
Fig. 4

Planar system of Fig. 1 with additional deflection grating to achieve arbitrary magnification.

Fig. 5
Fig. 5

Setup of Fig. 4 with additional deflection gratings at half the propagation distance before and behind the lens. (a) Side view, (b) top view.

Fig. 6
Fig. 6

Geometrical constraints of the planar imaging system.

Fig. 7
Fig. 7

Maximum number of resolvable points with respect to the propagation angle ϑ: (a) in the x direction, (b) in the y direction.

Equations (34)

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u˜(ν, μ)=-u(x, y)exp[-i2π(νx+μy)]dxdy.
r0=x0y0=z tan ϑcos ϕsin ϕ=zcos ϑ sin αsin β,
f0=ν0μ0=sin ϑλ cos ϕsin ϕ=1λ sin αsin β.
x=x-x0,y=y-y0,
ν=ν-ν0, μ=μ-μ0.
uα,β,z(x, y)=u(x-x0, y-y0)exp{-i2π[(x-x0)ν0+(y-y0)μ0]},
u˜α,β,z(ν, μ)=u˜(ν-ν0, μ-μ0)exp{i2π[(ν-ν0)x0+(μ-μ0)y0]}.
u˜α,β,z(ν, μ)=exp[-i2π(x0ν0+y0μ0)]-uα,β,z(x, y)×exp[-i2π(νx+μy)]dxdy.
u(x, y, z+Δz)=-u0(ξ, η, z)h(x-ξ, y-η, Δz)dxdy,
h(x, y, Δz)=1iλ expi 2πλ (x2+y2+Δz2)1/2(x2+y2+Δz2)1/2×cos[γ(x, y, z)].
hpα,β,Δz(x, y)=cos2 ϑiλz expi 2πλ Δzcos ϑ×expiπ cos ϑλz [x2(cos2 ϑ+sin2 β)+y2(cos2 ϑ+sin2 α)-2xy sin α sin β].
hpϑ,0,Δz(x, y)=cos2 ϑiλΔz expi 2πλ Δzcos ϑ×expiπ cos ϑλΔz (x2 cos2 ϑ+y2).
h˜(ν, μ, Δz)=expi 2πλ Δz(1-λ2ν2-λ2μ2)1/2.
h˜pα,β,Δz(ν, μ)=expi 2πλ Δz cos ϑexp-iπλzcos ϑ×1+sin2 αcos2 ϑν2+1+sin2 βcos2 ϑμ2+2 sin α sin βcos2 ϑ νμ.
h˜pϑ,0,Δz(ν, μ, Δz)=expi 2πλ Δz cos ϑ×exp-iπλ Δzcos3 ϑ ν2-iπλ Δzcos ϑ μ2.
πΔzν3λ2 sin ϑcos5(ϑ)+ν4λ3 1+sin2 ϑ(1+3 sin2 ϑ)4 cos7 ϑ<π100.
θ<λ100Δz cos5 ϑsin ϑ1/3,
θ<λ25Δz cos7 ϑ1+sin2 ϑ(1+3 sin2 ϑ)1/4.
hα,β,f(x, y)L(x, y)=1,
L(x, y)=exp-iπ cos ϑλf [x2(cos2 ϑ+sin2 β)+y2(cos2 ϑ+sin2 α)-2xy sin α sin β].
Lϑ,0(x, y)=exp-iπ cos ϑλf (x2 cos2 ϑ+y2).
P(x, y)=exp[i2π(xνp+yμp)],
uouta,β,z(x, y)=uinα+αp,β+βp,z(x, y),
Pl(x, y)=exp{i[(2π)/λ](sin ϑin-sin ϑout)x},
L(x, y)=expiπ2λzD [x2(cos3 ϑin+cos3 ϑout)+y2(cos ϑin+cos ϑout)].
Mx=cos3 ϑincos3 ϑout,My=cos ϑincos ϑout.
Pin/out(x, y)=exp{i[(2π)/λ]sin ϑin/out(x-y)}.
Mx=My=cos3 ϑincos3 ϑout 1+cos ϑout1+cos ϑin.
SBP=Dxdx×Dydy,
Ax+Dx4zD tan ϑ.
dx=2 2zDλAx cos3 ϑ,dy=2 2zDλAy cos ϑ,
Nx=Dxdx=DxAx cos3 ϑ4λzD,Ny=Dydy=DyAy cos ϑ4λzD.
Nx=zDλ sin2 ϑ cos ϑ,Ny=zDλ sin2 ϑ cos-1 ϑ.
SBP=Nx×Ny=zDλ sin2 ϑ2.

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