Abstract

The problem of the relation between the phase function and the thin microrelief profile of diffractive optical elements is considered in detail through the nonparaxial scalar diffraction theory for the case of a curved substrate surface and an astigmatic incident beam. Local-plane-wave expansions are used to calculate a coefficient of proportionality between microrelief height and phase jump as a function of microrelief orientation and of local slope angles of the incident and the output beams. Nonuniform transition heights between 2π-phase zones of the microrelief are calculated. Analytical and numerical data are presented for examples of high-numerical-aperture diffractive lenses on plane, spherical, or aspherical substrate surfaces.

© 1999 Optical Society of America

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References

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  1. D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1839 (1994).
    [CrossRef]
  2. R. Kingslake, Optical Systems Design (Academic, Orlando, Fla., 1983), Sect. 3.VII.
  3. M. Rossi, R. E. Kunz, H. P. Herzig, “Refractive and diffractive properties of planar micro-optical elements,” Appl. Opt. 34, 5996–6007 (1995).
    [CrossRef] [PubMed]
  4. Y. Han, L. N. Hazra, C. A. Delisle, “Exact surface-relief profile of a kinoform lens from its phase function,” J. Opt. Soc. Am. A 12, 524–529 (1995).
    [CrossRef]
  5. L. N. Hazra, Y. Han, C. A. Delisle, “Curved kinoform lenses for stigmatic imaging of axial objects,” Appl. Opt. 32, 4775–4794 (1993).
    [CrossRef] [PubMed]
  6. M. Francon, Holographie (Masson et Cje, Paris, 1969), Chap. 1, Sect. 1.
  7. J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform lenses,” Appl. Opt. 9, 1883–1887 (1970).
    [PubMed]
  8. G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” (MIT, Cambridge, Mass., 1991).
  9. M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper DTuD3.
  10. V. A. Soifer, M. A. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, Boca Raton, Fla., 1994).
  11. M. A. Golub, “Generic approach to relate surface-relief profile height and the phase function of diffractive optical elements,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 115–117.
  12. M. A. Golub, I. N. Sisakian, V. A. Soifer, “Computer generated optical elements in wavefront formation with intensity spatial modulation,” J. Mod. Opt. 38, 1067–1072 (1991).
    [CrossRef]
  13. D. Faklis, G. M. Morris, “Spectral properties of multi-order diffractive lenses,” Appl. Opt. 34, 2462–2468 (1995).
    [CrossRef] [PubMed]

1995 (3)

1994 (1)

1993 (1)

1991 (1)

M. A. Golub, I. N. Sisakian, V. A. Soifer, “Computer generated optical elements in wavefront formation with intensity spatial modulation,” J. Mod. Opt. 38, 1067–1072 (1991).
[CrossRef]

1970 (1)

Blough, C. G.

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper DTuD3.

Delisle, C. A.

Faklis, D.

Francon, M.

M. Francon, Holographie (Masson et Cje, Paris, 1969), Chap. 1, Sect. 1.

Golub, M. A.

M. A. Golub, I. N. Sisakian, V. A. Soifer, “Computer generated optical elements in wavefront formation with intensity spatial modulation,” J. Mod. Opt. 38, 1067–1072 (1991).
[CrossRef]

V. A. Soifer, M. A. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, Boca Raton, Fla., 1994).

M. A. Golub, “Generic approach to relate surface-relief profile height and the phase function of diffractive optical elements,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 115–117.

Grann, E. B.

Han, Y.

Hazra, L. N.

Herzig, H. P.

Hirsch, P. M.

Jordan, J. A.

Kingslake, R.

R. Kingslake, Optical Systems Design (Academic, Orlando, Fla., 1983), Sect. 3.VII.

Kunz, R. E.

Lesem, L. B.

Maystre, D.

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper DTuD3.

Moharam, M. G.

Morris, G. M.

Pommet, D. A.

Popov, E. K.

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper DTuD3.

Raguin, D. H.

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper DTuD3.

Rossi, M.

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

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper DTuD3.

Sisakian, I. N.

M. A. Golub, I. N. Sisakian, V. A. Soifer, “Computer generated optical elements in wavefront formation with intensity spatial modulation,” J. Mod. Opt. 38, 1067–1072 (1991).
[CrossRef]

Soifer, V. A.

M. A. Golub, I. N. Sisakian, V. A. Soifer, “Computer generated optical elements in wavefront formation with intensity spatial modulation,” J. Mod. Opt. 38, 1067–1072 (1991).
[CrossRef]

V. A. Soifer, M. A. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, Boca Raton, Fla., 1994).

Swanson, G. J.

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” (MIT, Cambridge, Mass., 1991).

Van Rooy, D. L.

Appl. Opt. (4)

J. Mod. Opt. (1)

M. A. Golub, I. N. Sisakian, V. A. Soifer, “Computer generated optical elements in wavefront formation with intensity spatial modulation,” J. Mod. Opt. 38, 1067–1072 (1991).
[CrossRef]

J. Opt. Soc. Am. A (2)

Other (6)

M. Francon, Holographie (Masson et Cje, Paris, 1969), Chap. 1, Sect. 1.

R. Kingslake, Optical Systems Design (Academic, Orlando, Fla., 1983), Sect. 3.VII.

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” (MIT, Cambridge, Mass., 1991).

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper DTuD3.

V. A. Soifer, M. A. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, Boca Raton, Fla., 1994).

M. A. Golub, “Generic approach to relate surface-relief profile height and the phase function of diffractive optical elements,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 115–117.

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Figures (10)

Fig. 1
Fig. 1

Illustrations for virtual splitting: (a) hybrid diffractive–refractive component, (b) thin diffractive microrelief pattern with a separated refractive lens. The virtual spacer S, with infinitesimal thickness, is introduced for separation of the diffractive microrelief and the refractive lens substrate. The surface Λ of the refractive lens also serves as a substrate surface for the diffractive microrelief.

Fig. 2
Fig. 2

Interaction of the incident beam with a transmitting diffractive microrelief pattern (located on the side opposite the incident beam). The microrelief of refractive index nM separates two homogeneous media of refractive indices n0 and n. The vector r describes a point of the plane or curved substrate surface Λ, rR represents a point on a thin microrelief surface, and h(r) expresses the microrelief height counted along the unit normal vector NS of the substrate surface. NR is the unit normal vector of the microrelief surface. S0(r), S(r), and SM(r) are the eikonals of the incident beam, the output beam, and the beam inside the microrelief material, respectively. N0, N, and NM are the unit vectors of the incident beam, the output beam, and the beam inside the microrelief material, respectively. The incidence angle θ0, the output angle θ, and the slope θM of the ray inside the microrelief media are measured relative to the normal of the substrate surface Λ.

Fig. 3
Fig. 3

Light beam interaction with the diffractive microrelief pattern working in the reflection mode. The microrelief of refractive index nM faces the homogeneous media of refractive index n0. The vector r describes a point on the plane or curved substrate surface Λ, rR represents a point on the thin microrelief surface, and h(r) expresses the microrelief height counted along the unit normal vector NS of the substrate surface. NR is the unit normal vector of the microrelief surface. N0 and N are the unit vectors of the incident and the output beams, respectively. The incidence angle θ0 and the output angle θ are measured relative to the normal NS to the substrate surface Λ. θM is a local reflection angle with respect to the microrelief surface normal NR.

Fig. 4
Fig. 4

Modulation factor μ(0)/μ(r) for the microrelief versus the relative coordinate r/rmax on the aperture of the f/2 lens. The coordinate r is the polar radius on the planar substrate surface of the lens, and rmax is the semidiameter of the lens. The function μ(0)/μ(r) describes the correction of the microrelief height as applied at the point with coordinate r. Solid curves, the microrelief on the side of the focus; dotted curve, the microrelief on the side of the incident beam.

Fig. 5
Fig. 5

Value of the modulation factor μ(0)/μ(rmax) on the edge of the diffractive lenses versus the f-number. The function μ(0)/μ(rmax) describes the correction of the microrelief height as applied at the point rmax that is the semidiameter of the lens aperture. Solid curves, the microrelief on the side of the focus; dotted curve, the microrelief on the side of the incident beam.

Fig. 6
Fig. 6

Diffraction efficiency Em (thick horizontal lines) and local efficiency m(r) distribution (curves) in diffraction order number m=6 versus the relative coordinate r/rmax on the aperture of the transmitting f/2 diffractive lens with a uniform zone transition height (correction not applied). The local diffraction efficiency of the corrected microrelief is represented by the constant 1. The coordinate r is the polar radius on the planar substrate surface of the lens, and rmax is the semidiameter of the lens. Solid curve, the microrelief on the side of the focus; dashed curve, the microrelief on the side of the incident beam; solid line, the microrelief on the side of the focus; dotted line, the microrelief on the side of the incident beam.

Fig. 7
Fig. 7

Diffraction efficiency Em (thin, upper curves) and local efficiency m(rmax) values (thick, lower curves) in diffraction order number m=6 versus the f-number of the transmitting diffractive lens with a uniform zone transition height (correction not applied). A well-corrected microrelief height corresponds to a constant value 1 in the phase-matching estimation used here. The coordinate rmax is the semidiameter of the lens aperture. Solid curves, the microrelief on the side of focus; dotted and dashed curves, the microrelief on the side of the incident beam.

Fig. 8
Fig. 8

Diffractive lens of focal length f0, working in the reflection mode under a plane incident wave. N0 and N are the unit vectors of the incident and the output beams, respectively. The incidence angle θ0=45° and the output angle θ are measured relative to the normal NS to the substrate surface Λ. u=(u, v) are Cartesian coordinates on the substrate surface such that the u axis lies in the plane of incidence. The z axis is orthogonal to the substrate surface and goes toward the beams, l=f0/2.

Fig. 9
Fig. 9

Focusing diffractive microrelief pattern located on a spherical substrate surface of radius R. N0 and N are the unit vectors of the incident and the output beams, respectively. NS is a unit normal vector of the substrate surface, and f0 is the focal length. The coordinate r is the distance from the point on the sphere to the optical axis.

Fig. 10
Fig. 10

Example of a diffractive microrelief profile with the correction applied (heavy curve). The thin curve illustrates a diffractive microrelief profile with a uniform zone transfer height. The relative height h(r)/hmax(0) of the microrelief is plotted versus the relative coordinate r/rmax on the lens aperture. r is the distance from the point on the lens to the optical axis, rmax is the semidiameter of the lens aperture, and hmax(0)=23.9 µm is the maximum height in the central zone of the diffractive microrelief.

Equations (42)

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μ0=nM-n0,h0max=λ/(nM-n0),
μ0=-2n0,h0max=λ/2n0,
kS(r)=kS0(r)+Φ(r).
rR=r+h(r)Nh,
Φ(r)=k[S(r)-S0(r)],
Φ(r)=kμ(r)h(r),
kS0(rR)=kS0(r)+kn0N0·(rR-r),
kS(rR)=kS(r)+knN·(rR-r),
kSM(rR)=kSM(r)+knMNM·(rR-r),
NM=n0nMN0+1-n0nM2N021/2NS,
N0=N0-(N0·NS)NS.
S(rR)=SM(rR),SM(r)=S0(r).
μ(r)=nMNM·Nh-nN·Nh.
μ(r)=n0[N0·Nh-(N0·NS)(NS·Nh)]+{nM2-n02[1-(N0·NS)2]}1/2(NS·Nh)-nN·Nh.
μ(r)={nM2-n02[1-(N0·NS)2]}1/2-nN·NS.
cos θ0=N0·NS,cos θ=N·NS.
μ=(nM2-n02 sin2 θ0)1/2-n cos θ,
μ=nM-n cos θ,
μ(r)=nMNM·Nh-n0N0·Nh,
μ(r)=n[N·Nh-(N·NS)(NS·Nh)]+{nM2-n2[1-(N·NS)2]}1/2(NS·Nh)-n0N0·Nh,
μ={nM2-n2[1-(N·NS)2]}1/2-n0N0·NS,
μ=(nM2-n2 sin2 θ)1/2-n0 cos θ0.
S0(rR)=S(rR)
μ(r)=n0(N0·Nh-N·Nh).
cos θ0=-N0·NS,cos θ=N·NS,
μ=-(n0 cos θ0+cos θ).
Φ(r)=mod2πm[φ(r)],
φ(rj)-φ(0)=j2πm.
h(r)=Φ(r)/kμ(r).
h(r)=hmax(r)mod2πm[φ(r)]/2πm,
hmax(r)=mλ/μ(r),
hmax(r)/hmax(0)=μ(0)/μ(r).
hmax(rj)=mλ/μ(rj)=hmax(0)[μ(0)/μ(rj)],
c(r)=h0max/hmax(r)=μ(r)/μ0.
=sinc2(mc-m),
m(r)=sinc2{m[c(r)-1]}.
Em=Gm(r)d2rGd2r,
cos θ0=1,cos θ=f/f02+r2,
μ(u, v)=-12+1+1+u2f02+v2f02-1/2,
μ0=-2.
μ(r)=nM2-r2R21/2-r2+[R-H(r)][f0-H(r)]R{r2+[f0-H(r)]2}1/2,
μ0=nM-1.

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