Abstract

Aspheric surfaces are increasingly used in the design of high-quality optical imaging systems. Therefore accurate testing methods for aspherics are also necessary. One possibility is to use a computer-generated hologram (CGH) as a part of a null lens in an interferometric testing device. However, CGHs normally have more than one diffraction order, thus causing disturbing areas in the interferogram. Here a simple approximative analytical expression is given for the spatial frequencies of the disturbing light in the interferogram coming from the different diffraction orders of the CGH. This expression also enables one to calculate the size and the shape of the disturbing areas in the interferogram. Some design examples for CGHs are given in an application of the expression.

© 2001 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 197–202.
  2. J. Schwider, O. Falkenstörfer, “Twyman–Green interferometer for testing micro spheres,” Opt. Eng. 34, 2972–2975 (1995).
    [CrossRef]
  3. A. J. MacGovern, J. C. Wyant, “Computer generated holograms for testing optical elements,” Appl. Opt. 10, 619–624 (1971).
    [CrossRef] [PubMed]
  4. J. C. Wyant, P. K. O’Neill, “Computer generated hologram: null lens test of aspheric wavefronts,” Appl. Opt. 13, 2762–2765 (1974).
    [CrossRef] [PubMed]
  5. H. Tiziani, “Prospects of testing aspheric surfaces with computer-generated holograms,” in Aspheric Optics: Design, Manufacture, Testing, P. Kuttner, T. L. Williams, eds., Proc. SPIE235, 72–78 (1980).
  6. S. M. Arnold, “How to test an asphere with a computer generated hologram,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 191–197 (1989).
  7. J. H. Burge, “Applications of computer-generated holograms for interferometric measurement of large aspheric optics,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 258–269 (1995).
    [CrossRef]
  8. J. Schwider, “Interferometric tests for aspherics,” in Fabrication and Testing of Aspheres, A. Lindquist, M. Piscotty, J. S. Taylor, eds., Vol. 24 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 103–114.
  9. V. Ronchi, “On the phase grating interferometer,” Appl. Opt. 4, 1041–1042 (1965).
    [CrossRef]
  10. W. T. Welford, “A vector raytracing equation for hologram lenses of arbitrary shape,” Opt. Commun. 14, 322–323 (1975).
    [CrossRef]
  11. H. Schreiber, J. Schwider, “A lateral shearing interferometer based on two Ronchi gratings in series,” Appl. Opt. 36, 5321–5324 (1997).
    [CrossRef] [PubMed]

1997 (1)

1995 (1)

J. Schwider, O. Falkenstörfer, “Twyman–Green interferometer for testing micro spheres,” Opt. Eng. 34, 2972–2975 (1995).
[CrossRef]

1975 (1)

W. T. Welford, “A vector raytracing equation for hologram lenses of arbitrary shape,” Opt. Commun. 14, 322–323 (1975).
[CrossRef]

1974 (1)

1971 (1)

1965 (1)

Arnold, S. M.

S. M. Arnold, “How to test an asphere with a computer generated hologram,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 191–197 (1989).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 197–202.

Burge, J. H.

J. H. Burge, “Applications of computer-generated holograms for interferometric measurement of large aspheric optics,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 258–269 (1995).
[CrossRef]

Falkenstörfer, O.

J. Schwider, O. Falkenstörfer, “Twyman–Green interferometer for testing micro spheres,” Opt. Eng. 34, 2972–2975 (1995).
[CrossRef]

MacGovern, A. J.

O’Neill, P. K.

Ronchi, V.

Schreiber, H.

Schwider, J.

H. Schreiber, J. Schwider, “A lateral shearing interferometer based on two Ronchi gratings in series,” Appl. Opt. 36, 5321–5324 (1997).
[CrossRef] [PubMed]

J. Schwider, O. Falkenstörfer, “Twyman–Green interferometer for testing micro spheres,” Opt. Eng. 34, 2972–2975 (1995).
[CrossRef]

J. Schwider, “Interferometric tests for aspherics,” in Fabrication and Testing of Aspheres, A. Lindquist, M. Piscotty, J. S. Taylor, eds., Vol. 24 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 103–114.

Tiziani, H.

H. Tiziani, “Prospects of testing aspheric surfaces with computer-generated holograms,” in Aspheric Optics: Design, Manufacture, Testing, P. Kuttner, T. L. Williams, eds., Proc. SPIE235, 72–78 (1980).

Welford, W. T.

W. T. Welford, “A vector raytracing equation for hologram lenses of arbitrary shape,” Opt. Commun. 14, 322–323 (1975).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 197–202.

Wyant, J. C.

Appl. Opt. (4)

Opt. Commun. (1)

W. T. Welford, “A vector raytracing equation for hologram lenses of arbitrary shape,” Opt. Commun. 14, 322–323 (1975).
[CrossRef]

Opt. Eng. (1)

J. Schwider, O. Falkenstörfer, “Twyman–Green interferometer for testing micro spheres,” Opt. Eng. 34, 2972–2975 (1995).
[CrossRef]

Other (5)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 197–202.

H. Tiziani, “Prospects of testing aspheric surfaces with computer-generated holograms,” in Aspheric Optics: Design, Manufacture, Testing, P. Kuttner, T. L. Williams, eds., Proc. SPIE235, 72–78 (1980).

S. M. Arnold, “How to test an asphere with a computer generated hologram,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 191–197 (1989).

J. H. Burge, “Applications of computer-generated holograms for interferometric measurement of large aspheric optics,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 258–269 (1995).
[CrossRef]

J. Schwider, “Interferometric tests for aspherics,” in Fabrication and Testing of Aspheres, A. Lindquist, M. Piscotty, J. S. Taylor, eds., Vol. 24 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 103–114.

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Figures (6)

Fig. 1
Fig. 1

Schematic setup of the interferometric device for testing aspherics by use of a CGH and a refractive system as null lens.

Fig. 2
Fig. 2

Relevant part of the testing system for the calculation of the spatial frequencies of the waves with different diffraction orders. A ray with direction vector e is diffracted by the CGH into several rays e m with diffraction order m (shown are only one ray with diffraction order 1 and one with a general order m) and reflected at the aspheric surface with the local surface normal N m . After this it is again diffracted by the CGH, now with the diffraction order m′. Ray e 1 is parallel to the surface normal N 1.

Fig. 3
Fig. 3

Ray-tracing scheme of the simulation setup for the best-fitting sphere as illuminating wave.

Fig. 4
Fig. 4

Illumination of the CGH with a sphere that fits best to the aspheric. Radius of curvature of the sphere R CGH = -236.8 mm. (a) Negative spatial frequency of the CGH -(∂ϕ/∂r)/(2π), which is also the approximative resulting spatial frequency calculated with relation (21) for m = 1 and m′ = 0 or m = 0 and m′ = 1 and also the exact resulting spatial frequency for m = 1 and m′ = 0 for a ray-tracing simulation. (b) Exact resulting spatial frequency for m = 0 and m′ = 1 (ray-tracing simulation). (c) Approximative resulting spatial frequency for m = -1 and m′ = 3 calculated with relation (21). (d) Exact resulting spatial frequency for m = -1 and m′ = 3 (ray-tracing simulation).

Fig. 5
Fig. 5

Illumination of the CGH with a sphere fitting the radius of curvature at the vertex of the aspheric, i.e., R CGH = -230 mm. (a) Negative spatial frequency of the CGH -(∂ϕ/∂r)/(2π), which is also the approximative resulting spatial frequency calculated with relation (21) for m = 1 and m′ = 0 or m = 0 and m′ = 1 and also the exact resulting spatial frequency for m = 1 and m′ = 0 for a ray-tracing simulation. (b) Exact resulting spatial frequency for m = 0 and m′ = 1 (ray-tracing simulation). (c) Approximative resulting spatial frequency for m = -1 and m′ = 3 calculated with relation (21). (d) Exact resulting spatial frequency for m = -1 and m′ = 3 (ray-tracing simulation).

Fig. 6
Fig. 6

Illumination of the CGH with a defocused sphere, i.e., R CGH = -300 mm. (a) Negative spatial frequency of the CGH -(∂ϕ/∂r)/(2π), which is also the approximative resulting spatial frequency calculated with relation (21) for m = 1 and m′ = 0 or m = 0 and m′ = 1 and also the exact resulting spatial frequency for m = 1 and m′ = 0 for a ray-tracing simulation. (b) Exact resulting spatial frequency for m = 0 and m′ = 1 (ray-tracing simulation). (c) Approximative resulting spatial frequency for m = -1 and m′ = 3 calculated with relation (21). (d) Exact resulting spatial frequency for m = -1 and m′ = 3 (ray-tracing simulation).

Equations (28)

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emx, y=em,xx, yem,yx, yem,zx, y=exx, y+mkΦx, yxeyx, y+mkΦx, yy1-exx, y+mkΦx, yx2-eyx, y+mkΦx, yy21/2,
ex, y=exx, yeyx, y1-ex2x, y-ey2x, y1/2,  k=2πλ.
emx, y=em,xx, yem,yx, yem,zx, yexx, y+mkΦx, yxeyx, y+mkΦx, yy1
ex, yexx, yeyx, y1.
N1x, y=e1x, y=exx, y+1kΦx, yxeyx, y+1kΦx, yy1.
e=e-2e·NN,
e1x, y=e1x, y-2e1x, y·e1x, ye1x, y=-e1x, y.
Nmx, y=N1x+Δx, y+Δy, Δx=m-1dx, ykΦx, yx,
Δy=m-1dx, ykΦx, yy,
Nmx, y=N1x+Δx, y+Δy=N1x, y+Δx N1x, yx+Δy N1x, yy,
Nmx, y=e1x, y+Δx e1x, yx+Δy e1x, yy=exx, y+Δx exx, yx+Δy exx, yy+1kΦx, yx+Δxk2Φx, yx2+Δyk2Φx, yxyeyx, y+Δx eyx, yx+Δy eyx, yy+1kΦx, yy+Δxk2Φx, yxy+Δyk2Φx, yy21.
emx, y=emx, y-2emx, y·Nmx, yNmx, y,
emx, y=-exx, y+2Δx exx, yx+2Δy exx, yyeyx, y+2Δx eyx, yx+2Δy eyx, yy1 +m-2kΦx, yx-2Δxk2Φx, yx2+Δyk2Φx, yxym-2kΦx, yy-2Δxk2Φx, yxy+Δyk2Φx, yy20.
em,mx+2Δx, y+2Δy=em,xx, y+mkΦx+2Δx, y+2Δyxem,yx, y+mkΦx+2Δx, y+2Δyy1=em,xx, y+mkΦx, yx+2Δx mk2Φx, yx2+2Δy mk2Φx, yxyem,yx, y+mkΦx, yy+2Δx mk2Φx, yxy+2Δy mk2Φx, yy21.
em,mx+2Δx, y+2Δy=-ex+2Δx, y+2Δy+m+m-2kΦx, yx+2 m-1kΔx 2Φx, yx2+Δy 2Φx, yxym+m-2kΦx, yy+2 m-1kΔx 2Φx, yxy+Δy 2Φx, yy20.
λνxx+2Δx, y+2Δy=m+m-2kΦx, yx+2 m-1m-1k2 dx, yΦx, yx2Φx, yx2+Φx, yy2Φx, yxy,λνyx+2Δx, y+2Δy=m+m-2kΦx, yy+2 m-1m-1k2 dx, yΦx, yx2Φx, yxy+Φx, yy2Φx, yy2,
λνxx, ym+m-2kΦx, yx+2 m-1m-1k2 dΦx, yx2Φx, yx2+Φx, yy2Φx, yxy,λνyx, ym+m-2kΦx, yy+2 m-1m-1k2 dΦx, yx2Φx, yxy+Φx, yy2Φx, yy2.
νx2+νy2>νcut-off2
Φx, y=Φoutx, y-Φinx, y=--Φoutx, y+Φinx, y.
νcut-off=N/2D=2.19/mm,
νxx, y=12πΦresx, yx,νyx, y=12πΦresx, yy.
νrr=12πΦresrr.
λνrrm+m-2kΦrr+2 m-1m-1k2 d Φrr2Φrr2.
Φdefr=ar2  Φdefrr=2ar  2Φdefrr2=2a.
λνrr2 m-1m-1k2 d4a2r  |νrr|8 λdπ2 a2r.
a=±πνcut-off8rcut-offλd1/2=±1.57π/mm2.
a=πλ1RCGH,0-1RCGH  1RCGH=1RCGH,0-aλπ.
RCGH=-298.1 mm.

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