Abstract

An optical setup for the testing of rotationally symmetric aspheres without a null optic is proposed. The optical setup is able to transfer the strongly curved wave fronts that stem from the reflection of a spherical testing wave front at a rotationally symmetric asphere. By simulation it is proved that the algorithms of the Shack–Hartmann sensor that is used can cope with the steep wave-front slopes (∼110λ/mm) in the detection plane. The systematic errors of the testing configuration are analyzed and separated. For all types of error, functionals are derived whose significance is proved by simulation. The maximum residual errors in the simulations are fewer than λ/500 (peak to valley).

© 2001 Optical Society of America

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References

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  1. A. Offner, D. Malacara, “Null test using compensators,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 427–454.
  2. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
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    [CrossRef]
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    [CrossRef] [PubMed]
  7. A. E. Lowman, J. E. Greivenkamp, “Modeling an interferometer for non-null testing of aspheres,” in Optical Manufacturing and Testing, V. J. Doherty, H. Stahl, eds., Proc. SPIE2536, 139–147 (1995).
    [CrossRef]
  8. N. Lindlein, F. Simon, J. Schwider, “Simulation of micro-optical systems with RAYTRACE,” Opt. Eng. 37, 1809–1816 (1998).
    [CrossRef]
  9. J. Pfund, N. Lindlein, J. Schwider, “Misalignment effects of the Shack–Hartmann sensor,” Appl. Opt. 37, 22–27 (1998).
    [CrossRef]
  10. J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  11. I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 10, pp. 323–349.
  12. N. Lindlein, J. Pfund, J. Schwider, “Expansion of the dynamic range of a Shack–Hartmann sensor by using astigmatic microlenses,” Opt. Eng. (to be published).
  13. W. H. Southwell, “Wave front estimation from wave front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  14. J. Pfund, N. Lindlein, J. Schwider, R. Burow, Th. Blümel, K.-E. Elssner, “Absolute sphericity measurement: a comparative study on the use of interferometry and a Shack–Hartmann sensor,” Opt. Lett. 23, 742–744 (1998).
    [CrossRef]
  15. J. Pfund, N. Lindlein, J. Schwider, “Dynamic range expansion of a Shack–Hartmann sensor by using a modified unwrapping algorithm,” Opt. Lett. 23, 995–997 (1998).
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  16. S. Groening, B. Sick, K. Donner, J. Pfund, N. Lindlein, J. Schwider, “Wave-front reconstruction with a Shack–Hartmann sensor with an iterative spline fitting method,” Appl. Opt. 39, 561–567 (2000).
    [CrossRef]
  17. H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
    [CrossRef]
  18. E. W. Young, “Optimal removal of all mislocation effects in interferometric tests,” in Optical Testing and Metrology, C. P. Grover, ed., Proc. SPIE661, 116–124 (1986).
    [CrossRef]
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2000

1998

1994

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

1987

1985

1980

1978

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1974

1972

1971

Blümel, Th.

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Donner, K.

Elssner, K.-E.

Falkenstörfer, O.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Gallagher, J. E.

Ghozeil, I.

I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 10, pp. 323–349.

Greivenkamp, J. E.

J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5258 (1987).
[CrossRef] [PubMed]

A. E. Lowman, J. E. Greivenkamp, “Modeling an interferometer for non-null testing of aspheres,” in Optical Manufacturing and Testing, V. J. Doherty, H. Stahl, eds., Proc. SPIE2536, 139–147 (1995).
[CrossRef]

Groening, S.

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Herriott, D. R.

Ichioka, Y.

Lindlein, N.

Lohmann, A. W.

Lowman, A. E.

A. E. Lowman, J. E. Greivenkamp, “Modeling an interferometer for non-null testing of aspheres,” in Optical Manufacturing and Testing, V. J. Doherty, H. Stahl, eds., Proc. SPIE2536, 139–147 (1995).
[CrossRef]

Malacara, D.

A. Offner, D. Malacara, “Null test using compensators,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 427–454.

McGovern, A. J.

Offner, A.

A. Offner, D. Malacara, “Null test using compensators,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 427–454.

Pfund, J.

Rimmer, M. P.

Rosenfeld, D. P.

Saito, H.

Schwider, J.

Sick, B.

Sickinger, H.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Simon, F.

N. Lindlein, F. Simon, J. Schwider, “Simulation of micro-optical systems with RAYTRACE,” Opt. Eng. 37, 1809–1816 (1998).
[CrossRef]

Southwell, W. H.

Wetherell, W. B.

White, A. D.

Wyant, J. C.

Yatagai, T.

Young, E. W.

E. W. Young, “Optimal removal of all mislocation effects in interferometric tests,” in Optical Testing and Metrology, C. P. Grover, ed., Proc. SPIE661, 116–124 (1986).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

Opt. Eng.

N. Lindlein, F. Simon, J. Schwider, “Simulation of micro-optical systems with RAYTRACE,” Opt. Eng. 37, 1809–1816 (1998).
[CrossRef]

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Opt. Lett.

Proc. IEEE

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Other

I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 10, pp. 323–349.

N. Lindlein, J. Pfund, J. Schwider, “Expansion of the dynamic range of a Shack–Hartmann sensor by using astigmatic microlenses,” Opt. Eng. (to be published).

A. E. Lowman, J. E. Greivenkamp, “Modeling an interferometer for non-null testing of aspheres,” in Optical Manufacturing and Testing, V. J. Doherty, H. Stahl, eds., Proc. SPIE2536, 139–147 (1995).
[CrossRef]

A. Offner, D. Malacara, “Null test using compensators,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 427–454.

E. W. Young, “Optimal removal of all mislocation effects in interferometric tests,” in Optical Testing and Metrology, C. P. Grover, ed., Proc. SPIE661, 116–124 (1986).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Optical setup for testing aspherical surfaces with spherical compensation.

Fig. 2
Fig. 2

(a) Ray diagram of a plane wave front. (b) Ray diagram of the wave front in the detection plane of the optical setup shown in Fig. 1 without a field stop. (c) Ray diagram of the wave front in the detection plane of the optical setup shown in Fig. 1 with a circular field stop of diameter d = 6.5 mm.

Fig. 3
Fig. 3

Ideal aspherical wave front W ideal. The peak-to-valley value is W = 63.9λ; the wave-front difference between two contours is 5λ.

Fig. 4
Fig. 4

Partial derivative of W ideal in the x direction; the wave-front slope difference between two contours is 10λ/mm.

Fig. 5
Fig. 5

Schematic of the Shack–Hartmann sensor.

Fig. 6
Fig. 6

Section of a simulated CCD distribution.

Fig. 7
Fig. 7

Geometry of the misalignment of the asphere.

Fig. 8
Fig. 8

Geometry of the misalignment of the detection plane. W is the local wave front, w′ is the optical path distance of the light ray of W from the correct to the actual detection plane, and z′ is the alignment distance.

Fig. 9
Fig. 9

ΔW OPD of the wave front W ideal for a misalignment error of z′ = 0.116 mm. The peak-to-valley value is 0.52λ; the difference between two contours is 0.05λ.

Fig. 10
Fig. 10

ΔW OPD after fitting and subtraction of the functional O OPD to the data in Fig. 9. The residual error is ∼2 × 10-3λ (peak to valley).

Fig. 11
Fig. 11

Geometry of the effect of an error in the imaging scale of the optical setup.

Fig. 12
Fig. 12

ΔW LME of the wave front W ideal for a misalignment error a = 1 × 10-3. The peak-to-valley value is 0.36λ; the difference between two contours is 0.05λ.

Fig. 13
Fig. 13

ΔW LME after fitting and subtraction of the functional O ISE to the data in Fig. 12. The residual error is ∼1 × 10-5λ (peak to valley).

Equations (35)

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fx, y=Cr21+1-k+1C2r21/2+a4r4+a6r6+a8r8+a10r10+a12r12+a14r14,
r=x2+y21/2, C=1Rvertexcurvature of the vertex of the asphere, kconic type.
Wβx=Wx.
Widealrmax=117λ/mm,
Wideal=A+P+N.
Wmeas-Wideal=M+O+F+S+2Σ.
Wmeas-Wideal=2Σ+M+O.
iW=σif2+σi12+σi221/2.
Wx, y=l=0Gm=0lclmxmyl-m
Mx, y=AB¯+BC¯=2AB¯=2BD¯ cos α=-2δd · nε · n.
δd=Dx+θyfDy-θxfDz+θxy-θyx,
n=11+fx2+fy21/2-fx-fy1,
fx=fx=Cx1-k+1C2r21/2+4a4xr2+6a6xr4+8a8xr6+10a10xr8,
fy=fy=Cy1-k+1C2r21/2+4a4yr2+6a6yr4+8a8yr6+10a10yr8.
ε=2n-e,
e=1RxyR2-x2-y21/2.
R2r, fr, Rvertex=r2+Rvertex-fr2, r=x2+y21/2,
ε · n=2--xfx-yfy+Rvertex-f1+fx2+fy21/2x2+y2+Rvertex-f21/2.
Mx, y=αfxDx+αfyDy-αDzTranslation-αy+ffyΘx+αx+ffxΘyTilt,
α=21+fx2+fy21/2×2--xfx-yfy+Rvertex-f1+fx2+fy21/2x2+y2+Rvertex-f21/2.
d=Dx, Dy, Dz, Θx, Θy
Φ=+αfx, +αfy, -α, -αy+ffy, αx+ffx
ΔWOPD=w-z=z1cos ϕ-1=z1+tan2 ϕ1/2-1,
cos2 γ=11+tan2 γ.
1+x1/21+12x,
ΔWOPDz tan2 ϕ2=z2Widealr2zOOPDr.
ΔWOPD=z12Widealr2zOOPDr.
zmax=2ΔWOPD,maxWidealr2=λ117λ/mm2=0.116 mm
ΔWOPD,max=2ΔΣ=λ/2,
r=1+ar,
WrWsimr=Wexp1+ar,
ΔWLME=Wexpr-Wsimr=Wr1+a-WrW1-ar-Wr.
Δr=ar
Wr=Wr-Wr-arar
ΔWLME=-arWr-aOLMEr.

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