Abstract

The dynamic range of a Ronchi test with a phase-shifted sinusoidal grating was investigated theoretically and experimentally. As the number of fringes in a Ronchi interferogram increases, the fringe visibility decreases, which results in a decrease of phase-measurement resolution. It is shown that in order to optimize the dynamic range the effective wavelength of the interferogram should be tuned to the characteristic wavelength of the object wave front. The maximum dynamic range achievable is estimated to be 16 times larger than that of a Fizeau interferometer. Suppressing higher-order diffraction components has achieved sheared interferograms with a signal-to-noise ratio in excess of 60:1. The effects of nonsinusoidal transmittance of the grating and the phase-shift errors were minimized by a seven-sample phase-shifting algorithm, and a phase measurement uncertainty of less than 1/700 has been achieved.

© 1997 Optical Society of America

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References

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  1. A. Cornejo–Rodriguez, Optical Shop Testing (Wiley, New York, 1978), Chap. 4, pp. 283–322.
  2. A. Cornejo, D. Malacara, “Ronchi test of aspherical surfaces, analysis, and accuracy,” Appl. Opt. 9, 1897–1901 (1970).
    [PubMed]
  3. R. S. Kasana, S. Boseck, K.-J. Rosenbruch, “Non-destructive collimation technique for measuring glass constants using a Ronchi grating shearing interferometer,” Opt. Laser Technol. 23, 101–105 (1984).
    [CrossRef]
  4. L. Carretero, C. Gonzalez, A. Fimia, I. Pascual, “Application of the Ronchi test to intraocular lenses: a comparison of theoretical and measured results,” Appl. Opt. 32, 4132–4137 (1993).
    [CrossRef] [PubMed]
  5. T. Yatagai, “Fringe scanning Ronchi test for aspherical surfaces,” Appl. Opt. 23, 3676–3679 (1984).
    [CrossRef] [PubMed]
  6. K. Omura, T. Yatagai, “Phase measuring Ronchi test,” Appl. Opt. 27, 523–528 (1988).
    [CrossRef]
  7. 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).
    [CrossRef]
  8. P. Hariharan, B. F. Oreb, Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer-controlled digital radial-shear interferometer,” Opt. Acta 31, 989–999 (1984).
    [CrossRef]
  9. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real time wavefront correction systems,” Appl. Opt. 14, 2622–2626 (1975).
    [CrossRef] [PubMed]
  10. K. A. Stetson, W. R. Brohinsky, “Electrooptic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631–3637 (1985).
    [CrossRef]
  11. D. Wan, D. Lin, “Ronchi test and a new phase reduction algorithm,” Appl. Opt. 29, 3255–3265 (1990).
    [CrossRef] [PubMed]
  12. J. Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]
  13. Z. Hegedus, Z. Zelenka, J. Gardner, “Interference patterns generated by a plane parallel plate,” Appl. Opt. 32, 2285–2288 (1993).
    [CrossRef] [PubMed]
  14. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  15. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  16. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
  17. K. R. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
    [CrossRef]
  18. H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  19. W. S. Meyers, “Contouring of a free oil surface,” in Thirty-sixth Annual Symposium of the International Society for Optical Engineering: Interferometry: Techniques and Analysis, Proc. Int. Soc. Opt. Eng.1755, 84–94 (1992).

1993 (2)

1992 (1)

1991 (1)

1990 (1)

1988 (2)

1986 (1)

1985 (1)

1984 (3)

P. Hariharan, B. F. Oreb, Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer-controlled digital radial-shear interferometer,” Opt. Acta 31, 989–999 (1984).
[CrossRef]

T. Yatagai, “Fringe scanning Ronchi test for aspherical surfaces,” Appl. Opt. 23, 3676–3679 (1984).
[CrossRef] [PubMed]

R. S. Kasana, S. Boseck, K.-J. Rosenbruch, “Non-destructive collimation technique for measuring glass constants using a Ronchi grating shearing interferometer,” Opt. Laser Technol. 23, 101–105 (1984).
[CrossRef]

1979 (1)

1977 (1)

1975 (1)

1974 (1)

1970 (1)

Boseck, S.

R. S. Kasana, S. Boseck, K.-J. Rosenbruch, “Non-destructive collimation technique for measuring glass constants using a Ronchi grating shearing interferometer,” Opt. Laser Technol. 23, 101–105 (1984).
[CrossRef]

Brangaccio, D. J.

Brohinsky, W. R.

Bruning, J. H.

Carretero, L.

Cornejo, A.

Cornejo–Rodriguez, A.

A. Cornejo–Rodriguez, Optical Shop Testing (Wiley, New York, 1978), Chap. 4, pp. 283–322.

Cubalchini, R.

Fimia, A.

Frankena, H. J.

Freischlad, K. R.

Fried, D. L.

Gallagher, J. E.

Gardner, J.

Gonzalez, C.

Hariharan, P.

P. Hariharan, B. F. Oreb, Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer-controlled digital radial-shear interferometer,” Opt. Acta 31, 989–999 (1984).
[CrossRef]

Hegedus, Z.

Herriott, D. R.

Kasana, R. S.

R. S. Kasana, S. Boseck, K.-J. Rosenbruch, “Non-destructive collimation technique for measuring glass constants using a Ronchi grating shearing interferometer,” Opt. Laser Technol. 23, 101–105 (1984).
[CrossRef]

Koliopoulos, C. L.

Larkin, K. G.

Lin, D.

Malacara, D.

Meyers, W. S.

W. S. Meyers, “Contouring of a free oil surface,” in Thirty-sixth Annual Symposium of the International Society for Optical Engineering: Interferometry: Techniques and Analysis, Proc. Int. Soc. Opt. Eng.1755, 84–94 (1992).

Omura, K.

Oreb, B. F.

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

P. Hariharan, B. F. Oreb, Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer-controlled digital radial-shear interferometer,” Opt. Acta 31, 989–999 (1984).
[CrossRef]

Pascual, I.

Rosenbruch, K.-J.

R. S. Kasana, S. Boseck, K.-J. Rosenbruch, “Non-destructive collimation technique for measuring glass constants using a Ronchi grating shearing interferometer,” Opt. Laser Technol. 23, 101–105 (1984).
[CrossRef]

Rosenfeld, D. P.

Smorenburg, C.

Stetson, K. A.

Takahashi, T.

Takajo, H.

Wan, D.

Wanzhi, Z.

P. Hariharan, B. F. Oreb, Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer-controlled digital radial-shear interferometer,” Opt. Acta 31, 989–999 (1984).
[CrossRef]

White, A. D.

Wingerden, J.

Wyant, J. C.

Yatagai, T.

Zelenka, Z.

Appl. Opt. (10)

L. Carretero, C. Gonzalez, A. Fimia, I. Pascual, “Application of the Ronchi test to intraocular lenses: a comparison of theoretical and measured results,” Appl. Opt. 32, 4132–4137 (1993).
[CrossRef] [PubMed]

T. Yatagai, “Fringe scanning Ronchi test for aspherical surfaces,” Appl. Opt. 23, 3676–3679 (1984).
[CrossRef] [PubMed]

K. Omura, T. Yatagai, “Phase measuring Ronchi test,” Appl. Opt. 27, 523–528 (1988).
[CrossRef]

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).
[CrossRef]

J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real time wavefront correction systems,” Appl. Opt. 14, 2622–2626 (1975).
[CrossRef] [PubMed]

K. A. Stetson, W. R. Brohinsky, “Electrooptic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631–3637 (1985).
[CrossRef]

D. Wan, D. Lin, “Ronchi test and a new phase reduction algorithm,” Appl. Opt. 29, 3255–3265 (1990).
[CrossRef] [PubMed]

J. Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
[CrossRef] [PubMed]

Z. Hegedus, Z. Zelenka, J. Gardner, “Interference patterns generated by a plane parallel plate,” Appl. Opt. 32, 2285–2288 (1993).
[CrossRef] [PubMed]

A. Cornejo, D. Malacara, “Ronchi test of aspherical surfaces, analysis, and accuracy,” Appl. Opt. 9, 1897–1901 (1970).
[PubMed]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

P. Hariharan, B. F. Oreb, Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer-controlled digital radial-shear interferometer,” Opt. Acta 31, 989–999 (1984).
[CrossRef]

Opt. Laser Technol. (1)

R. S. Kasana, S. Boseck, K.-J. Rosenbruch, “Non-destructive collimation technique for measuring glass constants using a Ronchi grating shearing interferometer,” Opt. Laser Technol. 23, 101–105 (1984).
[CrossRef]

Other (2)

A. Cornejo–Rodriguez, Optical Shop Testing (Wiley, New York, 1978), Chap. 4, pp. 283–322.

W. S. Meyers, “Contouring of a free oil surface,” in Thirty-sixth Annual Symposium of the International Society for Optical Engineering: Interferometry: Techniques and Analysis, Proc. Int. Soc. Opt. Eng.1755, 84–94 (1992).

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Figures (9)

Fig. 1
Fig. 1

Ronchi test for measuring the transmitted wave front of a lens: L1, test lens; G, quasi-sinusoidal grating; L2, imaging lens; D, rotating ground glass diffuser; C, CCD detector.

Fig. 2
Fig. 2

Irradiances of observed interferograms for measuring the spherical aberration of the transmitted wave front of a lens with two different grating periods: (a) d = 0.63 mm, (b) d = 0.20 mm.

Fig. 3
Fig. 3

Dynamic range and fringe visibility of a Ronchi test.

Fig. 4
Fig. 4

Recording geometry of a sinusoidal grating.

Fig. 5
Fig. 5

(a) Sheared interferogram of a defocused spherical wave front from a lens. (b) Measured phase of the interferogram modulus π.

Fig. 6
Fig. 6

(a) Sheared interferogram of an aberrated wave front from a lens. (b) Measured phase of the interferogram modulus π.

Fig. 7
Fig. 7

Measured phase distributions of the interferogram sheared in two orthogonal directions: (a) sheared in the x direction, (b) sheared in the y direction.

Fig. 8
Fig. 8

Calculated object wave front after integration by least-squares fitting.

Fig. 9
Fig. 9

Calculated wave aberration after subtraction of a parabolic wave front.

Tables (1)

Tables Icon

Table 1 Amplitudes tn and Relative Amplitudes sn for Quasi-sinusoidal, Perfect Sinusoidal, and Perfect Binary Gratings

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

tx, y=n=-tn expin2πxd-β,
Ix, y, β=n=- tn2+n=-m=-n-12tmtn cosφx-mΔ, y-φx-nΔ, y-n-mβ,
Ix, y, β=n=-tn2+k=1n=-2tn-ktn cos×φx-n-kΔ, y-φx-nΔ, y-kβ=s0+s0C cosα0-β+k=1 sk cosαk-kβ,
s0=n=- tn2,
α0=φx+Δ, y-φx-Δ, y2,
Cx, y=C0 cosφx+Δ, y+φx-Δ, y2-φx, y,
C0=4t0t1s0,
s1=4t1t2 cos ½φx+2Δ, y-φx+Δ, y-φx-Δ, y+φx-2Δ, y,
α1=½φx+2Δ, y-φx+Δ, y+φx-Δ, y-φx-2Δ, y,
φx, y=2πλcrm,
½φR+Δ, y-φR-Δ, y=πN.
λeff=Rd2f2.
Cx, y=C0 cosm2Δ21+m-2x2r2φx, yr2.
Cmin=C0 cosπ4m-1λλeffN
Cmin=C0 cosπ4mm-1λδmaxλeff2 for m>2.
δφran=2κ RpΔδσ,
κ=0.65581+0.4888 ln p.
D=2πλδmaxδφran,
D=pq2κδmaxΔλRCminC0=pq4κδmaxλeffCminC0
Dλeff=D00.797λ0λeffcosπ4.84λ02λeff2,
D0=0.181pqκδmaxmm-1λ1/2
λ0=1.10mm-1λδmax1/2.
D0=0.198pqmκN0,
s14i=1 titi+1,
sn2i=1n-1 titn-i+4 i=0titi+n for n>2,
α=arctan3I2+I3-I5-I6+I7-I1/3-I1-I2+I3+2I4+I5-I6-I7,
δα=α-α0=s1s0Csinα1-α0-s5s0C1/3sin α5 cos α0+cos α5 sin α0+terms involving s7 or higher orders,
α1-α098Δ33φx3x, y98mm-1ΔR32πλδmax for m>2=9π32mm-1λλeffλδmaxλeff2 for m>2,
3φx3av=1R0R3φx3dx=2πλmm-1δmaxR3.
δφ12RpΔs1s0Csinα1-α0p2RΔs1s0Cα1-α09π16mm-1λδmaxλeff2s1s0Cmin,
δφ52RpΔs5s0C231/216Np2136Rs5NΔs0Cmin,
1D=λ2πδφran+δφ1+δφ5δmax1Dran+981-1m×λλeff2s1s0Cmin+166RλπNΔδmaxs5s0Cmin,
1D=5.5κpqmm-1λδmax1/2+1λC0δmax×0.29s1s0+0.027RNΔs5s0,
1/D=0.66+0.44+0.33×10-3,
2Rpi+12,2Rpj+12 for i, j=1, 2, , p
φxi, j=RpΔφ2Rpi+12+Δ, 2Rpj+12-φ2Rpi+12-Δ, 2Rpj+12,
φyi, j=RpΔφ2Rpi+12, 2Rpj+12+Δ-φ2Rpi+12, 2Rpj+12-Δ,
gi, jφi, j-fi-12, j-12φi-1, j-1-fi-12, j+12φi-1, j+1-fi+12, j-12φi+1, j-1-fi+12, j+12φi+1, j+1=fi-12, j-12φxi-1, j-1+φyi-1, j-1+fi-12, j+12φxi-1, j-φyi-1, j+fi+12, j-12-φxi, j-1+φyi, j-1-fi+12, j+12φxi, j+φyi, j,
gi, j=fi-½, j-½+fi-½, j+½+fi+½, j-½+fi+½, j+½,
u-p/2-12+v-p/2-12p/22.
δφ2=2/s02C2MδI2,
δφtotδφ1+δφ5+δφran=0.0084π+0.0064π+0.013π=0.028π rad,

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