Abstract

We describe a new common-path, phase-shift, and shearing interferometric device capable of single-shot detection of optical phase profiles. It samples the input field and uses birefringent plates to fan out phase-shifted copies of the samples in the empty space between them. The phase shifts are given by the thickness of the plates and not by the relative position of the components, as in classical interferometers. This makes the device insensitive to vibrations. We recorded repeatability better than λ/100, even though strong shocks were applied to the air table in proximity to the system. We recorded better than λ/1000 repeatability under quiet conditions and estimated the accuracy to be better than λ/3000 at the shot-noise limit. In addition, the device is compact and easy to integrate in a variety of setups that require the measurement of optical phase profiles.

© 2002 Optical Society of America

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References

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  1. L. R. Dettmann, D. L. Modisett, “Interferogram acquisition using a high frame rate CCD camera,” in Optical Manufacturing and Testing II, P. H. Stahl, ed., Proc. SPIE3134, 429–437 (1997).
  2. A. L. Weijers, H. van Brug, H. J. Frankena, “Polarization phase stepping with a Savart element,” Appl. Opt. 37, 5150–5155 (1998).
    [CrossRef]
  3. R. Tumbar, R. A. Stack, D. J. Brady, “Wave-front sensing with a sampling field sensor,” Appl. Opt. 39, 72–84 (2000).
    [CrossRef]
  4. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  5. E. W. Rogala, H. H. Barrett, “Phase-shifting interferometry and maximum-likelihood estimation theory,” Appl. Opt. 36, 8871–8876 (1997).
    [CrossRef]
  6. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14, 918–930 (1997).
    [CrossRef]
  7. H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, New York, 1983).
  8. T. W. Stone, J. M. Battiato, “Optical array generation and interconnection using birefringent slabs,” Appl. Opt. 33, 182–191 (1994).
    [CrossRef] [PubMed]
  9. M. P. Kothiyal, C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–4442 (1985).
    [CrossRef] [PubMed]
  10. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  11. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.

2000 (1)

1998 (1)

1997 (2)

1996 (1)

1994 (1)

1985 (1)

Barrett, H. H.

Battiato, J. M.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Brady, D. J.

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, New York, 1983).

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.

Delisle, C.

Dettmann, L. R.

L. R. Dettmann, D. L. Modisett, “Interferogram acquisition using a high frame rate CCD camera,” in Optical Manufacturing and Testing II, P. H. Stahl, ed., Proc. SPIE3134, 429–437 (1997).

Farrant, D. I.

Frankena, H. J.

Hibino, K.

Kothiyal, M. P.

Larkin, K. G.

Modisett, D. L.

L. R. Dettmann, D. L. Modisett, “Interferogram acquisition using a high frame rate CCD camera,” in Optical Manufacturing and Testing II, P. H. Stahl, ed., Proc. SPIE3134, 429–437 (1997).

Oreb, B. F.

Rogala, E. W.

Stack, R. A.

Stone, T. W.

Surrel, Y.

Tumbar, R.

van Brug, H.

Weijers, A. L.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Appl. Opt. (6)

J. Opt. Soc. Am. A (1)

Other (4)

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, New York, 1983).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.

L. R. Dettmann, D. L. Modisett, “Interferogram acquisition using a high frame rate CCD camera,” in Optical Manufacturing and Testing II, P. H. Stahl, ed., Proc. SPIE3134, 429–437 (1997).

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Figures (11)

Fig. 1
Fig. 1

Principle of operation: (a) fan-out by imaging through birefringent crystals and (b) a SFS with birefringent fan-out. The sampled field is fanned out into a specific characteristic pattern.

Fig. 2
Fig. 2

Fan-out or characteristic pattern: (a) having both shear and phase diversity; and (b) having only shear diversity but no phase diversity.

Fig. 3
Fig. 3

Position of the image points for different output fields for the pattern in Fig. 2(b) generated with quartz plates.

Fig. 4
Fig. 4

System configuration used to match the sampling geometry to the detector geometry and the shear of the plates.

Fig. 5
Fig. 5

Photographs of the experimental setup: (a) the sampling mask and (b) the fan-out stage.

Fig. 6
Fig. 6

Output of the system for the pattern in Fig. 2(b) and a spherical wave input. Units on axis are pixels on the CCD.

Fig. 7
Fig. 7

Simultaneous multiple shear and phase shifts as shown by interferograms in different classes of pixels: (a) pixels 3, (b) pixels 7, (c) pixels 5, (d) pixels 2. Units are system pixels that we obtained by binning 3 × 3 CCD pixels.

Fig. 8
Fig. 8

Multiple shear but non-phase-shifted patterns: (a) pixels 3, (b) pixels 7, (c) pixels 5, (d) pixels 2. Units are system pixels.

Fig. 9
Fig. 9

Experimental estimation of measurement accuracy. A spherical lens is shifted laterally, and the phase shift of its associated fringe pattern is measured. The solid line is the phase estimate based on the knowledge of the lateral displacement. The rms fit is the standard deviation of the residual error of the linear (Lin.) fit, and the rms abs is the standard deviation of the difference between the measurements and the theoretical curve.

Fig. 10
Fig. 10

Spatial variation of the noise in the class of pixels labeled 3 in Fig. 2(b) when strong shocks are applied to the air table close to the system: (a) the average over time of the detected power (no interference); (b) the standard deviation (over time) of the detected power, normalized to the shot noise; (c) same as (a) but for an interference signal; (d) same as (b) but for an interference signal.

Fig. 11
Fig. 11

Spatial variation of the noise in the class of pixels labeled 3 in Fig. 2(b) in a quiet environment: (a) the average over time of the detected power (no interference); (b) the standard deviation (over time) of the detected power, normalized to the shot noise; (c) same as (a) but for an interference signal; (d) same as (b) but for an interference signal.

Tables (3)

Tables Icon

Table 1 Measurements on the Amplitude Pixels with and without Vibration Noise (Strong Shocks)

Tables Icon

Table 2 Internal Phase Error with and without Vibrations or Strong Shocks

Tables Icon

Table 3 Interference Measurements Normalized by the Intensity in Pixel 1 with and without Vibrations or Strong Shocks

Equations (23)

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EoEe=tan β,
cr=n=1N δr-rn Kn expiαn.
cr=δr-r0 K0 expiα0 +δr-r1x K11x expiα11x +δr+Δr-r1x K21x expiα21x +δr-r2x K12x expiα12x +δr+Δx-r2x K22x expiα22x +δr-r1y K11y expiα11y +δr+Δy-r1y K21y expiα21y +δr-r2y K12y expiα12y +δr+Δy-r2y K22y expiα22y.
Eer; d=expir2π/λF-d2  Eek; 0expikAdexpik0d1-d+d2 +Bed×exp-ik22k02Ce-1d+d1-d2Fd2-F×exp-ik2k02Ce-1d+d1-d2Fd2-Fexp-ik·rFd2-F d2k,
Eor; d=expir2π/λF-d2  Eek; 0expik0d1-d+d2+Bod ×exp-ik22k02Co-1d+d1-d2Fd2-Fexp-ik · rFd2-Fd2k.
A=12sin2δ1εe-1εo sin2δεe+cos2δεo-1, Be=sin2δεe+cos2δεo-1/2, Ce=1εesin2δεe+cos2δεo-1/2, Ce=1εeεosin2δεe+cos2δεo-3/2, Bo=εo, Co=1εo.
Eoutr; d; en; om=expik0enBen -1den+omBom-1domexpir2π/λd2-F× Eink; 0exp-ik2k0en denCenkˆ · zˆ×ĉn22 ·εo-εeεe · sin2δ×exp-ik22k0en2Cen-1 den+om2Com-1dom+d1-d2Fd2-F×expik · -r ·Fd2-F+enzˆ×ĉen×zˆ Aendend2k,
en2Cen-1den+om2Com-1dom +d1-d2Fd2-F=0.
hn=Andn.
Φen; om-Φep; oq=k0en Ben den+om Bomdom-ep Bepden-oq Boq doq.
Δzen=2M enCen-CoenAen hen,
maximum focus shift8λ1/2F-numberadjacent separation-Ma1.22λ.
α1=0+0+0+0,α2=ΔΦ1+0+0+0,α3=0+ΔΦ2+0+0,α4=ΔΦ1+ΔΦ2+0+0,α5=0+0+ΔΦ3+0,α6=ΔΦ1+0+ΔΦ3+0,α7=0+ΔΦ2+ΔΦ3+0,α8=ΔΦ1+ΔΦ2+ΔΦ3+0,α9=0+0+0+ΔΦ4,α10=ΔΦ1+0+0+ΔΦ4,α11=0+ΔΦ2+0+ΔΦ4,α12=ΔΦ1+ΔΦ2+0+ΔΦ4,α13=0+0+ΔΦ3+ΔΦ4,α14=ΔΦ1+0+ΔΦ3+ΔΦ4,α15=0+ΔΦ2+ΔΦ3+ΔΦ4,α16=ΔΦ1+ΔΦ2+ΔΦ3+ΔΦ4.
α3-α13=ΔΦ2-ΔΦ3-ΔΦ4, α7-α10=-ΔΦ1+ΔΦ2+ΔΦ3-ΔΦ4, α4-α14=ΔΦ2-ΔΦ3-ΔΦ4.
Eer; dexpiπ/λ2Ce-1d+d2-Fr+zˆ×ĉ×zˆAd2expik0dBe-1  Eek; 0×expik· r+zˆ×ĉ×zˆAdF2Ce-1d+d2-F×exp-ik22k0d1-F-F22Ce-1d+d2-Fd2k,
Eor; dexpiπ/λ2Co-1d+d2-Fr+zˆ×ĉ×zˆAd2expik0dBo-1  Eek; 0×expik ·rF2Co-1d+d2-Fexp-ik22k0d1-F-F22Co-1d+d2-Fd2k.
σΔϕ=0.5σK11A1K11A1σK11A1K21A2+σK21A1K12A1σK21A1K22A2+σ1P1K11A1σ1P1K21A21/2
IA1=|A1expiφ1 K0expiα0|2=I1K0, IA2=|A2expiφ2 K0expiα0|2=I2K0, IP1Δx=|A1expiφ1 K11xexpiα11x+A2expiφ2 K21xexpiα21x|2=I1K11x+I2K21x+2K11xK21xI1I21/2×cosφ1-φ2+α11x-α21x, IP2Δx=|A1expiφ1 K12xexpiα12x+A2expiφ2 K22xexpiα22x|2=I1K12x+I2K22x+2K12xK22xI1I21/2×cosφ1-φ2+α12x-α22x.
IA1=|A1expiφ1 K0expiα0|2=I1K0, IA3=|A3expiφ3 K0expiα0|2=I3K0, IP1Δx=|A1expiφ1 K11yexpiα11y+A3expiφ3 K21yexpiα21y|2=I1K11y+I3K21y+2K11yK21yI1I31/2×cosφ1-φ3+α11y-α21y, IP2Δx=|A1expiφ1 K12yexpiα12y+A3expiφ3 K22yexpiα22y|2=I1K12y+I3K22y+2K12yK22yI1I31/2× cosφ1-φ3+α12y-α22y.
A1=IA11/2K0; A2=IA21/2K0; A3=IA31/2K0, φ1-φ2=tan-1D1x cosα12x-α22x-D2x cosα11x-α21xD1x sinα12x-α22x-D2x sinα11x-α21x, φ1-φ3=tan-1D1y cosα12y-α22y-D2y cosα11y-α21yD1y sinα12y-α22y-D2y sinα11y-α21y,
D1x=K12xK22xK02IP1Δx-IA1K11x2-IA2K21x2, D2x=K11xK21xK02IP2Δx-IA1K12x2-IA2K22x2, D1y=K12yK22yK02IP1Δy-IA1K11y2-IA3K21y2, D2y=K11yK21yK02IP2Δy-IA1K12y2-IA3K22y2.
φ1-φ2=tan-1IP2Δx-I1K12x-I2K22xIP1Δx -I1K11x-I2K21xK11xK21xK12xK22x1/2, φ1-φ3=tan-1IP2Δy -I1K12y-I2K22yIP1Δy -I1K11y-I2K21yK11yK21yK12yK22y1/2.
φ1-φ2=sin-1IP2Δx-I1K12x-I2K22xK12xK22xI1I21/2, φ1-φ3=sin-1IP2Δy-I1K12y-I2K22yK12y K22yI1I21/2.

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