Abstract

We present a new type of optical wave-front sensor: the sampling field sensor (SFS). The SFS attempts to solve the problem of real-time optical phase detection. It has a high space-bandwidth product and can be made compact and vibration insensitive. We describe a particular implementation of this sensor and compare it, through numerical simulations, with a more mature technique based on the Shack–Hartmann wave-front sensor. We also present experimental results for SFS phase estimation. Finally, we discuss the advantages and drawbacks of this SFS implementation and suggest alternative implementations.

© 2000 Optical Society of America

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References

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  1. M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).
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1998

1997

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

1995

M. C. Roggemann, B. M. Welsh, P. J. Gardner, R. L. Johnson, B. L. Pedersen, “Sensing three-dimensional index-of-refraction variations by means of optical wavefront sensor measurements and tomographic reconstruction,” Opt. Eng. 34, 1374–1384 (1995).
[CrossRef]

R. C. Cannon, “Global wave-front reconstruction using Shack-Hartmann sensors,” J. Opt. Soc. Am. A 12, 2031–2039 (1995).
[CrossRef]

Y. Baharav, B. Spektor, J. Shamir, D. G. Crowe, W. Rhodes, R. Stroud, “Wave-front sensing by pseudo-phase-conjugate interferometry,” Appl. Opt. 34, 108–113 (1995).
[CrossRef] [PubMed]

1988

1981

1980

1979

R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
[CrossRef]

J. J. Knab, “Interpolation of band-limited functions using the approximate prolate series,” IEEE Trans. Inf. Theory IT-25, 717–720 (1979).
[CrossRef]

1978

1977

1975

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers (telescope testing),” Jpn. J. Appl. Phys. 14, 351–356 (1975).

1969

Baharav, Y.

Blumel, T.

Burow, R.

Cannon, R. C.

Crowe, D. G.

Cubalchini, R.

Dubik, B.

M. Zajac, B. Dubik, “Measurement of wavefront aberrations of diffractive imaging elements,” in Tenth Polish-Czech-Slovak Optical Conference: Wave and Quantum Aspects of Contemporary Optics, J. Nowak, M. Zajac, eds., Proc. SPIE3320, 237–241 (1998).
[CrossRef]

Elssner, K. E.

Fried, D. L.

Fugate, R. Q.

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

Gardner, P. J.

M. C. Roggemann, B. M. Welsh, P. J. Gardner, R. L. Johnson, B. L. Pedersen, “Sensing three-dimensional index-of-refraction variations by means of optical wavefront sensor measurements and tomographic reconstruction,” Opt. Eng. 34, 1374–1384 (1995).
[CrossRef]

Geary, J. M.

J. M. Geary, Introduction to Wavefront Sensors, Vol. TT18 of SPIE Tutorial Text (SPIE Press, Bellingham, Wash., 1995).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Herrmann, J.

Johnson, R. L.

M. C. Roggemann, B. M. Welsh, P. J. Gardner, R. L. Johnson, B. L. Pedersen, “Sensing three-dimensional index-of-refraction variations by means of optical wavefront sensor measurements and tomographic reconstruction,” Opt. Eng. 34, 1374–1384 (1995).
[CrossRef]

Klein, S. A.

Knab, J. J.

J. J. Knab, “Interpolation of band-limited functions using the approximate prolate series,” IEEE Trans. Inf. Theory IT-25, 717–720 (1979).
[CrossRef]

Lindlein, N.

Noll, R. J.

Pedersen, B. L.

M. C. Roggemann, B. M. Welsh, P. J. Gardner, R. L. Johnson, B. L. Pedersen, “Sensing three-dimensional index-of-refraction variations by means of optical wavefront sensor measurements and tomographic reconstruction,” Opt. Eng. 34, 1374–1384 (1995).
[CrossRef]

Pfund, J.

Reid, G. T.

D. W. Robinson, G. T. Reid, Interferogram Analysis (Institute of Physics Publishing, Philadelphia, Pa., 1993).

Rhodes, W.

Robinson, D. W.

D. W. Robinson, G. T. Reid, Interferogram Analysis (Institute of Physics Publishing, Philadelphia, Pa., 1993).

Roddier, F.

Roggemann, M. C.

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

M. C. Roggemann, B. M. Welsh, P. J. Gardner, R. L. Johnson, B. L. Pedersen, “Sensing three-dimensional index-of-refraction variations by means of optical wavefront sensor measurements and tomographic reconstruction,” Opt. Eng. 34, 1374–1384 (1995).
[CrossRef]

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Schwider, J.

Shamir, J.

Smartt, R. N.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers (telescope testing),” Jpn. J. Appl. Phys. 14, 351–356 (1975).

Southwell, W. H.

Spektor, B.

Steel, W. H.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers (telescope testing),” Jpn. J. Appl. Phys. 14, 351–356 (1975).

Stroud, R.

Toraldo di Francia, G.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1991).

Welsh, B.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Welsh, B. M.

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

M. C. Roggemann, B. M. Welsh, P. J. Gardner, R. L. Johnson, B. L. Pedersen, “Sensing three-dimensional index-of-refraction variations by means of optical wavefront sensor measurements and tomographic reconstruction,” Opt. Eng. 34, 1374–1384 (1995).
[CrossRef]

Zajac, M.

M. Zajac, B. Dubik, “Measurement of wavefront aberrations of diffractive imaging elements,” in Tenth Polish-Czech-Slovak Optical Conference: Wave and Quantum Aspects of Contemporary Optics, J. Nowak, M. Zajac, eds., Proc. SPIE3320, 237–241 (1998).
[CrossRef]

Appl. Opt.

IEEE Trans. Inf. Theory

J. J. Knab, “Interpolation of band-limited functions using the approximate prolate series,” IEEE Trans. Inf. Theory IT-25, 717–720 (1979).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers (telescope testing),” Jpn. J. Appl. Phys. 14, 351–356 (1975).

Opt. Eng.

M. C. Roggemann, B. M. Welsh, P. J. Gardner, R. L. Johnson, B. L. Pedersen, “Sensing three-dimensional index-of-refraction variations by means of optical wavefront sensor measurements and tomographic reconstruction,” Opt. Eng. 34, 1374–1384 (1995).
[CrossRef]

Opt. Lett.

Rev. Mod. Phys.

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

Other

M. Zajac, B. Dubik, “Measurement of wavefront aberrations of diffractive imaging elements,” in Tenth Polish-Czech-Slovak Optical Conference: Wave and Quantum Aspects of Contemporary Optics, J. Nowak, M. Zajac, eds., Proc. SPIE3320, 237–241 (1998).
[CrossRef]

J. M. Geary, Introduction to Wavefront Sensors, Vol. TT18 of SPIE Tutorial Text (SPIE Press, Bellingham, Wash., 1995).
[CrossRef]

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1991).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

D. W. Robinson, G. T. Reid, Interferogram Analysis (Institute of Physics Publishing, Philadelphia, Pa., 1993).

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Figures (10)

Fig. 1
Fig. 1

(a) SFS concept. (b) Physical implementation of the SFS.

Fig. 2
Fig. 2

Layout of overlapping adjacent receiver patterns. A receiver cell is the area covered by the corresponding receiver pattern.

Fig. 3
Fig. 3

Exact layout of the receiver mask, one receiver cell only. A marks the amplitude pixel and P1 and P2 are the two types of phase pixels.

Fig. 4
Fig. 4

Histogram of phase estimation error at maximum input bandwidth (1/220 µm-1). The size of the SFS sampling grid is 20 × 20, thus there are 760 phase estimation points. There are 14 points with an error within 2 rad of 2π.

Fig. 5
Fig. 5

Detection of a random input field with a spatial bandwidth of 1/220 µm-1. The input values are marked with circles and the estimated values are marked with crosses: (a) input field (real part) with B = 1/220 µm-1, (b) SHS amplitude detection, (c) SFS phase detection, and (d) SHS phase detection.

Fig. 6
Fig. 6

Detection of a random input field with a spatial bandwidth of 1/2200 µm-1. The input values are marked with circles and the estimated values are marked with crosses: (a) input field (real part) with B = 1/2200 µm-1, (b) SHS amplitude detection, (c) SFS phase detection, and (d) SHS phase detection.

Fig. 7
Fig. 7

SHS and SFS phase detection error for low input bandwidth. The input values are marked with circles and the estimated values are marked with crosses: (a) SFS detection and (b) SHS detection.

Fig. 8
Fig. 8

Experimental setup. Sampling mask is a 40 × 40 array of 10-µm-square holes in a chrome coating over a 1-mm-thick quartz substrate. The focal lengths of the lenses L 1, L 2, L 3, and L 4 are F 1 = F 2 = 50 mm and F 3 = F 4 = 100 mm. The sampling mask is placed approximately 1 mm in front of the front focal plane of L 1. The aperture of the stop in the Fourier plane of the first lens is 3 mm. The camera is a 576 × 384 CCD array with 22-µm-square pixels.

Fig. 9
Fig. 9

Experimental results: (a) signal in a phase pixel as a function of mirror tilt angle, (b) signal in two adjacent amplitude pixels as a function of mirror tilt angle, (c) signal in the same phase pixel for a smaller range of angles (working bandwidth) with a sinusoid fitting it, and (d) intensity (log 10) in the output plane of the device.

Fig. 10
Fig. 10

Experimental verification. (a) Change in the receiver pattern function with angle of incidence. The receiver pattern function is different for different values of the incidence angle. This is due to the failure of the SFS representation to model the output field at high input bandwidths (the model breaks down). (b) Histogram of the measured period for 300 phase pixels. The average period is approximately 150 mdeg with a standard deviation of 40 mdeg which is in agreement with the theoretical estimate.

Tables (2)

Tables Icon

Table 1 Error in Calibrating the SFS Parameters with a Least-Squares Fita

Tables Icon

Table 2 Comparison of Detection Errors of SFS and SHS

Equations (37)

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finx, y=m,n finmh, nhsincx-mhsincy-nh.
h12B.
freceiverx, y=m,n finmh, nhrx-mh, y-nh,
|rx, y|=>0, for 0|x|, |y|<h-ε/2=0, for |x|, |y|h-ε/2.
fsampledx, y=finx, yj,k s0x-jh, y-kh=m,n finmh, nhφx-mh, y-nh,
s0x=1 for |x|, |y|a20 fora2<|x|, |y|<h2.
φx, y=ψx, yj,k s0x-jh, y-kh,
freceiverx, y=fsampledx, yhdx, yh4-fx, y,
freceiverx, y=FT-1Fsampledu, vHdu, vH4fu, v=FT-1Fsampledu, vHSFSu, v,
freceiverx, y=fsampledx, yhSFS=m,n finmh, nhrx-mh, y-nh,
rx-mh, y-nh=φx-mh, y-nhhSFSx, y
rx, y=ψx, yj,k s0x-jh, y-khhSFSx, y.
rx, y=ψx, ys0x, yhdx, yh4fx, ys0x, yh4fx, yhdx, y.
rx, yexpj x2+y2λdFTs0ξ, ηexp×j ξ2+η2λdu=xλd,v=yλd,
BL=1a,
ah10.
h2λBLdqhλBL.
Γi,jxp, yp=σi,jxp, ypm,n finmh, nhrx-mh, y-nh2dxdy,
Γm,n0 ,0=|finmh, nh|2σm,n0, 0 |rx-mh, y-nh|2dxdy.
Γm,n0, 0=|finmh, nh|2Ka2,
Γm,nxPi, yPi=|finmh, nh|2σm,nxPi,yPi |rx-mh, y-nh|2dxdy+|finm+1h, nh|×σm,nxPi,yPi |rx-m+1h, y-nh|2dxdy+2 Refinmh, nh*finm+1h, nh×σm,nxPi,yPiRerx-mh, y-nh*rx-m+1h, y-nhdxdy,
Γm,nxPi, yPi=|finmh, nh|2K1i+|finm+1×h, nh|2K2i+KPi|finmh, nh|×|finm+1h, nh|cos×ϕmh, nh-ϕm+1h, nh-αi,
Ka=σ0,00, 0|rx, y|2dxdy1/2,  KPi=σ0,0xPi, yPi rx, y*rx-h, ydxdy for i=1, 2,  αi=-σ0, 0xPi, yPi rx, y*rx-h, ydxdy for i=1, 2,  K1i=σ0, 0xPi, yPi |rx, y|2dxdy with i=1, 2,  K2i=σ0, 0xPi, yPi |rx-h, y|2dxdy for i=1, 2.
|finestimatedih, jh|=Γi,j0, 0Ka.
|finm+1h, nhfinmh, nh|Kp1Kp2 sinφmh, nh-φm+1h, nhsinα1-α2=D1Kp2 cosα2-D2Kp1 cosα1  |finm+1h, nhfinmh, nh|Kp1Kp2 cosφmh, nh-φm+1h, nhsinα1-α2=-D1Kp2 sinα2+D2Kp1 sinα1,
D1=Γm,nxP1, yP1-K11|finmh, nh|2-K21|finm+1h, nh|2,  D2=Γm,nxP2, yP2-K12|finmh, nh|2-K22|finm+1h, nh|2.
|finestimatedmh, nh|=Γm,n0, 0Ka,  D1=Γm,nxP1, yP1-K11|finestimatedmh, nh|2-K21×|finestimatedm+1h, nh|2,  D2=Γm,nxP2, yP2-K12|finestimatedmh, nh|2-K22×|finestimatedm+1h, nh|2,  φestimatedmh, nh-φestimatedm+1h, nh=tan-1D1Kp2 cosα2-D2Kp1 cosα1-D1Kp2 sinα2+D2Kp1 sinα1.
rx, y=1-|x|qh1-|y|qh for |x|, |y|qh0 elsewhere.
K11KP1=qh-xP1q-1h+xP1,  K21KP1=q-1h+xP1qh-xP1.
K12KP2=qh-xP2q-1h+xP2,  K22KP2=q-1h+xP2qh-xP2.
|α1-α2|=2p+1π2,
α1-α2=σ0,0xP1, yP1 rx, y*rx-h, ydxdy-σ0,0xP2, yP2 rx, y*rx-h, ydxdy.
|α1-α2|=expj h2xP1-hλd-expj h2xP2-hλd.
|xP1-xP2|=λdπ4h.
h=110 μm; a=10 μm; d=1400 μm; BL=0.1 μm-1 with a Hanning window; xP1x=yP1y=55.0 μm, yP1x=xP1y=0.0 μm; xP2x=yP2y=55.0 μm, yP2x=xP2y=7.0 μm; Δx=Δy=3 μm.
Δfm,n=finestimatedmh, nh|-|finmh, nh,  Δφm,n;m,n+1=|φmh, nh-φmh, n+1h-φestimatedmh, nh-φestimatedmh, n+1h|,
T=λ2h180π 1000 mdeg.

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