Abstract

This paper proposes a sub-aperture correlation based numerical phase correction method for interferometric full field imaging systems provided the complex object field information can be extracted. This method corrects for the wavefront aberration at the pupil/ Fourier transform plane without the need of any adaptive optics, spatial light modulators (SLM) and additional cameras. We show that this method does not require the knowledge of any system parameters. In the simulation study, we consider a full field swept source OCT (FF SSOCT) system to show the working principle of the algorithm. Experimental results are presented for a technical and biological sample to demonstrate the proof of the principle.

©2013 Optical Society of America

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References

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  1. B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
    [PubMed]
  2. J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack-Hartmann wavefront sensor,” J. Microsc. 205(1), 61–75 (2002).
    [Crossref] [PubMed]
  3. M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
    [Crossref] [PubMed]
  4. M. Pircher and R. J. Zawadzki, “Combining adaptive optics with optical coherence tomography: Unveiling the cellular structure of the human retina in vivo,” Expert Rev. Ophthalmol. 2(6), 1019–1035 (2007).
    [Crossref]
  5. K. Sasaki, K. Kurokawa, S. Makita, and Y. Yasuno, “Extended depth of focus adaptive optics spectral domain optical coherence tomography,” Biomed. Opt. Express 3(10), 2353–2370 (2012).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  8. T. Haist, J. Hafner, M. Warber, and W. Osten, “Scene-based wavefront correction with spatial light modulators,” Proc. SPIE 7064, 70640M, 70640M-11 (2008).
    [Crossref]
  9. A. E. Tippie and J. R. Fienup, “Sub-Aperture Techniques Applied to Phase-Error Correction in Digital Holography,” in Digital Holography and Three-Dimensional Imaging, OSA Techinal Digest (CD) (Optical Society of America, 2011), paper DMA4. http://www.opticsinfobase.org/abstract.cfm?URI=DH-2011-DMA4
    [Crossref]
  10. S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A 25(4), 983–994 (2008).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  12. P. Hariharan, Optical Interferometry (Academic, 2003).
  13. D. Malacara, Optical Shop Testing (Wiley, 1992).
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    [Crossref] [PubMed]
  15. W. Drexler and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer, 2008).
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  17. A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19(13), 12027–12038 (2011).
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2012 (3)

2011 (1)

2010 (1)

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[Crossref] [PubMed]

2008 (3)

2007 (3)

2006 (1)

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[Crossref] [PubMed]

2003 (1)

2002 (1)

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack-Hartmann wavefront sensor,” J. Microsc. 205(1), 61–75 (2002).
[Crossref] [PubMed]

2001 (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
[PubMed]

Adie, S. G.

S. G. Adie, B. W. Graf, A. Ahmad, P. S. Carney, and S. A. Boppart, “Computational adaptive optics for broadband optical interferometric tomography of biological tissue,” Proc. Natl. Acad. Sci. U.S.A. 109(19), 7175–7180 (2012).
[Crossref] [PubMed]

Ahmad, A.

S. G. Adie, B. W. Graf, A. Ahmad, P. S. Carney, and S. A. Boppart, “Computational adaptive optics for broadband optical interferometric tomography of biological tissue,” Proc. Natl. Acad. Sci. U.S.A. 109(19), 7175–7180 (2012).
[Crossref] [PubMed]

Betzig, E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[Crossref] [PubMed]

Beverage, J. L.

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack-Hartmann wavefront sensor,” J. Microsc. 205(1), 61–75 (2002).
[Crossref] [PubMed]

Boppart, S. A.

S. G. Adie, B. W. Graf, A. Ahmad, P. S. Carney, and S. A. Boppart, “Computational adaptive optics for broadband optical interferometric tomography of biological tissue,” Proc. Natl. Acad. Sci. U.S.A. 109(19), 7175–7180 (2012).
[Crossref] [PubMed]

Carney, P. S.

S. G. Adie, B. W. Graf, A. Ahmad, P. S. Carney, and S. A. Boppart, “Computational adaptive optics for broadband optical interferometric tomography of biological tissue,” Proc. Natl. Acad. Sci. U.S.A. 109(19), 7175–7180 (2012).
[Crossref] [PubMed]

Dai, G. M.

Denk, W.

M. Rueckel and W. Denk, “Properties of coherence-gated wavefront sensing,” J. Opt. Soc. Am. A 24(11), 3517–3529 (2007).
[Crossref] [PubMed]

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[Crossref] [PubMed]

Descour, M. R.

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack-Hartmann wavefront sensor,” J. Microsc. 205(1), 61–75 (2002).
[Crossref] [PubMed]

Fienup, J. R.

Franke, G.

Graf, B. W.

S. G. Adie, B. W. Graf, A. Ahmad, P. S. Carney, and S. A. Boppart, “Computational adaptive optics for broadband optical interferometric tomography of biological tissue,” Proc. Natl. Acad. Sci. U.S.A. 109(19), 7175–7180 (2012).
[Crossref] [PubMed]

Guizar-Sicairos, M.

Hafner, J.

T. Haist, J. Hafner, M. Warber, and W. Osten, “Scene-based wavefront correction with spatial light modulators,” Proc. SPIE 7064, 70640M, 70640M-11 (2008).
[Crossref]

Haist, T.

T. Haist, J. Hafner, M. Warber, and W. Osten, “Scene-based wavefront correction with spatial light modulators,” Proc. SPIE 7064, 70640M, 70640M-11 (2008).
[Crossref]

Hillmann, D.

Hüttmann, G.

Ji, N.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[Crossref] [PubMed]

Koch, P.

Kumar, A.

Kurokawa, K.

Lührs, C.

Mack-Bucher, J. A.

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[Crossref] [PubMed]

Mahajan, V. N.

Makita, S.

Milkie, D. E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[Crossref] [PubMed]

Osten, W.

T. Haist, J. Hafner, M. Warber, and W. Osten, “Scene-based wavefront correction with spatial light modulators,” Proc. SPIE 7064, 70640M, 70640M-11 (2008).
[Crossref]

Pircher, M.

M. Pircher and R. J. Zawadzki, “Combining adaptive optics with optical coherence tomography: Unveiling the cellular structure of the human retina in vivo,” Expert Rev. Ophthalmol. 2(6), 1019–1035 (2007).
[Crossref]

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
[PubMed]

Poyneer, L. A.

Rueckel, M.

M. Rueckel and W. Denk, “Properties of coherence-gated wavefront sensing,” J. Opt. Soc. Am. A 24(11), 3517–3529 (2007).
[Crossref] [PubMed]

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[Crossref] [PubMed]

Sasaki, K.

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
[PubMed]

Shack, R. V.

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack-Hartmann wavefront sensor,” J. Microsc. 205(1), 61–75 (2002).
[Crossref] [PubMed]

Thurman, S. T.

Tippie, A. E.

Warber, M.

T. Haist, J. Hafner, M. Warber, and W. Osten, “Scene-based wavefront correction with spatial light modulators,” Proc. SPIE 7064, 70640M, 70640M-11 (2008).
[Crossref]

Yasuno, Y.

Zawadzki, R. J.

M. Pircher and R. J. Zawadzki, “Combining adaptive optics with optical coherence tomography: Unveiling the cellular structure of the human retina in vivo,” Expert Rev. Ophthalmol. 2(6), 1019–1035 (2007).
[Crossref]

Appl. Opt. (1)

Biomed. Opt. Express (1)

Expert Rev. Ophthalmol. (1)

M. Pircher and R. J. Zawadzki, “Combining adaptive optics with optical coherence tomography: Unveiling the cellular structure of the human retina in vivo,” Expert Rev. Ophthalmol. 2(6), 1019–1035 (2007).
[Crossref]

J. Microsc. (1)

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack-Hartmann wavefront sensor,” J. Microsc. 205(1), 61–75 (2002).
[Crossref] [PubMed]

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

J. Refract. Surg. (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
[PubMed]

Nat. Methods (1)

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Proc. Natl. Acad. Sci. U.S.A. (2)

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[Crossref] [PubMed]

S. G. Adie, B. W. Graf, A. Ahmad, P. S. Carney, and S. A. Boppart, “Computational adaptive optics for broadband optical interferometric tomography of biological tissue,” Proc. Natl. Acad. Sci. U.S.A. 109(19), 7175–7180 (2012).
[Crossref] [PubMed]

Proc. SPIE (1)

T. Haist, J. Hafner, M. Warber, and W. Osten, “Scene-based wavefront correction with spatial light modulators,” Proc. SPIE 7064, 70640M, 70640M-11 (2008).
[Crossref]

Other (4)

A. E. Tippie and J. R. Fienup, “Sub-Aperture Techniques Applied to Phase-Error Correction in Digital Holography,” in Digital Holography and Three-Dimensional Imaging, OSA Techinal Digest (CD) (Optical Society of America, 2011), paper DMA4. http://www.opticsinfobase.org/abstract.cfm?URI=DH-2011-DMA4
[Crossref]

P. Hariharan, Optical Interferometry (Academic, 2003).

D. Malacara, Optical Shop Testing (Wiley, 1992).

W. Drexler and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer, 2008).

Supplementary Material (1)

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Figures (11)

Fig. 1
Fig. 1 Schematic diagram for the interferometric setup. Lens L1 and L2 form a 4-f telecentric imaging system. Dotted rays show the imaging path of a point on the object in focus. Camera is at the focal plane of lens L1.

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Fig. 2
Fig. 2 Segmentation of Fourier data into KxK subapertures with K = 3. Green square dots represent sampling points of the average local slope data due to non-overlapping subapertures. Red dotted boxes represent subapertures with about 50% overlap in both directions. Blue circular dots represent sampling points of the slope data due to overlapping subapertures. With overlapping subapertures we can increase the sampling points to reduce the error in phase estimation.

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Fig. 3
Fig. 3 Schematic of the subaperture correlation based phase correction method.

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Fig. 4
Fig. 4 (a) Normalized power spectrum, (b) amplitude of spectral interferogram at a lateral pixel, (c) A-scan, (d) image slice at the peak location in red dotted box in (c), (e) phase error in radians applied at the pupil plane, (f) aberrated image due to phase error in (e).

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Fig. 5
Fig. 5 (a), (b), (c) and (g) show the phase corrected images for non-overlapping subaperture with values of K equal to 3, 5, 7 and 9 respectively, and (d), (e), (f) and (j) are the respective residual phase error in radians. (h) and (i) are the images for subapertures with 50 percent overlap for K equal to 3 and 5, and (k) and (l) are the respective residual phase error in radians.

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Fig. 6
Fig. 6 RMS residual phase error in radians for different values of K and different percentage of overlap between subapertures. For the same value of K the size of subapertures in non- overlapping and overlapping cases are the same.

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Fig. 7
Fig. 7 (a) Schematic of the experimental setup: M is the mirror, B.S. is the beam splitter, MO is the 5X NIR microscope objective, L1 and L2 are lens with focal length f = 200 mm, L3 and L4 are lens with f = 150 mm and 75 mm and P is the pupil places at the focal plane of L3 and L4, (b) sample consisting of layer of plastic sheet, film of dried milk and the RTT, (c) image of the RTT surface obtained with non-uniform plastic sheeting, (d) Fourier transform of the image shows it is band limited, and (e) zoomed in image of (c) showing 390×390 pixels . Focus was placed on the RTT.

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Fig. 8
Fig. 8 Phase corrected images of the one in Fig. 7, obtained using (a) non-overlapping subapertures with K = 3, (b) non-overlapping subapertures with K = 5 and (c) overlapping subapertures with K = 5. (d), (e) and (f) are the detected phase error across the aperture in radians in the case of (a), (b) and (c) respectively.

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Fig. 9
Fig. 9 (a) Aberrated image obtained using uniform plastic sheet, (b) phase corrected image using non-overlapping subapertures with K = 3, (c) image obtained by only defocus correction using two non-overlapping subapertures as in Fig. 7, (d) phase error in radians estimated for case (b) shows strong quadratic and fourth order terms, (e) quadratic phase error detected for case (c). Here defocus balances the spherical aberration.

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Fig. 10
Fig. 10 (a) A tomogram of the grape sample, (b) 3-D image volume with dotted line showing location of the tomogram shown in (a), (c) enface image obtained at the depth of 424.8 µm in the grape sample indicated by arrow in (a), (d) is the digitally focused image of (c).

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Fig. 11
Fig. 11 Comparison of coefficients of defocus error with depth z obtained theoretically and using the two subaperture method for a grape sample.

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Equations (20)

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

I d ( ξ;k )= | E s ( ξ;k ) | 2 + | E r ( ξ;k ) | 2 + E s * ( ξ;k ) E r ( ξ;k )+ E s ( ξ;k ) E r * ( ξ;k ).
E s ( ξ;k )=exp( i4kf ) ∫ E o ( u;k ) P( ξ−u;k ) d 2 u=exp( i4kf ) E s ' ( ξ;k )
E r ( ξ;k )=R( ξ;k )exp[ ik( 4f+Δz ) ]
I s ( ξ,z )=Δ ξ 2 I( mΔξ,nΔξ,z )= I m,n,z
D x,y =DFT[ I m,n,z ]= ∑ m=0 2M−1 ∑ n=0 2M−1 I m,n,z exp[ −i2π( mx 2M + ny 2M ) ]
L≈NΔx≈N λ 0 f 2MΔξ
s x po,qo = ∂ ϕ p ∂x − ∂ ϕ q ∂x = Δmπ M   and   s y po,qo = ∂ ϕ p ∂y − ∂ ϕ q ∂y = Δnπ M ,
I ˜ m,n,z = 1 4 M 2 ∑ x=0 2M−1 ∑ y=0 2M−1 D ˜ x,y exp[ −i ϕ e ( x,y ) ] exp[ i2π( mx 2M + ny 2M ) ].
a= πΔm MN
a T z = π λ 0 Δz N z 4 M 2 Δ ξ 2
Δz= λ 0 2 2nΔλ
D ˜ p = S ˜ p exp( i ϕ p )≈ S ˜ p exp{ i[ ϕ po +( x− x po ) ∂ ϕ p ∂x +( y− y po ) ∂ ϕ p ∂y ] }
D ˜ q = S ˜ q exp( i ϕ q )≈ S ˜ q exp{ i[ ϕ qo +( x− x qo ) ∂ ϕ q ∂x +( y− y qo ) ∂ ϕ q ∂y ] }
IDFT[ D ˜ p ]= 1 4 M 2 ∑ x=0 2M−1 ∑ y=0 2M−1 S ˜ p x,y exp( i ϕ p ) exp[ i2π( mx 2M + ny 2M ) ] ≈ I p ( m− M∂ ϕ p π∂x ,n− M∂ ϕ p π∂y )
IDFT[ D ˜ q ]= 1 4 M 2 ∑ x=0 2M−1 ∑ y=0 2M−1 S ˜ q x,y exp( i ϕ q ) exp[ i2π( mx 2M + ny 2M ) ] ≈ I q ( m− M∂ ϕ q π∂x ,n− M∂ ϕ q π∂y )
ϕ e = ∑ J=1 Ζ T J ( X,Y ) = ∑ J=2 Z ∑ j=0 J a Jj X j Y J−j ,
X=x−M ...0≤x≤2M−1 Y=y−M ...0≤y≤2M−1
∇ ϕ e = ∑ J=2 Z ∑ j=0 J a Jj ∇( X j Y J−j )
GA=S
A= ( G T G ) −1 G T S.

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