Abstract

We present an algorithm which exploits data redundancy to make computational, coherent, optical imaging more computationally efficient. This algorithm specifically addresses the computation of how light scattered by a sample is collected and coherently detected. It is of greatest benefit in the simulation of broadband optical systems employing coherent detection, such as optical coherence tomography. Although also amenable to time-harmonic data, the algorithm is designed to be embedded within time-domain electromagnetic scattering simulators such as the psuedo-spectral and finite-difference time domain methods. We derive the algorithm in detail as well as criteria which ensure accurate execution of the algorithm. We present simulations that verify the developed algorithm and demonstrate its utility. We expect this algorithm to be important to future developments in computational imaging.

© 2015 Optical Society of America

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References

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  1. P. Elbau, L. Mindrinos, and O. Scherzer, “Mathematical methods of optical coherence tomography,” in Handbook of Mathematical Methods in Imaging, O. Scherzer, ed. (Springer, 2015).
    [Crossref]
  2. T. Brenner, D. Reitzle, and A. Kienle, “An algorithm for simulating image formation in optical coherence tomography for cylinder scattering,” Proc. SPIE 9541, 95411F (2015)
    [Crossref]
  3. P. R. T. Munro, A. Curatolo, and D. D. Sampson, “Full wave model of image formation in optical coherence tomography applicable to general samples,” Opt. Express 23, 2541–2556 (2015).
    [Crossref] [PubMed]
  4. P. R. T. Munro and P. Török, “Vectorial, high numerical aperture study of Nomarski’s differential interference contrast microscope,” Opt. Express 13, 6833–6847 (2005).
    [Crossref] [PubMed]
  5. P. Török, P. R. T. Munro, and E. E. Kriezis, “High numerical aperture vectorial imaging in coherent optical microscopes,” Opt. Express 16, 507–523 (2008).
    [Crossref] [PubMed]
  6. I. R. Capoglu, J. D. Rogers, A. Taflove, and V. Backman, “Computational optical imaging using the finite-difference time-domain method,” in Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, A. Taflove, A. Oskooi, and S. G. Johnson, eds. (Artech House, 2013).
  7. R. L. Coe and E. J. Seibel, “Computational modeling of optical projection tomographic microscopy using the finite difference time domain method,” J. Opt. Soc. Am. A 29, 2696–2707 (2012).
    [Crossref]
  8. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems ii. structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959).
    [Crossref]
  9. V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” T. Opt. Inst. Petrograd 1, 1–36 (1919).
  10. R. Luneburg, Mathematical Theory of Optics (University of California Press, 1966).
  11. K. Yee, “Numerical solution of inital boundary value problems involving Maxwell’s equations in isotropic media,” IEEE T. Antenn. Propag. 14, 302–307 (1966).
    [Crossref]
  12. A. Taflove and S. Hagness, Computational Electrodynamics, 3 Edition (Artech House, 2005).
  13. P. Török, P. R. T. Munro, and E. E. Kriezis, “Rigorous near- to far-field transformation for vectorial diffraction calculations and its numerical implementation,” J. Opt. Soc. Am. A 23, 713–722 (2006).
    [Crossref]
  14. P. R. T. Munro and P. Török, “Calculation of the image of an arbitrary vectorial electromagnetic field,” Opt. Express 15, 9293–9307 (2007).
    [Crossref] [PubMed]
  15. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).
  16. T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23, 1027–1037 (2006).
    [Crossref]
  17. K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17, 19662–19673 (2009).
    [Crossref] [PubMed]
  18. P. R. T. Munro, D. Engelke, and D. D. Sampson, “A compact source condition for modelling focused fields using the pseudospectral time-domain method,” Opt. Express 22, 5599–5613 (2014).
    [Crossref] [PubMed]
  19. P. Török and P. R. T. Munro, “The use of gauss-laguerre vector beams in sted microscopy,” Opt. Express 12, 3605–3617 (2004).
    [Crossref] [PubMed]
  20. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
    [Crossref]
  21. M. Frigo and S. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
    [Crossref]
  22. V. Čížek, Discrete Fourier Transforms and Their Applications (Adam Hilger, 1986).

2015 (2)

T. Brenner, D. Reitzle, and A. Kienle, “An algorithm for simulating image formation in optical coherence tomography for cylinder scattering,” Proc. SPIE 9541, 95411F (2015)
[Crossref]

P. R. T. Munro, A. Curatolo, and D. D. Sampson, “Full wave model of image formation in optical coherence tomography applicable to general samples,” Opt. Express 23, 2541–2556 (2015).
[Crossref] [PubMed]

2014 (1)

2012 (1)

2009 (1)

2008 (1)

2007 (1)

2006 (2)

2005 (2)

2004 (1)

1966 (1)

K. Yee, “Numerical solution of inital boundary value problems involving Maxwell’s equations in isotropic media,” IEEE T. Antenn. Propag. 14, 302–307 (1966).
[Crossref]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems ii. structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959).
[Crossref]

1919 (1)

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” T. Opt. Inst. Petrograd 1, 1–36 (1919).

Backman, V.

I. R. Capoglu, J. D. Rogers, A. Taflove, and V. Backman, “Computational optical imaging using the finite-difference time-domain method,” in Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, A. Taflove, A. Oskooi, and S. G. Johnson, eds. (Artech House, 2013).

Boppart, S. A.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
[Crossref]

Brenner, T.

T. Brenner, D. Reitzle, and A. Kienle, “An algorithm for simulating image formation in optical coherence tomography for cylinder scattering,” Proc. SPIE 9541, 95411F (2015)
[Crossref]

Capoglu, I. R.

I. R. Capoglu, J. D. Rogers, A. Taflove, and V. Backman, “Computational optical imaging using the finite-difference time-domain method,” in Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, A. Taflove, A. Oskooi, and S. G. Johnson, eds. (Artech House, 2013).

Carney, P. S.

Cížek, V.

V. Čížek, Discrete Fourier Transforms and Their Applications (Adam Hilger, 1986).

Coe, R. L.

Curatolo, A.

Elbau, P.

P. Elbau, L. Mindrinos, and O. Scherzer, “Mathematical methods of optical coherence tomography,” in Handbook of Mathematical Methods in Imaging, O. Scherzer, ed. (Springer, 2015).
[Crossref]

Engelke, D.

Frigo, M.

M. Frigo and S. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

Hagness, S.

A. Taflove and S. Hagness, Computational Electrodynamics, 3 Edition (Artech House, 2005).

Ignatowsky, V. S.

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” T. Opt. Inst. Petrograd 1, 1–36 (1919).

Johnson, S.

M. Frigo and S. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
[Crossref]

Kienle, A.

T. Brenner, D. Reitzle, and A. Kienle, “An algorithm for simulating image formation in optical coherence tomography for cylinder scattering,” Proc. SPIE 9541, 95411F (2015)
[Crossref]

Kriezis, E. E.

Luneburg, R.

R. Luneburg, Mathematical Theory of Optics (University of California Press, 1966).

Marks, D. L.

Matsushima, K.

Mindrinos, L.

P. Elbau, L. Mindrinos, and O. Scherzer, “Mathematical methods of optical coherence tomography,” in Handbook of Mathematical Methods in Imaging, O. Scherzer, ed. (Springer, 2015).
[Crossref]

Munro, P. R. T.

Ralston, T. S.

Reitzle, D.

T. Brenner, D. Reitzle, and A. Kienle, “An algorithm for simulating image formation in optical coherence tomography for cylinder scattering,” Proc. SPIE 9541, 95411F (2015)
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems ii. structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959).
[Crossref]

Rogers, J. D.

I. R. Capoglu, J. D. Rogers, A. Taflove, and V. Backman, “Computational optical imaging using the finite-difference time-domain method,” in Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, A. Taflove, A. Oskooi, and S. G. Johnson, eds. (Artech House, 2013).

Sampson, D. D.

Scherzer, O.

P. Elbau, L. Mindrinos, and O. Scherzer, “Mathematical methods of optical coherence tomography,” in Handbook of Mathematical Methods in Imaging, O. Scherzer, ed. (Springer, 2015).
[Crossref]

Seibel, E. J.

Shimobaba, T.

Taflove, A.

I. R. Capoglu, J. D. Rogers, A. Taflove, and V. Backman, “Computational optical imaging using the finite-difference time-domain method,” in Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, A. Taflove, A. Oskooi, and S. G. Johnson, eds. (Artech House, 2013).

A. Taflove and S. Hagness, Computational Electrodynamics, 3 Edition (Artech House, 2005).

Török, P.

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems ii. structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
[Crossref]

Yee, K.

K. Yee, “Numerical solution of inital boundary value problems involving Maxwell’s equations in isotropic media,” IEEE T. Antenn. Propag. 14, 302–307 (1966).
[Crossref]

IEEE T. Antenn. Propag. (1)

K. Yee, “Numerical solution of inital boundary value problems involving Maxwell’s equations in isotropic media,” IEEE T. Antenn. Propag. 14, 302–307 (1966).
[Crossref]

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

Opt. Express (7)

Proc. IEEE (1)

M. Frigo and S. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
[Crossref]

Proc. Roy. Soc. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems ii. structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959).
[Crossref]

Proc. SPIE (1)

T. Brenner, D. Reitzle, and A. Kienle, “An algorithm for simulating image formation in optical coherence tomography for cylinder scattering,” Proc. SPIE 9541, 95411F (2015)
[Crossref]

T. Opt. Inst. Petrograd (1)

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” T. Opt. Inst. Petrograd 1, 1–36 (1919).

Other (7)

R. Luneburg, Mathematical Theory of Optics (University of California Press, 1966).

I. R. Capoglu, J. D. Rogers, A. Taflove, and V. Backman, “Computational optical imaging using the finite-difference time-domain method,” in Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, A. Taflove, A. Oskooi, and S. G. Johnson, eds. (Artech House, 2013).

P. Elbau, L. Mindrinos, and O. Scherzer, “Mathematical methods of optical coherence tomography,” in Handbook of Mathematical Methods in Imaging, O. Scherzer, ed. (Springer, 2015).
[Crossref]

A. Taflove and S. Hagness, Computational Electrodynamics, 3 Edition (Artech House, 2005).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
[Crossref]

V. Čížek, Discrete Fourier Transforms and Their Applications (Adam Hilger, 1986).

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

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Figures (6)

Fig. 1
Fig. 1 Schematic diagram of the optical system under study. Each position vector r1, r2 and r3 is a two dimensional vector representing transverse displacement relative to a location on the optical axis. The field is observed on plane z3 = zobs and there is an infinite half-space of material with refractive index ns beginning at z3 = −h. z3 is measured relative to the nominal focus of the right hand lens, in air. Angles θ1 and θ2 represent the semi-convergence angles of typical rays. Ra is the radius of the aperture.
Fig. 2
Fig. 2 Plots of the real part of K(λ0qx) in the continuous and discrete cases. K was sampled at the points denoted with dots and the dashed line added to enhance readability. The sensitivity functions transformed into the pupil have also been plotted for each simulated mode field diameter. The vertical black lines denote two NAs that were considered: the design NA of Ra/f2 and 0.35.
Fig. 3
Fig. 3 Plots of the lateral PSFs for each MFD calculated analytically (solid line) and numerically using a PSTD simulation (dot markers) for NA = Ra/f2 (a) and NA = 0.35 (b).
Fig. 4
Fig. 4 Plot of minimum simulation size as a function of zobs (a) and of NA (b) required to sample K correctly and to contain the image of ϕ1 = φ2 within the sample space.
Fig. 5
Fig. 5 Plots of the axial PSFs for each MFD calculated analytically (solid line) and numerically using a PSTD simulation (dot markers) for NA = Ra/f2.
Fig. 6
Fig. 6 a) Cross-section through the simulated scattering object which was λ0/6 thick and embedded in a material of refractive index 1.4. b) and c) show the magnitudes of the x and y components of the scattered field just above the scatterer. d) – e) show images when the scatterer was 0, 200μm and 400μm, respectively, from the focus.

Tables (1)

Tables Icon

Table 1 Base parameters of the numerical simulations, λ0 is the wavelength in air and MFD stands for mode field diameter.

Equations (33)

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U 3 ( r 3 , f ) = 1 N t n = 0 N t 1 u 3 ( r 3 , n Δ t ) exp ( i 2 π f n Δ t ) ,
M < 1 Δ t 1 2 f .
U 1 ( r 1 , f ) = ^ { K ( r 2 ) ^ { U 3 ( r 3 , f ) } }
= ^ { K ( r 2 ) ^ { 1 N t n = 0 N t 1 u 3 ( r 3 , n Δ t ) exp ( i 2 π f n Δ t ) } } = 1 N t n = 0 N t 1 exp ( i 2 π f n Δ t ) ^ { K ( r 2 ) ^ { u 3 ( r 3 , n Δ t ) } } ,
a = 2 ϕ 1 ( r 1 ) U 1 ( r 1 ) d 2 r 1
U 2 ( r 2 ) = 1 i λ f 2 U ˜ 3 ( r 2 λ f 2 , z obs ) K ( r 2 λ f 2 )
K ( q ) = exp ( i ( h + z obs ) n s 2 π λ 1 ( λ / n s ) 2 | q | 2 ) exp ( i h 2 π λ 1 λ 2 | q | 2 ) P ( q ) ,
P ( q ) = { 1 λ | q | NA 0 otherwise .
U 1 ( r 1 ) = 1 i λ f 1 U ˜ 2 ( r 1 λ f 1 ) = f 2 f 1 U 3 ( f 2 f 1 r 1 , z obs ) K ˜ ( f 2 f 1 r 1 ) ,
a = 2 ϕ 1 ( r 1 ) [ f 2 f 1 U 3 ( f 2 f 1 r 1 , z obs ) K ˜ ( f 2 f 1 r 1 ) ] d 2 r 1 = { ϕ 1 ( r 1 ) [ f 2 f 1 U 3 ( f 2 f 1 r 1 , z obs ) K ˜ ( f 2 f 1 r 1 ) ] } | q 1 = 0 = 2 ϕ ˜ 1 ( f 2 f 1 q 3 ) U ˜ 3 ( q 3 , z obs ) K ( q 3 ) d 2 q 3 ,
U ˜ { q } = 2 exp ( i 2 π r q ) U ( r ) d 2 r .
f ( x ) g * ( x ) d x = f ˜ ( q ) g ˜ * ( q ) d q ,
a = 2 ϕ ˜ 1 ( f 2 f 1 q 3 ) U ˜ 3 ( q 3 , z obs ) K ( q 3 ) d 2 q 3 = 2 ϕ ˜ 1 ( f 2 f 1 q 3 ) K ( q 3 ) U ˜ 3 ( q 3 , z obs ) d 2 q 3
= 2 ϕ ˜ 1 ( f 2 f 1 q 3 ) K ( q 3 ) ( ( U ˜ 3 ( q 3 , z obs ) ) * ) * d 2 q 3 = 2 { ϕ ˜ 1 ( f 2 f 1 q 3 ) K ( q 3 ) } ( { ( U ˜ 3 ( r 3 , z obs ) ) * } ) * d 2 r 3 = 2 { ϕ ˜ 1 ( f 2 f 1 q 3 ) K ( q 3 ) } U 3 ( r 3 , z obs ) d 2 r 3 .
a = 2 ϕ 1 ( r 1 r d ) U 1 ( r 1 ) d 2 r 1 .
U 3 ( ( i , j ) Δ 3 , f ) = 1 N t n = 0 N t 1 u 3 ( ( i , j ) Δ 3 , n Δ t ) exp ( i 2 π f n Δ t ) .
a = 1 N t n = 0 N t 1 exp ( i 2 π f n Δ t ) i , j ϕ ˜ 1 ( f 2 f 1 q 3 ) K ( q 3 ) u ˜ 3 ( q 3 , n Δ t ) 1 ( I Δ 3 ) 2
= 1 N t n = 0 N t 1 exp ( i 2 π f n Δ t ) i , j ^ { ϕ ˜ 1 ( f 2 f 1 q 3 ) K ( q 3 ) } u 3 ( r 3 , n Δ t ) Δ 3 2 ,
K ( q 3 ) = exp ( i Φ 1 ( q 3 ) ) ,
Φ 1 ( q 3 ) = ( h + z obs ) n s 2 π λ 1 ( λ / n s ) 2 | q 3 | 2 h 2 π λ 1 λ 2 | q 3 | 2 .
f q 3 x = 1 2 π Φ 1 q 3 x f q 3 y = 1 2 π Φ 1 q 3 y ,
f q i = ( h + z obs ) ( λ / n s ) q i 1 ( λ / n s ) 2 | q 3 | 2 + h λ q i 1 λ 2 | q 3 | 2 ,
Δ q 3 < 1 2 | f q i | ,
Δ q 3 < 1 2 NA | 1 1 NA 2 h + z obs n s 2 NA 2 |
max ( x 3 ) min ( x 3 ) > 2 NA | h 1 NA 2 h + z obs n s 2 NA 2 |
ϕ 3 ( r 3 ) = 0 sin 1 NA cos θ 1 sin θ 1 ( 1 + cos θ 2 ) J 0 ( 2 π λ | r 3 | sin θ 1 ) exp ( i 2 π λ h ( n s cos θ 2 cos θ 1 ) ) exp ( i n s 2 π λ z obs cos θ 2 ) exp ( ( F sin θ 1 / NA ) 2 ) d θ 1
| r 3 | < D | ϕ 3 ( r 3 ) | 2 d 2 r 3 > ( 1 ε ) 2 | ϕ 3 ( r 3 ) | 2 d 2 r 3
I 0 = 2 | ϕ 3 ( r 3 ) | 2 d 2 r 3 = 2 f 1 2 λ 4 f 2 2 π 3 W 2 ( 1 exp ( f 2 2 NA 2 π 2 W 2 2 f 1 2 λ 2 ) ) .
max ( x 3 ) min ( x 3 ) 2 D , where I ( D ) = ( 1 ε ) I 0 ,
I ( D ) = | r 3 | < D | ϕ 3 ( r 3 ) | 2 d 2 r 3 ,
0 N Δ x g ( x ) d x = Δ x n = 0 N 1 g ( n Δ x ) .
g ( x ) = 1 N k = 0 N 1 n = 0 N 1 exp ( i 2 π k x N Δ x ) exp ( i 2 π k n N ) g ( n Δ x ) d x .
0 N Δ x g ( x ) d x = 0 N Δ x 1 N k = 0 N 1 n = 0 N 1 exp ( i 2 π k x N Δ x ) exp ( i 2 π k n N ) g ( n Δ x ) d x = 1 N k = 0 N 1 n = 0 N 1 [ 0 N Δ x exp ( i 2 π k x N Δ x ) d x ] exp ( i 2 π k n N ) g ( n Δ x ) = 1 N k = 0 N 1 n = 0 N 1 δ k , 0 N Δ x exp ( i 2 π k n N ) g ( n Δ x ) = n = 0 N 1 g ( n Δ x ) Δ x as required .

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