## 2-D PSTD Simulation of optical phase conjugation for turbidity suppression

Optics Express, Vol. 15, Issue 24, pp. 16005-16016 (2007)

http://dx.doi.org/10.1364/OE.15.016005

Acrobat PDF (524 KB)

### Abstract

Turbidity Suppression via Optical Phase Conjugation (TS-OPC) is an optical phenomenon that uses the back propagation nature of optical phase conjugate light field to undo the effect of tissue scattering. We use the computationally efficient and accurate pseudospectral time-domain (PSTD) simulation method to study this phenomenon; a key adaptation is the volumetric inversion of the optical wavefront E-field as a means for simulating a phase conjugate mirror. We simulate a number of scenarios and demonstrate that TS-OPC deteriorates with increased scattering in the medium, or increased mismatch between the random medium and the phase conjugate wave during reconstruction.

© 2007 Optical Society of America

## 1. Introduction

7. H. F. Zhang, K. Maslov, G. Stoica, and L. H. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nature Biotechnology **24**, 848–851 (2006). [CrossRef] [PubMed]

9. M. Nieto-Vesperinas and E. Wolf, “Phase conjugation and symmetries with wave fields in free space containing evanescent components,” J. Opt. Soc. Am. A, 1429 (1985). [CrossRef]

*et al*. demonstrated that OPC could reverse optical scattering induced by a ground glass slide in 1966 [10

10. E. N. Leith and J. Upatnieks, “Holographic imagery through diffusing media,” J. Opt. Soc. Am. **56**, 523 (1966). [CrossRef]

*µ*) and anisotropy (

_{s}*g*) can be easier varied in a simulation. The angle dependent scattering profile of the scatterers can also be arbitrarily tailored—a flexibility that is difficult to implement experimentally. Specific spatial arrangements of scatterers can also be easier accommodated in simulations than in experiments. The third, and perhaps the most important, advantage for using simulation is that it allows access to details that cannot be readily observed/detected in the experimental context. The behavior of near-field components is an excellent example of a feature in TS-OPC that simulation can easily allow us to study, but which direct experimental study is difficult to implement.

## 2. Method

11. D. Boas, J. Culver, J. Stott, and A. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express **10**, 159–170 (2002). [PubMed]

12. X. X. Guo, M. F. G. Wood, and A. Vitkin, “Monte Carlo study of pathlength distribution of polarized light in turbid media,” Opt. Express **15**, 1348–1360 (2007). [CrossRef] [PubMed]

13. S. H. Tseng and B. Huang, “Comparing Monte Carlo simulation and pseudospectral time-domain numerical solutions of Maxwell’s equations of light scattering by a macroscopic random medium,” Appl. Phys. Lett.91 (2007). [CrossRef]

14. Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm,” IEEE Trans. Geosci. Remote Sens. **37**, 917–926 (1999). [CrossRef]

*D*dimensions that does not have geometric details or material inhomogeneities smaller than one-half wavelength, PSTD reduces the computer storage and the running-time by approximately 8

*: 1 relative to the conventional FDTD while achieving comparable accuracy [14*

^{D}14. Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm,” IEEE Trans. Geosci. Remote Sens. **37**, 917–926 (1999). [CrossRef]

*macroscopic*light scattering problem based on Maxwell’s equations, and enables accurate modeling of the PCM optical characteristics.

^{nd}-order finite-difference scheme as employed in the FDTD method. For the spatial derivatives, the PSTD method calculates the spatial derivatives in the frequency domain via discrete Fourier transform: Let {

**E**

*} denote values of the electric field or magnetic field, and {(∂*

_{i}**E**/∂x)

_{i}} denote the spatial derivative of

**E**along the x-direction. Based on the differentiation theorem of Fourier transform, we can write:

**F**and

**F**

^{-1}denote, respectively, the forward and inverse discrete Fourier transforms, and

*k̂*is the Fourier transform variable representing the

_{x}*x*-component of the numerical wave vector. The spatial derivatives in each direction can be obtained numerically. According to the Nyquist sampling theorem, the spatial derivatives calculated in (1) is of spectral accuracy (meaning as accurate as it can get with the given information), allowing the PSTD technique, with a coarse grid of two spatial samples per wavelength, to achieve similar accuracy as the FDTD technique (FDTD requires 20 spatial samples per wavelength.) Finally, an anisotropic perfectly matched layer absorbing boundary condition [16

16. S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE trans. Antennas Propag. **44**, 1630–1639 (1996). [CrossRef]

**E**-field and

**D**-field components throughout this region:

**H**and magnetic induction

**B**remain unchanged. The inversion of

**E**and

**D**in the inversion region causes the poynting vector

**S=E×H**to reverse direction without changing the amplitude. As a result, light with inverted phase will propagate in the opposite direction, exit the PCM interface from the right and effectively become an OPC wave that travels back towards the random medium. The inversion region must be large enough as to enclose the entire pulse when it has passed wholly through the PCM interface from the left to avoid discontinuity of the field components. If the OPC region only encloses a fraction of the light pulse, the discontinuity of the

**E**-field and

**D**-field will excite non-physical field oscillations as a result of the Gibb’s phenomenon.

*t*=0.05 fs. A macroscopic random medium consisting of a cluster of randomly positioned dielectric cylinders is placed in space. The random medium is illuminated by an incident light pulse; the light pulse is multiply scattered as it propagates through the cluster of cylinders. At time

*t*, the

**E**-field of the region to the right of the random medium is inverted, simulating OPC light field generation from a PCM that is placed adjacent to the random medium. Based on our understanding of the TS-OPC phenomenon, we expect the OPC light field to travel back through the sample and reconstruct the original incident light pulse. Depending upon the choice of the inversion region, a different placement of PCM can be simulated. The time

*t*should be chosen such that the pulse has left the random medium and has passed completely through the PCM interface. Each simulation takes typically ~12 hours with 4 computing cores of a Xeon Woodcrest 3.0GHz processor.

## 3. Results

*µ*, and the mismatch of the random medium and the OPC wavefront. These results are presented in Sections 3.C. and 3.D., respectively.

_{s}### 3.A. Light impinges a PCM in vacuum

### 3.B. Light propagates through a random medium then impinges a PCM

*t*=0.05 fs. Each cylinder has a diameter of 2.5 µm with a refractive index of 1.2. Three still images of the simulation movie clip (584KB movie) are shown in Fig. 3. The incident light pulse is a Gaussian pulse with a cross-sectional width of 13.4 µm. As the light pulse propagates through the cluster, it is multiply scattered by the randomly located dielectric cylinders, resulting in a reverberant wavefront, as shown in Fig. 3(b).

### 3.C. Scattering coefficient (µs)

*µ*on the TS-OPC efficiency. The simulation setup is the same as described in Section 3.B, except that the number of scatterers,

_{s}*N*, is varied among 500, 1000, 1500, 2000, and 2500. (For λ=1µm,

*N*=500 corresponds to a scattering coefficient

*µ*=0.0258 µm

_{s}^{-1}and

*N*=2500 corresponds to

*µ*=0.1291 µm

_{s}^{-1}.) The scattering coefficient is calculated by multiplying the extinction coefficient of a single dielectric cylinder and the number density of the cluster of dielectric cylinders [17]. The extinction coefficient of a single cylinder is determined by an analytical solution of Maxwell’s equations (Mie expansion), which also yields the phase function of a single cylinder [18

18. G. Mie, Ann. Phys. **25**, 377 (1908). [CrossRef]

*µ*) of the random medium can be varied as desired.

_{s}*N*is increased, the random medium becomes more complex; more scattering occurs and the TS-OPC efficiency drops. The refocused light pulse profile becomes blurry for larger

*N*, as shown in Fig. 4(a)–(f). In order to quantify the degradation of the TS-OPC effect, we calculate the total refocused energy (eq. (3)) that is focused into the vicinity of the original light pulse at

*t*=0.

*N*increases, the refocused energy decreases rapidly. This is anticipated since with more scattering, more light is lost as it scatters into other directions and never reaches the PCM, resulting in a less coherent refocused light pulse.

### 3.D. Misalignment of optics

## 4. Discussion

**D**- and

**E**-field inversion within the volume to the right of the PCM interface. Specifically, because the total duration of a light pulse is limited, such a time point can be chosen to be the time at which the bulk of the light pulse has passed through the random medium and the PCM interface. The task of simulating a CW light illumination is more complicated. In this case, it is difficult to assign a proper time point for the

**E**-field inversion to occur.

19. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. **32**, 2309–2311 (2007). [CrossRef] [PubMed]

## 5. Summary

## Acknowledgements

## References and links

1. | W. F. Cheong, S. A. Prahl, and A. J. Welch, “A Review of the Optical-Properties of Biological Tissues,” IEEE J. Quantum Electron. |

2. | A. Wax, C. H. Yang, R. R. Dasari, and M. S. Feld, “Measurement of angular distributions by use of low-coherence interferometry for light-scattering spectroscopy,” Opt. Lett. |

3. | L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van, J. M. Dam, M. S. Crawford, and Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett. |

4. | J. M. Schmitt, “Optical coherence tomography (OCT): A review,” IEEE J. Sel. Top. Quantum Electron. |

5. | G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science |

6. | W. Denk, J. H. Strickler, and W. W. Webb, “2-Photon Laser Scanning Fluorescence Microscopy,” Science |

7. | H. F. Zhang, K. Maslov, G. Stoica, and L. H. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nature Biotechnology |

8. | Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” ( |

9. | M. Nieto-Vesperinas and E. Wolf, “Phase conjugation and symmetries with wave fields in free space containing evanescent components,” J. Opt. Soc. Am. A, 1429 (1985). [CrossRef] |

10. | E. N. Leith and J. Upatnieks, “Holographic imagery through diffusing media,” J. Opt. Soc. Am. |

11. | D. Boas, J. Culver, J. Stott, and A. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express |

12. | X. X. Guo, M. F. G. Wood, and A. Vitkin, “Monte Carlo study of pathlength distribution of polarized light in turbid media,” Opt. Express |

13. | S. H. Tseng and B. Huang, “Comparing Monte Carlo simulation and pseudospectral time-domain numerical solutions of Maxwell’s equations of light scattering by a macroscopic random medium,” Appl. Phys. Lett.91 (2007). [CrossRef] |

14. | Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm,” IEEE Trans. Geosci. Remote Sens. |

15. | A. Taflove and S. C. Hagness, |

16. | S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE trans. Antennas Propag. |

17. | C. F. Bohren and D. R. Huffman, |

18. | G. Mie, Ann. Phys. |

19. | I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. |

**OCIS Codes**

(290.4210) Scattering : Multiple scattering

(290.7050) Scattering : Turbid media

**ToC Category:**

Scattering

**History**

Original Manuscript: October 5, 2007

Revised Manuscript: November 13, 2007

Manuscript Accepted: November 13, 2007

Published: November 19, 2007

**Virtual Issues**

Vol. 2, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

Snow H. Tseng and Changhuei Yang, "2-D PSTD Simulation of optical phase conjugation for turbidity suppression," Opt. Express **15**, 16005-16016 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-16005

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### References

- W. F. Cheong, S. A. Prahl, and A. J. Welch, "A Review of the Optical-Properties of Biological Tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990). [CrossRef]
- A. Wax, C. H. Yang, R. R. Dasari, and M. S. Feld, "Measurement of angular distributions by use of low-coherence interferometry for light-scattering spectroscopy," Opt. Lett. 26, 322-324 (2001). [CrossRef]
- L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, "Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution," Phys. Rev. Lett. 80, 627-630 (1998). [CrossRef]
- J. M. Schmitt, "Optical coherence tomography (OCT): A review," IEEE J. Sel. Top. Quantum Electron. 5, 1205-1215 (1999). [CrossRef]
- G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, "In vivo endoscopic optical biopsy with optical coherence tomography," Science 276, 2037-2039 (1997). [CrossRef] [PubMed]
- W. Denk, J. H. Strickler, and W. W. Webb, "2-Photon Laser Scanning Fluorescence Microscopy," Science 248, 73-76 (1990). [CrossRef] [PubMed]
- H. F. Zhang, K. Maslov, G. Stoica, and L. H. V. Wang, "Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging," Nature Biotechnology 24, 848-851 (2006). [CrossRef] [PubMed]
- Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, "Optical phase conjugation for turbidity suppression in biological samples," (in review).
- M. Nieto-Vesperinas, and E. Wolf, "Phase conjugation and symmetries with wave fields in free space containing evanescent components," J. Opt. Soc. Am. A 2, 1429 (1985). [CrossRef]
- E. N. Leith, and J. Upatnieks, "Holographic imagery through diffusing media," J. Opt. Soc. Am. 56, 523 (1966). [CrossRef]
- D. Boas, J. Culver, J. Stott, and A. Dunn, "Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head," Opt. Express 10, 159-170 (2002). [PubMed]
- X. X. Guo, M. F. G. Wood, and A. Vitkin, "Monte Carlo study of pathlength distribution of polarized light in turbid media," Opt. Express 15, 1348-1360 (2007). [CrossRef] [PubMed]
- S. H. Tseng, and B. Huang, "Comparing Monte Carlo simulation and pseudospectral time-domain numerical solutions of Maxwell's equations of light scattering by a macroscopic random medium," Appl. Phys. Lett. 91 (2007). [CrossRef]
- Q. H. Liu, "Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm," IEEE Trans. Geosci. Remote Sens. 37, 917-926 (1999). [CrossRef]
- A. Taflove, and S. C. Hagness, Computational Electrodynamics: the finite-difference time-domain method (Artech House, 2000).
- S. D. Gedney, "An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices," IEEE trans.Antennas Propag. 44, 1630-1639 (1996). [CrossRef]
- C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (A Wiley-Interscience Publication, 1983).
- G. Mie, Ann. Phys. 25, 377 (1908). [CrossRef]
- I. M. Vellekoop, and A. P. Mosk, "Focusing coherent light through opaque strongly scattering media," Opt. Lett. 32, 2309-2311 (2007). [CrossRef] [PubMed]

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