## Size and shape determination of spheroidal scatterers using two-dimensional angle resolved scattering |

Optics Express, Vol. 18, Issue 14, pp. 14616-14626 (2010)

http://dx.doi.org/10.1364/OE.18.014616

Acrobat PDF (3376 KB)

### Abstract

We demonstrate accurate determination of the size and shape of spherical and spheroidal scatterers through inverse analysis of two-dimensional solid-angle and depth resolved backscattered light intensities. Intensity of scattered light is measured over a wide range of solid angles using a novel scanning fiber optic interferometer from both individual and ensembles of scatterers. T-matrix based inverse analysis of these two-dimensional angular measurements yields completely unique size and aspect ratio determinations with subwavelength precision over a large range of possible scatterer geometries.

© 2010 OSA

## 1. Introduction

6. C. Xu, C. Vinegoni, T. S. Ralston, W. Luo, W. Tan, and S. A. Boppart, “Spectroscopic spectral-domain optical coherence microscopy,” Opt. Lett. **31**(8), 1079–1081 (2006). [CrossRef] [PubMed]

7. F. Robles, R. N. Graf, and A. Wax, “Dual window method for processing spectroscopic optical coherence tomography signals with simultaneously high spectral and temporal resolution,” Opt. Express **17**(8), 6799–6812 (2009). [CrossRef] [PubMed]

8. F. E. Robles and A. Wax, “Measuring morphological features using light-scattering spectroscopy and Fourier-domain low-coherence interferometry,” Opt. Lett. **35**(3), 360–362 (2010). [CrossRef] [PubMed]

9. J. W. Pyhtila, R. N. Graf, and A. Wax, “Determining nuclear morphology using an improved angle-resolved low coherence interferometry system,” Opt. Express **11**(25), 3473–3484 (2003). [CrossRef] [PubMed]

9. J. W. Pyhtila, R. N. Graf, and A. Wax, “Determining nuclear morphology using an improved angle-resolved low coherence interferometry system,” Opt. Express **11**(25), 3473–3484 (2003). [CrossRef] [PubMed]

10. Y. Zhu, N. G. Terry, and A. Wax, “Scanning fiber angle-resolved low coherence interferometry,” Opt. Lett. **34**(20), 3196–3198 (2009). [CrossRef] [PubMed]

## 2. Experimental setup

17. Y. Zhu, M. G. Giacomelli, and A. Wax, “Fiber-optic interferometric two-dimensional scattering-measurement system,” Opt. Lett. **35**(10), 1641–1643 (2010). [CrossRef] [PubMed]

_{21}) along (θ, φ) is then collected by Arm 1 by scanning the fiber in the x and y directions over the back focal plane of the lens. The other two signals, R

_{1}and R

_{2}, are generated by the 4% back reflections at the ends of the fiber. For low coherence interferometry, R

_{1}and S

_{21}are used as the reference and sample signals, respectively. R

_{2}simply provides a non-interferometric background intensity. At each point, the spectrometer reads out the interferogram produced by R

_{1}and S

_{21}, which is converted to a depth resolved A-scan via a Fourier transform. The choice of R

_{1}as the reference signal allows complete control over the incident and illumination polarizations, a precondition for two-dimensional measurements. Polarization controllers on the two arms are used to independently set the illumination and collection polarizations.

## 3. T-matrix simulation

### 3.1 T-matrix

20. K. J. Chalut, K. Kulangara, M. G. Giacomelli, A. Wax, and K. W. Leong, “Deformation of stem cell nuclei by nanotopographical cues,” Soft Matter **6**(8), 1675–1681 (2010). [CrossRef] [PubMed]

21. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: A review,” J. Quant. Spectrosc. Radiat. Transf. **55**(5), 535–575 (1996). [CrossRef]

_{11}, S

_{12}, S

_{21}and S

_{22}corresponding to the in-plane and cross-polarized scattered fields for each illumination polarization [Eq. (1)] were stored in a relational database. In the notation of Mishchenko these field quantities are the solutions to Eqs. (24-28) in [22

22. M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. **39**(6), 1026–1031 (2000). [CrossRef]

_{11}for a 10 μm spherical scatterer is presented in Fig. 2(a) . To build the database, 10,250 different scatterers were simulated, with diameters ranging from 8 μm to 18 μm in 80 nm increments, and aspect ratios between 0.7 (prolate spheroidal) and 1.1 (oblate spheroidal) in steps of 0.005. Refractive index was fixed at 1.58 for polystyrene scatterers, while 1.41 was used for the PDMS media.

### 3.2 Lens transformation and coordinate systems

^{I}represents the incident field components and E with no superscript represents the scattered electric field in the image plane with the first subscript referring to the illumination polarization while the second to the collected polarization. The resulting polarized field values were then resampled evenly in polar space [Fig. 2(b)]. We hereafter use these two-letter combinations to refer to each of these field components. PP and SS are co-polarization scattering signals along the y and x axis respectively while PS and SP are the cross-polarization signals.

### 3.3 Size distributions and database parameters

23. A. Wax, “Low-coherence light-scattering calculations for polydisperse size distributions,” J. Opt. Soc. Am. A **22**(2), 256–261 (2005). [CrossRef]

## 4. 2D angular scattering results

### 4.1 Comparison of simulated and experimental results

24. J. W. Pyhtila, H. Ma, A. J. Simnick, A. Chilkoti, and A. Wax, “Analysis of long range correlations due to coherent light scattering from in-vitro cell arrays using angle-resolved low coherence interferometry,” J. Biomed. Opt. **11**(3), 034022 (2006). [CrossRef]

## 5. Inverse analysis of spheroids

### 5.1 Image processing, registration and determination of angular range

### 5.2 Chi squared fitting

^{2}error:between the measured field and each simulated field in the database. The process involves three distinct steps which have been adapted from analysis of 1D a/LCI data [9

9. J. W. Pyhtila, R. N. Graf, and A. Wax, “Determining nuclear morphology using an improved angle-resolved low coherence interferometry system,” Opt. Express **11**(25), 3473–3484 (2003). [CrossRef] [PubMed]

- (1) The registered experimental data is transformed onto the simulated angular space using a projective transform and the edges of the GRIN lens are masked off to avoid spurious signals.
- (2) The experimental data is low pass filtered to remove high frequency oscillations that result from coherent interference of adjacent beads using a procedure developed previously [24]. While these oscillations carry information about the spatial distribution of scatterers within the illumination beam, they are removed to isolate the component of scattering due to individual scatterers.
24. J. W. Pyhtila, H. Ma, A. J. Simnick, A. Chilkoti, and A. Wax, “Analysis of long range correlations due to coherent light scattering from in-vitro cell arrays using angle-resolved low coherence interferometry,” J. Biomed. Opt.

**11**(3), 034022 (2006). [CrossRef] - (3) An angle-by-angle χ
^{2}error value [Eq. (2)] is computed between the experimental data and each simulated field. The lowest error value is determined and the corresponding ‘best fit’ yields the scatterer structure.

## 6. Discussion

### 6.1 *Solid angle resolved scattering*

*a priori*knowledge to narrow the range of expected scatterer geometries in order to reduce multiple inverse solutions to one, or at most a few, possible answers. In the absence of such knowledge, these techniques can misinterpret the analysis results by providing numerically optimal, but physically incorrect solutions. Given these limitations, light scattering methods have typically been focused on distinguishing a few possible tissue states over a limited range of geometries which are usually spherical.

## 7. Conclusion

## Acknowledgments

## References and links

1. | A. Dhar, K. S. Johnson, M. R. Novelli, S. G. Bown, I. J. Bigio, L. B. Lovat, and S. L. Bloom, “Elastic scattering spectroscopy for the diagnosis of colonic lesions: initial results of a novel optical biopsy technique,” Gastrointest. Endosc. |

2. | L. B. Lovat, K. Johnson, G. D. Mackenzie, B. R. Clark, M. R. Novelli, S. Davies, M. O’Donovan, C. Selvasekar, S. M. Thorpe, D. Pickard, R. Fitzgerald, T. Fearn, I. Bigio, and S. G. Bown, “Elastic scattering spectroscopy accurately detects high grade dysplasia and cancer in Barrett’s oesophagus,” Gut |

3. | V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, “Measuring cellular structure at submicrometer scale with light scattering spectroscopy,” IEEE J. Sel. Top. Quantum Electron. |

4. | M. S. Feld, V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, and J. Van Dam, “Detection of preinvasive cancer cells,” Nature |

5. | I. Itzkan, L. Qiu, H. Fang, M. M. Zaman, E. Vitkin, I. C. Ghiran, S. Salahuddin, M. Modell, C. Andersson, L. M. Kimerer, P. B. Cipolloni, K.-H. Lim, S. D. Freedman, I. Bigio, B. P. Sachs, E. B. Hanlon, and L. T. Perelman, “Confocal light absorption and scattering spectroscopic microscopy monitors organelles in live cells with no exogenous labels,” Proc. Natl. Acad. Sci. U.S.A. |

6. | C. Xu, C. Vinegoni, T. S. Ralston, W. Luo, W. Tan, and S. A. Boppart, “Spectroscopic spectral-domain optical coherence microscopy,” Opt. Lett. |

7. | F. Robles, R. N. Graf, and A. Wax, “Dual window method for processing spectroscopic optical coherence tomography signals with simultaneously high spectral and temporal resolution,” Opt. Express |

8. | F. E. Robles and A. Wax, “Measuring morphological features using light-scattering spectroscopy and Fourier-domain low-coherence interferometry,” Opt. Lett. |

9. | J. W. Pyhtila, R. N. Graf, and A. Wax, “Determining nuclear morphology using an improved angle-resolved low coherence interferometry system,” Opt. Express |

10. | Y. Zhu, N. G. Terry, and A. Wax, “Scanning fiber angle-resolved low coherence interferometry,” Opt. Lett. |

11. | M. G. Giacomelli, K. J. Chalut, J. H. Ostrander, and A. Wax, “Review of the Application of T-Matrix Calculations for Determining the Structure of Cell Nuclei With Angle-Resolved Light Scattering Measurements,” IEEE J. Sel. Top. Quantum Electron. |

12. | A. M. K. Nilsson, P. Alsholm, A. Karlsson, and S. Andersson-Engels, “T-matrix computations of light scattering by red blood cells,” Appl. Opt. |

13. | D. D. Duncan and M. E. Thomas, “Particle shape as revealed by spectral depolarization,” Appl. Opt. |

14. | J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, “Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures,” J. Biomed. Opt. |

15. | J. Ramachandran, T. M. Powers, S. Carpenter, A. Garcia-Lopez, J. P. Freyer, and J. R. Mourant, “Light scattering and microarchitectural differences between tumorigenic and non-tumorigenic cell models of tissue,” Opt. Express |

16. | Z. J. Smith and A. J. Berger, “Validation of an integrated Raman- and angular-scattering microscopy system on heterogeneous bead mixtures and single human immune cells,” Appl. Opt. |

17. | Y. Zhu, M. G. Giacomelli, and A. Wax, “Fiber-optic interferometric two-dimensional scattering-measurement system,” Opt. Lett. |

18. | M. I. Mishchenko, L. D. Travis, and J. W. Hovenier, |

19. | K. J. Chalut, M. G. Giacomelli, and A. Wax, “Application of Mie theory to assess structure of spheroidal scattering in backscattering geometries,” J. Opt. Soc. Am. A |

20. | K. J. Chalut, K. Kulangara, M. G. Giacomelli, A. Wax, and K. W. Leong, “Deformation of stem cell nuclei by nanotopographical cues,” Soft Matter |

21. | M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: A review,” J. Quant. Spectrosc. Radiat. Transf. |

22. | M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. |

23. | A. Wax, “Low-coherence light-scattering calculations for polydisperse size distributions,” J. Opt. Soc. Am. A |

24. | J. W. Pyhtila, H. Ma, A. J. Simnick, A. Chilkoti, and A. Wax, “Analysis of long range correlations due to coherent light scattering from in-vitro cell arrays using angle-resolved low coherence interferometry,” J. Biomed. Opt. |

25. | C. Amoozegar, M. G. Giacomelli, J. D. Keener, K. J. Chalut, and A. Wax, “Experimental verification of T-matrix-based inverse light scattering analysis for assessing structure of spheroids as models of cell nuclei,” Appl. Opt. |

**OCIS Codes**

(290.0290) Scattering : Scattering

(290.3200) Scattering : Inverse scattering

(290.5855) Scattering : Scattering, polarization

**ToC Category:**

Scattering

**History**

Original Manuscript: May 3, 2010

Revised Manuscript: June 20, 2010

Manuscript Accepted: June 21, 2010

Published: June 23, 2010

**Citation**

Michael Giacomelli, Yizheng Zhu, John Lee, and Adam Wax, "Size and shape determination of spheroidal scatterers using two-dimensional angle resolved scattering," Opt. Express **18**, 14616-14626 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14616

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### References

- A. Dhar, K. S. Johnson, M. R. Novelli, S. G. Bown, I. J. Bigio, L. B. Lovat, and S. L. Bloom, “Elastic scattering spectroscopy for the diagnosis of colonic lesions: initial results of a novel optical biopsy technique,” Gastrointest. Endosc. 63(2), 257–261 (2006). [CrossRef] [PubMed]
- L. B. Lovat, K. Johnson, G. D. Mackenzie, B. R. Clark, M. R. Novelli, S. Davies, M. O’Donovan, C. Selvasekar, S. M. Thorpe, D. Pickard, R. Fitzgerald, T. Fearn, I. Bigio, and S. G. Bown, “Elastic scattering spectroscopy accurately detects high grade dysplasia and cancer in Barrett’s oesophagus,” Gut 55(8), 1078–1083 (2005). [CrossRef]
- V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, “Measuring cellular structure at submicrometer scale with light scattering spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 7(6), 887–893 (2001). [CrossRef]
- M. S. Feld, V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, and J. Van Dam, “Detection of preinvasive cancer cells,” Nature 406(6791), 35–36 (2000). [CrossRef] [PubMed]
- I. Itzkan, L. Qiu, H. Fang, M. M. Zaman, E. Vitkin, I. C. Ghiran, S. Salahuddin, M. Modell, C. Andersson, L. M. Kimerer, P. B. Cipolloni, K.-H. Lim, S. D. Freedman, I. Bigio, B. P. Sachs, E. B. Hanlon, and L. T. Perelman, “Confocal light absorption and scattering spectroscopic microscopy monitors organelles in live cells with no exogenous labels,” Proc. Natl. Acad. Sci. U.S.A. 104(44), 17255–17260 (2007). [CrossRef] [PubMed]
- C. Xu, C. Vinegoni, T. S. Ralston, W. Luo, W. Tan, and S. A. Boppart, “Spectroscopic spectral-domain optical coherence microscopy,” Opt. Lett. 31(8), 1079–1081 (2006). [CrossRef] [PubMed]
- F. Robles, R. N. Graf, and A. Wax, “Dual window method for processing spectroscopic optical coherence tomography signals with simultaneously high spectral and temporal resolution,” Opt. Express 17(8), 6799–6812 (2009). [CrossRef] [PubMed]
- F. E. Robles and A. Wax, “Measuring morphological features using light-scattering spectroscopy and Fourier-domain low-coherence interferometry,” Opt. Lett. 35(3), 360–362 (2010). [CrossRef] [PubMed]
- J. W. Pyhtila, R. N. Graf, and A. Wax, “Determining nuclear morphology using an improved angle-resolved low coherence interferometry system,” Opt. Express 11(25), 3473–3484 (2003). [CrossRef] [PubMed]
- Y. Zhu, N. G. Terry, and A. Wax, “Scanning fiber angle-resolved low coherence interferometry,” Opt. Lett. 34(20), 3196–3198 (2009). [CrossRef] [PubMed]
- M. G. Giacomelli, K. J. Chalut, J. H. Ostrander, and A. Wax, “Review of the Application of T-Matrix Calculations for Determining the Structure of Cell Nuclei With Angle-Resolved Light Scattering Measurements,” IEEE J. Sel. Top. Quantum Electron. PP(99), 1–9 (2009).
- A. M. K. Nilsson, P. Alsholm, A. Karlsson, and S. Andersson-Engels, “T-matrix computations of light scattering by red blood cells,” Appl. Opt. 37(13), 2735–2748 (1998). [CrossRef]
- D. D. Duncan and M. E. Thomas, “Particle shape as revealed by spectral depolarization,” Appl. Opt. 46(24), 6185–6191 (2007). [CrossRef] [PubMed]
- J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, “Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures,” J. Biomed. Opt. 7(3), 378–387 (2002). [CrossRef] [PubMed]
- J. Ramachandran, T. M. Powers, S. Carpenter, A. Garcia-Lopez, J. P. Freyer, and J. R. Mourant, “Light scattering and microarchitectural differences between tumorigenic and non-tumorigenic cell models of tissue,” Opt. Express 15(7), 4039–4053 (2007). [CrossRef] [PubMed]
- Z. J. Smith and A. J. Berger, “Validation of an integrated Raman- and angular-scattering microscopy system on heterogeneous bead mixtures and single human immune cells,” Appl. Opt. 48(10), D109–D120 (2009). [CrossRef] [PubMed]
- Y. Zhu, M. G. Giacomelli, and A. Wax, “Fiber-optic interferometric two-dimensional scattering-measurement system,” Opt. Lett. 35(10), 1641–1643 (2010). [CrossRef] [PubMed]
- M. I. Mishchenko, L. D. Travis, and J. W. Hovenier, Light scattering by nonspherical particles: theory, measurements and applications (Academic, San Diego; London, 2000).
- K. J. Chalut, M. G. Giacomelli, and A. Wax, “Application of Mie theory to assess structure of spheroidal scattering in backscattering geometries,” J. Opt. Soc. Am. A 25(8), 1866–1874 (2008).
- K. J. Chalut, K. Kulangara, M. G. Giacomelli, A. Wax, and K. W. Leong, “Deformation of stem cell nuclei by nanotopographical cues,” Soft Matter 6(8), 1675–1681 (2010). [CrossRef] [PubMed]
- M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: A review,” J. Quant. Spectrosc. Radiat. Transf. 55(5), 535–575 (1996). [CrossRef]
- M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. 39(6), 1026–1031 (2000). [CrossRef]
- A. Wax, “Low-coherence light-scattering calculations for polydisperse size distributions,” J. Opt. Soc. Am. A 22(2), 256–261 (2005). [CrossRef]
- J. W. Pyhtila, H. Ma, A. J. Simnick, A. Chilkoti, and A. Wax, “Analysis of long range correlations due to coherent light scattering from in-vitro cell arrays using angle-resolved low coherence interferometry,” J. Biomed. Opt. 11(3), 034022 (2006). [CrossRef]
- C. Amoozegar, M. G. Giacomelli, J. D. Keener, K. J. Chalut, and A. Wax, “Experimental verification of T-matrix-based inverse light scattering analysis for assessing structure of spheroids as models of cell nuclei,” Appl. Opt. 48(10), D20–D25 (2009). [CrossRef] [PubMed]

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