Abstract

We develop a computational framework to examine the factors responsible for scattering-induced distortions of coherent anti-Stokes Raman scattering (CARS) signals in turbid samples. We apply the Huygens-Fresnel wave-based electric field superposition (HF-WEFS) method combined with the radiating dipole approximation to compute the effects of scattering-induced distortions of focal excitation fields on the far-field CARS signal. We analyze the effect of spherical scatterers, placed in the vicinity of the focal volume, on the CARS signal emitted by different objects (2μm diameter solid sphere, 2μm diameter myelin cylinder and 2μm diameter myelin tube). We find that distortions in the CARS signals arise not only from attenuation of the focal field but also from scattering-induced changes in the spatial phase that modifies the angular distribution of the CARS emission. Our simulations further show that CARS signal attenuation can be minimized by using a high numerical aperture condenser. Moreover, unlike the CARS intensity image, CARS images formed by taking the ratio of CARS signals obtained using x- and y-polarized input fields is relatively insensitive to the effects of spherical scatterers. Our computational framework provide a mechanistic approach to characterizing scattering-induced distortions in coherent imaging of turbid media and may inspire bottom-up approaches for adaptive optical methods for image correction.

© 2017 Optical Society of America

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References

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2017 (1)

I. N. Papadopoulos, J.-S Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

2016 (3)

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

H. B. de Aguiar, S. Gigan, and S. Brasselet, “Enhanced nonlinear imaging through scattering media using transmission-matrix-based wave-front shaping,” Phys. Rev. A 94, 043830 (2016).
[Crossref]

J. van der Kolk, A. Lesina, and L. Ramunno, “Effects of refractive index mismatch on SRS and CARS microscopy,” Opt. Express 24(22), 25752–25766 (2016).
[Crossref] [PubMed]

2015 (2)

H. B. de Aguiar, P. Gasecka, and S. Brasselet, “Quantitative analysis of light scattering in polarization-resolved nonlinear microscopy,” Opt. Express 23(7), 8960–8973 (2015).
[Crossref] [PubMed]

C. Zhang, D. Zhang, and J.-X. Cheng, “Coherent Raman scattering microscopy in biology and medicine,” Ann. Rev. Biomed. Eng. 17, 415–445 (2015).
[Crossref]

2014 (4)

2013 (3)

A. M. Barlow, K. Popov, M. Andreana, D. J. Moffatt, A. Ridsdale, A. D. Slepkov, J. L. Harden, L. Ramunno, and A. Stolow, “Spatial-spectral coupling in coherent anti-Stokes Raman scattering microscopy,” Opt. Express,  21(13), 15298–15307 (2013).
[Crossref] [PubMed]

C.-Y. Chung, J. Boik, and E. O. Potma, “Biomolecular imaging with coherent nonlinear vibrational microscopy,” Ann. Rev. Phys. Chem. 64, 77–99 (2013).
[Crossref]

C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. Biomed. Optics 18(5) 050902 (2013).
[Crossref]

2012 (3)

2011 (4)

Y. Fu, T. J. Frederick, T. B. Hu, G. E. Goings, S. D. Miller, and J.-X. Cheng, “Paranodal myelin retraction in relapsing experimental autoimmune encephalomyelitis visualized by coherent anti-Stokes Raman scattering microscopy,” J. Biomed. Opt. 16(10), 106006 (2011).
[Crossref] [PubMed]

J. Imitola, D. Cote, S. Rasmussen, X. S. Xie, Y. Liu, T. Chitnis, R. L. Sidman, C. P. Lin, and S. J. Khoury, “Multi-modal coherent anti-Stokes Raman scattering microscopy reveals microglia-associated myelin and axonal dysfunction in multiple sclerosis-like lesions in mice,” J. Biomed. Opt. 16, 021109 (2011).
[Crossref]

Y. Shi, D. Zhang, T. B. Hu, X. Wang, R. Shi, X.-M. Xu, and J.-X. Cheng, “Longitudinal in vivo coherent anti-Stokes Raman scattering imaging of demyelination and remyelination in injured spinal cord,” J. Biomed. Opt. 16, 106012 (2011).
[Crossref] [PubMed]

C. G. Koay, “A simple scheme for generating nearly uniform distribution of antipodally symmetric points on the unit sphere,” J. Comp. Sci. 2(4), 377–381 (2011).
[Crossref]

2010 (2)

2009 (3)

2008 (1)

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[Crossref] [PubMed]

2007 (3)

2006 (1)

2005 (2)

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[Crossref] [PubMed]

H. Wang, Y. Fu, P. Zickmund, R. Shi, and J.-X. Cheng, “Coherent anti-stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys J. 89, 581–591 (2005).
[Crossref] [PubMed]

2002 (1)

1987 (1)

T. L. Mazely and W. M. Hetherington, “Third-order susceptibility tensors of partially ordered systems,” J. Chem. Phys. 87(4), 1962–1966 (1987).
[Crossref]

1965 (1)

P. Maker and R. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137(3A) A801–A818 (1965).
[Crossref]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proceedings of the Royal Society of London A 253(1274), 358–379 (1959).
[Crossref]

Andreana, M.

Avezza, F.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

Barlow, A. M.

Bégin, S.

Bélanger, E.

Bifone, A.

Boik, J.

C.-Y. Chung, J. Boik, and E. O. Potma, “Biomolecular imaging with coherent nonlinear vibrational microscopy,” Ann. Rev. Phys. Chem. 64, 77–99 (2013).
[Crossref]

Bossi, M.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

Brasselet, S.

H. B. de Aguiar, S. Gigan, and S. Brasselet, “Enhanced nonlinear imaging through scattering media using transmission-matrix-based wave-front shaping,” Phys. Rev. A 94, 043830 (2016).
[Crossref]

H. B. de Aguiar, P. Gasecka, and S. Brasselet, “Quantitative analysis of light scattering in polarization-resolved nonlinear microscopy,” Opt. Express 23(7), 8960–8973 (2015).
[Crossref] [PubMed]

Canta, A.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

Cappello, V.

G. de Vito, V. Cappello, I. Tonazzini, M. Cecchini, and V. Piazza, “RP-CARS reveals molecular spatial order anomalies in myelin of an animal model of Krabbe disease,” J. Biophoton., 1–9 (2016).

Carozzi, V.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

Cavaletti, G.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

Cecchini, M.

G. de Vito, I. Tonazzini, M. Cecchini, and V. Piazza, “RP-CARS: label-free optical readout of the myelin intrinsic healthiness,” Opt. Express 22(11), 13733–13743 (2014).
[Crossref] [PubMed]

G. de Vito, V. Cappello, I. Tonazzini, M. Cecchini, and V. Piazza, “RP-CARS reveals molecular spatial order anomalies in myelin of an animal model of Krabbe disease,” J. Biophoton., 1–9 (2016).

Cheng, J.-X.

C. Zhang, D. Zhang, and J.-X. Cheng, “Coherent Raman scattering microscopy in biology and medicine,” Ann. Rev. Biomed. Eng. 17, 415–445 (2015).
[Crossref]

Y. Shi, D. Zhang, T. B. Hu, X. Wang, R. Shi, X.-M. Xu, and J.-X. Cheng, “Longitudinal in vivo coherent anti-Stokes Raman scattering imaging of demyelination and remyelination in injured spinal cord,” J. Biomed. Opt. 16, 106012 (2011).
[Crossref] [PubMed]

Y. Fu, T. J. Frederick, T. B. Hu, G. E. Goings, S. D. Miller, and J.-X. Cheng, “Paranodal myelin retraction in relapsing experimental autoimmune encephalomyelitis visualized by coherent anti-Stokes Raman scattering microscopy,” J. Biomed. Opt. 16(10), 106006 (2011).
[Crossref] [PubMed]

H. Wang, Y. Fu, P. Zickmund, R. Shi, and J.-X. Cheng, “Coherent anti-stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys J. 89, 581–591 (2005).
[Crossref] [PubMed]

J.-X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19(6), 1363–1375 (2002).
[Crossref]

J.-X. Cheng and X. S. Xie, Coherent Raman Scattering Microscopy (CRC Press, 2013).

Chiorazzi, A.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

Chitnis, T.

J. Imitola, D. Cote, S. Rasmussen, X. S. Xie, Y. Liu, T. Chitnis, R. L. Sidman, C. P. Lin, and S. J. Khoury, “Multi-modal coherent anti-Stokes Raman scattering microscopy reveals microglia-associated myelin and axonal dysfunction in multiple sclerosis-like lesions in mice,” J. Biomed. Opt. 16, 021109 (2011).
[Crossref]

Chung, C.-Y.

C.-Y. Chung, J. Boik, and E. O. Potma, “Biomolecular imaging with coherent nonlinear vibrational microscopy,” Ann. Rev. Phys. Chem. 64, 77–99 (2013).
[Crossref]

Cote, D.

J. Imitola, D. Cote, S. Rasmussen, X. S. Xie, Y. Liu, T. Chitnis, R. L. Sidman, C. P. Lin, and S. J. Khoury, “Multi-modal coherent anti-Stokes Raman scattering microscopy reveals microglia-associated myelin and axonal dysfunction in multiple sclerosis-like lesions in mice,” J. Biomed. Opt. 16, 021109 (2011).
[Crossref]

Côté, D.

Crippa, L.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

Davis, M. A.

de Aguiar, H. B.

H. B. de Aguiar, S. Gigan, and S. Brasselet, “Enhanced nonlinear imaging through scattering media using transmission-matrix-based wave-front shaping,” Phys. Rev. A 94, 043830 (2016).
[Crossref]

H. B. de Aguiar, P. Gasecka, and S. Brasselet, “Quantitative analysis of light scattering in polarization-resolved nonlinear microscopy,” Opt. Express 23(7), 8960–8973 (2015).
[Crossref] [PubMed]

de Vito, G.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

G. de Vito, I. Tonazzini, M. Cecchini, and V. Piazza, “RP-CARS: label-free optical readout of the myelin intrinsic healthiness,” Opt. Express 22(11), 13733–13743 (2014).
[Crossref] [PubMed]

G. de Vito, A. Bifone, and V. Piazza, “Rotating-polarization CARS microscopy: combining chemical and molecular orientation sensitivity,” Opt. Express 20(28), 29369–29377 (2012).
[Crossref]

G. de Vito, V. Cappello, I. Tonazzini, M. Cecchini, and V. Piazza, “RP-CARS reveals molecular spatial order anomalies in myelin of an animal model of Krabbe disease,” J. Biophoton., 1–9 (2016).

Denk, W.

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[Crossref] [PubMed]

Djaker, N.

Duncan, D. D.

S. A. Prahl, D. D. Duncan, and D. G. Fischer, “Monte Carlo propagation of spatial coherence,” Proc. SPIE,  718771870G (2010).
[Crossref]

Dunn, A. K.

Evans, C. L.

Feld, M. S.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[Crossref] [PubMed]

Fink, M.

A. P. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[Crossref]

Fischer, D. G.

S. A. Prahl, D. D. Duncan, and D. G. Fischer, “Monte Carlo propagation of spatial coherence,” Proc. SPIE,  718771870G (2010).
[Crossref]

Frederick, T. J.

Y. Fu, T. J. Frederick, T. B. Hu, G. E. Goings, S. D. Miller, and J.-X. Cheng, “Paranodal myelin retraction in relapsing experimental autoimmune encephalomyelitis visualized by coherent anti-Stokes Raman scattering microscopy,” J. Biomed. Opt. 16(10), 106006 (2011).
[Crossref] [PubMed]

Freudiger, C. W.

Fu, Y.

Y. Fu, T. J. Frederick, T. B. Hu, G. E. Goings, S. D. Miller, and J.-X. Cheng, “Paranodal myelin retraction in relapsing experimental autoimmune encephalomyelitis visualized by coherent anti-Stokes Raman scattering microscopy,” J. Biomed. Opt. 16(10), 106006 (2011).
[Crossref] [PubMed]

H. Wang, Y. Fu, P. Zickmund, R. Shi, and J.-X. Cheng, “Coherent anti-stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys J. 89, 581–591 (2005).
[Crossref] [PubMed]

Gachet, D.

Gasecka, P.

Gigan, S.

H. B. de Aguiar, S. Gigan, and S. Brasselet, “Enhanced nonlinear imaging through scattering media using transmission-matrix-based wave-front shaping,” Phys. Rev. A 94, 043830 (2016).
[Crossref]

Girkin, J. M.

Goings, G. E.

Y. Fu, T. J. Frederick, T. B. Hu, G. E. Goings, S. D. Miller, and J.-X. Cheng, “Paranodal myelin retraction in relapsing experimental autoimmune encephalomyelitis visualized by coherent anti-Stokes Raman scattering microscopy,” J. Biomed. Opt. 16(10), 106006 (2011).
[Crossref] [PubMed]

Guan, Y.

Harden, J. L.

Hayakawa, C. K.

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

Helmchen, F.

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[Crossref] [PubMed]

Hetherington, W. M.

T. L. Mazely and W. M. Hetherington, “Third-order susceptibility tensors of partially ordered systems,” J. Chem. Phys. 87(4), 1962–1966 (1987).
[Crossref]

Hokr, B. H.

B. H. Hokr, V. V. Yakovlev, and M. O. Scully, “Efficient time-dependent Monte Carlo simulations of stimulated Raman scattering in a turbid medium,” ACS Photonics 1(12), 1322–1329 (2014).
[Crossref]

Hu, T. B.

Y. Shi, D. Zhang, T. B. Hu, X. Wang, R. Shi, X.-M. Xu, and J.-X. Cheng, “Longitudinal in vivo coherent anti-Stokes Raman scattering imaging of demyelination and remyelination in injured spinal cord,” J. Biomed. Opt. 16, 106012 (2011).
[Crossref] [PubMed]

Y. Fu, T. J. Frederick, T. B. Hu, G. E. Goings, S. D. Miller, and J.-X. Cheng, “Paranodal myelin retraction in relapsing experimental autoimmune encephalomyelitis visualized by coherent anti-Stokes Raman scattering microscopy,” J. Biomed. Opt. 16(10), 106006 (2011).
[Crossref] [PubMed]

Huang, Z.

Imitola, J.

J. Imitola, D. Cote, S. Rasmussen, X. S. Xie, Y. Liu, T. Chitnis, R. L. Sidman, C. P. Lin, and S. J. Khoury, “Multi-modal coherent anti-Stokes Raman scattering microscopy reveals microglia-associated myelin and axonal dysfunction in multiple sclerosis-like lesions in mice,” J. Biomed. Opt. 16, 021109 (2011).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (John Wiley and Sons, Inc., 1975).

Jouhanneau, J.-S

I. N. Papadopoulos, J.-S Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

Judkewitz, B.

I. N. Papadopoulos, J.-S Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

Katz, O.

Khoury, S. J.

J. Imitola, D. Cote, S. Rasmussen, X. S. Xie, Y. Liu, T. Chitnis, R. L. Sidman, C. P. Lin, and S. J. Khoury, “Multi-modal coherent anti-Stokes Raman scattering microscopy reveals microglia-associated myelin and axonal dysfunction in multiple sclerosis-like lesions in mice,” J. Biomed. Opt. 16, 021109 (2011).
[Crossref]

Koay, C. G.

C. G. Koay, “A simple scheme for generating nearly uniform distribution of antipodally symmetric points on the unit sphere,” J. Comp. Sci. 2(4), 377–381 (2011).
[Crossref]

Koninck, Y. De

Krishnamachari, V. V.

Laffray, S.

Lagendijk, A.

A. P. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[Crossref]

Lenne, P. F.

Lerosey, G.

A. P. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[Crossref]

Lesina, A.

Lin, C. P.

J. Imitola, D. Cote, S. Rasmussen, X. S. Xie, Y. Liu, T. Chitnis, R. L. Sidman, C. P. Lin, and S. J. Khoury, “Multi-modal coherent anti-Stokes Raman scattering microscopy reveals microglia-associated myelin and axonal dysfunction in multiple sclerosis-like lesions in mice,” J. Biomed. Opt. 16, 021109 (2011).
[Crossref]

Lin, J.

Liu, Q.

C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. Biomed. Optics 18(5) 050902 (2013).
[Crossref]

Liu, Y.

J. Imitola, D. Cote, S. Rasmussen, X. S. Xie, Y. Liu, T. Chitnis, R. L. Sidman, C. P. Lin, and S. J. Khoury, “Multi-modal coherent anti-Stokes Raman scattering microscopy reveals microglia-associated myelin and axonal dysfunction in multiple sclerosis-like lesions in mice,” J. Biomed. Opt. 16, 021109 (2011).
[Crossref]

Lombardi, R.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

Lu, F.

Maker, P.

P. Maker and R. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137(3A) A801–A818 (1965).
[Crossref]

Marmiroli, P.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

Mazely, T. L.

T. L. Mazely and W. M. Hetherington, “Third-order susceptibility tensors of partially ordered systems,” J. Chem. Phys. 87(4), 1962–1966 (1987).
[Crossref]

Meregalli, C.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

Miller, S. D.

Y. Fu, T. J. Frederick, T. B. Hu, G. E. Goings, S. D. Miller, and J.-X. Cheng, “Paranodal myelin retraction in relapsing experimental autoimmune encephalomyelitis visualized by coherent anti-Stokes Raman scattering microscopy,” J. Biomed. Opt. 16(10), 106006 (2011).
[Crossref] [PubMed]

Moffatt, D. J.

Mosk, A. P.

Mosk, A. P. P.

A. P. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

Oggioni, N.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

Papadopoulos, I. N.

I. N. Papadopoulos, J.-S Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

Pegoraro, A. F.

Piazza, V.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

G. de Vito, I. Tonazzini, M. Cecchini, and V. Piazza, “RP-CARS: label-free optical readout of the myelin intrinsic healthiness,” Opt. Express 22(11), 13733–13743 (2014).
[Crossref] [PubMed]

G. de Vito, A. Bifone, and V. Piazza, “Rotating-polarization CARS microscopy: combining chemical and molecular orientation sensitivity,” Opt. Express 20(28), 29369–29377 (2012).
[Crossref]

G. de Vito, V. Cappello, I. Tonazzini, M. Cecchini, and V. Piazza, “RP-CARS reveals molecular spatial order anomalies in myelin of an animal model of Krabbe disease,” J. Biophoton., 1–9 (2016).

Poland, S. P.

Popov, K.

Popov, K. I.

Potma, E. O.

Poulet, J. F. A.

I. N. Papadopoulos, J.-S Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

Prahl, S. A.

S. A. Prahl, D. D. Duncan, and D. G. Fischer, “Monte Carlo propagation of spatial coherence,” Proc. SPIE,  718771870G (2010).
[Crossref]

Psaltis, D.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[Crossref] [PubMed]

Ramunno, L.

Ranasinghesagara, J. C.

Rasmussen, S.

J. Imitola, D. Cote, S. Rasmussen, X. S. Xie, Y. Liu, T. Chitnis, R. L. Sidman, C. P. Lin, and S. J. Khoury, “Multi-modal coherent anti-Stokes Raman scattering microscopy reveals microglia-associated myelin and axonal dysfunction in multiple sclerosis-like lesions in mice,” J. Biomed. Opt. 16, 021109 (2011).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proceedings of the Royal Society of London A 253(1274), 358–379 (1959).
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Ridsdale, A.

Rigneault, H.

Rodriguez-Menendez, V.

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

Sandeau, N.

Scully, M. O.

B. H. Hokr, V. V. Yakovlev, and M. O. Scully, “Efficient time-dependent Monte Carlo simulations of stimulated Raman scattering in a turbid medium,” ACS Photonics 1(12), 1322–1329 (2014).
[Crossref]

Sheppard, C.

Shi, R.

Y. Shi, D. Zhang, T. B. Hu, X. Wang, R. Shi, X.-M. Xu, and J.-X. Cheng, “Longitudinal in vivo coherent anti-Stokes Raman scattering imaging of demyelination and remyelination in injured spinal cord,” J. Biomed. Opt. 16, 106012 (2011).
[Crossref] [PubMed]

H. Wang, Y. Fu, P. Zickmund, R. Shi, and J.-X. Cheng, “Coherent anti-stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys J. 89, 581–591 (2005).
[Crossref] [PubMed]

Shi, Y.

Y. Shi, D. Zhang, T. B. Hu, X. Wang, R. Shi, X.-M. Xu, and J.-X. Cheng, “Longitudinal in vivo coherent anti-Stokes Raman scattering imaging of demyelination and remyelination in injured spinal cord,” J. Biomed. Opt. 16, 106012 (2011).
[Crossref] [PubMed]

Sidman, R. L.

J. Imitola, D. Cote, S. Rasmussen, X. S. Xie, Y. Liu, T. Chitnis, R. L. Sidman, C. P. Lin, and S. J. Khoury, “Multi-modal coherent anti-Stokes Raman scattering microscopy reveals microglia-associated myelin and axonal dysfunction in multiple sclerosis-like lesions in mice,” J. Biomed. Opt. 16, 021109 (2011).
[Crossref]

Silberberg, Y.

Slepkov, A. D.

Small, E.

Starosta, M. S.

Stolow, A.

Terhune, R.

P. Maker and R. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137(3A) A801–A818 (1965).
[Crossref]

Tonazzini, I.

G. de Vito, I. Tonazzini, M. Cecchini, and V. Piazza, “RP-CARS: label-free optical readout of the myelin intrinsic healthiness,” Opt. Express 22(11), 13733–13743 (2014).
[Crossref] [PubMed]

G. de Vito, V. Cappello, I. Tonazzini, M. Cecchini, and V. Piazza, “RP-CARS reveals molecular spatial order anomalies in myelin of an animal model of Krabbe disease,” J. Biophoton., 1–9 (2016).

Vallée, R.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley and Sons Inc, 1957).

van der Kolk, J.

Vellekoop, I. M.

Venugopalan, V.

Volkmer, A.

Wang, H.

Wang, X.

Y. Shi, D. Zhang, T. B. Hu, X. Wang, R. Shi, X.-M. Xu, and J.-X. Cheng, “Longitudinal in vivo coherent anti-Stokes Raman scattering imaging of demyelination and remyelination in injured spinal cord,” J. Biomed. Opt. 16, 106012 (2011).
[Crossref] [PubMed]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proceedings of the Royal Society of London A 253(1274), 358–379 (1959).
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Wright, A. J.

Xie, X. S.

J. Imitola, D. Cote, S. Rasmussen, X. S. Xie, Y. Liu, T. Chitnis, R. L. Sidman, C. P. Lin, and S. J. Khoury, “Multi-modal coherent anti-Stokes Raman scattering microscopy reveals microglia-associated myelin and axonal dysfunction in multiple sclerosis-like lesions in mice,” J. Biomed. Opt. 16, 021109 (2011).
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A. J. Wright, S. P. Poland, J. M. Girkin, C. W. Freudiger, C. L. Evans, and X. S. Xie, “Adaptive optics for enhanced signal in CARS microscopy,” Opt. Express 15(26), 18209–18219 (2007).
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J.-X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19(6), 1363–1375 (2002).
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Xu, X.-M.

Y. Shi, D. Zhang, T. B. Hu, X. Wang, R. Shi, X.-M. Xu, and J.-X. Cheng, “Longitudinal in vivo coherent anti-Stokes Raman scattering imaging of demyelination and remyelination in injured spinal cord,” J. Biomed. Opt. 16, 106012 (2011).
[Crossref] [PubMed]

Yakovlev, V. V.

B. H. Hokr, V. V. Yakovlev, and M. O. Scully, “Efficient time-dependent Monte Carlo simulations of stimulated Raman scattering in a turbid medium,” ACS Photonics 1(12), 1322–1329 (2014).
[Crossref]

Yang, C.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[Crossref] [PubMed]

Yaqoob, Z.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[Crossref] [PubMed]

Zhang, C.

C. Zhang, D. Zhang, and J.-X. Cheng, “Coherent Raman scattering microscopy in biology and medicine,” Ann. Rev. Biomed. Eng. 17, 415–445 (2015).
[Crossref]

Zhang, D.

C. Zhang, D. Zhang, and J.-X. Cheng, “Coherent Raman scattering microscopy in biology and medicine,” Ann. Rev. Biomed. Eng. 17, 415–445 (2015).
[Crossref]

Y. Shi, D. Zhang, T. B. Hu, X. Wang, R. Shi, X.-M. Xu, and J.-X. Cheng, “Longitudinal in vivo coherent anti-Stokes Raman scattering imaging of demyelination and remyelination in injured spinal cord,” J. Biomed. Opt. 16, 106012 (2011).
[Crossref] [PubMed]

Zheng, W.

Zhu, C.

C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. Biomed. Optics 18(5) 050902 (2013).
[Crossref]

Zickmund, P.

H. Wang, Y. Fu, P. Zickmund, R. Shi, and J.-X. Cheng, “Coherent anti-stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys J. 89, 581–591 (2005).
[Crossref] [PubMed]

ACS Photonics (1)

B. H. Hokr, V. V. Yakovlev, and M. O. Scully, “Efficient time-dependent Monte Carlo simulations of stimulated Raman scattering in a turbid medium,” ACS Photonics 1(12), 1322–1329 (2014).
[Crossref]

Ann. Rev. Biomed. Eng. (1)

C. Zhang, D. Zhang, and J.-X. Cheng, “Coherent Raman scattering microscopy in biology and medicine,” Ann. Rev. Biomed. Eng. 17, 415–445 (2015).
[Crossref]

Ann. Rev. Phys. Chem. (1)

C.-Y. Chung, J. Boik, and E. O. Potma, “Biomolecular imaging with coherent nonlinear vibrational microscopy,” Ann. Rev. Phys. Chem. 64, 77–99 (2013).
[Crossref]

Appl. Opt. (1)

Biophys J. (1)

H. Wang, Y. Fu, P. Zickmund, R. Shi, and J.-X. Cheng, “Coherent anti-stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys J. 89, 581–591 (2005).
[Crossref] [PubMed]

J. Biomed. Opt. (3)

Y. Fu, T. J. Frederick, T. B. Hu, G. E. Goings, S. D. Miller, and J.-X. Cheng, “Paranodal myelin retraction in relapsing experimental autoimmune encephalomyelitis visualized by coherent anti-Stokes Raman scattering microscopy,” J. Biomed. Opt. 16(10), 106006 (2011).
[Crossref] [PubMed]

J. Imitola, D. Cote, S. Rasmussen, X. S. Xie, Y. Liu, T. Chitnis, R. L. Sidman, C. P. Lin, and S. J. Khoury, “Multi-modal coherent anti-Stokes Raman scattering microscopy reveals microglia-associated myelin and axonal dysfunction in multiple sclerosis-like lesions in mice,” J. Biomed. Opt. 16, 021109 (2011).
[Crossref]

Y. Shi, D. Zhang, T. B. Hu, X. Wang, R. Shi, X.-M. Xu, and J.-X. Cheng, “Longitudinal in vivo coherent anti-Stokes Raman scattering imaging of demyelination and remyelination in injured spinal cord,” J. Biomed. Opt. 16, 106012 (2011).
[Crossref] [PubMed]

J. Biomed. Optics (1)

C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. Biomed. Optics 18(5) 050902 (2013).
[Crossref]

J. Chem. Phys. (1)

T. L. Mazely and W. M. Hetherington, “Third-order susceptibility tensors of partially ordered systems,” J. Chem. Phys. 87(4), 1962–1966 (1987).
[Crossref]

J. Comp. Sci. (1)

C. G. Koay, “A simple scheme for generating nearly uniform distribution of antipodally symmetric points on the unit sphere,” J. Comp. Sci. 2(4), 377–381 (2011).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Nat. Methods (1)

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[Crossref] [PubMed]

Nat. Photonics (3)

A. P. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[Crossref]

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[Crossref] [PubMed]

I. N. Papadopoulos, J.-S Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

Neurobio. Aging (1)

A. Canta, A. Chiorazzi, V. Carozzi, C. Meregalli, N. Oggioni, M. Bossi, V. Rodriguez-Menendez, F. Avezza, L. Crippa, R. Lombardi, G. de Vito, V. Piazza, G. Cavaletti, and P. Marmiroli, “Age-related changes in the function and structure of the peripheral sensory pathway in mice,” Neurobio. Aging,  45136–148 (2016).
[Crossref]

Opt. Express (10)

E. Bélanger, S. Bégin, S. Laffray, Y. De Koninck, R. Vallée, and D. Côté, “Quantitative myelin imaging with coherent anti-Stokes Raman scattering microscopy: alleviating the excitation polarization dependence with circularly polarized laser beams,” Opt. Express 17(21), 18419–18432 (2009).
[Crossref]

G. de Vito, I. Tonazzini, M. Cecchini, and V. Piazza, “RP-CARS: label-free optical readout of the myelin intrinsic healthiness,” Opt. Express 22(11), 13733–13743 (2014).
[Crossref] [PubMed]

A. J. Wright, S. P. Poland, J. M. Girkin, C. W. Freudiger, C. L. Evans, and X. S. Xie, “Adaptive optics for enhanced signal in CARS microscopy,” Opt. Express 15(26), 18209–18219 (2007).
[Crossref] [PubMed]

J. Lin, H. Wang, W. Zheng, F. Lu, C. Sheppard, and Z. Huang, “Numerical study of effects of light polarization, scatterer sizes and orientations on near-field coherent anti-Stokes Raman scattering microscopy,” Opt. Express 17(4), 2423–2434 (2009).
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J. Lin, W. Zheng, H. Wang, and Z. Huang, “Effects of scatterers’ sizes on near-field coherent anti-Stokes Raman scattering under tightly focused radially and linearly polarized light excitation,” Opt. Express 18(10), 10888–10895 (2010).
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J. van der Kolk, A. Lesina, and L. Ramunno, “Effects of refractive index mismatch on SRS and CARS microscopy,” Opt. Express 24(22), 25752–25766 (2016).
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M. S. Starosta and A. K. Dunn, “Three-dimensional computation of focused beam propagation through multiple biological cells,” Opt. Express 17(15), 12455–12469 (2009).
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H. B. de Aguiar, P. Gasecka, and S. Brasselet, “Quantitative analysis of light scattering in polarization-resolved nonlinear microscopy,” Opt. Express 23(7), 8960–8973 (2015).
[Crossref] [PubMed]

G. de Vito, A. Bifone, and V. Piazza, “Rotating-polarization CARS microscopy: combining chemical and molecular orientation sensitivity,” Opt. Express 20(28), 29369–29377 (2012).
[Crossref]

A. M. Barlow, K. Popov, M. Andreana, D. J. Moffatt, A. Ridsdale, A. D. Slepkov, J. L. Harden, L. Ramunno, and A. Stolow, “Spatial-spectral coupling in coherent anti-Stokes Raman scattering microscopy,” Opt. Express,  21(13), 15298–15307 (2013).
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Opt. Lett. (2)

Optica (1)

Phys. Rev. (1)

P. Maker and R. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137(3A) A801–A818 (1965).
[Crossref]

Phys. Rev. A (1)

H. B. de Aguiar, S. Gigan, and S. Brasselet, “Enhanced nonlinear imaging through scattering media using transmission-matrix-based wave-front shaping,” Phys. Rev. A 94, 043830 (2016).
[Crossref]

Proc. SPIE (1)

S. A. Prahl, D. D. Duncan, and D. G. Fischer, “Monte Carlo propagation of spatial coherence,” Proc. SPIE,  718771870G (2010).
[Crossref]

Proceedings of the Royal Society of London A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proceedings of the Royal Society of London A 253(1274), 358–379 (1959).
[Crossref]

Other (6)

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley and Sons Inc, 1957).

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

J. D. Jackson, Classical Electrodynamics (John Wiley and Sons, Inc., 1975).

G. de Vito, V. Cappello, I. Tonazzini, M. Cecchini, and V. Piazza, “RP-CARS reveals molecular spatial order anomalies in myelin of an animal model of Krabbe disease,” J. Biophoton., 1–9 (2016).

J.-X. Cheng and X. S. Xie, Coherent Raman Scattering Microscopy (CRC Press, 2013).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Illustration of focused beam propagation, CARS signal generation in the focal volume and signal emission. (a) The HF plane waves of pump and Stokes beams propagate separately in a medium with scatterers. (b) The spatially dependent polarization is computed in the focal volume. (c) Dipole radiation and the far-field detection. The lens are geometrically represented by reference spherical surfaces. Numerical aperture of the excitation and detection lenses are NAex and NAdet, respectively.

Fig. 2
Fig. 2

Simulation Setup. (a) A 2 μm diameter spherical scatterer (gold) is placed at different locations of yz grid (x = 0) to obtain its effect on the CARS intensity. (b)–(d) The lens system is scanned in the xy plane while keeping the object (green) and the spherical scatterer stationary. We consider CARS imaging in a (b) non-scattering medium; and in systems containing a spherical scatterer placed at (c) (x, y, z) = (0, 0, −5) μm and (d) (x, y, z) = (0, 1.5, −5) μm.

Fig. 3
Fig. 3

Far-field CARS radiation patterns (from L to R) from a 2 μm diameter solid sphere, 2 μm diameter myelin cylinder, and 2 μm diameter myelin tube (centered and shifted by 0.875 μm left of the optical axis) in a (a,d) non-scattering medium and (b,e) medium with scatterer placed at (x, y, z) = (0, 0, −5)μm. NAex = 0.825 in (a,b,c) and NAex = 0.55 in (d,e,f). Insets to the left of each radiation pattern show yz cross-sections of the amplitude (upper) and phase (lower) of P(3)(r). Insets in rows (c) and (f) show the amplitude (left) and phase (right) differences of (b) and (e) relative to the corresponding non-scattering cases, (a) and (d), respectively. Each inset spans 2μm × 2μm. Each radiation profile was multiplied by the number in the bracket to provide same maximum radiance. The percentages in (b) and (e) indicate the CARS intensity relative to the corresponding non-scattering case. Detection numerical aperture is fixed at NAdet = 0.95.

Fig. 4
Fig. 4

The far-field CARS radiation patterns from a 2 μm diameter solid sphere (blue) located at the focal point in a medium with a single scatterer (gold) placed 5μm below the optical plane at y locations of y = −2, −1, 0, 1, 2 μm as shown. The effect of scatterer size is shown for diameters of (a) 1 μm, (b) 2 μm, (c) 3 μm, and (d) 4 μm. Each radiation profile was multiplied by the number in the bracket to provide same maximum radiance. Insets to the bottom of each radiation pattern show yz cross-sections of amplitude difference (left) and phase difference (right). Amplitude/phase differences are calculated by subtracting amplitude/phase of P(3)(r) induced in a non scattering medium. Excitation and detection numerical apertures are fixed at NAex = 0.825 and NAdet = 0.95.

Fig. 5
Fig. 5

The far-field CARS intensity as a function of the yz particle location grid. The object that is placed at the focus is 2 μm sphere, 2 μm myelin cylinder, and 2 μm myelin tube (centered and shifted by 0.875 μm left of the optical axis). Scatterer diameter is 2 μm. NAdet is (a) 0.95 and (b) 0.55. (c) Intensity ratio after dividing (b) by (a). The center value (white color) of the ratio color bar represents the ratio obtained for the non scattering medium. Excitation numerical aperture is fixed at NAex = 0.825.

Fig. 6
Fig. 6

CARS images (xy scan) of 2 μm sphere, 2 μm myelin cylinder, and 2 μm myelin tube located at the focus for (a) x-polarized incident and (b) y-polarized incident upon the lens in a non-scattering medium. (c) The polarization ratio is calculated by dividing (a) by (b). Size of each image is 4.05μm × 4.05μm. Small arrows (yellow) show the orientation of the input polarization. Excitation and detection numerical apertures are fixed at NAex = 0.825 and NAdet = 0.95.

Fig. 7
Fig. 7

CARS images (xy scan) of 2 μm sphere, 2 μm myelin-type cylinder, and 2 μm myelin tube located at the focus for (a) x-polarized incident and (b) y-polarized incident upon the lens. The results are shown for a spherical scatterer placed at (x, y, z) = (0, 0, −5) μm (left) and (x, y, z) = (0, 1.5, −5) μm (right). Dashed circle (white) shows the location of the scatterer. (c) The polarization ratio is calculated by dividing (a) by (b). Size of each image is 4.05μm × 4.05μm. Small arrows (yellow) show the orientation of the input polarization. Excitation and detection numerical apertures are fixed at NAex = 0.825 and NAdet = 0.95.

Equations (8)

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| E inc ( x , y ) | = E 0 exp [ ( x 2 + y 2 ) / ω 0 2 ] ,
( E E ) = | E inc ( x , y ) | ( cos ϕ sin ϕ sin ϕ cos ϕ ) ( JV ) n inc n ( cos θ ) 1 2 ,
( E unscat E unscat ) = ( E E ) exp ( i k d ) ,
( E scat E scat ) = 1 k r s ( S 2 ( r s , θ s ) 0 0 S 1 ( r s , θ s ) ) ( cos ϕ s sin ϕ s sin ϕ s cos ϕ s ) ( E D E D ) ,
( E x ( r ) E y ( r ) E z ( r ) ) = ( E x unscat ( r ) + E x scat ( r ) E y unscat ( r ) + E y scat ( r ) E z unscat ( r ) + E z scat ( r ) ) ,
P i ( 3 ) ( r ) = j , k , l χ i j k l ( 3 ) ( r ) E p j ( r ) E p k ( r ) E S l * ( r ) ,
E C ( R ; r ) = V e i k C | R r | 4 π | R r | 3 [ ( R r ) × P ( 3 ) ( r ) ] × ( R r )  d V ,
I C θ = 0 α m a x ϕ = 0 2 π | E C ( R ) | 2 | R | 2 sin θ d ϕ d θ

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