Abstract

Photoacoustic remote sensing microscopy (PARS) represents a new paradigm within the optical imaging community by providing high sensitivity (>50 dB in vivo) non-contact optical absorption contrast in scattering media with a reflection-mode configuration. Unlike contact-based photoacoustic modalities which can acquire complete A-scans with a single excitation pulse due to slow acoustic propagation facilitating the use of time-gated collection of returning acoustic signals, PARS provides depth resolution only through optical sectioning. Here we introduce a new approach for providing coherence-gated depth-resolved PARS imaging using a difference between pulsed-interrogation optical coherence tomography scan-lines with and without excitation pulses. Proposed methods are validated using simulations which account for pulsed-laser induced initial-pressures and accompanying refractive index changes. The changes in refractive index are shown to be proportional to optical absorption. It is demonstrated that to achieve optimal image quality, several key parameters must be selected including interrogation pulse duration and delay. The proposed approach offers the promise of non-contact depth-resolved optical absorption contrast at optical-resolution scales and may complement the scattering contrast offered by optical coherence tomography.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (2)

L. Snider, K. Bell, P. Reza, and R. J. Zemp, “Toward wide-field high-speed photoacoustic remote sensing microscopy,” Photons Plus Ultrasound: Imag. Sens. 10494, 1049423 (2018).

K. Bell, P. Hajireza, and R. Zemp, “Scattering cross-sectional modulation in photoacoustic remote sensing microscopy,” Opt. Lett. 43, 146–149 (2018).
[Crossref] [PubMed]

2017 (4)

K. L. Bell, P. Hajireza, W. Shi, and R. J. Zemp, “Temporal evolution of low-coherence reflectrometry signals in photoacoustic remote sensing microscopy,” Appl. Opt. 56, 5172–5181 (2017).
[Crossref] [PubMed]

M. Almasian, T. G. Leeuwen, and D. J. Faber, “Oct amplitude and speckle statistics of discrete random media,” Sci. Rep. 7, 14873 (2017).
[Crossref] [PubMed]

P. Hajireza, W. Shi, K. Bell, R. J. Paproski, and R. J. Zemp, “Non-interferometric photoacoustic remote sensing microscopy,” Light. Sci. Appl. 6, e16278 (2017).
[Crossref]

J. Yao, “When pressure meets light: detecting the photoacoustic effect at the origin,” Light. Sci. Appl. 6, e17062 (2017).
[Crossref]

2016 (1)

E. Leiss-Holzinger, J. Bauer-Marschallinger, A. Hochreiner, P. Hollinger, and T. Berer, “Dual modality noncontact photoacoustic and spectral domain oct imaging,” Ultrason. Imag. 38, 19–31 (2016).
[Crossref]

2015 (7)

T. Berer, E. Leiss-Holzinger, A. Hochreiner, J. Bauer-Marschallinger, and A. Buchsbaum, “Multimodal noncontact photoacoustic and optical coherence tomography imaging using wavelength-division multiplexing,” J. Biomed. Opt. 20, 046013 (2015).
[Crossref]

J. A. Izatt, M. A. Choma, and A.-H. Dhalla, “Theory of optical coherence tomography,” Opt. Coherence Tomog.: Technol. Appl. 2015 pp. 65–94 (2015).
[Crossref]

X. L. Deán-Ben, G. A. Pang, F. Montero de Espinosa, and D. Razansky, “Non-contact optoacoustic imaging with focused air-coupled transducers,” Appl. Phys. Lett. 107, 051105 (2015).
[Crossref]

J. Horstmann, H. Spahr, C. Buj, M. Münter, and R. Brinkmann, “Full-field speckle interferometry for non-contact photoacoustic tomography,” Phys. Med. Biol. 60, 4045 (2015).
[Crossref] [PubMed]

A. Zhang, Q. Zhang, Y. Huang, Z. Zhong, and R. K. Wang, “Multifunctional 1050 nm spectral domain oct system at 147 khz for posterior eye imaging,” Sovremennye tekhnologii v meditsine 71 (2015).
[PubMed]

B. H. Hokr, J. N. Bixler, G. Elpers, B. Zollars, R. J. Thomas, V. V. Yakovlev, and M. O. Scully, “Modeling focusing gaussian beams in a turbid medium with monte carlo simulations,” Opt. Express 23, 8699–8705 (2015).
[Crossref] [PubMed]

T. Wang, T. Pfeiffer, E. Regar, W. Wieser, H. van Beusekom, C. T. Lancee, G. Springeling, I. Krabbendam, A. F. van der Steen, R. Huber, and et al., “Heartbeat oct: in vivo intravascular megahertz-optical coherence tomography,” Biomed. Opt. Express 6, 5021–5032 (2015).
[Crossref] [PubMed]

2013 (1)

2012 (2)

G. Rousseau, A. Blouin, and J.-P. Monchalin, “Non-contact photoacoustic tomography and ultrasonography for tissue imaging,” Biomed. Opt. Express 3, 16–25 (2012).
[Crossref] [PubMed]

S. G. Resink, W. Steenbergen, and A. C. Boccara, “State-of-the art of acoust-optic sensing and imaging of turbid media,” J. Biomed. Opt. 17, 040901 (2012).
[Crossref]

2011 (2)

Y. Sun, M. Y. Sy, Y.-X. J. Wang, A. T. Ahuja, Y.-T. Zhang, and E. Pickwell-MacPherson, “A promising diagnostic method: Terahertz pulsed imaging and spectroscopy,” World J. Radiol. 3, 55 (2011).
[Crossref] [PubMed]

Y. Wang, C. Li, and R. K. Wang, “Noncontact photoacoustic imaging achieved by using a low-coherence interferometer as the acoustic detector,” Opt. Lett. 36, 3975–3977 (2011).
[Crossref] [PubMed]

2010 (2)

A. F. Fercher, “Optical coherence tomography–development, principles, applications,” Zeitschrift für Medizinische Physik 20, 251–276 (2010).
[Crossref]

V. Ntziachristos, J. S. Yoo, and G. M. van Dam, “Current concepts and future perspectives on surgical optical imaging in cancer,” J. Biomed. Opt. 15, 066024 (2010).
[Crossref]

2008 (1)

S. J. Kavros, D. A. Liedl, A. J. Boon, J. L. Miller, J. A. Hobbs, and K. L. Andrews, “Expedited wound healing with noncontact, low-frequency ultrasound therapy in chronic wounds: a retrospective analysis,” Adv. Ski. Wound Care 21, 416–423 (2008).
[Crossref]

2005 (2)

S. Sakadžić and L. V. Wang, “Modulation of multiply scattered coherent light by ultrasonic pulses: an analytical model,” Phys. Rev. E 72, 036620 (2005).
[Crossref]

B. Karamata, K. Hassler, M. Laubscher, and T. Lasser, “Speckle statistics in optical coherence tomography,” J. Opt. Soc. Am. A 22, 593–596 (2005).
[Crossref]

1999 (1)

L. F. Eichenfield and C. A. Hardaway, “Neonatal dermatology,” Curr. Opin. Pediat. 11, 471–474 (1999).
[Crossref]

1995 (1)

M. B. Malloy-McDonald, “Skin care for high-risk neonates,” J. Wound Oostomy Cont. Nurs. 22, 177–182 (1995).
[Crossref]

1991 (1)

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Anten. Propag. 14, 302–307 (1966).
[Crossref]

Ahuja, A. T.

Y. Sun, M. Y. Sy, Y.-X. J. Wang, A. T. Ahuja, Y.-T. Zhang, and E. Pickwell-MacPherson, “A promising diagnostic method: Terahertz pulsed imaging and spectroscopy,” World J. Radiol. 3, 55 (2011).
[Crossref] [PubMed]

Almasian, M.

M. Almasian, T. G. Leeuwen, and D. J. Faber, “Oct amplitude and speckle statistics of discrete random media,” Sci. Rep. 7, 14873 (2017).
[Crossref] [PubMed]

Andrews, K. L.

S. J. Kavros, D. A. Liedl, A. J. Boon, J. L. Miller, J. A. Hobbs, and K. L. Andrews, “Expedited wound healing with noncontact, low-frequency ultrasound therapy in chronic wounds: a retrospective analysis,” Adv. Ski. Wound Care 21, 416–423 (2008).
[Crossref]

Bauer-Marschallinger, J.

E. Leiss-Holzinger, J. Bauer-Marschallinger, A. Hochreiner, P. Hollinger, and T. Berer, “Dual modality noncontact photoacoustic and spectral domain oct imaging,” Ultrason. Imag. 38, 19–31 (2016).
[Crossref]

T. Berer, E. Leiss-Holzinger, A. Hochreiner, J. Bauer-Marschallinger, and A. Buchsbaum, “Multimodal noncontact photoacoustic and optical coherence tomography imaging using wavelength-division multiplexing,” J. Biomed. Opt. 20, 046013 (2015).
[Crossref]

A. Hochreiner, J. Bauer-Marschallinger, P. Burgholzer, B. Jakoby, and T. Berer, “Non-contact photoacoustic imaging using a fiber based interferometer with optical amplification,” Biomed. Opt. Express 4, 2322–2331 (2013).
[Crossref] [PubMed]

Bell, K.

K. Bell, P. Hajireza, and R. Zemp, “Scattering cross-sectional modulation in photoacoustic remote sensing microscopy,” Opt. Lett. 43, 146–149 (2018).
[Crossref] [PubMed]

L. Snider, K. Bell, P. Reza, and R. J. Zemp, “Toward wide-field high-speed photoacoustic remote sensing microscopy,” Photons Plus Ultrasound: Imag. Sens. 10494, 1049423 (2018).

P. Hajireza, W. Shi, K. Bell, R. J. Paproski, and R. J. Zemp, “Non-interferometric photoacoustic remote sensing microscopy,” Light. Sci. Appl. 6, e16278 (2017).
[Crossref]

Bell, K. L.

Berer, T.

E. Leiss-Holzinger, J. Bauer-Marschallinger, A. Hochreiner, P. Hollinger, and T. Berer, “Dual modality noncontact photoacoustic and spectral domain oct imaging,” Ultrason. Imag. 38, 19–31 (2016).
[Crossref]

T. Berer, E. Leiss-Holzinger, A. Hochreiner, J. Bauer-Marschallinger, and A. Buchsbaum, “Multimodal noncontact photoacoustic and optical coherence tomography imaging using wavelength-division multiplexing,” J. Biomed. Opt. 20, 046013 (2015).
[Crossref]

A. Hochreiner, J. Bauer-Marschallinger, P. Burgholzer, B. Jakoby, and T. Berer, “Non-contact photoacoustic imaging using a fiber based interferometer with optical amplification,” Biomed. Opt. Express 4, 2322–2331 (2013).
[Crossref] [PubMed]

Bixler, J. N.

Blouin, A.

Boccara, A. C.

S. G. Resink, W. Steenbergen, and A. C. Boccara, “State-of-the art of acoust-optic sensing and imaging of turbid media,” J. Biomed. Opt. 17, 040901 (2012).
[Crossref]

Boon, A. J.

S. J. Kavros, D. A. Liedl, A. J. Boon, J. L. Miller, J. A. Hobbs, and K. L. Andrews, “Expedited wound healing with noncontact, low-frequency ultrasound therapy in chronic wounds: a retrospective analysis,” Adv. Ski. Wound Care 21, 416–423 (2008).
[Crossref]

Bouma, B.

B. Bouma, Handbook of Optical Coherence Tomography (CRC, 2001).
[Crossref]

Brinkmann, R.

J. Horstmann, H. Spahr, C. Buj, M. Münter, and R. Brinkmann, “Full-field speckle interferometry for non-contact photoacoustic tomography,” Phys. Med. Biol. 60, 4045 (2015).
[Crossref] [PubMed]

Buchsbaum, A.

T. Berer, E. Leiss-Holzinger, A. Hochreiner, J. Bauer-Marschallinger, and A. Buchsbaum, “Multimodal noncontact photoacoustic and optical coherence tomography imaging using wavelength-division multiplexing,” J. Biomed. Opt. 20, 046013 (2015).
[Crossref]

Buj, C.

J. Horstmann, H. Spahr, C. Buj, M. Münter, and R. Brinkmann, “Full-field speckle interferometry for non-contact photoacoustic tomography,” Phys. Med. Biol. 60, 4045 (2015).
[Crossref] [PubMed]

Burgholzer, P.

Chen, Z.

Z. Chen, S. Yang, and D. Xing, “All-optically integrated multimodality imaging system: combined photoacoustic microscopy, optical coherence tomography, and fluorescence imaging,” (2016), SPIE/COS Photonics Asia, pp. 100240H.

Choma, M. A.

J. A. Izatt, M. A. Choma, and A.-H. Dhalla, “Theory of optical coherence tomography,” Opt. Coherence Tomog.: Technol. Appl. 2015 pp. 65–94 (2015).
[Crossref]

Deán-Ben, X. L.

X. L. Deán-Ben, G. A. Pang, F. Montero de Espinosa, and D. Razansky, “Non-contact optoacoustic imaging with focused air-coupled transducers,” Appl. Phys. Lett. 107, 051105 (2015).
[Crossref]

Dhalla, A.-H.

J. A. Izatt, M. A. Choma, and A.-H. Dhalla, “Theory of optical coherence tomography,” Opt. Coherence Tomog.: Technol. Appl. 2015 pp. 65–94 (2015).
[Crossref]

Eichenfield, L. F.

L. F. Eichenfield and C. A. Hardaway, “Neonatal dermatology,” Curr. Opin. Pediat. 11, 471–474 (1999).
[Crossref]

Elpers, G.

Faber, D. J.

M. Almasian, T. G. Leeuwen, and D. J. Faber, “Oct amplitude and speckle statistics of discrete random media,” Sci. Rep. 7, 14873 (2017).
[Crossref] [PubMed]

Fercher, A. F.

A. F. Fercher, “Optical coherence tomography–development, principles, applications,” Zeitschrift für Medizinische Physik 20, 251–276 (2010).
[Crossref]

Gan, X.

Gu, M.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech house, 2005).

Hajireza, P.

Hardaway, C. A.

L. F. Eichenfield and C. A. Hardaway, “Neonatal dermatology,” Curr. Opin. Pediat. 11, 471–474 (1999).
[Crossref]

Hassler, K.

Hobbs, J. A.

S. J. Kavros, D. A. Liedl, A. J. Boon, J. L. Miller, J. A. Hobbs, and K. L. Andrews, “Expedited wound healing with noncontact, low-frequency ultrasound therapy in chronic wounds: a retrospective analysis,” Adv. Ski. Wound Care 21, 416–423 (2008).
[Crossref]

Hochreiner, A.

E. Leiss-Holzinger, J. Bauer-Marschallinger, A. Hochreiner, P. Hollinger, and T. Berer, “Dual modality noncontact photoacoustic and spectral domain oct imaging,” Ultrason. Imag. 38, 19–31 (2016).
[Crossref]

T. Berer, E. Leiss-Holzinger, A. Hochreiner, J. Bauer-Marschallinger, and A. Buchsbaum, “Multimodal noncontact photoacoustic and optical coherence tomography imaging using wavelength-division multiplexing,” J. Biomed. Opt. 20, 046013 (2015).
[Crossref]

A. Hochreiner, J. Bauer-Marschallinger, P. Burgholzer, B. Jakoby, and T. Berer, “Non-contact photoacoustic imaging using a fiber based interferometer with optical amplification,” Biomed. Opt. Express 4, 2322–2331 (2013).
[Crossref] [PubMed]

Hokr, B. H.

Hollinger, P.

E. Leiss-Holzinger, J. Bauer-Marschallinger, A. Hochreiner, P. Hollinger, and T. Berer, “Dual modality noncontact photoacoustic and spectral domain oct imaging,” Ultrason. Imag. 38, 19–31 (2016).
[Crossref]

Horstmann, J.

J. Horstmann, H. Spahr, C. Buj, M. Münter, and R. Brinkmann, “Full-field speckle interferometry for non-contact photoacoustic tomography,” Phys. Med. Biol. 60, 4045 (2015).
[Crossref] [PubMed]

Huang, Y.

A. Zhang, Q. Zhang, Y. Huang, Z. Zhong, and R. K. Wang, “Multifunctional 1050 nm spectral domain oct system at 147 khz for posterior eye imaging,” Sovremennye tekhnologii v meditsine 71 (2015).
[PubMed]

Huber, R.

Izatt, J. A.

J. A. Izatt, M. A. Choma, and A.-H. Dhalla, “Theory of optical coherence tomography,” Opt. Coherence Tomog.: Technol. Appl. 2015 pp. 65–94 (2015).
[Crossref]

Jakoby, B.

Karamata, B.

Kavros, S. J.

S. J. Kavros, D. A. Liedl, A. J. Boon, J. L. Miller, J. A. Hobbs, and K. L. Andrews, “Expedited wound healing with noncontact, low-frequency ultrasound therapy in chronic wounds: a retrospective analysis,” Adv. Ski. Wound Care 21, 416–423 (2008).
[Crossref]

Krabbendam, I.

Lancee, C. T.

Lasser, T.

Laubscher, M.

Leeuwen, T. G.

M. Almasian, T. G. Leeuwen, and D. J. Faber, “Oct amplitude and speckle statistics of discrete random media,” Sci. Rep. 7, 14873 (2017).
[Crossref] [PubMed]

Leigh, W. B.

W. B. Leigh, Devices for Optoelectronics (Marcel Dekker, 1996).

Leiss-Holzinger, E.

E. Leiss-Holzinger, J. Bauer-Marschallinger, A. Hochreiner, P. Hollinger, and T. Berer, “Dual modality noncontact photoacoustic and spectral domain oct imaging,” Ultrason. Imag. 38, 19–31 (2016).
[Crossref]

T. Berer, E. Leiss-Holzinger, A. Hochreiner, J. Bauer-Marschallinger, and A. Buchsbaum, “Multimodal noncontact photoacoustic and optical coherence tomography imaging using wavelength-division multiplexing,” J. Biomed. Opt. 20, 046013 (2015).
[Crossref]

Li, C.

Liedl, D. A.

S. J. Kavros, D. A. Liedl, A. J. Boon, J. L. Miller, J. A. Hobbs, and K. L. Andrews, “Expedited wound healing with noncontact, low-frequency ultrasound therapy in chronic wounds: a retrospective analysis,” Adv. Ski. Wound Care 21, 416–423 (2008).
[Crossref]

Malloy-McDonald, M. B.

M. B. Malloy-McDonald, “Skin care for high-risk neonates,” J. Wound Oostomy Cont. Nurs. 22, 177–182 (1995).
[Crossref]

Miller, J. L.

S. J. Kavros, D. A. Liedl, A. J. Boon, J. L. Miller, J. A. Hobbs, and K. L. Andrews, “Expedited wound healing with noncontact, low-frequency ultrasound therapy in chronic wounds: a retrospective analysis,” Adv. Ski. Wound Care 21, 416–423 (2008).
[Crossref]

Monchalin, J.-P.

Montero de Espinosa, F.

X. L. Deán-Ben, G. A. Pang, F. Montero de Espinosa, and D. Razansky, “Non-contact optoacoustic imaging with focused air-coupled transducers,” Appl. Phys. Lett. 107, 051105 (2015).
[Crossref]

Münter, M.

J. Horstmann, H. Spahr, C. Buj, M. Münter, and R. Brinkmann, “Full-field speckle interferometry for non-contact photoacoustic tomography,” Phys. Med. Biol. 60, 4045 (2015).
[Crossref] [PubMed]

Ntziachristos, V.

V. Ntziachristos, J. S. Yoo, and G. M. van Dam, “Current concepts and future perspectives on surgical optical imaging in cancer,” J. Biomed. Opt. 15, 066024 (2010).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Several key aspects of the detection mechanism are highlighted. (a) Shows of the central idea behind CG-PARS detection where two OCT acquisitions are subtracted from each other to highlight regions of optical absorption. In this example, OCT1 represents an unperturbed acquisition of the optical scattering distribution, and OCT2 represents the same region directly following an excitation event where regions have been modulated through the PARS mechanism. The difference then highlights these modulated regions. Scale bars: 50 μm. (b) Shows the relative timing between the excitation pulse (green), interrogation pulses (red), and the activity of the OCT detection array. Of particular importance is the relative delay, and the duration of the second interrogation pulse relative to the excitation pulse.
Fig. 2
Fig. 2 An example of simulated contrast performance. Here three targets are placed inside a scattering medium where (i) is a purely absorbing target, (ii) is both absorbing and scattering, and (iii) is a purely scattering target. OCT has sensitivity to the optically scattering regions, and CG-PARS has sensitivity to the optical absorption. Note that since CG-PARS is also reliant on the optical scattering for signal, the absorption of target (i) must be larger than that of target (ii) to produce similar CG-PARS signal. Scale bars: 100 μm.
Fig. 3
Fig. 3 A comparison between the scanning patterns of the newly proposed CG-PARS and conventional PARS microscopy. Since CG-PARS is capable of acquiring full depth-resolved A-scans within two laser pulse events only 2M laser shots are required to perform a B-scans. Conventional PARS however, requires MN laser shots to characterize the same region.
Fig. 4
Fig. 4 An example of a CG-PARS interaction with a optically absorbing and optically scattering regions. (a) Highlights the reflectivity signal in depth with time. (b) shows the change in the reflectivity profile.
Fig. 5
Fig. 5 The effects of various interrogation timing parameters are highlighted. The target is a homogeneous absorbing region between 40 and 60 μm in depth.
Fig. 6
Fig. 6 The effects of various receive SNR values for (a) The recovered SNR of the produced scattering A-scans and (b) The recovered SNR of the produced absorption A-scans.
Fig. 7
Fig. 7 A comparison between PARS experimental B-Scans and simulation results. (a) PARS C-Scan maximum intensity projection of 7μm carbon fibers along with a (inset) representative experimental B-Scan (scale bar: 100 μm) and (b) a representative cross-section of a single fiber (scale bar: 25 μm). B-scans were pulled from volumetric data, where multiple maximum-amplitude C-scans were acquired at various depths. (c) A PARS LSV simulation of a cylindrical absorber with diameter 7 μm situated within a non-scattering medium (using n = 1.33). Scale bar: 25 μm.
Fig. 8
Fig. 8 A comparison between (a) the CG-PARS 1D simulation (Sec. 2.2), (b) the CG-PARS LSV model (Sec. 2.3.3), and (c–d) the PARS LSV model (Sec. 2.3.2). The 1D simulation and the LSV model in (a) and (b) agree well with each other in terms of predicted axial and lateral resolutions. Conventional PARS performs extremely poorly with the same optical setup as it can only provide axial resolution as defined by the optical section. If a higher numerical aperture is used (NA = 0.2) the axial resolution would improve, however, alignment issues and optical scattering may confound this. In the above simulations, phantoms consist of two point targets which are situated within a non-scattering medium (n = 1.33). The first numerical aperture (NA = 0.02) is typical of commercially available OCT systems and is assuming a effective focal length of 100 mm and a input beam diameter of 4 mm. Interrogation is performed using a pulsed source with 1310 nm central wavelength and 85 nm spectral bandwidth. Scale bars represent 25 μm.
Fig. 9
Fig. 9 A more complex phantom a simulated using the LSV model detailed in section 2.3. (a) Shows the optical scattering distribution of the phantom and the recovery provided by (b) conventional OCT. (c) shows the optical absorption distribution of the phantom and the recovery provided by (d) CG-PARS. Scale bars: 50μm.

Equations (25)

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

b ( z ) = | | { r ^ s , 1 } | | { r ^ s , 2 } | | .
2 p ( x , t ) p i 1 , j , k n 2 p i , j , k n + p i + 1 , j , k n ( Δ x ) 2 + p i , j 1 , k n 2 p i , j , k n + p i , j + 1 , k n ( Δ y ) 2 + p i , j , k 1 n 2 p i , j , k n + p i , j , k + 1 n ( Δ z ) 2
2 p ( x , t ) t 2 p i , j , k n + 1 2 p i , j , k n + p i , j , k n 1 ( Δ t ) 2
H ( x , t ) t H i , j , k n + 1 H i , j , k n Δ t
p i , j , k n + 1 = ( Δ t ) 2 [ β C p H i , j , k n + 1 H i , j , k n Δ t + c a 2 ( p i 1 , j , k n 2 p i , j , k n + p i + 1 , j , k n ( Δ x ) 2 + p i , j 1 , k n 2 p i , j , k n + p i , j + 1 , k n ( Δ y ) 2 + p i , j , k 1 n 2 p i , j , k n + p i , j , k 1 n ( Δ z ) 2 ) ] + 2 p i , j , k n p i , j , k n 1 .
n * ( x , t ) = n 0 ( x ) + δ n ( x , t ) = n 0 + n 0 3 p 2 ρ m c a 2
n ¯ * ( z , t ) = x Ψ ( z ) I int ( x , t ) n * ( x , t ) x Ψ ( z ) I int ( x , t )
Ψ ( z ) = { x | ϕ ( z ) ϕ ( x ) < ϕ ( z + Δ z ) }
E r ( ν , t ) = E 0 , r ( ν ) e i [ 4 π ν Δ r / c ] ,
S OCT ( x ) = h OCT ( x , x ) r ( x ) d x .
h OCT ( x , x ) = 2 E R E S 0 h E , int 2 ( x , x ) R z z ( z z ) .
h E , int ( x , x ) = w 0 w ( z f ) exp ( ρ 2 ( x , x ) w ( z f ) 2 ) exp ( i ( k ( z f ) + k ρ 2 ( x , x ) 2 R ( z f ) ψ ( z f ) ) )
R z z ( z ) = exp ( z 2 2 σ z 2 ) .
S OCT , LSI ( x ) = h OCT , LSI ( x ) * r ( x )
h OCT , LSI ( x ) = 2 E R E S 0 exp ( 2 ρ 2 w 0 2 ) exp ( i 2 k z ) exp ( z 2 2 σ z 2 ) .
S PARS ( x ) = h PARS , int ( x , x ) Δ r I ( x , x ) d x
h PARS , int ( x , x ) = | E S 0 | 2 | h E , int ( x , x ) | 4
δ n ( x , x ) = Γ η n 3 2 ρ m ν s 2 μ a ( x ) ϕ ex ( x , x )
S PARS ( x ) | E S 0 | 2 I ex , 0 | h E , int ( x , x ) | 4 | h E , ex ( x , x ) | 2 μ a ( x ) r ( x ) d x .
S PARS , LSI ( x ) | h E , int ( x ) | 4 | h E , ex ( x ) | 2 * μ a ( x ) r ( x ) .
S CG PARS ( x ) = | | S OCT ( x ) | | S OCT + Δ S OCT | |
Δ S OCT ( x ) = E R E S 0 I ex , 0 h E , int 2 ( x , x ) R z z ( x , x ) | h E , ex ( x , x ) | 2 μ a ( x ) r ( x ) d x .
S CG PARS ( x ) Δ S OCT ( x ) .
S CG PARS , LSI ( x ) h OCT , LSI ( x ) h EX , LSI ( x ) * μ a ( x ) r ( x )
h EX , LSI ( x ) = I ex , 0 exp ( 2 ρ 2 w 0 2 )

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