Abstract

We developed a technique to simultaneously measure self-phase modulation and two-photon absorption using shaped femtosecond laser pulses. In the conventional Z-scan measurement technique the amount of nonlinearity is determined by measuring the change in shape and intensity of a transmitted laser beam. In contrast, our method sensitively measures nonlinearity-induced changes in the pulse spectrum. In this work we demonstrate the technique in nonlinear absorptive and dispersive samples, quantify the obtained signal, and compare the measurements with traditional Z-scans. This technique is capable of measuring these nonlinearities in highly scattering samples.

© 2008 Optical Society of America

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  1. D. Fu, T. Ye, T. E. Matthews, B. J. Chen, G. Yurtserver, and W. S. Warren, "High-resolution in vivo imaging of blood vessels without labeling," Opt. Lett. 32, 2641-2643 (2007).
    [CrossRef] [PubMed]
  2. D. Fu, T. Ye, T. E. Matthews, G. Yurtsever, and W. S. Warren, "Two-color, two-photon, and excited-state absorption microscopy," J. Biomed. Opt. 12, 054004 (2007).
    [CrossRef] [PubMed]
  3. M. C. Fischer, H. Liu, I. R. Piletic, Y. Escobedo-Lozoya, R. Yasuda, and W. S. Warren, "Self-phase modulation signatures of neuronal activity," Opt. Lett. 33219-221 (2008).
    [CrossRef]
  4. M. Sheik-Bahae, A. A. Said, and E. W. van Stryland, "High-sensitivity, single-beam n2 measurements," Opt. Lett. 14, 955-957 (1989).
    [CrossRef] [PubMed]
  5. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E.W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
    [CrossRef]
  6. T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. van Stryland, "Eclipsing Z-scan measurement of ?/104 wavefront distortion," Opt. Lett. 19, 317-319 (1994).
    [CrossRef] [PubMed]
  7. G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, "Z-scan technique through beam radius measurements," Appl. Phys. B: Lasers and Optics 76, 83-86 (2003).
    [CrossRef]
  8. H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurements of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666-2668 (1991).
    [CrossRef]
  9. J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E.W. Van Stryland, "Time-resolved Z-scan measurements of optical nonlinearities," J. Opt. Soc. Am. B 11, 1009-1017 (1994).
    [CrossRef]
  10. I. Guedes, L. Misoguti, L. De Boni, and S. C. Zilio, "Heterodyne Z-scan measurements of slow absorbers," J. Appl. Phys. 101, 063112 (2007).
    [CrossRef]
  11. J.-M. Menard, M. Betz, I. Sigal, and H. M. Van Driel, "Single-beam differential Z-scan technique," Appl. Opt. 46, 2119-2122 (2007).
    [CrossRef] [PubMed]
  12. P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, "Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 253-293 (1997).
    [CrossRef]
  13. R. E. Bridges, G. L. Fischer, and R. W. Boyd, "Z-scan measurement technique for non-Gaussian beams and arbitrary sample thicknesses," Opt. Lett. 20, 1821-1823 (1995).
    [CrossRef] [PubMed]
  14. M. C. Fischer, T. Ye, G. Yurtsever, A. Miller, M. Ciocca, W. Wagner, and W. S. Warren, "Two-photon absorption and self-phase modulation measurements with shaped femtosecond laser pulses," Opt. Lett. 30, 1551-1553 (2005).
    [CrossRef] [PubMed]
  15. C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, "Femtosecond laser pulse shaping by use of microsecond radio-frequency pulses," Opt. Lett. 19, 737-739 (1994).
    [CrossRef] [PubMed]
  16. R. W. Boyd, Nonlinear Optics, (Academic Press, 2003).
  17. G. P. Agrawal, Nonlinear Fiber Optics, (Elsevier / Academic Press, 2007).
  18. S. T. Hendow and S. A. Shakir, "Recursive numerical solution for nonlinear wave propagation in fibers and cylindrically symmetrical systems," Appl. Opt. 25, 1759-1764 (1986).
    [CrossRef] [PubMed]
  19. J. A. Hermann and R. G. McDuff, "Analysis of spatial scanning with thick optically nonlinear media," J. Opt. Soc. Am. B 10, 2056-2064 (1993).
    [CrossRef]
  20. J. A. Hermann, "Nonlinear optical absorption in thick media," J. Opt. Soc. Am. B 14, 814-823 (1997).
    [CrossRef]
  21. P. Tian and W. S. Warren, "Ultrafast measurement of two-photon absorption by loss modulation," Opt. Lett. 27, 1634-1636 (2002).
    [CrossRef]
  22. W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl Phys B 76, 215-229 (2003).
    [CrossRef]

2008 (1)

2007 (4)

J.-M. Menard, M. Betz, I. Sigal, and H. M. Van Driel, "Single-beam differential Z-scan technique," Appl. Opt. 46, 2119-2122 (2007).
[CrossRef] [PubMed]

D. Fu, T. Ye, T. E. Matthews, B. J. Chen, G. Yurtserver, and W. S. Warren, "High-resolution in vivo imaging of blood vessels without labeling," Opt. Lett. 32, 2641-2643 (2007).
[CrossRef] [PubMed]

D. Fu, T. Ye, T. E. Matthews, G. Yurtsever, and W. S. Warren, "Two-color, two-photon, and excited-state absorption microscopy," J. Biomed. Opt. 12, 054004 (2007).
[CrossRef] [PubMed]

I. Guedes, L. Misoguti, L. De Boni, and S. C. Zilio, "Heterodyne Z-scan measurements of slow absorbers," J. Appl. Phys. 101, 063112 (2007).
[CrossRef]

2005 (1)

2003 (2)

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl Phys B 76, 215-229 (2003).
[CrossRef]

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, "Z-scan technique through beam radius measurements," Appl. Phys. B: Lasers and Optics 76, 83-86 (2003).
[CrossRef]

2002 (1)

1997 (2)

J. A. Hermann, "Nonlinear optical absorption in thick media," J. Opt. Soc. Am. B 14, 814-823 (1997).
[CrossRef]

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, "Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 253-293 (1997).
[CrossRef]

1995 (1)

1994 (3)

1993 (1)

1991 (1)

H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurements of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666-2668 (1991).
[CrossRef]

1990 (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E.W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

1989 (1)

1986 (1)

Becker, A.

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl Phys B 76, 215-229 (2003).
[CrossRef]

Betz, M.

Boyd, R. W.

Bridges, R. E.

Chapple, P. B.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, "Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 253-293 (1997).
[CrossRef]

Chen, B. J.

Chin, S. L.

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl Phys B 76, 215-229 (2003).
[CrossRef]

Ciocca, M.

de Araujo, C. B.

H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurements of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666-2668 (1991).
[CrossRef]

De Boni, L.

I. Guedes, L. Misoguti, L. De Boni, and S. C. Zilio, "Heterodyne Z-scan measurements of slow absorbers," J. Appl. Phys. 101, 063112 (2007).
[CrossRef]

Escobedo-Lozoya, Y.

Fakis, M.

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, "Z-scan technique through beam radius measurements," Appl. Phys. B: Lasers and Optics 76, 83-86 (2003).
[CrossRef]

Fischer, G. L.

Fischer, M. C.

Fu, D.

D. Fu, T. Ye, T. E. Matthews, B. J. Chen, G. Yurtserver, and W. S. Warren, "High-resolution in vivo imaging of blood vessels without labeling," Opt. Lett. 32, 2641-2643 (2007).
[CrossRef] [PubMed]

D. Fu, T. Ye, T. E. Matthews, G. Yurtsever, and W. S. Warren, "Two-color, two-photon, and excited-state absorption microscopy," J. Biomed. Opt. 12, 054004 (2007).
[CrossRef] [PubMed]

Giannetas, V.

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, "Z-scan technique through beam radius measurements," Appl. Phys. B: Lasers and Optics 76, 83-86 (2003).
[CrossRef]

Golubtsov, I. S.

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl Phys B 76, 215-229 (2003).
[CrossRef]

Gomes, A. S. L.

H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurements of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666-2668 (1991).
[CrossRef]

Goswami, D.

Guedes, I.

I. Guedes, L. Misoguti, L. De Boni, and S. C. Zilio, "Heterodyne Z-scan measurements of slow absorbers," J. Appl. Phys. 101, 063112 (2007).
[CrossRef]

Hagan, D. J.

Hendow, S. T.

Hermann, J. A.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, "Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 253-293 (1997).
[CrossRef]

J. A. Hermann, "Nonlinear optical absorption in thick media," J. Opt. Soc. Am. B 14, 814-823 (1997).
[CrossRef]

J. A. Hermann and R. G. McDuff, "Analysis of spatial scanning with thick optically nonlinear media," J. Opt. Soc. Am. B 10, 2056-2064 (1993).
[CrossRef]

Hillegas, C. W.

Iwasaki, A.

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl Phys B 76, 215-229 (2003).
[CrossRef]

Kandidov, V. P.

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl Phys B 76, 215-229 (2003).
[CrossRef]

Kosareva, O.

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl Phys B 76, 215-229 (2003).
[CrossRef]

Liu, H.

Liu, W.

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl Phys B 76, 215-229 (2003).
[CrossRef]

Ma, H.

H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurements of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666-2668 (1991).
[CrossRef]

Matthews, T. E.

D. Fu, T. Ye, T. E. Matthews, B. J. Chen, G. Yurtserver, and W. S. Warren, "High-resolution in vivo imaging of blood vessels without labeling," Opt. Lett. 32, 2641-2643 (2007).
[CrossRef] [PubMed]

D. Fu, T. Ye, T. E. Matthews, G. Yurtsever, and W. S. Warren, "Two-color, two-photon, and excited-state absorption microscopy," J. Biomed. Opt. 12, 054004 (2007).
[CrossRef] [PubMed]

McDuff, R. G.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, "Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 253-293 (1997).
[CrossRef]

J. A. Hermann and R. G. McDuff, "Analysis of spatial scanning with thick optically nonlinear media," J. Opt. Soc. Am. B 10, 2056-2064 (1993).
[CrossRef]

McKay, T. J.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, "Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 253-293 (1997).
[CrossRef]

Menard, J.-M.

Miller, A.

Misoguti, L.

I. Guedes, L. Misoguti, L. De Boni, and S. C. Zilio, "Heterodyne Z-scan measurements of slow absorbers," J. Appl. Phys. 101, 063112 (2007).
[CrossRef]

Persephonis, P.

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, "Z-scan technique through beam radius measurements," Appl. Phys. B: Lasers and Optics 76, 83-86 (2003).
[CrossRef]

Piletic, I. R.

Polyzos, I.

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, "Z-scan technique through beam radius measurements," Appl. Phys. B: Lasers and Optics 76, 83-86 (2003).
[CrossRef]

Said, A. A.

Shakir, S. A.

Sheik-Bahae, M.

Sigal, I.

Staromlynska, J.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, "Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 253-293 (1997).
[CrossRef]

Strickland, D.

Tian, P.

Tsigaridas, G.

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, "Z-scan technique through beam radius measurements," Appl. Phys. B: Lasers and Optics 76, 83-86 (2003).
[CrossRef]

Tull, J. X.

Van Driel, H. M.

van Stryland, E. W.

Van Stryland, E.W.

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E.W. Van Stryland, "Time-resolved Z-scan measurements of optical nonlinearities," J. Opt. Soc. Am. B 11, 1009-1017 (1994).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E.W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Wagner, W.

Wang, J.

Warren, W. S.

Wei, T.-H.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E.W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Xia, T.

Yasuda, R.

Ye, T.

Yurtserver, G.

Yurtsever, G.

Zilio, S. C.

I. Guedes, L. Misoguti, L. De Boni, and S. C. Zilio, "Heterodyne Z-scan measurements of slow absorbers," J. Appl. Phys. 101, 063112 (2007).
[CrossRef]

Appl Phys B (1)

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl Phys B 76, 215-229 (2003).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B: Lasers and Optics (1)

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, "Z-scan technique through beam radius measurements," Appl. Phys. B: Lasers and Optics 76, 83-86 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurements of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666-2668 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E.W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

J. Appl. Phys. (1)

I. Guedes, L. Misoguti, L. De Boni, and S. C. Zilio, "Heterodyne Z-scan measurements of slow absorbers," J. Appl. Phys. 101, 063112 (2007).
[CrossRef]

J. Biomed. Opt. (1)

D. Fu, T. Ye, T. E. Matthews, G. Yurtsever, and W. S. Warren, "Two-color, two-photon, and excited-state absorption microscopy," J. Biomed. Opt. 12, 054004 (2007).
[CrossRef] [PubMed]

J. Nonlinear Opt. Phys. Mater. (1)

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, "Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 253-293 (1997).
[CrossRef]

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

Opt. Lett. (8)

P. Tian and W. S. Warren, "Ultrafast measurement of two-photon absorption by loss modulation," Opt. Lett. 27, 1634-1636 (2002).
[CrossRef]

M. C. Fischer, T. Ye, G. Yurtsever, A. Miller, M. Ciocca, W. Wagner, and W. S. Warren, "Two-photon absorption and self-phase modulation measurements with shaped femtosecond laser pulses," Opt. Lett. 30, 1551-1553 (2005).
[CrossRef] [PubMed]

D. Fu, T. Ye, T. E. Matthews, B. J. Chen, G. Yurtserver, and W. S. Warren, "High-resolution in vivo imaging of blood vessels without labeling," Opt. Lett. 32, 2641-2643 (2007).
[CrossRef] [PubMed]

M. C. Fischer, H. Liu, I. R. Piletic, Y. Escobedo-Lozoya, R. Yasuda, and W. S. Warren, "Self-phase modulation signatures of neuronal activity," Opt. Lett. 33219-221 (2008).
[CrossRef]

R. E. Bridges, G. L. Fischer, and R. W. Boyd, "Z-scan measurement technique for non-Gaussian beams and arbitrary sample thicknesses," Opt. Lett. 20, 1821-1823 (1995).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, and E. W. van Stryland, "High-sensitivity, single-beam n2 measurements," Opt. Lett. 14, 955-957 (1989).
[CrossRef] [PubMed]

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. van Stryland, "Eclipsing Z-scan measurement of ?/104 wavefront distortion," Opt. Lett. 19, 317-319 (1994).
[CrossRef] [PubMed]

C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, "Femtosecond laser pulse shaping by use of microsecond radio-frequency pulses," Opt. Lett. 19, 737-739 (1994).
[CrossRef] [PubMed]

Other (2)

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

G. P. Agrawal, Nonlinear Fiber Optics, (Elsevier / Academic Press, 2007).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of the experimental setup.

Fig. 2.
Fig. 2.

Sketch of an incoming pulse with a spectral hole in the frequency domain (a) and the time domain (b). Part (c) illustrates A h 2 A h ˜ in Eq. 2 (the nonlinear polarization term). Note the non-zero component at the hole location, which in principle allows for a background-free detection. Part (d) indicates a pulse spectrum with a square hole (as used in the experiments) with a local oscillator component.

Fig. 3.
Fig. 3.

Scaling of the hole refilling and homodyne signal as a function of input power. The symbols are experimental data, the lines are linear fit on a double-log scale. The fit results for the slope is indicated in the graph. For the R6G solution, only the first 6 points (with lower power) were used for fitting. The background data was measured when the focus was outside the sample and was subtracted from the sample data at each power level.

Fig. 4.
Fig. 4.

SPM and TPA measurements in a quartz cuvette filled with 30mM R6G in methanol. The shaded areas indicate different materials (glass thickness 1.25 mm, gap 1 mm; both thicknesses are scaled by their respective index of refraction). The input power in this experiment was 60µW. Part (a) shows the transmission through the sample measured in the far field without aperture (conventional open-aperture Z-scan). The boxcar averager was set to average 300 consecutive laser pulses. The lower parts show the homodyne signal measured without aperture (b) and with aperture (c). The lock-in time constant was set to 30 ms and the glass region was used to set the reference phase. The portion of transmitted power onto the photo detector in (a) was attenuated to reach a similar electrical signal level as the level from the homodyne photo detector in (b) for comparison.

Fig. 5.
Fig. 5.

(a) Simulations of SPM and TPA measurements in a medium with SPM (left) and TPA (right). The sample thickness was taken as 200λ and the Rayleigh length as 12λ. The solid lines indicate spectral analysis in the center of the far field distribution (on-axis), the dashed lines are averages over the entire radial distribution. (b) Real and imaginary parts of the integral expression Q as defined in the text. The ratio of zR to the sample length L was taken as 1/20. The solid lines are on-axis values, the dashed lines are radial averages.

Fig. 6.
Fig. 6.

Numerical values of the scaling factor g as a function of the ratio between the hole width and the bandwidth of the input pulse. A square hole in a transform-limited pulse spectrum was assumed.

Fig. 7.
Fig. 7.

TPA and SPM measurements of R6G solutions of various concentrations in a glass cuvette. The solid lines connect the first 2 points (and the origin in the case of TPA). Saturation effects due to pulse reshaping are visible for higher concentrations.

Equations (20)

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

[ 1 2 k 0 2 + i z β 2 2 τ 2 + i α 0 2 + i ( α 2 i η 2 ) A ( z , τ ) 2 ] A ( z , τ ) = 0 .
Δ A ˜ ( ω ) = z ( i η 2 α 2 ) A h 2 A h ˜ ,
A ˜ hom = A ˜ LO exp ( i ϕ ) + Δ A ˜ ,
A ˜ hom 2 = A ˜ LO 2 + Δ A ˜ 2 + 2 z A ˜ LO A h 2 A h ˜ ( η 2 sin ϕ α 2 cos ϕ ) ,
A A lin = 1 1 8 B Q with Q = E i ( λ a x ) E i ( λ b x ) .
T closed 1 1 4 Re ( B Q 0 )
T open 1 2 π 4 Re ( B Q I lin ρ d ρ ) P lin ,
S hom A lin * e i Φ Δ A ρ d ρ e i Φ B Q I lin ρ d ρ .
A ( ρ , z , τ ) = A 0 ( z , τ ) w 0 w ( z ) exp ( ρ 2 w ( z ) 2 + i Φ ( ρ , z ) )
z A 0 ( z , τ ) = 1 2 ( i η 2 α 2 ) w 0 2 w ( z ) 2 A 0 ( z , τ ) 2 A 0 ( z , τ ) .
Δ A out ( τ ) ( i η 2 α 2 ) π 2 z R A in ( τ ) 2 A in ( τ ) .
A out ( τ ) = A LO ( τ ) e + ( i η 2 α 2 ) π 2 z R A h ( τ ) 2 A h ( τ ) .
P det = 1 2 π π w 0 2 2 n 0 c 2 π slit A ˜ out ( ω ) 2 d ω .
P det , ϕ P ref = { π 2 α 2 z R r LO A 0 2 g ( cos ϕ ) π 2 η 2 z R r LO A 0 2 g ( sin ϕ ) ,
g = a h ( τ ) 2 a h ( τ ) d τ ,
Δ T = { 1 2 π 2 α I z R I 0 ( TPA ) 1 2 ln 3 2 π λ n I z R I 0 ( SPM ) ,
P det , ϕ P ref = { 1 π r LO g Δ T ( TPA ) π ln 3 r LO g Δ T ( SPM ) ,
SNR Z = S Δ T P in 𝓡 ( 2 e Δ f 𝓡 S P in ) 1 2 = ( 𝓡 2 e Δ f ) 1 2 Δ T S P in and
SNR H = P sig 𝓡 ( 2 e Δ f 𝓡 P LO ) 1 2 = ( 𝓡 2 e Δ f ) 1 2 P sig r LO ( R slit P in ) 1 2 ,
SNR H SNR Z = { 1 π g R slit   ( TPA ) π ln 3 g R slit S ( SPM ) .

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