Abstract

The theory for a sensitive spectroscopy based on the photothermal deflection of a laser beam is developed. We consider cw and pulsed cases of both transverse and collinear photothermal deflection spectroscopy for solids, liquids, gases, and thin films. The predictions of the theory are experimentally verified, its implications for imaging and microscopy are given, and the sources of noise are analyzed. The sensitivity and versatility of photothermal deflection spectroscopy are compared with thermal lensing and photoacoustic spectroscopy.

© 1981 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. See, for example, J. Stone, J. Opt. Soc. Am. 62, 327 (1972); Appl. Opt. 12, 1828 (1973).
    [CrossRef] [PubMed]
  2. J. R. Whinnery, Acc. Chem. Res. 7, 225 (1974) and references therein.
    [CrossRef]
  3. R. L. Swofford, J. A. Morrell, J. Appl. Phys. 49, 3667 (1978) and references therein.
    [CrossRef]
  4. For an overview of photoacoustic spectroscopy, see Y.-H. Pao, Ed., Optoacoustic Spectroscopy and Detection (Academic, New York, 1977).
  5. D. Fournier, A. C. Boccara, J. Badoz, in Digest of Topical Meeting on Photoacoustic Spectroscopy (Optical Society of America, Washington, D.C., 1979), paper ThA1.
  6. A. C. Boccara, D. Fournier, W. Jackson, N. M. Amer, Opt. Lett. 5, 377 (1980).
    [CrossRef] [PubMed]
  7. D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
    [CrossRef]
  8. A. C. Boccara, D. Fournier, J. Badoz, Appl. Phys. Lett. 36, 130 (1980).
    [CrossRef]
  9. J. C. Murphy, L. C. Aamodt, J. Appl. Phys. 51, 4580 (1980).
    [CrossRef]
  10. M. Billardon, J. M. Ortega, E.S.P.C.I., private communication.
  11. F. A. McDonald, G. C. Wetsel, J. Appl. Phys. 49, 2313 (1978).
    [CrossRef]
  12. W. Jackson, N. M. Amer, J. Appl. Phys. 51, 3343 (1980).
    [CrossRef]
  13. H. S. Bennett, R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
    [CrossRef]
  14. L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 49, 3036 (1978).
    [CrossRef]
  15. A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
    [CrossRef]
  16. A. Mandalis, B. S. Royce, J. Appl. Phys. 50, 4331 (1979).
  17. D. C. Smith, IEEE J. Quantum Electron. QE-5, 600 (1969).
    [CrossRef]
  18. D. Fournier, A. C. Boccara, in Scanned Image Microscopy, E. A. Ash, Ed. (Academic, London, 1980).
  19. L. W. Casperson, Appl. Opt. 12, 2434 (1973).
    [CrossRef] [PubMed]
  20. For some high frequency modulation experiments with very tight focusing, we observed a small secondary maximum with a 180° phase shift. Because the pump and probe beams intersect at an angle and the probe beam deflects slightly in region 0, a mechanism similar to that discussed in Sec. IV.A.2.3, could be present.
  21. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford, Clarendon, 1959).
  22. Silicon Detector Corp., Newbury Park, Calif.
  23. In some cases, it is experimentally advantageous to use a liquid in region 0 to make use of the typically higher dn/dt for liquids. In this case, it is obvious that the role of regions 0 and 2 should be interchanged.
  24. S. R. Brueck, H. Kildal, L. J. Belanger, Opt. Commun. 34, 199 (1980).
    [CrossRef]
  25. A. Van der Ziel, Noise in Measurements (Wiley, New York, 1976).
  26. R. L. Thomas, J. J. Pouch, Y. H. Wong, L. D. Favro, P. K. Kuo, A. Rosencwaig, J. Appl. Phys. 51, 1152 (1980) and references therein.
    [CrossRef]
  27. H. K. Wickramasinghe, R. C. Bray, V. Jipson, C. F. Quate, J. R. Salcedo, Appl. Phys. Lett. 33, 923 (1978).
    [CrossRef]
  28. G. Busse, A. Ograbeck, J. Appl. Phys. 51, 3576 (1980) and references therein.
    [CrossRef]

1980 (8)

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

A. C. Boccara, D. Fournier, J. Badoz, Appl. Phys. Lett. 36, 130 (1980).
[CrossRef]

J. C. Murphy, L. C. Aamodt, J. Appl. Phys. 51, 4580 (1980).
[CrossRef]

W. Jackson, N. M. Amer, J. Appl. Phys. 51, 3343 (1980).
[CrossRef]

S. R. Brueck, H. Kildal, L. J. Belanger, Opt. Commun. 34, 199 (1980).
[CrossRef]

R. L. Thomas, J. J. Pouch, Y. H. Wong, L. D. Favro, P. K. Kuo, A. Rosencwaig, J. Appl. Phys. 51, 1152 (1980) and references therein.
[CrossRef]

G. Busse, A. Ograbeck, J. Appl. Phys. 51, 3576 (1980) and references therein.
[CrossRef]

A. C. Boccara, D. Fournier, W. Jackson, N. M. Amer, Opt. Lett. 5, 377 (1980).
[CrossRef] [PubMed]

1979 (1)

A. Mandalis, B. S. Royce, J. Appl. Phys. 50, 4331 (1979).

1978 (4)

H. K. Wickramasinghe, R. C. Bray, V. Jipson, C. F. Quate, J. R. Salcedo, Appl. Phys. Lett. 33, 923 (1978).
[CrossRef]

L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 49, 3036 (1978).
[CrossRef]

F. A. McDonald, G. C. Wetsel, J. Appl. Phys. 49, 2313 (1978).
[CrossRef]

R. L. Swofford, J. A. Morrell, J. Appl. Phys. 49, 3667 (1978) and references therein.
[CrossRef]

1977 (1)

H. S. Bennett, R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
[CrossRef]

1976 (1)

A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
[CrossRef]

1974 (1)

J. R. Whinnery, Acc. Chem. Res. 7, 225 (1974) and references therein.
[CrossRef]

1973 (1)

1972 (1)

1969 (1)

D. C. Smith, IEEE J. Quantum Electron. QE-5, 600 (1969).
[CrossRef]

Aamodt, L. C.

J. C. Murphy, L. C. Aamodt, J. Appl. Phys. 51, 4580 (1980).
[CrossRef]

L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 49, 3036 (1978).
[CrossRef]

Amer, N. M.

A. C. Boccara, D. Fournier, W. Jackson, N. M. Amer, Opt. Lett. 5, 377 (1980).
[CrossRef] [PubMed]

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

W. Jackson, N. M. Amer, J. Appl. Phys. 51, 3343 (1980).
[CrossRef]

Badoz, J.

A. C. Boccara, D. Fournier, J. Badoz, Appl. Phys. Lett. 36, 130 (1980).
[CrossRef]

D. Fournier, A. C. Boccara, J. Badoz, in Digest of Topical Meeting on Photoacoustic Spectroscopy (Optical Society of America, Washington, D.C., 1979), paper ThA1.

Belanger, L. J.

S. R. Brueck, H. Kildal, L. J. Belanger, Opt. Commun. 34, 199 (1980).
[CrossRef]

Bennett, H. S.

H. S. Bennett, R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
[CrossRef]

Billardon, M.

M. Billardon, J. M. Ortega, E.S.P.C.I., private communication.

Boccara, A. C.

A. C. Boccara, D. Fournier, W. Jackson, N. M. Amer, Opt. Lett. 5, 377 (1980).
[CrossRef] [PubMed]

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

A. C. Boccara, D. Fournier, J. Badoz, Appl. Phys. Lett. 36, 130 (1980).
[CrossRef]

D. Fournier, A. C. Boccara, in Scanned Image Microscopy, E. A. Ash, Ed. (Academic, London, 1980).

D. Fournier, A. C. Boccara, J. Badoz, in Digest of Topical Meeting on Photoacoustic Spectroscopy (Optical Society of America, Washington, D.C., 1979), paper ThA1.

Bray, R. C.

H. K. Wickramasinghe, R. C. Bray, V. Jipson, C. F. Quate, J. R. Salcedo, Appl. Phys. Lett. 33, 923 (1978).
[CrossRef]

Brueck, S. R.

S. R. Brueck, H. Kildal, L. J. Belanger, Opt. Commun. 34, 199 (1980).
[CrossRef]

Busse, G.

G. Busse, A. Ograbeck, J. Appl. Phys. 51, 3576 (1980) and references therein.
[CrossRef]

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford, Clarendon, 1959).

Casperson, L. W.

Favro, L. D.

R. L. Thomas, J. J. Pouch, Y. H. Wong, L. D. Favro, P. K. Kuo, A. Rosencwaig, J. Appl. Phys. 51, 1152 (1980) and references therein.
[CrossRef]

Forman, R. A.

H. S. Bennett, R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
[CrossRef]

Fournier, D.

A. C. Boccara, D. Fournier, W. Jackson, N. M. Amer, Opt. Lett. 5, 377 (1980).
[CrossRef] [PubMed]

A. C. Boccara, D. Fournier, J. Badoz, Appl. Phys. Lett. 36, 130 (1980).
[CrossRef]

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

D. Fournier, A. C. Boccara, in Scanned Image Microscopy, E. A. Ash, Ed. (Academic, London, 1980).

D. Fournier, A. C. Boccara, J. Badoz, in Digest of Topical Meeting on Photoacoustic Spectroscopy (Optical Society of America, Washington, D.C., 1979), paper ThA1.

Gerlach, R.

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

Gersho, A.

A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
[CrossRef]

Jackson, W.

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford, Clarendon, 1959).

Jipson, V.

H. K. Wickramasinghe, R. C. Bray, V. Jipson, C. F. Quate, J. R. Salcedo, Appl. Phys. Lett. 33, 923 (1978).
[CrossRef]

Kildal, H.

S. R. Brueck, H. Kildal, L. J. Belanger, Opt. Commun. 34, 199 (1980).
[CrossRef]

Kuo, P. K.

R. L. Thomas, J. J. Pouch, Y. H. Wong, L. D. Favro, P. K. Kuo, A. Rosencwaig, J. Appl. Phys. 51, 1152 (1980) and references therein.
[CrossRef]

Mandalis, A.

A. Mandalis, B. S. Royce, J. Appl. Phys. 50, 4331 (1979).

McDonald, F. A.

F. A. McDonald, G. C. Wetsel, J. Appl. Phys. 49, 2313 (1978).
[CrossRef]

Morrell, J. A.

R. L. Swofford, J. A. Morrell, J. Appl. Phys. 49, 3667 (1978) and references therein.
[CrossRef]

Murphy, J. C.

J. C. Murphy, L. C. Aamodt, J. Appl. Phys. 51, 4580 (1980).
[CrossRef]

L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 49, 3036 (1978).
[CrossRef]

Ograbeck, A.

G. Busse, A. Ograbeck, J. Appl. Phys. 51, 3576 (1980) and references therein.
[CrossRef]

Ortega, J. M.

M. Billardon, J. M. Ortega, E.S.P.C.I., private communication.

Pouch, J. J.

R. L. Thomas, J. J. Pouch, Y. H. Wong, L. D. Favro, P. K. Kuo, A. Rosencwaig, J. Appl. Phys. 51, 1152 (1980) and references therein.
[CrossRef]

Quate, C. F.

H. K. Wickramasinghe, R. C. Bray, V. Jipson, C. F. Quate, J. R. Salcedo, Appl. Phys. Lett. 33, 923 (1978).
[CrossRef]

Rosencwaig, A.

R. L. Thomas, J. J. Pouch, Y. H. Wong, L. D. Favro, P. K. Kuo, A. Rosencwaig, J. Appl. Phys. 51, 1152 (1980) and references therein.
[CrossRef]

A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
[CrossRef]

Royce, B. S.

A. Mandalis, B. S. Royce, J. Appl. Phys. 50, 4331 (1979).

Salcedo, J. R.

H. K. Wickramasinghe, R. C. Bray, V. Jipson, C. F. Quate, J. R. Salcedo, Appl. Phys. Lett. 33, 923 (1978).
[CrossRef]

Smith, D. C.

D. C. Smith, IEEE J. Quantum Electron. QE-5, 600 (1969).
[CrossRef]

Stone, J.

Swofford, R. L.

R. L. Swofford, J. A. Morrell, J. Appl. Phys. 49, 3667 (1978) and references therein.
[CrossRef]

Thomas, R. L.

R. L. Thomas, J. J. Pouch, Y. H. Wong, L. D. Favro, P. K. Kuo, A. Rosencwaig, J. Appl. Phys. 51, 1152 (1980) and references therein.
[CrossRef]

Van der Ziel, A.

A. Van der Ziel, Noise in Measurements (Wiley, New York, 1976).

Wetsel, G. C.

F. A. McDonald, G. C. Wetsel, J. Appl. Phys. 49, 2313 (1978).
[CrossRef]

Whinnery, J. R.

J. R. Whinnery, Acc. Chem. Res. 7, 225 (1974) and references therein.
[CrossRef]

Wickramasinghe, H. K.

H. K. Wickramasinghe, R. C. Bray, V. Jipson, C. F. Quate, J. R. Salcedo, Appl. Phys. Lett. 33, 923 (1978).
[CrossRef]

Wong, Y. H.

R. L. Thomas, J. J. Pouch, Y. H. Wong, L. D. Favro, P. K. Kuo, A. Rosencwaig, J. Appl. Phys. 51, 1152 (1980) and references therein.
[CrossRef]

Acc. Chem. Res. (1)

J. R. Whinnery, Acc. Chem. Res. 7, 225 (1974) and references therein.
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

A. C. Boccara, D. Fournier, J. Badoz, Appl. Phys. Lett. 36, 130 (1980).
[CrossRef]

H. K. Wickramasinghe, R. C. Bray, V. Jipson, C. F. Quate, J. R. Salcedo, Appl. Phys. Lett. 33, 923 (1978).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. C. Smith, IEEE J. Quantum Electron. QE-5, 600 (1969).
[CrossRef]

J. Appl. Phys. (10)

G. Busse, A. Ograbeck, J. Appl. Phys. 51, 3576 (1980) and references therein.
[CrossRef]

J. C. Murphy, L. C. Aamodt, J. Appl. Phys. 51, 4580 (1980).
[CrossRef]

F. A. McDonald, G. C. Wetsel, J. Appl. Phys. 49, 2313 (1978).
[CrossRef]

W. Jackson, N. M. Amer, J. Appl. Phys. 51, 3343 (1980).
[CrossRef]

H. S. Bennett, R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
[CrossRef]

L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 49, 3036 (1978).
[CrossRef]

A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
[CrossRef]

A. Mandalis, B. S. Royce, J. Appl. Phys. 50, 4331 (1979).

R. L. Thomas, J. J. Pouch, Y. H. Wong, L. D. Favro, P. K. Kuo, A. Rosencwaig, J. Appl. Phys. 51, 1152 (1980) and references therein.
[CrossRef]

R. L. Swofford, J. A. Morrell, J. Appl. Phys. 49, 3667 (1978) and references therein.
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

S. R. Brueck, H. Kildal, L. J. Belanger, Opt. Commun. 34, 199 (1980).
[CrossRef]

Opt. Lett. (1)

Other (9)

For an overview of photoacoustic spectroscopy, see Y.-H. Pao, Ed., Optoacoustic Spectroscopy and Detection (Academic, New York, 1977).

D. Fournier, A. C. Boccara, J. Badoz, in Digest of Topical Meeting on Photoacoustic Spectroscopy (Optical Society of America, Washington, D.C., 1979), paper ThA1.

A. Van der Ziel, Noise in Measurements (Wiley, New York, 1976).

D. Fournier, A. C. Boccara, in Scanned Image Microscopy, E. A. Ash, Ed. (Academic, London, 1980).

For some high frequency modulation experiments with very tight focusing, we observed a small secondary maximum with a 180° phase shift. Because the pump and probe beams intersect at an angle and the probe beam deflects slightly in region 0, a mechanism similar to that discussed in Sec. IV.A.2.3, could be present.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford, Clarendon, 1959).

Silicon Detector Corp., Newbury Park, Calif.

In some cases, it is experimentally advantageous to use a liquid in region 0 to make use of the typically higher dn/dt for liquids. In this case, it is obvious that the role of regions 0 and 2 should be interchanged.

M. Billardon, J. M. Ortega, E.S.P.C.I., private communication.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1

Geometry for theory. Heat deposited diffuses into regions 0 and 2 as well as radially. For transverse PDS, the probe beam axis may be displaced along the y axis by a distance y0.

Fig. 2
Fig. 2

Scattering geometry. Scattering region may focus the beam differently in the S1 and S2 directions (elliptical Gaussian beams).

Fig. 3
Fig. 3

Effective interaction length in collinear PDS. For simplicity, beams are assumed to be parallel but interact only over a distance li.

Fig. 4
Fig. 4

Experimental apparatus: (a) transverse PDS; (b) collinear PDS.

Fig. 5
Fig. 5

(a) Probe spot on detector; (b) continuous detector. Maximum distance from probe focal spot to the detector is dmax.

Fig. 6
Fig. 6

Collinear PDS: (a) signal amplitude vs beam displacement x0/a; (b) signal phase vs beam displacement x0/a.

Fig. 7
Fig. 7

Collinear PDS: (a) signal amplitude vs frequency; (b) signal phase vs frequency.

Fig. 8
Fig. 8

Transverse PDS. Signal vs frequency for various offsets z0. Beam radius is 140 μm.

Fig. 9
Fig. 9

Transverse PDS. Signal amplitude vs beam radius a for different beam offsets z0. Frequency is 48 Hz, and tilt angle is 0°.

Fig. 10
Fig. 10

Transverse PDS. Signal amplitude vs off-axis displacement y0 for various z0 offsets. Material is 600-nm edge filter glass, frequency is 48 Hz, tilt angle is 0°, and beam radius is 70 μm. Inset shows direction of heat flow and origin of the second maximum.

Fig. 11
Fig. 11

Transverse PDS. Signal amplitude vs tilt angle ψ. Frequency is 48 Hz, and the sample is 600-nm edge filter glass.

Fig. 12
Fig. 12

Transverse PDS. Signal amplitude vs beam offset y0 at 48 Hz. Tilt angle is 0°.

Fig. 13
Fig. 13

Signal vs time for benzene at 607 nm. Collinear PDS with the probe beam at x0 = a/2.

Fig. 14
Fig. 14

(a) Absorption vs wavelength for 0.1% benzene in distilled CCl4. (b) Signal vs time. Horizontal scale is 1 msec/div, wavelength is 606 nm, and 1024 averages were taken.

Tables (1)

Tables Icon

Table I Summary of Some Photothermally Based Spectroscopies

Equations (46)

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

2 T 0 1 k 0 T 0 t = 0 region 0 ,
2 T 1 1 k 1 T 1 t = Q ( r , t ) κ 1 region 1 ,
2 T 2 1 k 2 T 2 t = 0 region 2 ,
T 0 | z = 0 = T 1 | z = 0 , T 1 | z = l = T 2 | z = l ,
κ 0 T 0 z | z = 0 = κ 1 T 1 z | z = 0 , κ 1 T 1 z | z = l = κ 2 T 2 z | z = l ,
Q ( r , t ) = 1 2 4 P α π 2 a 2 exp ( α z ) exp ( 2 r 2 / a 2 ) exp ( i ω t ) + c . c .
T 0 ( r , t ) = 1 2 0 δ d δ J 0 ( δ r ) E ( δ ) exp ( β 0 z ) exp ( i ω t ) + c . c . ;
T 2 ( r , t ) = 1 2 0 δ d δ J 0 ( δ r ) D ( δ ) exp [ β 2 ( z l ) ] exp ( i ω t ) + c . c . ;
T 1 ( r , t ) = 1 2 0 δ d δ J 0 ( δ r ) [ Γ ( δ ) exp ( α z ) + A ( δ ) exp ( β 1 z ) + B ( δ ) exp ( β 1 z ) ] exp ( i ω t ) + c . c .,
Γ ( δ ) = P α π 2 κ 1 exp [ ( δ a ) 2 / 8 ] β 1 2 α 2 ,
β i 2 = δ 2 + i ω / k i .
A ( δ ) = [ ( 1 g ) ( b r ) exp ( α l ) + ( g + r ) ( 1 + b ) × exp ( β 1 l ) ] Γ ( δ ) / H ( δ ) , B ( δ ) = [ ( 1 + g ) ( b r ) exp ( α l ) + ( g + r ) ( 1 b ) × exp ( β 1 l ) ] Γ ( δ ) / H ( δ ) , D ( δ ) = Γ ( δ ) exp ( α l ) + A ( δ ) exp ( β 1 l ) + B ( δ ) exp ( β 1 l ) , E ( δ ) = Γ ( δ ) + A ( δ ) + B ( δ ) , H ( δ ) = [ ( 1 + g ) ( 1 + b ) exp ( β 1 l ) ( 1 g ) ( 1 b ) exp ( β 1 l ) ] ,
g = κ 0 β 0 / κ 1 β 1 , b = κ 2 β 2 / κ 1 β 1 , r = α / β 1 .
T ¯ 1 ( z ) = 2 π [ Γ ( 0 ) exp ( α z ) + A ( 0 ) exp ( K 1 z ) + B ( 0 ) exp ( K 1 z ) ] ,
K i 2 = i ω / k i and T ¯ 1 ( z ) = 2 π 0 r d r T 1 ( r , z ) .
T ¯ 1 ( z ) = P α H ( 0 ) π 2 κ 1 ( K 1 2 α 2 ) { ( 1 + g ) ( 1 + b ) exp ( K 1 l α z ) ( 1 g ) ( 1 b ) exp ( K 1 l α z ) ( 1 g ) ( b r ) × exp ( α l K 1 z ) ( g + r ) ( 1 + b ) exp [ K 1 ( l z ) ] ( 1 + g ) ( b r ) exp ( α l + K 1 z ) ( g + r ) ( 1 b ) × exp ( K 1 l + K 1 z ) } .
0 l d z T 1 ( r , t ) = 1 2 P [ 1 exp ( α l ) ] π 2 κ 1 × 0 δ J 0 ( δ r ) exp [ ( δ a ) 2 / 8 ] δ 2 + K 1 2 d δ exp ( i ω t ) + c . c .
0 l d z T 1 ( r , t ) = 1 2 P [ 1 exp ( α l ) ] 4 π 2 i ω ( ρ C ) 1 a 2 exp ( 2 r 2 / a 2 ) exp ( i ω t ) + c . c .
0 l d z T 1 ( r , t ) = 1 2 P [ 1 exp ( α l ) ] π 2 κ 1 r [ 1 exp ( 2 r 2 / a 2 ) ] exp ( i ω t ) + c . c .
n ( r , t ) = n 0 + Δ n ( r , t ) = n 0 + n T | T ambient T ( r , t ) ,
d d s ( n 0 d r 0 d s ) = n ( r , t ) ,
d d s ( 1 / q S i ) = ( 1 q S i ) 2 2 n n 0 S i 2 i = 1,2 ,
d r 0 d s = 1 n 0 path n ( r , t ) d s ,
1 / q S i | end of interaction 1 / q S i | beginning of interaction = path d s ( 1 2 2 n n 0 S i 2 q S i ) i = 1,2.
d r 0 d s ϕ = 1 n 0 n T path T ( r , t ) d s ,
1 / F i = 1 n 0 path 2 n S i 2 d s = 1 n 0 n T path 2 T S i 2 d s i = 1,2.
ϕ = exp ( i ω t ) 2 n 0 n T { 0 l d z 0 δ 2 d δ J 1 ( δ x 0 ) × [ Γ ( δ ) exp ( α z ) + A ( δ ) exp ( β 1 z ) + B ( δ ) exp ( β 1 z ) ] + 0 l i d z 0 δ 2 J 1 ( δ x 0 ) d δ D ( δ ) exp [ β 2 ( z l ) ] } + c . c . l i > l ,
ϕ = 1 2 n 0 n T exp ( i ω t ) 0 l i d z 0 δ 2 d δ J 1 ( δ x 0 ) [ Γ ( δ ) exp ( α z ) + A ( δ ) exp ( β 1 z ) B ( δ ) exp ( β 1 z ) ] + c . c . l i < l
y = y 0 z = ( tan ψ ) x + z 0 .
ϕ = 1 2 exp ( i ω t ) n 0 n T z 0 / ( tan ψ ) d x × 0 δ J 0 ( δ y 0 2 + x 2 ) β 0 E ( δ ) exp { ( tan ψ ) x + z 0 ] β 0 } d δ + c . c .
ϕ ( t ) = 1 2 π i c i c + i ϕ ( p ) exp ( p t ) d t .
T ( r , t ) = 0 t d t 0 2 π r d r Q ( r , t ) G ( r , r , t t ) ,
Q ( r , t ) = { ( 2 α E 0 / π a 2 t 0 ) exp ( 2 r 2 / a 2 ) 0 t t 0 , 0 t > t 0 , G ( r , r , t t ) = 1 4 π κ 1 ( t t ) exp [ ( r 2 + r 2 ) 4 k 1 ( t t ) ] × I 0 [ r r 2 k 1 ( t t ) ] ;
T r = α E 0 π κ 1 t 0 2 r { exp [ 2 r 2 / ( a 2 + 8 k 1 t ) ] exp ( 2 r 2 / a 2 ) } 0 t t 0 , T r = α E 0 π κ 1 t 0 2 r ( exp [ 2 r 2 / ( a 2 + 8 k 1 t ) ] exp { 2 r 2 / [ a 2 + 8 k 1 ( t t 0 ) ] } ) t > t 0 .
Δ V V = Δ I I 0 = 4 Δ x 0 2 π w 2 2 exp ( 2 r 2 / w 2 2 ) d r = ( 4 / 2 π ) Δ x w 2 ,
w 2 ( λ d ) / ( π w 0 n 0 ) ,
Δ V = 4 2 π ϕ π w 0 n 0 λ V ,
d max = ( l 1 π w 0 n 0 ) / λ ;
Δ V V = 0.55 ϕ 2 l 1 π w 0 n 0 λ .
i rms = ( 4 k B T F 1 ) 1 / 2 ( 1 R L + e I 0 2 k B T + i n 2 4 k B T + e n 2 4 R L 2 k B T + e n 2 ω 2 C 2 12 k B T ) 1 / 2 ,
( Δ I / I ) TL = 0.8 w 0 2 λ n t λ P l TL a 2 κ 1 ( 1 1 + t c / 2 t ) ,
( Δ I / I ) PDS = 4 α P 2 π w 0 λ n T l PDS a κ 1 { exp [ 1 / 2 ( 1 + 2 t / t c ) ] exp ( 1 / 2 ) } ;
( Δ I / I ) PDS ( Δ I / I ) TL = 0.78 l PDS l TL .
l PDS = l TL = 0.2 π w 0 2 n 0 / λ .
( acoustic / thermal ) = ( k 0 B T 2 ω T 0 / C ρ ) exp ( z 0 / l t ) ,
acoustic / thermal 10 11 ω exp ( z 0 / l t ) ,

Metrics