Abstract

Optical Gaussian beam interaction with a one-dimensional temperature field in the form of a thermal wave in the Raman–Nath configuration is analyzed. For the description of the Gaussian beam propagation through the nonstationary temperature field the complex geometric optics method was used. The influence of the refractive coefficient modulation by thermal wave on the complex ray phase, path, and amplitude was taken into account. It was assumed that for detection of the modulated Gaussian beam parameters two types of detector can be used: quadrant photodiodes or centroidal photodiodes. The influence of such parameters as the size and position of the Gaussian beam waist, the laser–screen (detector) distance, the thermal wave beam position and width, as well as thermal wave frequency and the distance between the probing optical beam axis and source of thermal waves on the so-called normal signal was taken into account.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 51, 4580-4588 (1980).
    [CrossRef]
  2. L. C. Aamodt and C. Murphy, “Photothermal measurement using localised excitation source,” J. Appl. Phys. 52, 4903-4914 (1981).
    [CrossRef]
  3. F. A. McDonald and G. C. Wetsel, Jr., “Resolution and definition in thermal imaging,” Proceedings of the 1984 Ultrasonics Symposium (IEEE, 1984), pp. 622-628.
    [CrossRef]
  4. F. A. McDonald, G. C. Wetsel, Jr., and G. E. Jamieson, “Photothermal beam-deflection imaging of vertical interfaces of solids,” Can. J. Phys. 64, 1265-1268 (1986).
    [CrossRef]
  5. E. L. Lasalle, F. Lepoutre, and J. P. Roger, “Probe beam size effects in photothermal deflection experiments,” J. Appl. Phys. 64, 1-5 (1988).
    [CrossRef]
  6. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford U. Press, Oxford, 1959).
  7. R. J. Bukowski, “Optical gaussian beam in acoustooptics. Theoretical description of noncollinear isotropic interactions,” Proc. SPIE 5828, 1-15 (2005).
    [CrossRef]
  8. A. Glazov and K. Muratikov, “Photodeflection signal formation in thermal wave spectroscopy and microscopy of solids within the framework of wave optics. “Mirage” effect geometry,” Opt. Commun. 84, 283-289 (1991).
    [CrossRef]
  9. A. Glazov and K. Muratikov, “Calculation of photodeflection signal in the framework of wave optics,” Tech. Phys. 38, 344-352 (1993).
  10. R. J. Bukowski, “Mirage effect description in the frame of complex rays optics,” Proc. SPIE 3581, 285-292 (1998).
    [CrossRef]
  11. R. J. Bukowski and D. Korte, “Gaussian beam phase change and deflection in temperature field of thermal wave,” presented at the Workshop 2001--Photoacoustics and Photothermics, Ebernburg, Germany, 26-28 September 2001.
  12. D. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, “The complex ray theory of photodeflection signal formation--comparison with the ray theory and experimental results,” J. Appl. Phys. 100, 063501 (2006).
    [CrossRef]
  13. D. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, “Photodeflection signal formation in photothermal measurements--comparison of the complex ray theory, the ray theory, the wave theory, and experimental results,” Appl. Opt. 46, 5216-5227 (2007).
    [CrossRef]
  14. D. K. Kobylińska, R. J. Bukowski, J. Bodzenta, S. Kochowski, and A. Kaźmierczak-Bałata, “Detector effects in photothermal deflection experiments,” Appl. Opt. 47, 1559-1566 (2008).
    [CrossRef] [PubMed]
  15. J. Petykiewicz, Wave Optics (Kluwer, 1992).
  16. Yu. A. Kravcov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (WNT, 1993) [Polish edition].
  17. Yu. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719-730 (1967).
    [CrossRef]
  18. J. B. Keller and W. Streifer, “Complex rays with applications to Gaussian beams,” J. Opt. Soc. Am. 61, 40-43 (1971).
    [CrossRef]
  19. J. C. Power and M. A. Schweitzer, “Diffraction theory of the impulse mirage effect,” Opt. Eng. 36, 521-534 (1997).
    [CrossRef]
  20. J. Bodzenta, “Thermal wave method in investigation of thermal properties of solids,” Eur. Phys. J. Spec. Top. 154, 305-311(2008).
    [CrossRef]

2008 (2)

2007 (1)

2006 (1)

D. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, “The complex ray theory of photodeflection signal formation--comparison with the ray theory and experimental results,” J. Appl. Phys. 100, 063501 (2006).
[CrossRef]

2005 (1)

R. J. Bukowski, “Optical gaussian beam in acoustooptics. Theoretical description of noncollinear isotropic interactions,” Proc. SPIE 5828, 1-15 (2005).
[CrossRef]

1998 (1)

R. J. Bukowski, “Mirage effect description in the frame of complex rays optics,” Proc. SPIE 3581, 285-292 (1998).
[CrossRef]

1997 (1)

J. C. Power and M. A. Schweitzer, “Diffraction theory of the impulse mirage effect,” Opt. Eng. 36, 521-534 (1997).
[CrossRef]

1993 (1)

A. Glazov and K. Muratikov, “Calculation of photodeflection signal in the framework of wave optics,” Tech. Phys. 38, 344-352 (1993).

1991 (1)

A. Glazov and K. Muratikov, “Photodeflection signal formation in thermal wave spectroscopy and microscopy of solids within the framework of wave optics. “Mirage” effect geometry,” Opt. Commun. 84, 283-289 (1991).
[CrossRef]

1988 (1)

E. L. Lasalle, F. Lepoutre, and J. P. Roger, “Probe beam size effects in photothermal deflection experiments,” J. Appl. Phys. 64, 1-5 (1988).
[CrossRef]

1986 (1)

F. A. McDonald, G. C. Wetsel, Jr., and G. E. Jamieson, “Photothermal beam-deflection imaging of vertical interfaces of solids,” Can. J. Phys. 64, 1265-1268 (1986).
[CrossRef]

1981 (1)

L. C. Aamodt and C. Murphy, “Photothermal measurement using localised excitation source,” J. Appl. Phys. 52, 4903-4914 (1981).
[CrossRef]

1980 (1)

C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 51, 4580-4588 (1980).
[CrossRef]

1971 (1)

1967 (1)

Yu. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719-730 (1967).
[CrossRef]

Aamodt, L. C.

L. C. Aamodt and C. Murphy, “Photothermal measurement using localised excitation source,” J. Appl. Phys. 52, 4903-4914 (1981).
[CrossRef]

C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 51, 4580-4588 (1980).
[CrossRef]

Bodzenta, J.

D. K. Kobylińska, R. J. Bukowski, J. Bodzenta, S. Kochowski, and A. Kaźmierczak-Bałata, “Detector effects in photothermal deflection experiments,” Appl. Opt. 47, 1559-1566 (2008).
[CrossRef] [PubMed]

J. Bodzenta, “Thermal wave method in investigation of thermal properties of solids,” Eur. Phys. J. Spec. Top. 154, 305-311(2008).
[CrossRef]

D. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, “Photodeflection signal formation in photothermal measurements--comparison of the complex ray theory, the ray theory, the wave theory, and experimental results,” Appl. Opt. 46, 5216-5227 (2007).
[CrossRef]

D. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, “The complex ray theory of photodeflection signal formation--comparison with the ray theory and experimental results,” J. Appl. Phys. 100, 063501 (2006).
[CrossRef]

Bukowski, R. J.

D. K. Kobylińska, R. J. Bukowski, J. Bodzenta, S. Kochowski, and A. Kaźmierczak-Bałata, “Detector effects in photothermal deflection experiments,” Appl. Opt. 47, 1559-1566 (2008).
[CrossRef] [PubMed]

D. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, “Photodeflection signal formation in photothermal measurements--comparison of the complex ray theory, the ray theory, the wave theory, and experimental results,” Appl. Opt. 46, 5216-5227 (2007).
[CrossRef]

D. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, “The complex ray theory of photodeflection signal formation--comparison with the ray theory and experimental results,” J. Appl. Phys. 100, 063501 (2006).
[CrossRef]

R. J. Bukowski, “Optical gaussian beam in acoustooptics. Theoretical description of noncollinear isotropic interactions,” Proc. SPIE 5828, 1-15 (2005).
[CrossRef]

R. J. Bukowski, “Mirage effect description in the frame of complex rays optics,” Proc. SPIE 3581, 285-292 (1998).
[CrossRef]

R. J. Bukowski and D. Korte, “Gaussian beam phase change and deflection in temperature field of thermal wave,” presented at the Workshop 2001--Photoacoustics and Photothermics, Ebernburg, Germany, 26-28 September 2001.

Burak, B.

D. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, “Photodeflection signal formation in photothermal measurements--comparison of the complex ray theory, the ray theory, the wave theory, and experimental results,” Appl. Opt. 46, 5216-5227 (2007).
[CrossRef]

D. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, “The complex ray theory of photodeflection signal formation--comparison with the ray theory and experimental results,” J. Appl. Phys. 100, 063501 (2006).
[CrossRef]

Carslaw, H. S.

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

Glazov, A.

A. Glazov and K. Muratikov, “Calculation of photodeflection signal in the framework of wave optics,” Tech. Phys. 38, 344-352 (1993).

A. Glazov and K. Muratikov, “Photodeflection signal formation in thermal wave spectroscopy and microscopy of solids within the framework of wave optics. “Mirage” effect geometry,” Opt. Commun. 84, 283-289 (1991).
[CrossRef]

Jaeger, J. C.

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

Jamieson, G. E.

F. A. McDonald, G. C. Wetsel, Jr., and G. E. Jamieson, “Photothermal beam-deflection imaging of vertical interfaces of solids,” Can. J. Phys. 64, 1265-1268 (1986).
[CrossRef]

Kazmierczak-Balata, A.

Keller, J. B.

Kobylinska, D.

D. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, “Photodeflection signal formation in photothermal measurements--comparison of the complex ray theory, the ray theory, the wave theory, and experimental results,” Appl. Opt. 46, 5216-5227 (2007).
[CrossRef]

D. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, “The complex ray theory of photodeflection signal formation--comparison with the ray theory and experimental results,” J. Appl. Phys. 100, 063501 (2006).
[CrossRef]

Kobylinska, D. K.

Kochowski, S.

Korte, D.

R. J. Bukowski and D. Korte, “Gaussian beam phase change and deflection in temperature field of thermal wave,” presented at the Workshop 2001--Photoacoustics and Photothermics, Ebernburg, Germany, 26-28 September 2001.

Kravcov, Yu. A.

Yu. A. Kravcov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (WNT, 1993) [Polish edition].

Kravtsov, Yu. A.

Yu. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719-730 (1967).
[CrossRef]

Lasalle, E. L.

E. L. Lasalle, F. Lepoutre, and J. P. Roger, “Probe beam size effects in photothermal deflection experiments,” J. Appl. Phys. 64, 1-5 (1988).
[CrossRef]

Lepoutre, F.

E. L. Lasalle, F. Lepoutre, and J. P. Roger, “Probe beam size effects in photothermal deflection experiments,” J. Appl. Phys. 64, 1-5 (1988).
[CrossRef]

McDonald, F. A.

F. A. McDonald, G. C. Wetsel, Jr., and G. E. Jamieson, “Photothermal beam-deflection imaging of vertical interfaces of solids,” Can. J. Phys. 64, 1265-1268 (1986).
[CrossRef]

F. A. McDonald and G. C. Wetsel, Jr., “Resolution and definition in thermal imaging,” Proceedings of the 1984 Ultrasonics Symposium (IEEE, 1984), pp. 622-628.
[CrossRef]

Muratikov, K.

A. Glazov and K. Muratikov, “Calculation of photodeflection signal in the framework of wave optics,” Tech. Phys. 38, 344-352 (1993).

A. Glazov and K. Muratikov, “Photodeflection signal formation in thermal wave spectroscopy and microscopy of solids within the framework of wave optics. “Mirage” effect geometry,” Opt. Commun. 84, 283-289 (1991).
[CrossRef]

Murphy, C.

L. C. Aamodt and C. Murphy, “Photothermal measurement using localised excitation source,” J. Appl. Phys. 52, 4903-4914 (1981).
[CrossRef]

C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 51, 4580-4588 (1980).
[CrossRef]

Orlov, Yu. I.

Yu. A. Kravcov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (WNT, 1993) [Polish edition].

Petykiewicz, J.

J. Petykiewicz, Wave Optics (Kluwer, 1992).

Power, J. C.

J. C. Power and M. A. Schweitzer, “Diffraction theory of the impulse mirage effect,” Opt. Eng. 36, 521-534 (1997).
[CrossRef]

Roger, J. P.

E. L. Lasalle, F. Lepoutre, and J. P. Roger, “Probe beam size effects in photothermal deflection experiments,” J. Appl. Phys. 64, 1-5 (1988).
[CrossRef]

Schweitzer, M. A.

J. C. Power and M. A. Schweitzer, “Diffraction theory of the impulse mirage effect,” Opt. Eng. 36, 521-534 (1997).
[CrossRef]

Streifer, W.

Wetsel, G. C.

F. A. McDonald, G. C. Wetsel, Jr., and G. E. Jamieson, “Photothermal beam-deflection imaging of vertical interfaces of solids,” Can. J. Phys. 64, 1265-1268 (1986).
[CrossRef]

F. A. McDonald and G. C. Wetsel, Jr., “Resolution and definition in thermal imaging,” Proceedings of the 1984 Ultrasonics Symposium (IEEE, 1984), pp. 622-628.
[CrossRef]

Appl. Opt. (2)

Can. J. Phys. (1)

F. A. McDonald, G. C. Wetsel, Jr., and G. E. Jamieson, “Photothermal beam-deflection imaging of vertical interfaces of solids,” Can. J. Phys. 64, 1265-1268 (1986).
[CrossRef]

Eur. Phys. J. Spec. Top. (1)

J. Bodzenta, “Thermal wave method in investigation of thermal properties of solids,” Eur. Phys. J. Spec. Top. 154, 305-311(2008).
[CrossRef]

J. Appl. Phys. (4)

D. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, “The complex ray theory of photodeflection signal formation--comparison with the ray theory and experimental results,” J. Appl. Phys. 100, 063501 (2006).
[CrossRef]

E. L. Lasalle, F. Lepoutre, and J. P. Roger, “Probe beam size effects in photothermal deflection experiments,” J. Appl. Phys. 64, 1-5 (1988).
[CrossRef]

C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 51, 4580-4588 (1980).
[CrossRef]

L. C. Aamodt and C. Murphy, “Photothermal measurement using localised excitation source,” J. Appl. Phys. 52, 4903-4914 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

A. Glazov and K. Muratikov, “Photodeflection signal formation in thermal wave spectroscopy and microscopy of solids within the framework of wave optics. “Mirage” effect geometry,” Opt. Commun. 84, 283-289 (1991).
[CrossRef]

Opt. Eng. (1)

J. C. Power and M. A. Schweitzer, “Diffraction theory of the impulse mirage effect,” Opt. Eng. 36, 521-534 (1997).
[CrossRef]

Proc. SPIE (2)

R. J. Bukowski, “Mirage effect description in the frame of complex rays optics,” Proc. SPIE 3581, 285-292 (1998).
[CrossRef]

R. J. Bukowski, “Optical gaussian beam in acoustooptics. Theoretical description of noncollinear isotropic interactions,” Proc. SPIE 5828, 1-15 (2005).
[CrossRef]

Radiophys. Quantum Electron. (1)

Yu. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719-730 (1967).
[CrossRef]

Tech. Phys. (1)

A. Glazov and K. Muratikov, “Calculation of photodeflection signal in the framework of wave optics,” Tech. Phys. 38, 344-352 (1993).

Other (5)

R. J. Bukowski and D. Korte, “Gaussian beam phase change and deflection in temperature field of thermal wave,” presented at the Workshop 2001--Photoacoustics and Photothermics, Ebernburg, Germany, 26-28 September 2001.

F. A. McDonald and G. C. Wetsel, Jr., “Resolution and definition in thermal imaging,” Proceedings of the 1984 Ultrasonics Symposium (IEEE, 1984), pp. 622-628.
[CrossRef]

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

J. Petykiewicz, Wave Optics (Kluwer, 1992).

Yu. A. Kravcov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (WNT, 1993) [Polish edition].

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 (6)

Fig. 1
Fig. 1

Experimental setup for the solid state photothermal investigation with photodeflectional detection. The uniformly gas-heated region has a width of Δ z = z p z l , and its left edge is at a distance z l from the system beginning (the light beam “input”). The light beam waist has a radius of b 0 and is placed at distance L from the beam “input.” b s and b D are the probing beam radii over the sample center and at the detector, respectively. The probing beam axis runs at height h over the sample surface and is perpendicular to the thermal wave column (Raman–Nath configuration). The detector plane coordinate is equal to z D .

Fig. 2
Fig. 2

Dependence of the apparatus function amplitude on the sample position for L / z D = 0.33 and for different modulation frequencies f and for different probing beam radii b 0 . Left column, quadrant diode; right column, centroid diode ( z D = 1.5 m , Δ z = 5 mm ).

Fig. 3
Fig. 3

Dependence of the apparatus function amplitude on the probing beam radius b 0 in the beam waist for different modulation frequencies f and for different probing beam waist positions L. Left column, quadrant diode; right column, centroid diode ( z D = 1.5 m , Δ z = 5 mm ).

Fig. 4
Fig. 4

Optimal probing beam radii b opt obtained on the basis of maxima observed in Fig. 3. Left column, quadrant diode; right column, centroid diode. From the upper row it appears that the b opt values practically do not depend on probing beam waist position L, but from the bottom row it is clear that they depend on sample position z l .

Fig. 5
Fig. 5

Dependence of the apparatus function amplitude on the probing beam height h over the sample surface for different modulation frequencies f and for different probing beam radii b 0 in the beam waist. Left column, quadrant diode; right column, centroid diode ( z D = 1.5 m , Δ z = 5 mm , L = 0.5 m , z l = 0.5 m ).

Fig. 6
Fig. 6

Optimal probing beam heights h opt obtained on the base of maxima observed in Fig. 5. Left column, quadrant diode; right column, centroidal diode.

Equations (46)

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

Δ u ( r ) + k 0 2 ε ( r ) u ( r ) = 0 ,
u ( r ) = A ( r ) exp ( i k 0 ψ ( r ) ) , A ( r ) = m = 0 A m ( r ) ( i k 0 ) m .
{ ( ψ ) 2 = n 2 ( r ) 2 ( A 0 ) ( ψ ) + A 0 Δ ψ = 0 2 ( A m ) ( ψ ) + A m Δ ψ = Δ A m 1 , m = 1 , 2 , .
{ d r d τ = p d p d τ = 1 2 n 2 ( r ) , { p = ψ d ψ d τ = p 2 = n 2 ( r ) .
ψ ( τ ) = ψ 0 ( ξ , η ) + 0 τ n 2 [ r ( τ ) ] d τ , A 0 ( τ ) = A 0 ( 0 ) [ D ( 0 ) D ( τ ) ] 1 / 2 ,
u ( ξ , η , τ ) E 0 z R c z R c + i n 0 τ exp [ i k 0 n 0 ( ( n 0 τ L ) + i ξ 2 + η 2 2 z R z R c z R ) ] ,
{ x = ξ ( 1 + i n 0 τ z R c ) y = η ( 1 + i n 0 τ z R c ) z = n 0 τ 1 + ξ 2 + η 2 z R c 2 .
T ( x , z ) T 0 = ϑ ( x , z ) = [ θ g + ϑ g exp ( k g ( x + h ) ) cos ( Ω t k g ( x + h ) + γ g ) ] Δ H ( z ; z l , z p ) .
n ( T ) = n 0 + d n d T | T 0 ( T T 0 ) = n 0 + n 0 s T ( T T 0 ) ,
ε ( T ) = n 2 ( T ) = n 0 2 + ν ( r ) , ν ( r ) 2 n 0 2 s T ϑ ( x , z ) .
ν ( r ) 2 n 0 2 s T ϑ c ( x , z ) ,
ϑ c ( x , z ) = [ θ g + ϑ g exp ( φ t ) exp ( k g c x ) ] Δ H ( z ; z l , z p ) ,
k g c = ( 1 + i ) k g , φ t = i ( Ω t + γ g ) k g c h .
r 1 = 0 τ ( τ τ ) 1 2 ν ( r 0 ( τ ) ) d τ .
x 1 = s T ϑ g z R c k g c ξ 2 exp ( φ t ) × { [ 1 k g c ξ i n 0 ( τ τ p ) z R c ] exp [ k g c ξ ( 1 + i n 0 τ p z R c ) ] [ 1 k g c ξ i n 0 ( τ τ l ) z R c ] exp [ k g c ξ ( 1 + i n 0 τ l z R c ) ] } ,
y 1 = 0 ,
z 1 = s T ϑ g n 0 exp ( φ t ) 1 + ( ξ 2 + η 2 ) / z R c 2 { ( τ τ l ) exp [ k g c ξ ( 1 + i n 0 τ l z R c ) ] ( τ τ p ) exp [ k g c ξ ( 1 + i n 0 τ p z R c ) ] } .
ψ 1 ( ξ , η , τ ) = 1 2 0 τ ν ( r ( τ ) ) d τ .
ψ 1 ( ξ , η , τ ) = s T ϑ g i n 0 z R c k g c ξ exp ( φ t ) { exp [ k g c ξ ( 1 + i n 0 τ p z R c ) ] exp [ k g c ξ ( 1 + i n 0 τ l z R c ) ] } .
D ( τ ) x 0 ξ y 0 η z 0 τ + x 1 ξ y 0 η z 0 τ + x 0 ξ y 0 η z 1 τ = D 0 ( τ ) + D 1 ( τ ) ,
A ( τ ) A 0 ( τ ) [ 1 + a 1 ( τ ) ] , a 1 ( τ ) = 1 2 [ x 1 / ξ x 0 / ξ + z 1 / τ z 0 / τ ] ,
a 1 ( τ ) = 1 2 s T ϑ g exp ( φ t ) { a p 1 ( τ ) exp [ k g c ξ ( 1 + i n 0 τ p z R c ) ] a l 1 ( τ ) exp [ k g c ξ ( 1 + i n 0 τ l z R c ) ] } ,
a r 1 ( τ ) = 1 z R c 2 k g c ξ ( 1 + i n 0 τ z R c ) [ k g c 2 ( 1 + i n 0 τ r z R c ) i n 0 ( τ τ r ) z R c 1 ξ ( k g c ( 1 i n 0 ( τ 2 τ r ) z R c ) + 2 ξ ) ] , r = l , p .
u ( ξ , η , τ ) = A 0 ( τ ) [ 1 + a 1 ( τ ) ] exp { i k 0 [ ψ 0 ( ξ , η , τ ) + ψ 1 ( ξ , η , τ ) ] } ,
ψ 0 ( ξ , η , τ ) = n 0 [ ( n 0 τ L ) + i ξ 2 + η 2 2 z R z R c z R ] .
{ x D = x 0 ( ξ D , η D , τ D ) + x 1 ( ξ D , η D , τ D ) y D = y 0 ( ξ D , η D , τ D ) z D = z 0 ( ξ D , η D , τ D ) + z 1 ( ξ D , η D , τ D ) ,
x 1 ( ξ D , η D , τ D ) x 1 ( ξ D 0 , η D 0 , τ D 0 ) = x 10 , z 1 ( ξ D , η D , τ D ) z 1 ( ξ D 0 , η D 0 , τ D 0 ) = z 10 ,
ξ D = ξ D 0 + ξ D 1 , η D = η D 0 + η D 1 , τ D = τ D 0 + τ D 1 ,
{ τ D 1 x D z D n 0 z R D 2 x 10 1 n 0 ( 1 + i x D 2 + y D 2 z R D 2 z D z R D ) z 10 η D 1 i x D y D z D z R c z R D 4 x 10 + i y D z R c z R D 2 z 10 ξ D 1 z R c z R D ( 1 i x D 2 z R D 2 z D z R D ) x 10 + i x D z R c z R D 2 z 10 ,
ψ ( ξ D , η D , τ D ) ψ 0 ( ξ D , η D , τ D ) + ψ 1 ( ξ D 0 , η D 0 , τ D 0 ) = ψ D 0 ( r D ) = n 0 ( z D L ) + i n 0 x D 2 + y D 2 2 z R D .
A ( r D ) E 0 z R c z R D [ 1 + a D 1 ( r D ) ] ,
a D 1 ( r D ) = s T ϑ g 1 2 k g c x D 3 z R D 2 exp ( φ t ) [ a l 1 ( r D ) exp ( k g c x D z R p z R D ) a p 1 ( r D ) exp ( k g c x D z R l z R D ) ] ,
a r 1 ( r D ) = k g c [ z R D 2 2 z R c ( z R D z R r ) ] x D 3 + z R D [ k g c 2 z R D z R r ( z R D z R r ) 2 ( z R D z R c ) ] x D 2 + k g c z R D 3 ( z R D 2 z R r ) x D 2 z R D 4 , r = l , p .
I ( r D ) I G 0 ( r D ) + 2 Re ( a D 1 ( r D ) ) I G 0 ( r D ) = I G 0 ( r D ) + I V ( r D ) ,
F ( q ) = i F i ( q ) = i σ i K q S D i I V ( r D ) d S D ,
F n ( q ) = K q + d y D ( 0 + h D 0 ) d x D I V ( r D ) = 2 K q Re [ h D + sgn ( x D ) a D 1 ( x D , z D ) I G 0 x ( x D ) d x D ] ,
I G 0 x ( x D ) = + I G 0 ( r D ) d y D = I 0 D + exp [ ( x D 2 + y D 2 ) b D 2 ] d y D = P G π b D exp ( x D 2 b D 2 ) .
F n ( c ) = K c S x D I V ( r D ) d S D = 2 K c Re [ h D + x D a D 1 ( x D , z D ) I G 0 x ( x D ) d x D ] ,
F n ( p ) = K n ( p ) Re [ ϑ s ( t ) f a p ( p ) ( z D , z l , z p , h , h D , Ω , b 0 , L ) ] , ( p ) = ( q ) , ( c ) ,
h D = h z R 2 ( z D L ) 2 z R 2 ( z s L ) 2 , z s = z l + z p 2 .
f a p ( c ) = π exp ( k g c h ) [ f l 4 k g c z R D 3 exp ( b D 2 k g c 2 4 z R D 2 z R l 2 ) f p 4 k g c z R D 3 exp ( b D 2 k g c 2 4 z R D 2 z R p 2 ) + f p l ] ,
f r = b D 4 k g c 2 z R r ( z R D 2 2 z R D z R c + 2 z R c z R r ) + 2 b D 2 z R D 2 [ k g c 2 z R D z R r ( z R D z R r ) + 2 ( z R D z R c ) + 8 z R D 5 ] ; r = l , p ,
f p l = i π b D z R D 2 [ exp ( i b D k g c 2 z R D z R l ) exp ( i b D k g c 2 z R D z R p ) ] ,
k g c = ( 1 + i ) k g , b D = b 0 1 + ( z D L ) 2 z R 2 ,
z R r = z R c + i z r , r = l , p , D ,
z R c = z R iL , z R = k 0 n 0 b 0 2 , k g = Ω / ( 2 κ g ) .

Metrics