Abstract

We report on a new method of inspecting deep microstructures manufactured in transparent media. Although their lateral dimension (tens of microns) do not exceed the diffraction limit for optical microscopy resolution, their deepness makes the nondestructive measurements practically impossible with presently available methods. We show that the optical vortex interferometer with a vortex generator can be used to differentiate between the samples of good and poor quality. The measurement system is simple and the interpretation of the results is straightforward.

© 2008 Optical Society of America

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References

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  1. M. S. Soskin and M. V. Vasnetsov, Singular Optics, Prog. Opt. (Elsevier Science, 2001). Vol. 42, pp. 219-276 (2001).
    [CrossRef]
  2. M. Vasnetsov and K. Staliunas, eds., Optical Vortices, (Nova Science, New York, 1999).
  3. J. Masajada, "The interferometry based on the regular lattice of optical vortices," Opt. Appl. 37,167-185 (2007).
  4. J. Masajada, A Popio?ek-Masajada, and D Wieliczka, "The interferometric system using optical vortices as phase markers," Opt. Commun. 207,85-93 (2002).
    [CrossRef]
  5. O. V. Angelsky, R. N. Besha, and I. I. Mokhun, "Appearance of wave front dislocations under interference among beams with simple wave fronts," Opt. Appl. 37,273-278 (1997).
  6. P. Kurzynowski, W. A. Wo?niak, and E. Fr?czek, "Optical vortices generation using Wollastone prism," Appl. Opt. 45, 7898-7903 (2006).
    [CrossRef] [PubMed]
  7. P. Kurzynowski and M. Borwi?ska, "Generation of vortex-type markers in a one-way setup," Appl. Opt. 46,676-679 (2007).
    [CrossRef] [PubMed]
  8. J. Masajada, A. Popio?ek-Masajada, and M. Leniec, "Creation of vortex lattice by a wavefront division," Opt. Express 15, 5196-5207 (2007), http://www.opticsexpress.org/abstract.cfm? URI=oe-15-8-5196
    [CrossRef] [PubMed]
  9. S. Velghe and J. Primot, "Nondiffracting array generation using an N-wave interferometer," J. Opt. Soc. Am. A 16, 293-298 (1999).
    [CrossRef]
  10. S. Vyas and P. Senthilkumaran, "Interferometric optical vortex array generator," Appl. Opt. 46,2893-2898 (2007).
    [CrossRef] [PubMed]
  11. N. R. Heckenberg, R. G. Mcduff, C. P. Smith, and A. G. White, "Generation of optical phase singularities by computer-generated holograms," Opt. Lett. 17, 221-223 (1992).
    [CrossRef] [PubMed]
  12. V. A Soifer and M. Golub, Laser beam mode selection by computer generated holograms, (CRC Press, London, 1999).
  13. D. Ganic, X. Gan, and M. Gu, "Generation of doughnut laser beams by use of a liquid-crystal cell with a conversion efficiency near 100%," Opt. Lett. 27,1351-1353 (2002).
    [CrossRef]
  14. W. Wang, N Ishii, S. Hanson, Y Miyamoto, and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248,59-68 (2005).
    [CrossRef]
  15. W. Wang, T. Yokozeki, R. Ishjima, A Wada, Y. Miyamoto, M. Takeda, and S. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express, 14, 120-127 (2006), http://www.opticsexpress.org/abstract.cfm? URI=oe-14-1-120
    [CrossRef] [PubMed]
  16. W. Wang, T. Yokozeki, R. Ishijima, S. G. Hanson, and M. Takeda, "Optical vortex metrology based on the core structures of phase singularities in Laguerre-Gauss transform of a speckle pattern," Opt. Express 14, 10195-10206 (2006), http://www.opticsexpress.org/abstract.cfm? URI=oe-14-22-10195
    [CrossRef] [PubMed]
  17. V. P. Tychinsky and C. H. F. Velzel, Super-resolution in Microscopy, in: Current trends in optics, (Academic Press, 1994), Chap. 18.
  18. M. Totzeck and H. J. Tiziani, "Phase-singularities in 2D diffraction fields and interference microscopy," Opt. Commun. 138,365-382 (1997).
    [CrossRef]
  19. B. Sektor, A. Normatov, and J. Shamir, "Experimetnal validation of 20nm sensitivity of Singular Beam Microscopy," Proc. SPIE 6616,661622 (2007).
    [CrossRef]
  20. B. Sektor, A. Normatov, and J. Shamir, "Singular Beam Microscopy," Appl. Opt. 47, 78-87 (2008).
    [CrossRef]
  21. J. Masajada, Optical vortices and their application to interferometry, (Oficyna Wydawnicza Politechniki Wroc?awskiej, Wroc?aw 2004), monographs of Wroclaw University of Technology.
  22. W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems, Vol. 2, Physical Image Formation, (Wiley, New York 2005) Chap. 23
  23. J. T. Sheridan and C. J. R. Sheppard, "An examination of the theories for the calculation of diffraction by square wave gratings: 1. Thickness and period variations for normal incidence," Optik 85, 25-32 (1990).
  24. D. P. Biss, K. S. Youngworth, and T. G. Brown, "Dark-field imaging with cylindrical-vector beams," Appl. Opt. 45,470-479 (2006).
    [CrossRef] [PubMed]
  25. I. Freund andV. Freilikher, "Parametrization of anisotropic vortices," J. Opt. Soc. Am. A 14, 1902-1910 (1997).
    [CrossRef]
  26. W. Wang, M. R. Dennis, R. Ishijima, T. Yokozeki, A. Matsuda, S. G. Hanson, and M. Takeda, "Poincaré sphere representation for the anisotropy of phase singularities and its applications to optical vortex metrology for fluid mechanical analysis," Opt. Express 15, 11008-11019 (2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-17-11008
    [CrossRef] [PubMed]
  27. A. Erdèlyi, ed., Tables of integral transforms, (McGraw-Hill, New York, 1953).

2008

B. Sektor, A. Normatov, and J. Shamir, "Singular Beam Microscopy," Appl. Opt. 47, 78-87 (2008).
[CrossRef]

2007

J. Masajada, "The interferometry based on the regular lattice of optical vortices," Opt. Appl. 37,167-185 (2007).

B. Sektor, A. Normatov, and J. Shamir, "Experimetnal validation of 20nm sensitivity of Singular Beam Microscopy," Proc. SPIE 6616,661622 (2007).
[CrossRef]

P. Kurzynowski and M. Borwi?ska, "Generation of vortex-type markers in a one-way setup," Appl. Opt. 46,676-679 (2007).
[CrossRef] [PubMed]

S. Vyas and P. Senthilkumaran, "Interferometric optical vortex array generator," Appl. Opt. 46,2893-2898 (2007).
[CrossRef] [PubMed]

2006

2005

W. Wang, N Ishii, S. Hanson, Y Miyamoto, and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248,59-68 (2005).
[CrossRef]

2002

J. Masajada, A Popio?ek-Masajada, and D Wieliczka, "The interferometric system using optical vortices as phase markers," Opt. Commun. 207,85-93 (2002).
[CrossRef]

D. Ganic, X. Gan, and M. Gu, "Generation of doughnut laser beams by use of a liquid-crystal cell with a conversion efficiency near 100%," Opt. Lett. 27,1351-1353 (2002).
[CrossRef]

1999

1997

I. Freund andV. Freilikher, "Parametrization of anisotropic vortices," J. Opt. Soc. Am. A 14, 1902-1910 (1997).
[CrossRef]

O. V. Angelsky, R. N. Besha, and I. I. Mokhun, "Appearance of wave front dislocations under interference among beams with simple wave fronts," Opt. Appl. 37,273-278 (1997).

M. Totzeck and H. J. Tiziani, "Phase-singularities in 2D diffraction fields and interference microscopy," Opt. Commun. 138,365-382 (1997).
[CrossRef]

1992

1990

J. T. Sheridan and C. J. R. Sheppard, "An examination of the theories for the calculation of diffraction by square wave gratings: 1. Thickness and period variations for normal incidence," Optik 85, 25-32 (1990).

Angelsky, O. V.

O. V. Angelsky, R. N. Besha, and I. I. Mokhun, "Appearance of wave front dislocations under interference among beams with simple wave fronts," Opt. Appl. 37,273-278 (1997).

Besha, R. N.

O. V. Angelsky, R. N. Besha, and I. I. Mokhun, "Appearance of wave front dislocations under interference among beams with simple wave fronts," Opt. Appl. 37,273-278 (1997).

Biss, D. P.

Borwinska, M.

Brown, T. G.

Fraczek, E.

Freilikher, V.

Freund, I.

Gan, X.

Ganic, D.

Gu, M.

Hanson, S.

W. Wang, N Ishii, S. Hanson, Y Miyamoto, and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248,59-68 (2005).
[CrossRef]

Heckenberg, N. R.

Ishii, N

W. Wang, N Ishii, S. Hanson, Y Miyamoto, and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248,59-68 (2005).
[CrossRef]

Kurzynowski, P.

Masajada, J.

J. Masajada, "The interferometry based on the regular lattice of optical vortices," Opt. Appl. 37,167-185 (2007).

J. Masajada, A Popio?ek-Masajada, and D Wieliczka, "The interferometric system using optical vortices as phase markers," Opt. Commun. 207,85-93 (2002).
[CrossRef]

Mcduff, R. G.

Miyamoto, Y

W. Wang, N Ishii, S. Hanson, Y Miyamoto, and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248,59-68 (2005).
[CrossRef]

Mokhun, I. I.

O. V. Angelsky, R. N. Besha, and I. I. Mokhun, "Appearance of wave front dislocations under interference among beams with simple wave fronts," Opt. Appl. 37,273-278 (1997).

Normatov, A.

B. Sektor, A. Normatov, and J. Shamir, "Singular Beam Microscopy," Appl. Opt. 47, 78-87 (2008).
[CrossRef]

B. Sektor, A. Normatov, and J. Shamir, "Experimetnal validation of 20nm sensitivity of Singular Beam Microscopy," Proc. SPIE 6616,661622 (2007).
[CrossRef]

Popiolek-Masajada, A

J. Masajada, A Popio?ek-Masajada, and D Wieliczka, "The interferometric system using optical vortices as phase markers," Opt. Commun. 207,85-93 (2002).
[CrossRef]

Primot, J.

Sektor, B.

B. Sektor, A. Normatov, and J. Shamir, "Singular Beam Microscopy," Appl. Opt. 47, 78-87 (2008).
[CrossRef]

B. Sektor, A. Normatov, and J. Shamir, "Experimetnal validation of 20nm sensitivity of Singular Beam Microscopy," Proc. SPIE 6616,661622 (2007).
[CrossRef]

Senthilkumaran, P.

Shamir, J.

B. Sektor, A. Normatov, and J. Shamir, "Singular Beam Microscopy," Appl. Opt. 47, 78-87 (2008).
[CrossRef]

B. Sektor, A. Normatov, and J. Shamir, "Experimetnal validation of 20nm sensitivity of Singular Beam Microscopy," Proc. SPIE 6616,661622 (2007).
[CrossRef]

Sheppard, C. J. R.

J. T. Sheridan and C. J. R. Sheppard, "An examination of the theories for the calculation of diffraction by square wave gratings: 1. Thickness and period variations for normal incidence," Optik 85, 25-32 (1990).

Sheridan, J. T.

J. T. Sheridan and C. J. R. Sheppard, "An examination of the theories for the calculation of diffraction by square wave gratings: 1. Thickness and period variations for normal incidence," Optik 85, 25-32 (1990).

Smith, C. P.

Takeda, M.

W. Wang, N Ishii, S. Hanson, Y Miyamoto, and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248,59-68 (2005).
[CrossRef]

Tiziani, H. J.

M. Totzeck and H. J. Tiziani, "Phase-singularities in 2D diffraction fields and interference microscopy," Opt. Commun. 138,365-382 (1997).
[CrossRef]

Totzeck, M.

M. Totzeck and H. J. Tiziani, "Phase-singularities in 2D diffraction fields and interference microscopy," Opt. Commun. 138,365-382 (1997).
[CrossRef]

Velghe, S.

Vyas, S.

Wang, W.

W. Wang, N Ishii, S. Hanson, Y Miyamoto, and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248,59-68 (2005).
[CrossRef]

White, A. G.

Wieliczka, D

J. Masajada, A Popio?ek-Masajada, and D Wieliczka, "The interferometric system using optical vortices as phase markers," Opt. Commun. 207,85-93 (2002).
[CrossRef]

Wozniak, W. A.

Youngworth, K. S.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Appl.

O. V. Angelsky, R. N. Besha, and I. I. Mokhun, "Appearance of wave front dislocations under interference among beams with simple wave fronts," Opt. Appl. 37,273-278 (1997).

J. Masajada, "The interferometry based on the regular lattice of optical vortices," Opt. Appl. 37,167-185 (2007).

Opt. Commun.

J. Masajada, A Popio?ek-Masajada, and D Wieliczka, "The interferometric system using optical vortices as phase markers," Opt. Commun. 207,85-93 (2002).
[CrossRef]

W. Wang, N Ishii, S. Hanson, Y Miyamoto, and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248,59-68 (2005).
[CrossRef]

M. Totzeck and H. J. Tiziani, "Phase-singularities in 2D diffraction fields and interference microscopy," Opt. Commun. 138,365-382 (1997).
[CrossRef]

Opt. Lett.

Optik

J. T. Sheridan and C. J. R. Sheppard, "An examination of the theories for the calculation of diffraction by square wave gratings: 1. Thickness and period variations for normal incidence," Optik 85, 25-32 (1990).

Proc. SPIE

B. Sektor, A. Normatov, and J. Shamir, "Experimetnal validation of 20nm sensitivity of Singular Beam Microscopy," Proc. SPIE 6616,661622 (2007).
[CrossRef]

Other

J. Masajada, A. Popio?ek-Masajada, and M. Leniec, "Creation of vortex lattice by a wavefront division," Opt. Express 15, 5196-5207 (2007), http://www.opticsexpress.org/abstract.cfm? URI=oe-15-8-5196
[CrossRef] [PubMed]

V. A Soifer and M. Golub, Laser beam mode selection by computer generated holograms, (CRC Press, London, 1999).

W. Wang, M. R. Dennis, R. Ishijima, T. Yokozeki, A. Matsuda, S. G. Hanson, and M. Takeda, "Poincaré sphere representation for the anisotropy of phase singularities and its applications to optical vortex metrology for fluid mechanical analysis," Opt. Express 15, 11008-11019 (2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-17-11008
[CrossRef] [PubMed]

A. Erdèlyi, ed., Tables of integral transforms, (McGraw-Hill, New York, 1953).

J. Masajada, Optical vortices and their application to interferometry, (Oficyna Wydawnicza Politechniki Wroc?awskiej, Wroc?aw 2004), monographs of Wroclaw University of Technology.

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems, Vol. 2, Physical Image Formation, (Wiley, New York 2005) Chap. 23

W. Wang, T. Yokozeki, R. Ishjima, A Wada, Y. Miyamoto, M. Takeda, and S. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express, 14, 120-127 (2006), http://www.opticsexpress.org/abstract.cfm? URI=oe-14-1-120
[CrossRef] [PubMed]

W. Wang, T. Yokozeki, R. Ishijima, S. G. Hanson, and M. Takeda, "Optical vortex metrology based on the core structures of phase singularities in Laguerre-Gauss transform of a speckle pattern," Opt. Express 14, 10195-10206 (2006), http://www.opticsexpress.org/abstract.cfm? URI=oe-14-22-10195
[CrossRef] [PubMed]

V. P. Tychinsky and C. H. F. Velzel, Super-resolution in Microscopy, in: Current trends in optics, (Academic Press, 1994), Chap. 18.

M. S. Soskin and M. V. Vasnetsov, Singular Optics, Prog. Opt. (Elsevier Science, 2001). Vol. 42, pp. 219-276 (2001).
[CrossRef]

M. Vasnetsov and K. Staliunas, eds., Optical Vortices, (Nova Science, New York, 1999).

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Figures (10)

Fig. 1.
Fig. 1.

Cross section of a well and a pillar as examples of HARMS structure. The sidewalls are expected to be perpendicular to the base; however, manufacturing process makes them more or less inclined (dashed lines). Red lines indicated the incident focused Gaussian beam. In our analysis the top surface of the sample defines an object plane.

Fig. 2.
Fig. 2.

(a). HARMS sample diameter (dark circle) is large when compared with a focused spot of Gaussian beam (light circle), so we have assumed that edge to be straight; (b). the description of the slope. The sample is scanned along y-axis.With these assumptions, we can write the diffraction integral for our problem as

Fig. 3.
Fig. 3.

(a). Maximum value of the Gaussian envelope of the formula (10) depends periodically on the parameter h. In this example h 1=0.0505mm, h 2=0.05061mm, h 3=0.05073mm, h 4=0.0584mm, f=30mm, λ=632nm; b) the diameter of the Gaussian envelope of the expression (10) depends on the optical power of the focusing element. In this example f 1=80mm, f 2=40mm, f 3=20mm, f 4=5mm.

Fig. 4.
Fig. 4.

Phase map in a grayscale (values range from -π to π) of the formula (12) for a) q=-1µm; b) q=0; c) q=1µm. The solid lines are lines of constant amplitude. The vortex point is in the center of the beam. In case of q=0, the line of π-phase jump is well seen. The jump line is shifted from the beam center. For smaller and grater values of q, the phase distribution becomes more smooth. When adding the plane wave-like term (10), two vortices can be generated (for q=0). Such possible vortices are marked by stars in Fig. 4(b). The area of these images has a diameter of 5mm and represents the CCD camera detector. The whole example is calculated for the objective with a focal length of 50mm and He-Ne laser line of 632.8nm.

Fig. 5.
Fig. 5.

(a). Phase portrait of formula (13b); (b) the phase portrait of expression (13) for q=-4µm, qs-q=8µm. The beam center is at the center of the shadow area (slope center); (c) the phase portrait of formula (13) for q=-5µm, qs-q=8µm. The beam center is shifted one micron towards higher part of the sample as compared to the position in case (b). Red circles mark the position of vortex points. The objective has focal length 50mm, the step height is h=400µm and refractive index of the sample n=1.4. The y-axis is perpendicular to the edge and shows scanning direction. The area of the image has a diameter of 5mm. Values range from -π to π.

Fig. 6.
Fig. 6.

(a). Fringe pattern of the synthetic hologram for optical vortex generation and its diffraction pattern (b) the zero and both first diffraction orders are shown; (c) the interference pattern formed by the first order of the diffraction image Fig. 6(b) and coaxial reference wave. A characteristic spiral-like fringe is visible.

Fig. 7.
Fig. 7.

(a). The image of the hologram used as a vortex generator. The first order beams were oversaturated to make visible the weak second order beam (on the right). The weak second order beam of the diffraction pattern shows that we have manufactured a good quality sinusoidal fringe pattern. The propagation angle of the first order beam is about 20 times bigger than in case of the hologram shown in Fig. 6(a); (b) the image of the vortex beam (first order beam) from our hologram after it has passed through the measurement system shown in Fig. 8. The beam quality is not perfect, nevertheless satisfactory for our measurements.

Fig. 8.
Fig. 8.

The scheme of a measurement system. M-mirrors, L-laser, BS-beamsplitters, O-a sample on the scanning stage, OB-objectives, H-hologram generating an optical vortex.

Fig. 9.
Fig. 9.

Scanning the sample in an ideal position. (a) the beam center is before the edge. The beam starts to feel the presence of the edge and vortex point, indicated by fork-like fringes and marked by a red circle, moves a bit off the beam center; (b) the beam center is close to the edge, two vortices are present in the diffraction image; (c) the beam center crosses the edge, still two vortices in a slightly different position are present; (d) the beam center goes beyond the edge (q≈2µm), there is only one vortex point going back to the beam center.

Fig. 10.
Fig. 10.

Scanning the sample tilted at an angle of about 1.5°. (a) the beam center is before the starting point of the slope. The beam starts to feel the presence of the edge. Vortex point, indicated by fork-like fringes and marked by a red circle, moves slightly off the beam center; (b) the beam center is close to the slope center, three vortices are present at the diffraction image; (c) the beam center crosses the slope center. Now five vortices can be detected; d) the beam center goes a bit beyond the upper edge – there are three vortices in the image. The expected value of the slope width is qs-q≈7µm.

Equations (32)

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

t ( x , y ) = exp { i k n d } ( exp { i k h } + exp { i k ( n d + h n a q + a q ) } exp { i k a y ( n 1 ) } + exp { i k n h } )
t ( x , y ) = exp { i k n d } ( exp { i k h } + exp { i k n h } )
U G ( x , y ) = ( x + i sgn σ y ) m exp { x 2 + y 2 b }
U ( x i , y i ) = Ω exp { i k n d } ( In 1 + In 2 + In 3 )
In 1 = exp { i k h } q U G ( x , y ) exp { i k 2 zet ( x 2 + y 2 2 x x i 2 y y i ) } d x d y
In 2 = ts q qs U G ( x , y ) exp { ik 2 z ( x 2 + y 2 2 x x i ) + ik ( 1 z y i + n a ) } d x d y
In 3 = exp { i k n h } q U G ( x , y ) exp { ik 2 zet ( x 2 + y 2 2 x x i 2 y y i ) } d x d y
U ( x i , y i ) = 1 2 β Ξ x [ i exp { q ( β q + 2 i k z y i ) } sgn σ c 1 + i k z Ξ y ( x + i sgn σ y ) ( c 1 + c 2 erf ( β q i k z y i β ) ) ]
i k z = ik 2 z ; β = 1 b + i k z ; Ξ v = π β exp { i k z 2 β v i 2 } and v { x , y }
c 1 = exp { i k h } exp { i k n h } ; c 2 = exp { i k h } + exp { i k n h }
U ( x i , y i ) = 1 4 β Ξ x ( f 1 + f 2 + f 3 )
f 1 = 2 i sgn σ ( α 1 + α 2 + α 3 + α 4 )
α 1 = exp { q ( β q + 2 ikz y i ) } exp { i k h } ;
α 2 = exp { qs ( β qs + 2 ikz y i ) } exp { i k n h }
α 3 = exp { q ( i k n a β q + 2 ikz y i ) } ts ;
α 4 = exp { qs ( i k n a β q s + 2 ikz y i ) } ts
f 2 = μ 1 μ 2 μ 3
μ 1 = exp { ( i k n a + 2 i k z y i ) 2 4 β } ts
μ 2 = k n a + 2 ikz ( x i + I sgn σ y i )
μ 3 = erf ( i k n a 2 β q s + 2 i k z y i 2 β ) erf ( i k n a 2 β q + 2 i k z y i 2 β )
f 3 = 2 ikz Ξ y ( x i + i sgn σ y i ) η
η = c 1 + c 2 [ exp { i k n h } erf ( β q ikz y i β ) exp { i k n h } erf ( β qs ikz y i β ) ]
exp { q 2 b } exp { i ikz ( 2 q y i ) } sgn σ c 1
h = h 0 + λ n
i k z Ξ y ( x i + i sgn σ y i ) ( c 1 + c 2 erf ( β q i k z y i β ) )
U ( x i , y i ) = 1 2 β Ξ x ( P 1 + P 2 )
P 1 = i exp { q ( β q + 2 i k z y i ) } sgn σ exp { i k h } +
i exp { q s ( β q s + 2 i k z y i ) } sgn σ exp { i k n h }
P 2 = i k z Ξ y ( x + i sgn σ y ) ( c 1 + exp { i k h } erf ( β q i k z y i β ) exp { i k n h } erf ( β q s i k z y i β ) )
U ( x i , y i ) = Ξ x y T
Ξ x y = π β exp { i k z 2 β ( x i 2 + y i 2 ) }
T = c 1 + exp { i k h } erf ( β q i k z y i β ) exp { i k n h } erf ( β q s i k z y i β )

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