Abstract

The diffraction of electromagnetic waves for light focused by a high numerical aperture lens from a first material into a second material is treated. The second material has a different refractive index from that of the first material and introduces spherical aberration. We solve the diffraction problem for the case of a planar interface between two isotropic and homogeneous materials with this interface perpendicular to the optical axis. The solution is obtained in a rigorous mathematical manner, and it satisfies the homogeneous wave equation. The electric and magnetic strength vectors are determined in the second material. The solution is in a simple form that can be readily used for numerical computation. A physical interpretation of the results is given, and the paraxial approximation of the solution is derived.

© 1995 Optical Society of America

Full Article  |  PDF Article

Corrections

P. Török, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation: errata," J. Opt. Soc. Am. A 12, 1605-1605 (1995)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-12-7-1605

References

  • View by:
  • |
  • |
  • |

  1. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [CrossRef]
  2. P. Debye, “Das Verhalten von Lichtwellen in der Näha eines Brennpunktes oder einer Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).
  3. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  4. R. K. Luneburg, Mathematical Theory of Optics, 2nd ed. (U. of California Press, Berkeley, Calif., 1966).
  5. R. Kant, “Analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
    [CrossRef]
  6. J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface,”J. Opt. Soc. Am. 66, 955–961 (1976).
    [CrossRef]
  7. J. J. Stamnes, G. C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,”J. Opt. Soc. Am. 66, 780–788 (1976).
    [CrossRef]
  8. J. J. Stamnes, G. C. Sherman, “Radiation of electromagnetic fields in biaxially anisotropic media,”J. Opt. Soc. Am. 68, 502–508 (1978).
    [CrossRef]
  9. H. Ling, S.-W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  10. Y. Ji, K. Hongo, “Analysis of electromagnetic waves refracted by a spherical dielectric interface,” J. Opt. Soc. Am. A 8, 541–548 (1991).
    [CrossRef]
  11. J. J. Stamnes, Waves in Focal Regions, 1st ed. (Adam Hilger, Bristol, UK, 1986).
  12. S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970).
  14. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, San Francisco, 1968).

1993 (2)

R. Kant, “Analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
[CrossRef]

1991 (1)

1984 (1)

1981 (1)

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

1978 (1)

1976 (2)

1959 (2)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

1909 (1)

P. Debye, “Das Verhalten von Lichtwellen in der Näha eines Brennpunktes oder einer Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970).

Cremer, C.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
[CrossRef]

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Näha eines Brennpunktes oder einer Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).

Gasper, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, San Francisco, 1968).

Hell, S.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
[CrossRef]

Hongo, K.

Ji, Y.

Kant, R.

R. Kant, “Analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

Lee, S.-W.

Li, Y.

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Ling, H.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics, 2nd ed. (U. of California Press, Berkeley, Calif., 1966).

Reiner, G.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Sherman, G. C.

Stamnes, J. J.

Stelzer, E. H. K.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
[CrossRef]

Wolf, E.

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970).

Ann. Phys. (Leipzig) (1)

P. Debye, “Das Verhalten von Lichtwellen in der Näha eines Brennpunktes oder einer Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).

J. Microsc. (Oxford) (1)

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
[CrossRef]

J. Mod. Opt. (1)

R. Kant, “Analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

Other (4)

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, San Francisco, 1968).

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970).

R. K. Luneburg, Mathematical Theory of Optics, 2nd ed. (U. of California Press, Berkeley, Calif., 1966).

J. J. Stamnes, Waves in Focal Regions, 1st ed. (Adam Hilger, Bristol, UK, 1986).

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 (2)

Fig. 1
Fig. 1

Diagram showing light focused by a lens into a single medium.

Fig. 2
Fig. 2

Diagram showing light focused by a lens into two media separated by a planar interface.

Equations (58)

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

E ˜ ( P , t ) = Re [ E ( P ) exp ( - i ω t ) ] , H ˜ ( P , t ) = Re [ H ( P ) exp ( - i ω t ) ] ,
E ( P ) = - i k 2 π Ω a ( s x , s y ) s z exp { i k [ Φ ( s x , s y ) + s ^ · r p ] } × d s x d s y ,
H ( P ) = - i k 2 π Ω b ( s x , s y ) s z exp { i k [ Φ ( s x , s y ) + s ^ · r p ] } × d s x d s y ,
E 1 ( x , y , - d ) = - i k 1 2 π Ω 1 W ( e ) ( s ^ 1 ) exp [ i k 1 ( s 1 x x + s 1 y y - s 1 z d ) ] d s 1 x d s 1 y ,
W ( e ) ( s ^ 1 ) = a ( s 1 x , s 1 y ) s 1 z .
T ( 2 ) W ( e ) ,
E 2 ( x , y , - d ) = - i k 1 2 π Ω 1 T ( e ) W ( e ) ( s ^ 1 ) exp [ i k 1 ( s 1 x x + s 1 y y - s 1 z d ) ] d s 1 x d s 1 y
E 2 ( r p ) = - i k 2 2 π Ω 2 F ( e ) ( s ^ 2 ) exp ( i k 2 s ^ 2 · r p ) d s 2 x d s 2 y .
k 2 s ^ 2 - k 1 s ^ 1 = ( k 2 cos ϕ 2 - k 1 cos ϕ 1 ) u ,
k 2 s 2 x = k 1 s 1 x ,             k 2 s 2 y = k 1 s 1 y .
E 2 ( r p ) = - i k 2 2 π Ω 1 F ( e ) ( s ^ 2 ) exp ( i k 2 s ^ 2 · r p ) × J 0 ( s 1 x , s 1 y ; s 2 x , s 2 y ) d s 1 x d s 1 y ,
J 0 = ( k 1 k 2 ) 2 ,
F ( e ) ( s ^ 1 , s ^ 2 ) = ( k 2 k 1 ) T ( e ) a ( s 1 x , s 1 y ) s 1 z exp [ - i d ( k 1 s 1 z - k 2 s 2 z ) ] .
E 2 ( x , y , z ) = - i k 2 2 2 π k 1 Ω 1 T ( e ) a ( s 1 x , s 1 y ) s 1 z × exp [ - i d ( k 1 s 1 z - k 2 s 2 z ) ] exp ( i k 2 s 2 z z ) × exp [ i k 1 ( s 1 x x + s 1 y y ) ] d s 1 x d s 1 y .
H 2 ( x , y , z ) = - i k 2 2 2 π k 1 Ω 1 T ( m ) b ( s 1 x , s 1 y ) s 1 z × exp [ - i d ( k 1 s 1 z - k 2 s 2 z ) ] exp ( i k 2 s 2 z z ) × exp [ i k 1 ( s 1 x x + s 1 y y ) ] d s 1 x d s 1 y .
s 2 z = ( 1 - n 1 2 n 2 2 + n 1 2 n 2 2 s 1 z 2 ) 1 / 2 - [ 1 - n 1 2 n 2 2 ( s 1 x 2 + s 1 y 2 ) ] 1 / 2 .
s ^ 1 = ( sin ϕ 1 ) ( cos θ ) i ^ + ( sin ϕ 1 ) ( sin θ ) j ^ + ( cos ϕ 1 ) k ^ ,
s ^ 2 = ( sin ϕ 2 ) ( cos θ ) i ^ + ( sin ϕ 2 ) ( sin θ ) j ^ + ( cos ϕ 2 ) k ^ ,
r p = r p [ ( sin ϕ p ) ( cos θ p ) i ^ + ( sin ϕ p ) ( sin θ p ) j + ( cos ϕ p ) k ^ ] ,
E ( 0 ) = ( E 0 0 0 ) .
E ( p , s , ζ ) ( 1 ) = A ( ϕ 1 ) P ( 1 ) L R E ( x , y , z ) ( 0 ) ,
R = [ cos θ sin θ 0 - sin θ cos θ 0 0 0 1 ] ,
L = [ cos ϕ 1 0 sin ϕ 1 0 1 0 - sin ϕ 1 0 cos ϕ 1 ] ,
P ( 1 ) = [ cos ϕ 1 0 - sin ϕ 1 0 1 0 sin ϕ 1 0 cos ϕ 1 ] .
E ( x , y , z ) ( 2 ) = R - 1 [ P ( 2 ) ] - 1 I E ( p , s , ζ ) ( 1 ) ,
I = [ τ p 0 0 0 τ s 0 0 0 τ p ] ,
τ s = 2 sin ϕ 2 cos ϕ 1 sin ( ϕ 1 + ϕ 2 ) ,             τ p = 2 sin ϕ 2 cos ϕ 1 sin ( ϕ 1 + ϕ 2 ) cos ( ϕ 1 - ϕ 2 ) ,
[ P ( 2 ) ] - 1 = [ cos ϕ 2 0 sin ϕ 2 0 1 0 - sin ϕ 2 0 cos ϕ 2 ] ,
E ( x , y , z ) ( 2 ) = A ( ϕ 1 ) ( τ p cos ϕ 2 cos 2 θ + τ s sin 2 θ τ p cos ϕ 2 sin θ cos θ - τ s sin θ cos θ - τ p sin ϕ 2 cos θ ) .
A ( ϕ 1 ) = f l 0 cos 1 / 2 ϕ 1 ,
c = T ( e ) a = E ( 2 ) .
d = ( 2 μ 2 ) 1 / 2 s ^ 2 × c .
d = ( 2 μ 2 ) 1 / 2 A ( ϕ 1 ) × ( - τ p cos θ + τ s sin θ cos θ cos ϕ 2 τ p cos 2 θ + τ s sin 2 θ cos ϕ 2 - τ s sin θ cos ϕ 2 ) .
J p = sin ϕ 1 cos ϕ 1 ;
d s 1 x d s 1 y = sin ϕ 1 cos ϕ 1 d ϕ 1 d θ .
κ = n 1 sin ϕ 1 sin ϕ p cos ( θ - θ p ) + n 2 cos ϕ 2 cos ϕ p ,
Ψ ( ϕ 1 , ϕ 2 , - d ) = - d ( n 1 cos ϕ 1 - n 2 cos ϕ 2 ) .
E 2 ( r p , - d ) = - i k 2 2 2 π k 1 Ω 1 c ( ϕ 1 , ϕ 2 , θ ) × exp { i k 0 [ r p κ + Ψ ( ϕ 1 , ϕ 2 , - d ) ] } × sin ϕ 1 d ϕ 1 d θ ,
H 2 ( r p , - d ) = - i k 2 2 2 π k 1 Ω 1 d ( ϕ 1 , ϕ 2 , θ ) × exp { i k 0 [ r p κ + Ψ ( ϕ 1 , ϕ 2 , - d ) ] } × sin ϕ 1 d ϕ 1 d θ ,
e 2 x = + i K 2 π 0 α 0 2 π ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) × [ ( τ p cos ϕ 2 + τ s ) + ( cos 2 θ ) ( τ p cos ϕ 2 - τ s ) ] × exp { i k 0 [ r p κ + Ψ ( ϕ 1 , ϕ 2 , - d ) ] } d ϕ 1 d θ , e 2 y = + i K 2 π 0 α 0 2 π ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) ( sin 2 θ ) × ( τ p cos ϕ 2 - τ s ) × exp { i k 0 [ r p κ + Ψ ( ϕ 1 , ϕ 2 , - d ) ] } d ϕ 1 d θ , e 2 z = - i K π 0 α 0 2 π ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) τ p sin ϕ 2 cos θ × exp { i k 0 [ r p κ + Ψ ( ϕ 1 , ϕ 2 , - d ) ] } d ϕ 1 d θ ,
K = k 2 2 f l 0 2 k 1 = π n 2 2 f l 0 λ n 1 .
h 2 x = + i K n 2 2 π 0 α 0 2 π ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) ( sin 2 θ ) × ( τ s cos ϕ 2 - τ p ) × exp { i k 0 [ r p κ + Ψ ( ϕ 1 , ϕ 2 , - d ) ] } d ϕ 1 d θ , h 2 y = + i K n 2 2 π 0 α 0 2 π ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) × [ ( τ p + τ s cos ϕ 2 ) + ( τ p + τ s cos ϕ 2 ) ( cos 2 θ ) ] × exp { i k 0 [ r p κ + Ψ ( ϕ 1 , ϕ 2 , - d ) ] } d ϕ 1 d θ , h 2 z = - i K n 2 π 0 α 0 2 π ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) τ s sin ϕ 2 sin θ × exp { i k 0 [ r p κ + Ψ ( ϕ 1 , ϕ 2 , - d ) ] } d ϕ 1 d θ ,
e 2 x = - i K [ I 0 ( e ) + I 2 ( e ) cos ( 2 θ p ) ] , e 2 y = - i K I 2 ( e ) sin ( 2 θ p ) , e 2 z = - 2 K I 1 ( e ) cos θ p ,
h 2 x = - i K n 2 I 2 ( h ) sin ( 2 θ p ) , h 2 y = - i K n 2 [ I 0 ( h ) - I 2 ( h ) cos ( 2 θ p ) ] , h 2 z = - 2 K n 2 I 1 ( h ) sin θ p ,
v = k 1 ( x 2 + y 2 ) 1 / 2 sin α = k 1 r p sin ϕ p sin α , u = k 2 z sin 2 α = k 2 r p cos ϕ p sin 2 α
I 0 ( e ) = 0 α ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) exp [ i k 0 Ψ ( ϕ 1 , ϕ 2 , - d ) ] × ( τ s + τ p cos ϕ 2 ) J 0 ( v sin ϕ 1 sin α ) × exp ( i u cos ϕ 2 sin 2 α ) d ϕ 1 , I 1 ( e ) = 0 α ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) exp [ i k 0 Ψ ( ϕ 1 , ϕ 2 , - d ) ] × τ p ( sin ϕ 2 ) J 1 ( v sin ϕ 1 sin α ) exp ( i u cos ϕ 2 sin 2 α ) d ϕ 1 , I 2 ( e ) = 0 α ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) exp [ i k 0 Ψ ( ϕ 1 , ϕ 2 , - d ) ] × ( τ s - τ p cos ϕ 2 ) J 2 ( v sin ϕ 1 sin α ) × exp ( i u cos ϕ 2 sin 2 α ) d ϕ 1 ;
I 0 ( h ) = 0 α ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) exp [ i k 0 Ψ ( ϕ 1 , ϕ 2 , - d ) ] × ( τ p + τ s cos ϕ 2 ) J 0 ( v sin ϕ 1 sin α ) × exp ( i u cos ϕ 2 sin 2 α ) d ϕ 1 , I 1 ( h ) = 0 α ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) exp [ i k 0 Ψ ( ϕ 1 , ϕ 2 , - d ) ] × τ s ( sin ϕ 2 ) J 1 ( v sin ϕ 1 sin α ) exp ( i u cos ϕ 2 sin 2 α ) d ϕ 1 , I 2 ( h ) = 0 α ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) exp [ i k 0 Ψ ( ϕ 1 , ϕ 2 , - d ) ] × ( τ p - τ s cos ϕ 2 ) J 2 ( v sin ϕ 1 sin α ) × exp ( i u cos ϕ 2 sin 2 α ) d ϕ 1 ,
E 2 ( x , y , z ) = K Ω 1 W 0 ( s 1 x , s 1 y ) × exp [ i 2 π ( s 1 x x , s 1 y y ) ] d s 1 x d s 1 y ,
s 1 x = n 1 s 1 x λ ,             s 1 y = n 1 s 1 y λ ,
W 0 = c ( s 1 x , s 1 y ) s 1 z exp [ - i ( k 1 s 1 z - k 2 s 2 z ) d ] exp ( i k 2 s 2 z z ) = c ( s 1 x , s 1 y ) s 1 z exp ( - i k 1 s 1 z d ) exp [ i k 2 s 2 z ( z + d ) ] ,
( k 1 k 2 ) 2 ( s 1 x 2 + s 1 y 2 ) < 1
E 2 ( x , y , z ) = K Ω 1 W 0 ( s 1 x , s 1 y ) exp ( i k 2 s 2 z z ) × exp [ i 2 π ( s 1 x x + s 1 y y ) ] d s 1 x d s 1 y ,
W 0 = W 0 exp ( i k 2 s 2 z z ) .
E 2 ( x , y , z ) = - i k 2 2 2 π k 1 Ω 1 T ( e ) a ( s 1 x , s 1 y ) s 1 z × exp [ - i d ( k 1 s 1 z - k 2 s 2 z ) ] × exp ( i k 2 s 2 z z ) exp [ i k 1 ( s 1 x x + s 1 y y ) ] d s 1 x d s 1 y .
s 1 z = [ 1 - ( s 1 x 2 + s 1 y 2 ) ] 1 / 2 1 - s 1 x 2 + s 1 y 2 2 = 1 - σ 2 2 , s 2 z = [ 1 - ( s 2 x 2 + s 2 y 2 ) ] 1 / 2 1 - s 2 x 2 + s 2 y 2 2 = 1 - ( n 1 n 2 ) 2 σ 2 2 .
E 2 ( par ) ( 0 , 0 , z ) = i k 2 2 k 1 B exp { - k 1 [ d ( 1 - n 2 n 1 ) + n 2 n 1 z ] } × 0 β exp ( - 1 2 i k 1 D σ 1 2 ) σ d σ ,
D = ( 1 - n 1 n 2 ) d + n 1 n 2 z .
I ( 0 , 0 , z ) = E 2 ( par ) ( 0 , 0 , z ) 2 = I 0 | exp ( - 1 2 i k 1 D β 2 ) - 1 1 4 k 1 D β 2 | 2 = 4 I 0 sin 2 ( 1 4 k 1 D β 2 ) ( 1 4 k 1 D β 2 ) .

Metrics