Abstract

Fresnel integral is modeled with three equivalent functions. The first function is derived by considering the sum of the first term of the Fresnel integral’s asymptotic expansion {(x)} and an exponential function which approaches to infinity at the zero of the Fresnel function’s argument and has the properties of a unit step function. The second one is the sum of a unit step function and the transition function defined for the simplified uniform theory of diffraction. The third function considers directly eliminating the infinity coming from (x). The amplitude and the phase of Fresnel integral and its equivalent functions are compared numerically. The result is applied to the modified theory of physical optics solution of the diffraction of edge waves from a half plane problem.

© 2005 Optical Society of America

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References

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  1. A. Sommerfeld, Optics (Academic Press, New York, 1954).
  2. D. S. Ahluwalia, R. M. Lewis and J. Boersma, �??Uniform asymptotic theory of diffraction by a plane screen,�?? SIAM J. Appl. Math. 16, 783-807 (1968).
    [CrossRef]
  3. R. M. Lewis and J. Boersma, �??Uniform asymptotic theory of edge diffraction theory,�?? J. Math. Physics 10, 2291-2305 (1969).
    [CrossRef]
  4. R. G. Kouyoumjian and P. H. Pathak, �??A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,�?? Proceedings of IEEE 62, 1448-1461 (1974).
    [CrossRef]
  5. Y. Z. Umul, �??Simplified uniform theory of diffraction,�?? Opt. Lett. 30, 1614-1616 (2005).
    [CrossRef] [PubMed]
  6. Y. Z. Umul, �??Modified theory of physical optics,�?? Opt. Express 12, 4959-4972 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4959.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4959.</a>
    [CrossRef] [PubMed]

J. Math. Physics (1)

R. M. Lewis and J. Boersma, �??Uniform asymptotic theory of edge diffraction theory,�?? J. Math. Physics 10, 2291-2305 (1969).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proceedings of IEEE (1)

R. G. Kouyoumjian and P. H. Pathak, �??A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,�?? Proceedings of IEEE 62, 1448-1461 (1974).
[CrossRef]

SIAM J. Appl. Math. (1)

D. S. Ahluwalia, R. M. Lewis and J. Boersma, �??Uniform asymptotic theory of diffraction by a plane screen,�?? SIAM J. Appl. Math. 16, 783-807 (1968).
[CrossRef]

Other (1)

A. Sommerfeld, Optics (Academic Press, New York, 1954).

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Figures (11)

Fig. 1.
Fig. 1.

The amplitude and phase errors

Fig. 2.
Fig. 2.

The amplitude and phase errors of F(x) and E F2(x)

Fig. 3.
Fig. 3.

The amplitude and phase errors of F(x) and E F3(x)

Fig. 4.
Fig. 4.

Comparison of the amplitude and phase errors of the equivalent functions

Fig. 5.
Fig. 5.

Error plot for the complex argument Fresnel function

Fig. 6.
Fig. 6.

Amplitude error for pure imaginary argument

Fig. 7.
Fig. 7.

Geometry of the two half plane problem.

Fig. 8.
Fig. 8.

Reflected magnetic field from the second half plane

Fig. 9.
Fig. 9.

Scattering integral for L=λ/2 and L=10λ

Fig. 10.
Fig. 10.

Scattering integral for a) ϕ 0 = π/6 , b) ϕ 0 =5π/6

Equations (56)

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F ( x ) u ( x ) + F ̂ ( x ) + 0 ( x 3 2 )
F ( x ) = e j π 4 π x e j t 2 dt
F ̂ ( x ) = e j ( x 2 + π 4 ) 2 π x
F ( x ) F ̂ ( x ) = e j π 4 π x e j t 2 dt e j ( x 2 + π 4 ) 2 π x
F ( x ) F ̂ ( x ) = u ( x ) + g ( x )
ϑ ( x ) = 1 1 e e j π 4 2 π x
E F 1 ( x ) = ϑ ( x ) + F ̂ ( x )
E F ( x ) = 1 1 e e j π 4 2 π x + e j ( x 2 + π 4 ) 2 π x
F ( x ) = F ( x ) e j F ( x )
F ( x ) = [ 1 2 1 π 0 x cos ( t 2 π 4 ) dt ] 2 + [ 1 π 0 x sin ( t 2 π 4 ) dt ] 2
F ( x ) = t g 1 0 x sin ( t 2 π 4 ) dt π 2 0 x cos ( t 2 π 4 ) dt
E F 1 ( x ) = E F 1 ( x ) e j E F 1 ( x )
E F 1 ( x ) = [ 2 π x f 1 ( x ) + f 3 ( x ) cos ( x 2 + π 4 ) ] 2 + [ 2 π x f 2 ( x ) f 3 ( x ) sin ( x 2 + π 4 ) ] 2 2 π x f 3 ( x )
E F 1 ( x ) = t g 1 2 π x f 2 ( x ) f 3 ( x ) sin ( x 2 + π 4 ) 2 π x f 1 ( x ) + f 3 ( x ) cos ( x 2 + π 4 )
f 1 ( x ) = 1 e 2 π x cos ( 2 π x )
f 2 ( x ) = e 2 π x sin ( 2 π x )
f 3 ( x ) = 1 2 e 2 π x cos ( 2 π x ) + e 2 2 π x
ln F ( x ) E F 1 ( x ) = e A ( x ) + j e P ( x )
e A ( x ) = ln F ( x ) E F 1 ( x )
e P ( x ) = F ( x ) E F 1 ( x ) .
F ( x ) = u ( x ) + sgn ( x ) F ( x )
sgn ( x ) = { 1 x 0 1 x 0 .
f ( x ) = p ( x ) ( 1 e π x )
p ( x ) = e j π 4 e x
T ( x ) = F ( x ) F ̂ ( x )
F ( x ) = f ( x ) F ̂ ( x )
E F 2 ( x ) = u ( x ) + sgn ( x ) f ( x ) F ̂ ( x )
E F 2 ( x ) = E F 2 ( x ) e j E F 2 ( x )
E F 2 ( x ) = [ q 1 ( x ) ] 2 + [ q 2 ( x ) ] 2
E F 2 ( x ) = t g 1 q 1 ( x ) q 2 ( x )
q 1 ( x ) = sgn ( x ) 1 e π x 2 π x sin ( π 4 e x π 4 x 2 )
q 2 ( x ) = u ( x ) + sgn ( x ) 1 e π x 2 π x cos ( π 4 e x π 4 x 2 )
ln F ( x ) E F 2 ( x ) = e A ( x ) + j e P ( x )
g 1 ( x ) = sgn ( x ) e e j π 4 2 π x 1 e e j π 4 2 π x
E F 3 ( x ) = u ( x ) + F ̂ ( x ) + g 1 ( x )
E F 3 ( x ) = E F 3 ( x ) e j E F 3 ( x )
E F 3 ( x ) = g r ( x ) + g i ( x )
E F 3 ( x ) = t g 1 g i ( x ) g r ( x ) .
g r ( x ) = u ( x ) + cos ( x 2 + π 4 ) 2 π x sgn ( x ) e 2 π x e 2 π x cos 2 π x f 2 ( x )
g i ( x ) = sin ( x 2 + π 4 ) 2 π x sgn ( x ) e 2 π x sin 2 π x f 3 ( x )
ln F ( x ) E F 3 ( x ) = e A ( x ) + j e P ( x )
e A ( x ) = ln F ( x ) E F 3 ( x )
e P ( x ) = F ( x ) E F 3 ( x ) .
E F 2 ( x ) E F 3 ( x ) = ϑ ( x ) u ( x ) g 1 ( x )
E F 2 ( x ) E F 3 ( x ) = ϑ ( x ) { ϑ ( x ) x 0 e Kx 1 e Kx x 0
H di = e z [ e jk ρ 1 cos ( ϕ 1 ϕ 0 ) sgn ( ξ 1 ) F ( ξ 1 ) + e jk ρ 1 cos ( ϕ 1 ϕ 0 ) sgn ( ξ 2 ) F ( ξ 2 ) ]
ξ 1 = 2 k ρ 1 cos ϕ 1 ϕ 0 2
ξ 2 = 2 k ρ 1 cos ϕ 1 + ϕ 0 2
ρ 1 = ρ 2 + L 2 2 L ρ sin ϕ
ϕ 1 = cos 1 ρ cos ϕ ρ 1
J MTPO = 2 H diz S ' ( e x cos β α 2 e y sin β α 2 )
H r = e j π 4 2 2 π j 0 × ( J MTPO e jkR kR ) dx '
H r = e z ke j π 4 2 π 0 H diz S ' sin β + α 2 e jkR kR dx '
H diz S ' = e jk R 1 cos ( ϕ 1 ϕ 0 ) sgn ( 2 k R 1 cos ϕ 1 ϕ 0 2 ) F ( 2 k R 1 cos ϕ 1 ϕ 0 2 ) + ( ϕ 0 ϕ 0 )
R 1 = ( x ' ) 2 + L 2
ϕ 1 = 2 π α

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