1. Introduction
It is the aim of this paper to define three equivalent functions instead of the Fresnel integral by using a function which will cancel the infinity coming from the pole of the first term in Fresnel function’s asymptotic expansion. The amplitude and the phase of the related functions will be compared numerically. The result will be applied to the modified theory of diffraction (MTPO) solution of the diffraction of edge waves from a half plane problem. This solution contains the integration of Fresnel function.
A time factor e^{jwt} is assumed and suppressed throughout the paper.
2. Derivation of the methods
Three equivalent functions for the Fresnel integral will be derived by considering the asymptotic expansion of
for x → ∞. F(x) is the Fresnel integral, defined by
and F̂(x) is the first term in its asymptotic expansion, which can be given as
for x → ∞. u(x) is the unit step function, which is equal to 1 for x<0 and 0 for x>0.
2.1 First method
The method relies on canceling the pole of F̂(x) by defining a unit step function which approaches to infinity for x = 0. This is a similar approach with the transition function of UAT, which can be given as
for 
x → ∞. Equation (
4) has a pole at
x = 0 and is equal to
u(
x) everywhere except
x = 0 according to Eq. (
1). Equation (
1) can be written as
where g(x) is an unknown function which goes to infinity at x = 0 and is equal to zero, otherwise. A function can be introduced as
which has the same properties with Eq. (
4) and represents
u(
x)+
g(
x). An equivalent function can be defined for the Fresnel integral as
by considering Eqs. (
5) and (
6). The infinity of
ϑ(
x) is canceled by adding
F̂(
x).
A ∞  ∞ indeterminacy is created as a result of this summation. As a result, the equivalent function can be defined by the equation of
and can be used instead of the Fresnel integral.
The phase and amplitude of the Fresnel function will be compared with E_{F1}(x). The Fresnel integral can be written as
where F(x) is defined as √F(x)F
^{*}(x). The amplitude and the phase functions are equal to
and
respectively. The Matlab codes of the Fresnel function and the integrals, written in Eq. (
10) is given in the Appendix. It is also possible to express the equivalent function as
for the amplitude and the phase functions can be written as
and
respectively. The functions, defined in Eqs. (
13) and (
14) can be expressed as
by considering Eq. (
8). The amplitude and phase functions of
F(
x) and
E_{F}(
x) will be compared in order to test their identity.
Fig. 1. The amplitude and phase errors
where e_{A}(x) shows the amplitude error of
and e_{P}(x) represents the phase error as
Such a representation gives a physical understanding of the error. Equation (
17) shows the deviation in amplitude. Equation (
18) expresses the phase error directly in degrees and puts forward the amount of the point to point phase error. This representation is more meaningful from a logarithmic expression.
It is important to note that the ripples, seen in
Fig. 1(a) for
x ≥ 4 are the result of the step size N, given in the Appendix. When N increases, the amplitude of the ripples will decrease, but it is apparent that computing time will increase. N is taken as 20.000 for the plots, given in
Fig. 1.
2.2 Second method
The property of
will be taken into account in order to derive the second equivalent function. sgn(x) is the signum function, which is defined as
The transition function of SUTD can be written as
where
which is given in Ref. [
5
5
.
Y. Z.
Umul
, “
Simplified uniform theory of diffraction
,”
Opt. Lett.
30
,
1614

1616
(
2005
). [CrossRef] [PubMed]
]. The transition function of UTD can be expressed as
and the term of F(x) can be evaluated as
by equating Eq. (
23) to Eq. (
21). As a result one obtains
when Eq. (
24) is combined with Eq. (
19).
The equivalent function in Eq. (
25) can be written as
where E
_{F2}(x) and ∠E
_{F2}(x) are the amplitude and phase functions of E
_{F2}(x), respectively. E
_{F2}(x) can be represented as
and the phase function can be written as
where
will be used in determining the error comparison of the equivalent function and the Fresnel integral.
e_{A}(
x) and
e_{p}(
x) were defined by Eqs. (
17) and (
18) and are valid for the present case when the Eqs. (
27) and (
28) are considered.
Fig. 2. The amplitude and phase errors of F(x) and E
_{F2}(x)
2.3 Third method
An equivalent representation for the function
g(x), given in Eq. (
5), will be defined as a third step. As mentioned before,
g(x) is equal to zero everywhere except
x=
0, where it approaches to infinity. Such a function can be defined by
and the equivalent Fresnel function can be written as
by using Eq. (
31) in Eq. (
5).
There is no uniform theory in literature that has the same approach with the third equivalent function. A function of g(x), which gives the exact Fresnel function when added to the term of u( x)+F̂(x), is described. Since its properties are known (although the function
g
_{1}(x) which shows a similar variation is defined. This function eliminates the infinity of F̂(x) and when added to u(x), gives an equivalent function for the Fresnel integral. A new uniform approach for diffraction can be derived by using this concept.
E
_{F3}(x) can be expressed as
for the amplitude function is equal to
and the phase function can be obtained as
The functions in Eqs.(
34) and (
35) can be defined as
and
where
f
_{3}(
x) was given in Eq. (
15.c). The error functions can be derived by taking the natural logarithm of the proportion of the Fresnel integral and the equivalent function which can be written as
where e_{A}(x) shows the amplitude error of
and e_{P}(x) represents the phase error as
Fig. 3. The amplitude and phase errors of F(x) and E
_{F3}(x)
Fig. 4. Comparison of the amplitude and phase errors of the equivalent functions
It can be seen from
Fig. 4 that the error variation of
E
_{F1}(
x) and
E
_{F3}(
x) are about the
from their definitions. The equation of
can be obtained for
K = exp (
jπ/4)2√
π. It is apparent that Eq. (
42) is equal to zero for x<0 and approaches to zero for x>0.
Fig. 5. Error plot for the complex argument Fresnel function
The first equivalent function will be compared with the Fresnel integral for pure imaginary argument as a second step.
Fig. 6. Amplitude error for pure imaginary argument
3. Numerical example: Scattering of edge waves from a PEC half plane
A physical optics scattering problem will be observed in this section in order to the scattering of edge diffracted waves from a half plane will be examined by using the method of modified theory of diffraction. The geometry of the problem is given in
Fig. 7.
Fig. 7. Geometry of the two half plane problem.
A magnetic polarized plane wave with unit amplitude is illuminating the first half plane. The second half plane is lying in the shadow region of the first half plane. It is obvious that only the edge diffracted waves reach the second plane. The main objective of this analysis is to obtain an integral which contains Fresnel function. The computation time of the integral is expected to increase, but it will be shown that this defeat can be eliminated by using equivalent functions instead of Fresnel integral.
The MTPO integral of reflected fields will be considered. The edge diffracted waves of the first half plane can be written as
where
which are the detour parameters of UTD. ρ
_{1} and ϕ
_{1} are the cylindrical coordinate quantities of the first half plane and are related to the second half plane coordinates as
for L is the distance between the half planes. MTPO surface current can be found by considering the boundary condition of J→es=n→×H→t
∣_{S´} as
where the modified unit vector of the second surface is equal to n→1=cos(u+α)e→x+sin(u+α)e→yforu=π2−α+β2. The integral of the reflected magnetic field can be written as
where the curl operation will be applied according to the observation point coordinates [
6]. The resultant integral can be expressed as
where H_{diz}∣_{S´} represents the value of the incident field on the reflection surface and is equal to
for R
_{1} is the value of ρ
_{1} on the surface. The related quantities are equal to
on the surface of reflection.
α is equal to
tg
^{1}(
L/
x´). Equation (
48) will be plotted for the Fresnel integral and the first equivalent function, used in Eq. (
49).
Fig. 8. Reflected magnetic field from the second half plane
Fig. 9. Scattering integral for L=λ/2 and L=10λ
Fig. 10. Scattering integral for a) ϕ
_{0} = π/6 , b) ϕ
_{0} =5π/6
4. Conclusion
In this work, three equivalent functions are derived for the Fresnel integral. It is stated that the first two functions are related with UAT and UTD approaches. The third equivalent function attempts to eliminate the infinity coming from F̂(x) and can be thought as a variation of UTD since UTD makes uniform the diffraction field by multiplying the diffraction coefficient by a transition function which has a zero at the transition region. This approach creates a 0/0 indeterminacy. The concept of the third equivalent function is to eliminate the related infinity by creating a ∞  ∞ indeterminacy.
The numerical comparisons show that the functions represent the Fresnel integral with a very good degree of correctness. The equivalent functions consist of the sum of two or three basic functions. It is always easy to deal with these functions for numerical or analytic evaluations. Fresnel integrals can be found in many simulation programs, but there is always a need of integral evaluation and this creates a problem when dealing with the diffraction of complex bodies.