Wedge diffraction model in gravitational lensing by cosmic strings

Optik 157 (2018) 1227–1234

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Original research article

Wedge diffraction model in gravitational lensing by cosmic strings Yusuf Ziya Umul Electronic and Communication Department, Cankaya University, Eskis¸ehir Yolu 29. Km, Yenimahalle, Ankara, 06810, Turkey

a r t i c l e

i n f o

Article history: Received 7 November 2017 Accepted 18 December 2017 Keywords: Cosmic strings Gravitational lensing Diffraction

a b s t r a c t A cosmic string forms a conical topology, which enables a diffraction phenomenon by a wedge with an inner angle, equals to the double of the deficit angle. A wedge, on the edge of which the string is located, is defined in the flat space-time. The radiated wave from a star is represented by two plane waves, propagating on the two surfaces of the wedge for grazing incidence. The uniform expressions of the diffracted waves are obtained and investigated numerically. © 2017 Elsevier GmbH. All rights reserved.

1. Introduction Since the matter has not been distributed homogenously in the universe, it is generally thought in the literature that two, one or zero dimensional defects occurred in the early stages [1]. These topological defects are the results of phase transitions with broken symmetry. Theoretically three kinds of discontinuities are proposed. The first one is the domain wall, which has a two dimensional structure [2]. The second defect is monopole and has zero dimension [3]. Domain walls and monopoles have some contradictions with the actual cosmological models. Only the existence of the cosmic strings which are one dimensional is allowed theoretically as being in infinite length and/or loop [4]. Vilenkin [5] proposed the observation of cosmic strings with the aid of gravitational lensing. He suggested that the strings might be responsible for the double images of some quasars and galaxies. Hogan and Narayan discussed the possible cases for the image formation of quasars, related with the gravitational lensing of strings [6]. Exact interior and exterior solutions of Einstein’s field equations were obtained by Gott [7] for vacuum strings and the lensing effects were analyzed with the aid of these solutions. Yamamoto and Tsunoda [8] solved the wave equations in the background of a straight and static cosmic string and examined the lensing effect with the obtained wave expressions. Suyama et al. [9] defined the inhomogeneous wave equation on the space-time of a cosmic string in the cylindrical coordinates. Their solution has the form of a diffracted wave, since it includes the Fresnel integral in the form of an error function. However, the possible diffraction phenomenon by a cosmic string was proposed by Linet [10], who solved the inhomogeneous wave equation in the topology of a cosmic string. Osipov expressed the observed luminosity of a star, which is located behind a cosmic string, in terms of the Fresnel integral [11]. Recently, Fernandez-Nunez and Bulashenko proposed the usage of the geometrical theory of diffraction (GTD) for the lensing phenomenon of a cosmic string [12]. They considered the surfaces of the wedge, which occur in the flat space-time of the string, as independent half-planes and obtained the scattered fields for grazing incidence in the exact solution of Sommerfeld. The authors also investigated the same problem for a line source radiating on the face of the half-plane and discussed the locations of the Fresnel zones [13].

E-mail address: [email protected] https://doi.org/10.1016/j.ijleo.2017.12.077 0030-4026/© 2017 Elsevier GmbH. All rights reserved.

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Y.Z. Umul / Optik 157 (2018) 1227–1234

Fig. 1. Diffraction geometry by a wedge.

The aim of this letter is to propose a wedge diffraction based model for the gravitational lensing by a cosmic string. The inner angle of the wedge will be taken as the double of the deficit angle as shown in [12]. First of all, the theory of wedge diffraction for grazing incidence will be reviewed. A new form that satisfies the boundary conditions will be proposed for the diffracted wave expressions. The obtained diffracted field representation will be applied to the cosmic string problem. The uniform expressions of the diffraction waves will be obtained by the method of the uniform theory of diffraction (UTD) [14,15]. The analysis of the scattered fields in terms of gravitational lensing will be performed numerically. A time factor of exp(jωt) will be suppressed throughout the paper. ω is the angular frequency. 2. Wedge diffraction for grazing incidence In this section, the diffraction of waves by a soft (total field is equal to zero on the surface) or hard (normal derivative of the total field is equal to zero on the surface) wedge will be studied for grazing incidence (the angle of incidence is equal to zero). The geometry is given in Fig. 1. P is the observation point. The cylindrical coordinates are given by (, , z). The edge of the wedge is located on the z axis. P shows the observation point. 0 is the angle of incidence. represents the outer angle of the wedge. The incident field has the expression of ui = u0 ejk cos(−0 )

(1)

where u0 is the complex amplitude. u is a component of the electromagnetic field. k shows the wave-number. The highfrequency expression of the diffracted field can be written as 

u0 e−j 4 sin ud = √ n 2

 n



1 cos

 n

− cos

−0 n

∓



1 cos

 n

− cos

+0 n

e−jk



(2)

k

for n is equal to / [16]. The minus and plus signs, in the brackets, are valid for soft and hard surfaces respectively. When the angle of incidence is equal to zero, the incident field has the expression of ui = u0 ejk cos 

(3)

Fig. 2. The flat space-time geometry for the cosmic string.

Y.Z. Umul / Optik 157 (2018) 1227–1234

1229

Fig. 3. Total diffracted field with its sub-components for the observation distance is equal to 750.

and the diffracted field becomes ud = 0

(4)

and 

ud =

2u0 e−j 4 sin √ n 2

 n

e−jk

1 cos

 n

− cos

 n



(5)

k

for soft and hard wedges respectively. However, these expressions of the diffracted field are not compatible with the physical reality. In order to explain this situation, the total field of uT = uGO + ud

(6)

is taken into account for grazing incidence. The geometrical optics (GO) (uGO ) wave can be defined by uGO = u0 eU ( − )

(7)

where U(x) is the unit step function, which is equal to one for x > 0 and zero otherwise. It is apparent that the GO field has a discontinuity at  =  for both of the soft and hard surfaces. If the diffracted wave becomes zero for the grazing incidence, a discontinuous wave will occur in space. For the hard wedge, the value of the diffracted field is doubled. This means that the total field, in Eq. (6), is also discontinuous in this case. Thus the diffraction fields, in Eqs. (4) and (5), are not correct. The literature [12,14] accepts Eq. (4) and proposes the division of Eq. (5) by 2. However, this approach is also problematic. The diffracted field can be taken as 

ud =

u0 e−j 4 sin √ n 2

 n

e−jk

1 cos

 n

− cos

 n



(8)

k

for both of the soft and hard wedges, since the reflected GO wave does not exist for grazing incidence. In this case, Eq. (8) does not satisfy the boundary conditions at  = 0 and  = 2 for the soft wedge. The diffracted field also does not satisfy the boundary condition at  = 2 for the hard wedge. Thus the consideration of Eq. (8) as the diffracted wave does not solve the problem. As a result, a diffracted wave must exist in space in order to compensate the discontinuity of the GO field. Furthermore, it must also satisfy the required boundary conditions on the faces of the wedge. In order to obtain such an expression for the diffracted field, Eq. (2) can be arranged as ud = udi + udr

(9)

where udi and udr can be introduced by 

udi

u0 e−j 4 =− √ n2 2



cos sin

−+0 2n −+0 2n

∓

cos sin

++0 2n ++0 2n



e−jk



k

(10)

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Y.Z. Umul / Optik 157 (2018) 1227–1234

Fig. 4. Total diffracted field with its sub-components for the observation distance is equal to 2000.

and 

udr

u0 e−j 4 =− √ n2 2

 cos sin

+−0 2n +−0 2n

∓

cos sin

−−0 2n −−0 2n



e−jk



(11)

k

respectively [17,18]. The subscripts i and r represents the incident and reflected, since udi and udr have asymptotes at  =  + 0 and  =  − 0 , which are the locations of the shadow and reflection boundaries respectively. It is apparent that these two diffracted field components satisfy the boundary conditions on the soft and hard wedges independently. If the angle of incidence is equal to zero, the reflected field will cease to exist. For this reason, we propose to use udi for the diffracted field, in Eq. (6). Thus ud can be introduced as 

ud = −

u0 e−j 4 √ n2 2



cos sin

− 2n − 2n

∓

cos sin

+ 2n + 2n



e−jk



(12)

k

for soft and hard wedges. 3. Waves in the space-time of a cosmic string An infinitely long cosmic string is taken into account. Thus the geometry is symmetric according to . The linear mass density of the string is given by . The space-time metric can be expressed by the equation [7,12] ds2 = −dt 2 + d2 + (1 − 4G)2 2 d2 + dz 2 ,

(13)

which can be reduced to the form of ds2 = −dt 2 + d2 + 2 d2 + dz 2

(14)

by the transform of  = (1 − 4G). This metric defines a globally conical topology, which is locally flat [12]. As shown in Fig. 2, Eq. (14) shows a wedge, which is excluded from the real space with an inner angle of 8G. The string is located along the z axis of the wedge. The angle delta reads = 4G.

(15)

The light, coming from a source propagates parallel to the surfaces of the wedge. The incident rays, coming to the edge of the wedge will be diffracted according to Fig. 2. The outer angle of the wedge can be defined by the relation = 2 ( − ) when Eq. (15) is considered with the geometry, in Fig. 2.

(16)

Y.Z. Umul / Optik 157 (2018) 1227–1234

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Fig. 5. Amplitude error versus the observation angle.

According to the metric, in Eq. (14), the waves satisfy the equation of 1 ∂  ∂



∂u  ∂



2

+

1 ∂ u + k2 u = 0 2 ∂2

(17)

for time harmonic variation. The incident waves, in Fig. 2, can be expressed by ui1 = u0 ejk cos(− )

(18)

ui2 = u0 ejk cos(+ ) ,

(19)

and

which satisfy the wave equation, in Eq. (17). Following [12], the wedge is supposed to be hard. According to Eq. (12), the diffracted fields can be introduced as 

ud1 = −

u0 e−j 4 √ n2 2

ud2 = −

u0 e−j 4 √ n2 2

and 



cos sin



cos sin

−+ 2n −+ 2n

3−− 2n 3−− 2n

+

cos sin

−

+− 2n +− 2n

cos sin



−− 2n −− 2n

e−jk



(20)

k



e−jk



(21)

k

for the first and second incident waves respectively. n is equal to



n=2 1−





(22)

Thus the total field reads uT = uGO1 + uGO2 + ud1 + ud2

(23)

according to Eq. (6). The GO waves uGO1 and uGO2 can be defined by uGO1 = u0 ejk cos(− ) U ( + − )

(24)

uGO2 = u0 ejk cos(+ ) U ( −  + )

(25)

and

respectively. ud1 and ud2 have asymptotes at + and −, which are in the physical space of  ∈ [, 2−], since these fields expressions are high-frequency approximations. In order to obtain the uniform diffracted fields, the method of UTD

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Y.Z. Umul / Optik 157 (2018) 1227–1234

Fig. 6. Phase error versus the observation angle.

will be applied. In this technique, the non-uniform diffracted field expression is multiplied by a transition function, which is defined by F [|x|] Fˆ [|x|]

T (x) =

(26)

where the Fresnel function F[x] and Fˆ [x] are introduced as 

ej 4 F [x] = √ 

∞ 2

e−jv dv

(27)

x

and  2 e−j( 4 +x ) √ 2 x

Fˆ [x] =

(28)

respectively [15]. Thus the uniform diffracted waves read 

ud1 = −

u0 e−j 4 √ n2 2

and 

ud2

u0 e−j 4 =− √ n2 2



cos sin



cos sin

−+ 2n −+ 2n



T − +

3−− 2n 3−− 2n

−

cos sin

cos sin

+− 2n +− 2n

−− 2n −− 2n





T +

e−jk

 

(29)

k

e−jk



(30)

k

for the parameter ± can be defined by



± = ±

2k cos

± 2

(31)

It is also possible to rewrite the total field as uT = uT 1 + uT 2

(32)

where uT 1 and uT 2 have the expressions uT 1 = uGO1 + ud1

(33)

uT 2 = uGO2 + ud2

(34)

and

respectively. This representation provides a graphical representation of the GO waves and the related diffracted fields that compensate their discontinuities at the transition boundaries.

Y.Z. Umul / Optik 157 (2018) 1227–1234

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4. Numerical results In this section, the behavior of the fields will be investigated numerically. The total fields and their sub-components will be plotted for different values of the observation distance () and deficit angle (). The angle of observation () varies between  and 2−. Fig. 3 shows the variation of the total field, given in Eq. (32), versus the observation angle for the deficit angle is equal to 1.8◦ . The distance of observation is taken as 750 where  is the wavelength. The total GO and total diffracted waves are uGO1 +uGO2 and ud1 +ud2 respectively. The GO waves, coming from the two separate faces of the wedge interfere with each other in the region of 178.2◦ < < 181.8◦ . In this region, the maximum value of the intensity is equal to 2. The GO wave has two discontinuities, namely at 178.2◦ and 181.8◦ . The addition of the diffracted field to the GO wave eliminates these discontinuities. The total field is continuous everywhere. The effect of the diffracted wave on the total field is clearly seen from the figure. At 178.8◦ and 181.2◦ the intensity of the GO waves decrease from 2 to 1.677. At 180◦ , the intensity increases from 2 to 2.341. Also a phase shift is observed between the GO and total fields because of the diffracted wave. Fig. 4 plots the variation of the total field and its sub-components with respect to the observation angle. The distance of observation is equal to 2000. The deficit angle of the wedge is the same with the previous figure. It can be seen from the figure that the number of the interference fringes increases as the observation distance increases. The remaining behaviors of the fields are the same with the ones in Fig. 3. Only the maxima and minima of the total field, between 178.2o and 181.8◦ approaches to 2. As a last step, the diffracted field, obtained in this letter, will be compared with the one, used in [12]. Fernandez-Nunez and Bulashenko considered the incident scattered wave in the solution of Sommerfeld [19]. The total diffracted field can be expressed by

 

 

− + u0 ejk cos(+ ) sign + F +

uTdl = u0 ejk cos(− ) sign − F

(35)

when the GO waves are excluded. The total diffracted wave can be defined as uTd = ud1 + ud2

(36)

according to this letter. Two error functions can be introduced by



uTdl

eA () = ln

uTd

(37)

and eP () = ln

∠uTdl ∠uTd

(38)

for amplitude and phase respectively. ln is the natural logarithm. The sign ∠ represents the phase. Fig. 5 shows the variation of the amplitude error function, given by Eq. (37), versus the observation angle. The values of the observation distance and deficit angle of wedge are the same with the ones in Fig. 3. The amplitude error is negative in the double illuminated region where both of uGO1 and uGO2 exist. Its value discontinuously jumps to positive values at the transition boundaries located at 178.2◦ and 181.8◦ where the diffracted fields have their maxima. The error increases with the increasing value of the observation angle. In Fig. 6, the variation of the phase error function, in Eq. (38), is plotted with respect to the observation angle. The values of the other parameters are the same with the previous figure. The difference between the two diffracted field expressions gets its maxima at the transition boundaries in terms of phase. 5. Conclusions In this letter, the diffraction phenomenon of waves by a cosmic string is investigated. Novel expression for the diffracted waves are introduced for soft and hard wedge for the grazing incidence. The uniform diffracted fields are constructed with the method of UTD. The numerical results are given for total field and its sub-components. The diffracted fields, obtained in this study, are compared numerically with the one, used in [12]. For small deficit angles of wedge, the two field expressions are in harmony, since the wedge reduces to a half-plane when the inner angle of the wedge approaches to zero. However, the usage of the model, introduced in this letter, is more appropriate physically, since the diffraction geometry is based on a wedge. References [1] [2] [3] [4] [5] [6]

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