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A formula to investigate the wave effect in a multi-lens system is presented on the basis of a path integral formalism by generalizing the work by Nakamura and Deguchi (1999). The wave effect of a system with two lenses is investigated in an analytic way as a simple application to demonstrate usefulness of the formula and variety of wave effect in multi-lens system.

Recently several authors have investigated wave effect in gravitational lensing phenomenon, motivated by the possibility that the wave effect might be detected in future gravitational wave observatories [

Nakamura and Deguchi developed an elegant formalism for gravitational lens using the path integral approach [

We start by reviewing the path integral formalism for gravitational lens phenomenon [

d s 2 = − ( 1 + 2 U ( r , θ , φ ) ) d t 2 + ( 1 − 2 U ( r , θ , φ ) ) ( d r 2 + r 2 d θ 2 + r 2 s i n 2 θ d φ 2 ) , (1)

where U ( r , θ , φ ) is the Newtonian potential. Propagation of massless field ϕ is described by the wave equation

∂ ∂ x μ ( − g g μ ν ∂ ∂ x ν ) ϕ ( r , t ) = 0 , (2)

where g is the determinant of the metric and r represents the spatial coordinates ( r , θ , φ ) . We consider a monochromatic wave from a point source with the wave number k. We set

ϕ ( r , t ) = A r F ( r ) e − i k ( t − r ) , (3)

where A is a constant, then Equation (2) yields

∂ 2 F ∂ r 2 + 2 i k ∂ F ∂ r + 1 r 2 1 s i n θ ∂ ∂ θ ( s i n θ ∂ F ∂ θ ) + 1 r 2 1 s i n 2 θ ∂ 2 F ∂ φ 2 − 4 k 2 U ( r , θ , φ ) F = 0. (4)

Assuming that the first term is negligible compared with the second term and θ ≪ 1 , Equation (4) reduces to

i ∂ F ∂ r = − 1 2 k r 2 [ 1 θ ∂ ∂ θ ( θ ∂ F ∂ θ ) + 1 θ 2 ∂ 2 F ∂ φ 2 ] + 2 k U ( r , θ , φ ) F . (5)

Due to analogy of Equation (5) with the Schrödinger equation, using the path integral formulation of quantum mechanics, the solution can be written as follows,

F ( r , θ , φ ) = ∫ D Θ exp [ i k ∫ 0 r d r ( 1 2 r 2 Θ ˙ 2 ( r ) − 2 U ( r , Θ ( r ) ) ) ] , (6)

where the dot denotes the differentiation with respect to r and Θ is used to represent the variables ( θ , φ ) , which are related by Θ = ( θ c o s φ , θ s i n φ ) . The expression (6) means the sum of all possible path Θ ( r ) , fixing the initial point (source’s position) and the final point (observer’s position).

Let us consider multi-lens system, including n lenses as shown in

explicit expression of Equation (6) is written as Equation (24) in Appendix (see also [

#Math_24# (7)

with

ψ ( Θ l m ) = 2 ∫ r l m − δ r r l m + δ r d r U ( r , Θ ) , (8)

where r l m − 1 , l m = r l m − r l m − 1 , r l n + 1 = r N , and Θ l m is the variable on the mth lens plane. As expected, stationary condition of the phase of the integrand in Equation (7) reproduces the lens equation in the multi-lens system [

∇ Θ l m ′ ∑ m = 1 n ( r l m r l m + 1 2 r l m , l m + 1 | Θ l m − Θ l m + 1 | 2 − ψ ( Θ l m ) ) = 0 , (9)

for each m ′ = 1 , ⋯ , n , which hints at a way to the geometrical optics limit. The Gaussian approximation around a stationary solution yields the result in the geometrical optics limit (see also [

In this section we consider a simple case with two lenses.

F = k 2 π i r 1 r 2 r 1 , 2 ∫ d 2 Θ 1 exp [ i k ( r 1 r 2 r 1 , 2 | Θ 1 − Θ 2 | 2 − ψ ( Θ 1 ) ) ] × k 2 π i r 2 r 3 r 2 , 3 ∫ d 2 Θ 2 exp [ i k ( r 2 r 3 2 r 2 , 3 | Θ 2 − Θ 3 | 2 − ψ ( Θ 2 ) ) ] (10)

with

ψ ( Θ j ) = 4 G M j l n ( | Θ j | ) , (11)

for j = 1 , 2 , where r i , j = r j − r i and the position of an observer is ( r 3 , Θ 3 ) . From Eqution (10) we have

F = k i r 1 r 2 r 1 , 2 ∫ 0 ∞ d θ 1 θ 1 1 − 4 i k G M 1 exp [ i k r 1 r 2 2 r 1 , 2 ( θ 1 2 + θ 2 2 ) ] J 0 ( k r 1 r 2 r 1 , 2 θ 1 θ 2 ) × k i r 2 r 3 r 2 , 3 ∫ 0 ∞ d θ 2 θ 2 1 − 4 i k G M 2 exp [ i k r 2 r 3 2 r 2 , 3 ( θ 2 2 + θ 3 2 ) ] J 0 ( k r 2 r 3 r 2 , 3 θ 2 θ 3 ) , (12)

where | Θ j | = θ j and J 0 ( y ) is the Bessel function of the first kind. Integration with respect to θ 1 can be performed (see [

#Math_44# (13)

where α is a real number which represents a constant phase and 1 F 1 ( a , b ; y ) is the Kummer’s function. With the use of the definition of the Bessel function

J 0 ( z ) = ∑ L = 0 ∞ ( − 1 ) L ( L ! ) 2 ( z 2 ) 2 L , (14)

we have (see [

F = e i α ′ e π k G ( M 1 + M 2 ) Γ ( 1 − 2 i k G M 1 ) z ∑ L = 0 ∞ ( − i ) L ( L ! ) 2 Γ ( 1 + L − 2 i k G M 2 ) × ( x z ) L F 2 1 ( 1 − 2 i k G M 1 , 1 + L − 2 i k G M 2 , 1 ; 1 − z ) , (15)

where we defined

z = r 3 ( r 2 − r 1 ) r 2 ( r 3 − r 1 ) , (16)

x = k r 2 r 3 θ 3 2 2 ( r 3 − r 2 ) , (17)

α ′ is a real constant and 2 F 1 ( a , b , c ; y ) is the Hypergeometric function. In the limit θ 3 = 0 ( x = 0 ) , Eqution (15) reduces to

F = e i α ′ e π k G ( M 1 + M 2 ) Γ ( 1 − 2 i k G M 1 ) Γ ( 1 − 2 i k G M 2 ) z × F 2 1 ( 1 − 2 i k G M 1 , 1 − 2 i k G M 2 , 1 ; 1 − z ) , (18)

and we have

| F | 2 = 4 π k G M 1 1 − e − 4 π k G M 1 4 π k G M 2 1 − e − 4 π k G M 2 z 2 | 2 F 1 ( 1 − 2 i k G M 1 , 1 − 2 i k G M 2 , 1 ; 1 − z ) | 2 . (19)

We consider the coincidence limit that the distance between the two lenses becomes zero, i.e., r 1 = r 2 . In this case, from previous investigation (e.g., [

F ( x ) ≡ e π k G ( M 1 + M 2 ) Γ ( 1 − 2 i k G M 1 ) F 1 1 ( 1 − 2 i k G ( M 1 + M 2 ) , 1 ; − i x ) , (20)

excepting a constant phase factor. We compare our result with Eqution (20). First, let us consider the limit r 1 = r 2 , i.e., z = 0 of Eqution (15). Using the mathematical formula

#Math_61# (21)

in the case θ 3 = 0 ( x = 0 ) , we can easily have

l i m z → 0 | F | 2 = 4 π k G ( M 1 + M 2 ) 1 − e − 4 π k G ( M 1 + M 2 ) = | F ( 0 ) | 2 . (22)

This is the expected result. Note that | F | 2 is regarded as the magnification factor.

k G M 1 = k G M 2 = 1 , but result depends significantly on the parameters k G M 1 , k G M 2 and z , as shown in

In the present paper, we have presented a general formula to investigate wave effect in multi-lens system, which has been derived with the use of path integral approach. The formula is expressed in terms of integration with respect to variables of lens planes. It is difficult to perform the integration in general cases, but a system with two Schwarzschild lenses is an example for which the integration can be performed in an analytic way. The model considered in the present paper is simplified and limited, however, it suggests variety of wave effect in multi-lens phenomenon. It is required to develop a numerical method to perform integration in general lens configuration in future work.

This work is supported in part by Grant-in-Aid for Scientific research of Japanese Ministry of Education, Culture, Sports, Science and Technology, No. 15740155.

Yamamoto, K. (2017) Path Integral Formulation for Wave Effect in Multilens System. International Journal of Astronomy and Astrophysics, 7, 221-229. https://doi.org/10.4236/ijaa.2017.73018

In this Appendix, we review derivation of Equation (7) from Equation (6). We consider the configuration depicted as

ψ ( Θ l m ) = 2 ∫ r l − δ r r l + δ r d r U ( r , Θ ) , (23)

for m = 1 , ⋯ , n , respectively. In this case, the path integral formula (6) can be written as

F = [ ∏ j = 1 N − 1 ∫ d 2 Θ j A j ] exp [ i k ( ϵ ∑ j = 1 N − 1 r j r j + 1 2 | Θ j + 1 − Θ j ϵ | 2 − ∑ m = 1 n ψ ( Θ l m ) ) ] , (24)

where the normalization is chosen

A j = 2 π i ϵ k r j r j + 1 , (25)

so that F = 1 in the limit ψ = 0 . Then, Equation (24) is rephrased as

F = [ ∏ j = 1 l 1 − 1 ∫ d 2 Θ j A j ] exp [ i k ( ϵ ∑ j = 1 l 1 − 1 r j r j + 1 2 | Θ j + 1 − Θ j ϵ | 2 ) ] × [ ∏ j = l 1 l 2 − 1 ∫ d 2 Θ j A j ] exp [ i k ( ϵ ∑ j = l 1 l 2 − 1 r j r j + 1 2 | Θ j + 1 − Θ j ϵ | 2 − ψ ( Θ l 1 ) ) ] × ⋯ × [ ∏ j = l n N − 1 ∫ d 2 Θ j A j ] exp [ i k ( ϵ ∑ j = l n N − 1 r j r j + 1 2 | Θ j + 1 − Θ j ϵ | 2 − ψ ( Θ l n ) ) ] , (26)

where ϵ is the separation between two neighboring planes. With the use of the following equality, which can be proven by the mathematical induction,

∑ j = l m l m + 1 − 1 r j r j + 1 | Θ j + 1 − Θ j | 2 = ϵ r l m r l m + 1 r l m + 1 − r l m | Θ l m + 1 − Θ l m | 2 + ∑ j = l m + 1 l m + 1 − 1 r j 2 r j + 1 − r l m r j − r l m | Θ j − u l m , j | 2

with

u l m , j = r l m Θ l m + ( j − l m ) r j + 1 Θ j + 1 j ( r j + 1 − r l m ) , (27)

we have

#Math_98# (28)

which is equivalent to Eqution (7).

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