Quasi-geometrical Optics Approximation in

Gravitational Lensing

Ryuichi Takahashi Department of Physics, Kyoto University

Kyoto, Japan

Ray path in geometrical optics

Ray path formed by diffraction effect

Diffracted ray passing through the lens center is newly formed

source

observer lens

§１．Ｉｎｔｒｏｄｕｃｔｉｏｎ

Ｍ

lens

Eｒ

１/２Ｅ ＭＤｒ Ｅｉｎｓｔｅｉｎ radius :

●Gravitational lensing

D

observer

Gravitational lensing of light is usually treated in geometrical optics

・double slit

1

2Eｒ

Ｄ

double slit

path length difference : Ｍ 21

２１Ｅ ＭＤｒ/

Ｍ Wave effect (diffraction)

monochromatic waves

（Nakamura 1998）

●Wave effects in gravitational lensing （Schneider et al. 1992; Nakamura & Deguchi 1999 ）

・long wavelength

gravitational waves λ 1AU

for space interferometer

screen

The wave does not feel the existence of the lens

lens

・λ ＞M Diffraction effect ＊ source

・λ ≪M Geometrical optics approximation

・λ ＜M Quasi-geometrical optics approximation

Geometrical 1st order perturbation optics in powers of λ /M

(λ →0) (correction term arising from diffraction )

Difference between geometrical optics and wave optics

1

8101

M

M

AUM

§２.Wave Optics in Gravitational Lensing

(Misner et al. 1973; Schneider et al. 1992)

Background metric

dxdxgdUdtUdsB)(222 2121 r

：gravitational potential of the lens )(rU 1U

Wave propagation

02 )(;;

hRhB :Background

Riemann tensor

)(BR

Lensed waveform )(~

Lh

)(~

)()(~

hFhL

Unlensed waveform

),(exp2),( 2 ηξξη d

LSL

S tidiDD

DF

)(2

),(

2

ξηξ

ηξ

SLLS

SLd

DDD

DDt 8, 2ξ

Amplification factor

),(exp2),( 2 yxxy wTid

i

wwF

Amplification factor

Ｌ

S

O ηLDLSD

SD

ξ

SL DD 00 , ηyξx

20LLS

S

DD

Dw

（ξ ：normalization constant of the length）

Nondimensional quantities

：nondimensional frequency

)(2

1),(

2xyxyx T : nondimensional time delay

d

S

LLS tD

DDT 20

Er0

Expanding F (w,y) in powers of 1/w

1,1 Tyx

Mw Mλ1/w

Setting (Einstein radius)

●Geometrical optics approximation w

Stationary points of the T (x,y) contribute to

the above integral :

Image positions are determined

j

jjgeo iwTF exp2/1

j

jdj

L tthth ),(),( ,2/1

rr

Amplification factor

0),( xyxT

：lens equation

jx

Lensed wave in time domain

),(exp2),( 2 yxxy wTid

i

wwF

（Fermat’s principle）

（Schneider et al. 1992; Nakamura & Deguchi 1999 ）

)(xxy

Expanding T(x,y) around jx

)~(~~),(),( 321 xxxTTT bajbaj yxyx

jjj TT xxxyx ~),,(

●Quasi-geometrical optics approximation 1w

j

tt

jj

L

geo

L thtdththd

),(~

),(),(2/1

rrr

Amplification factor

),(exp2),( 2 yxxy wTid

i

wwF

Expanding T(x,y) to higher order

)~(~~~~),(

~~~),(

~~),(),(

5

241

61

21

xxxxxT

xxxT

xxTTT

dcbajdcba

cbajcba

bajbaj

yx

yx

yxyx

Effects on magnifications of the images

)(

exp

2/3

241

6

12122

21

wxxxxT

xxxTxxTideF

dcbajdcbaw

cbajcbawbajba

iwT

ji

j x

)(1 22/1

we

w

ijiwT

jjj

2

)3(

2

2)3(

3

)4(

2

11

12

5

2

1

16

1

jjj

jj

j

jj

j

j

j

j

j

xx

jjjjj x 1,1 2121jiwTjj e

w

idF

2/1

Lensed wave in time domain

Amplification factor

),(exp2),( 2 yxxy wTid

i

wwF

Central cusp of the lens

)0(2 rrInner density profile

x ,1

)(),(2

21 xyxyx T

x0)( x

)(

1 22/32

0

2

2/2

wyw

edF

iwy

)/1(exp2

2

2

2

101

22/2

wxxxyidwi

eF

iwy

x

observer

lens

source

(e.g. Singular Isothermal Sphere)

･Point mass lens

ｗ≫１： geometrical optics limit

Twi

geo eiF

2/12/1

TwFgeo sin22/12

)()( 2 ξξ M

§３.Results

･Singular isothermal sphere

TwFgeo sin22/12

2/)( 2v ξｗ≫１： geometrical optics limit

･Isothermal sphere with a core

x

Deflection potential 22

cxx

Contribution from the lens center

)/1()(exp2

22

2

2

101

22/2

wwxxxxyidwi

edF c

iwy

x

For ｗｘ ＜1, the behavior is similar to that in SIS

§4.Conclusion

We discussed the effects of diffraction in

wave optics of gravitational lensing. We

studied first order perturbation in powers

of λ /M .

１．The magnification μ is changed to

μ + dμ , where dμ is of the order of

λ /M .

２． Diffracted ray passing through the lens center is newly formed .