Gravitational lensing for interstellar power transmission

arXiv:2310.17578v2
[gr-qc]
31
Oct
2023
Gravitational lensing for interstellar power transmission
Slava G. Turyshev
Jet Propulsion Laboratory, California Institute of Technology,
4800 Oak Grove Drive, Pasadena, CA 91109-0899, USA
(Dated: November 1, 2023)
We investigate light propagation in the gravitational field of multiple gravitational lenses. Assum-
ing these lenses are sufficiently spaced to prevent interaction, we consider a linear alignment for the
transmitter, lenses, and receiver. Remarkably, in this axially-symmetric configuration, we can solve
the relevant diffraction integrals – result that offers valuable analytical insights. We show that the
point-spread function (PSF) is affected by the number of lenses in the system. Even a single lens is
useful for transmission either it is used as a part of the transmitter or it acts on the receiver’s side.
We show that power transmission via a pair of lenses benefits from light amplification on both ends
of the link. The second lens plays an important role by focusing the signal to a much tighter spot;
but in practical lensing scenarios, that lens changes the structure of the PSF on scales much smaller
than the telescope, so that additional gain due to the presence of the second lens is independent of its
properties and is govern solely by the transmission geometry. While evaluating the signal-to-noise
ratio (SNR) in various transmitting scenarios, we see that a single-lens transmission performs on par
with a pair of lenses. The fact that the second lens amplifies the brightness of the first one, creates a
challenging background for signal reception. Nevertheless, in all the cases considered here, we have
found practically-relevant SNR values. As a result, we were able to demonstrate the feasibility of
establishing interstellar power transmission links relying on gravitational lensing – a finding with
profound implications for applications targeting interstellar power transmission.
I. INTRODUCTION
Recently, we have explored the optical properties of the solar gravitational lens (SGL) [1, 2] and have shown that
the SGL is characterized by a significant light amplification and angular resolution. As such, the SGL provides unique
capabilities for direct imaging and spectroscopy of faint targets such as exoplanets in our stellar neighborhood [3, 4].
It can be assumed that pairs of stellar gravitational lenses could facilitate energy transmission across interstellar
distances, utilizing equipment of a scale and power akin to that currently employed for interplanetary communications
[5]. There is a prevailing expectation that such a configuration would benefit from the light amplification by both
lenses, thus, enabling significant increases in the signal-to-noise ratio (SNR) of the transmitted signal [6–11]. However,
a comprehensive analysis of these transmission scenarios remains to be undertaken.
Our objective here is to examine light propagation in multi-lens systems and evaluate the associated light ampli-
fication. To do that, we will rely on the wave-theoretical treatment of the gravitational lensing phenomena and will
use analytical tools that we developed in our prior studies of the SGL [2–4, 12–14]. As such, these methods can be
seamlessly adapted to explore power transmission within the multi-lens configurations.
This paper is organized as follows: In Section II, we present the wave-theoretical tools to describe the propagation
of EM waves in a gravitational field. In Section III, we consider various lensing geometries that involve one and two
gravitational lenses, and derive the relevant light amplification factors. In Section IV, we discuss power transmission
with lensing configurations utilizing both a single lens and a pair of lenses. In Section V we evaluate detection
sensitivity in various cases considered and evaluate the relevant SNRs. Our conclusions are presented in Section VI.
To streamline the discussion, we moved some material to Appendices. Appendix A presents an alternative evaluation
of the diffraction integral in the case of a two-lens transmission. Appendix B presents a path integral formulation.
II. EM WAVES IN A GRAVITATIONAL FIELD
As electromagnetic (EM) wave travels through a gravitational field, interaction with gravity causes the wave to
scatter and diffract [2]. In Ref. [12, 13], while studying the Maxwell equations on the background a weak gravity
space-time, we developed a solution to the Mie problem for the diffraction of the EM waves on a large gravitating
body (see [2, 15]) and found the EM field at an image plane located in any of the optical regions behind the lens.

2
FIG. 1: A lens-centric geometry for power transmission via gravitational lensing showing the transmitter, the lens, and the
receiver. Also shown is the distance from the lens to the transmitter plane, z0, and that from the lens to the receiver plane, z.
A. Diffraction of light
We consider a stellar gravitational lens with the Schwarzschild radius of rg = 2GM/c2
with M being lens’ mass. We
use a cylindrical coordinate system centered at the lens (ρ, φ, z) with its z-coordinate oriented along the wavevector
k, a unit vector in the unperturbed direction of the propagation of the incident wave that originated at the source
positioned at (−z0, x′
). We also introduce a light ray’s impact parameter, b, and coordinates of the receiver (z, x) on
the image plane located in the strong interference region at distance z from the lens. These quantities are given as:
k = (0, 0, 1), b = b(cos φξ, sin φξ, 0), x′
= ρ′
(cos φ′
, sin φ′
, 0), x = ρ(cos φ, sin φ, 0). (1)
With these definitions, the EM field on an image plane takes the following form (see details in [12, 13]):

Eρ
Hρ

=

Hφ
−Eφ

= A(x′
, x)e−iωt

cos φ
sin φ

+ O

r2
g, ρ2
/z2

, (2)
with the remaining components being small, i.e., (Ez, Hz) ∝ O(ρ/z).
Assuming the validity of eikonal and the thin lens approximation, the Fresnel-Kirchhoff diffraction formula yields
the following expression for the wave’s amplitude at the observer (receiver) location
A(x′
, x) = E0
k
iz0z
1
2π
ZZ
d2
b A0(b)eikS(x′
,b,x)
, (3)
where S(x′
, x, b) is the effective path length (eikonal) along a path from the source position (−z0, x′
) to the observer
position (z, x) via a point (0, b) on the lens plane (Fig. 1 shows overall geometry of the gravitational lensing system)
S(x′
, b, x) =
q
(b − x′)2 + z2
0 +
p
(b − x)2 + z2 − ψ(b) =
= z0 + z +
(x − x′
)2
2(z0 + z)
+
z0 + z
2z0z

b −
z0
z0 + z
x −
z
z0 + z
x′
2
− ψ(b) + O
b4
z4
0
,
b4
z4

, (4)
where the last term in this expression is the gravitational phase shift, ψ(b), that is acquired by the EM wave as it
propagates along its geodetic path from the source to the image plane on the background of the gravitational field
with potential, U, that has the form (see discussion in [12, 16]):
ψ(b) =
2
c2
Z z
z0
dzU(b, z) = krg ln 4k2
zz0 + 2rg ln kb + O(Jn), (5)
where Jn, n ∈ 2, 3, 4... are the spherical harmonics coefficients representing the mass distribution inside the stellar
lens [12]. Then, the wave amplitude on the observer plane can be written as
A(x′
, x) = A0(x′
, x)F(x′
, x), (6)
where A0(x, x) is the wave amplitude at the receiver (observer) in the absence of the gravitational potential U:
A0(x′
, x) =
E0
z0 + z
eikS0(x′
,x)
, S0(x′
, x) = z0 + z +
(x − x′
)2
2(z0 + z)
, (7)

3
with S0(x′
, x) is the path length along a straight path from x′
to x. In the case of a monopole lens, the amplification
factor F(x, x) is given by the following form of a diffraction integral
F(x′
, x) =
z0 + z
z0z
keiφG
2πi
ZZ
d2
b A0(b)eikS1(x′
,b,x)
,
S1(x′
, b, x) =
z0 + z
2z0z

b −
z0
z0 + z

x +
z
z0
x′
2
− 2rg ln kb, (8)
where the phase factor is given as φG = krg ln 4k2
zz0 and S1(x′
, x) is the Fermat potential along a path from the
source position x′
to the observer position x via a point b on the lens plane. The first term in S1(x′
, x) is the difference
of the geometric time delay between a straight path from the source to the observer and a deflected path. The second
term is due to the time delay in the gravitational potential of the lens object (i.e., the Shapiro time delay).
In the case of an isolated spherically-symmetric gravitational lens positioned on the optical axis, x′
= 0, collecting
all the relevant terms, we present the amplification factor on the receiver (observer) plane
F(x) =
z0 + z
z0z
keiφG
2πi
ZZ
d2
b A0(b) exp
h
ik
z0 + z
2z0z

b −
z0
z0 + z
x
2
− 2rg ln kb
i
. (9)
We consider the case of two non-interacting thin gravitational lenses. We denote zt to be the transmitter’s distance
from the first lens, z12 is the distance between the lenses, and zr is the distance between the second lens and the
receiver. We assume that the lenses are at a very large distance, z12 ≫ zt, zr, from each other; also zt and zr are in
the focal regions of the respective lenses. This allows us to treat the light propagation independently for each lens.
Furthermore, as we are interested to evaluate the largest light amplification, we consider the most favorable trans-
mission geometry: we assume that all participants – the source (or transmitter), the first and the second monopole
lenses, and the observer (or receiver) – are all situated on the same line – the primary optical axis of the lensing
system. Clearly, any deviation from this axially-symmetric geometry will reduce the energy transmitting efficiency.
In this scheme, the light emitted by the source is diffracted by the gravitational field of the first lens, that focuses
and amplifies light which now becomes the source for the second lens. This EM field encounters the second lens, then
the third lens and so on and ultimately it reaches the observer that is positioned in the focal region of the last lens.
B. Transmission geometry
We observe that there is no formal solution for the scenario in which a transmitter is positioned at a large but finite
distance from a lens, emitting a spherical or Gaussian beam in the direction of the lens. Instead, we will employ the
tools developed for the situation in which it is permissible to approximate the incident waves as plane waves. This
approach uses results obtained from the study of diffraction by a monopole gravitational lens, such as the SGL.
It is known that when an EM wave, originating at infinity, travels near a gravitational body, its wavefront experiences
bending. In general relativity, this deflection angle is θgr = 2rg/b. As a result, a massive body acts as a lens by focusing
the EM radiation (i.e., the light rays intersect the optical axis) at the distance, z, that is determined as
b
z
= θgr ⇒ z =
b2
2rg
. (10)
In [17], we have shown that expression for z from (10) is modified when the light rays are coming from a source
located at a finite distance, z0, from the lens. In this case, defining α = b/z0 and using small angle approximation,
the new expression reads
b
z
= θgr − α ⇒ z =
b2
2rg
1
1 − b2/2rgz0
. (11)
Clearly, the transmitter’s distance with respect to the lens, determines four signal transmission regimes, namely
i). For θgr α or when 0 ≤ z0 b2
/2rg there is no transmission. In this case, the light rays either are completely
absorbed by the lens or are not able to focus, e.g., do not reach the optical axis on the other side behind the lens.
ii). For θgr = α or when z0 = b2
/2rg, the focal distance z is infinite, implying that after passing by the lens the light
rays are collimated, never reaching the optical axis. In this case, there will be shadow behind the lens except for
the presence of the bright sport of Arago that is weakening with distance from the lens [18, 19].

4
iii). For θgr α or when z0 b2
/2rg, the light will begin to focus at a finite distance from the lens, forming all the
regions relevant for the diffraction problem including the shadow, interference region, and that of the geometric
optics (see Fig. 4 in [2]). According to (11), these regions will be formed farther from the lens, compared to (10).
iv). For θgr ≫ α or when z0 ≫ b2
/2rg, the point source is effectively at infinity and, depending on the impact
parameter, the light rays will form all the typical diffraction regions, beginning at the distance given by (10).
As a result, for optimal reception, an observer needs to be positioned in the focal region of a lens at the distance
given by (11). Same logic works when a transmission link includes another lens which is then followed by a receiver.
III. LIGHT AMPLIFICATION
We are now in a position to explore the light propagation in the gravitational field and to describe light amplification
in various lensing configurations, including scenarios with one or two linearly-aligned gravitational lenses.
A. One lens transmission
We consider a stellar monopole gravitational lens with mass M1, Schwarzschild radius rg1 = 2GM1/c2
, and physical
radius R1. We assume that transmitter is positioned in the lens’ focal region and placed on the optical axis, which is
defined to be the line connecting the transmitter, the lens, and the receiver. So, that in (6)–(9) we can set x′
= 0.
Note that there are two distinct architectures to form a transmission link that are determined by where transmitter
and receiver are placed with respect to the lens: i). In one case, the transmitter is positioned close to the lens but
at the distance larger than the beginning of the focal region, namely zt ≥ R2
1/2rg1 = 547.8 (R1/R⊙)2
(M⊙/M1) AU.
(Note that in the case of the Sun, the solar corona increases the distance from where practical transmissions may
occur. Thus, for λ ∼ 1 µm, these ranges are beyond ∼ 650 AU from the Sun.) In this case, the receiver, is placed at
an interstellar distance from the lens, so that zr ≫ zt. ii). In another case, the positions are switched, so that the
transmitter is now placed at an interstellar distance from the lens, so that zt ≫ R2
1/2rg1 , but the receiver is at the
lens’ focal region, zr ≥ R2
1/2rg1 . Our formulation below will cover both of these cases.
Considering diffraction on a single lens, with the help of (9) and taking A0(b) = 1, we determine the amplification
factor of the EM wave at the observer’s plane that is positioned a distance zr from the lens as below
F1GL(x) =
keiφG1
iz̃1
1
2π
ZZ
d2
b1 exp
h
ik
1
2z̃1

b1 −
z̃1
zr
x
2
− 2rg1 ln kb1
i
, where z̃1 =
ztzr
zt + zr
, (12)
where φG1 = krg1 ln 4k2
ztzr. After re-arranging the terms and removing the spherical wave [15], we present (12) as
F1GL(x) =
keiφG1
iz̃1
1
2π
Z ∞
0
b1db1 exp
h
ik
b2
1
2z̃1
− 2rg1 ln kb1
i Z 2π
0
dφξ1 exp
h
− i

k
b1ρ
zr
cos[φξ1 − φ]
i
=
=
keiφG1
iz̃1
Z ∞
0
b1db1J0

k
b1ρ
zr

exp
h
ik
b2
1
2z̃1
− 2rg1 ln kb1
i
. (13)
Considering the four scenarios detailed in Sec. II B: Case i): This case is trivial and does not require a formal
treatment. Case ii): When θ = α, the following condition is met: b2
1/2zt − 2rg1 ln kb1 = 0. For this situation, the
amplification factor (13) simplifies to (see derivation details in [19]):
F0
1GL(x) =

1 +
zr
zt

ei φG1+
kR2
1
2zr

J0

k
R1ρ
zr

, (14)
which has the properties of the bright spot of Arago. This is a faint spot of light with brightness that is proportional
to the ratio (zr/zt). Note that stellar atmospheres will make the edges of a spherical lens softer, thus severely affecting
formation of the spot to the point of washing it out. This scenario is not very useful for power transmission.
Next, we examine the scenarios, Case iii) and Case iv), as highlighted in Sec. II B. These cases are similar and can
be characterized in the same manner. For that, we take the last integral in (13) with the method of stationary phase
[2] to determine the impact parameter for which the phase is stationary:
b1 =
p
2rg1 z̃1, (15)

5
yielding the following result for the amplification factor:
F1GL(x) =
p
2πkrg1 eiϕ1
J0

k
p
2rg1 z̃1
zr
ρ

, (16)
where ϕ1 is given as ϕ1 = φG1 + k rg1 − rg1 ln krg1 − rg1 ln 2kz̃1

− 1
4 π.
To evaluate the light amplification of a single lens, we first determine its point-spread function (PSF). For that,
we use the generic solution for the EM field (2) with solution (7) together with (16), and study the Poynting vector,
S = (c/4π) [ReE × ReH] , that describes the energy flux in the image plane [15]. Normalizing this flux to the time-
averaged value that would be observed if the gravitational field of the first lens were absent, |S0| = (c/8π)E2
0 /(zt+zr)2
,
we determine the PSF of a single lens PSF1GL = |S|/|S0| as below:
PSF1GL(x) = 2πkrg1 J2
0

k
p
2rg1 z̃1
zr
ρ

. (17)
Therefore, the largest value for the light amplification factor of a single lens is realized when ρ = 0, yielding [1, 2]
µ0
1GL = 2πkrg1 ≃ 1.17 × 1011
M1
M⊙
1 µm
λ

. (18)
Normalized
PSF
-600 -400 -200 0 200 400 600
0.0
0.2
0.4
0.6
0.8
1.0
Distance from the optical axis, ρ, [m]
FIG. 2: Normalized PSF of a single lens in the trans-
mitting lens scenario (19). Note that the first zero here
is much farther out from the optical axis.
Nominally, amplification (18) is realized for a telescope with
the diameter that is less then the characteristic pattern of the
Bessel function J0(kρ
p
2rg1 z̃1/zr). In practice, if the telescope
diameter d is larger than that of the first zero of the projected
Airy pattern (17), it would average the light amplification over
that aperture (see discussion in [2].) We shall discuss this fact
when we look at different transmission scenarios below.
Coming back to the transmission scenarios iii) and iv) dis-
cussed in Sec. II B, we note that both of these cases are de-
scribed by the results obtained. The difference is just the
meaning of the distances of transmitter and receiver in these
situations. Both of these distances led to the effective distance
z̃1 in (12). In the case of scenario iii), the transmitter is at the
distance zt ≪ zr. In the case iv), these distances are reversed
with the transmitter now at the large distance compared to the
receiver, zt ≪ zr from it.
With solution (17), we may now consider the two cases with
drastically different positions of the transmitter and the re-
ceiver that were discussed at the beginning of this section.
1. Transmitting lens scenario
In the case, when zt ≪ zr, the effective distance z̃1 from (12) reduces to z̃1 = ztzr/(zt + zr) ≃ zt. Therefore, the
PSF of the transmission with a single lens (17) takes the form
PSFt
1GL(x) ≃ 2πkrg1 J2
0

k
p
2rg1 zt
zr
ρ

. (19)
We shall call this case the transmitting lens scenario, thus there will be superscript {}t
on the relevant quantiles.
Note that in this case, an observer in the focal region of the lens, positioned on the optical axis at zr ≫ zt from it,
will see an Einstein ring around the lens with the radius θt
1 given as (see details in [20]) below:
θt
1 =
p
2rg1 zt
zr
≃ 2.46 × 10−9
M1
M⊙
1
2
zt
650 AU
1
2
10 pc
zr

rad. (20)
We observe that the first zero of the projected Airy pattern in (19) occurs at the distance of ρt
1GL = 2.40483 /(kθt
1) ≃
155.82 m (λ/1 µm)(M⊙/M1)
1
2 (650 AU/zt)
1
2 (zr/10 pc) from the optical axis, which is large and aperture averaging
may not be important. Fig. 2 shows the relevant behavior of this PSF. Therefore, while considering the relevant
transmission links, one may have to use the entire PSF from (19) with its maximal value µ0
1GL given by (18).

6
2. Receiving lens scenario
In the case, when zt ≫ zr, the effective distance z̃1 from (12) reduces to z̃1 = ztzr/(zt + zr) ≃ zr. Therefore, the
PSF of a single lens transmission (17) takes the form
PSFr
1GL(x) ≃ 2πkrg1 J2
0

k
r
2rg1
zr
ρ

. (21)
We shall call this case the receiving lens scenario, thus there will be superscript {}r
on the relevant quantiles.
Note that in this case, an observer in the focal region of the lens, positioned on the optical axis at zr ≫ zt from it,
will see an Einstein ring around the lens with the radius θr
1 given as (see details in [20]):
θr
1 =
r
2rg1
zr
≃ 7.80 × 10−6
M1
M⊙
1
2
650 AU
zr
1
2
rad, (22)
which is much larger than that obtained for the transmitting lens scenario (20).
We observe that, in this case the first zero of the projected Airy pattern (21) occurs at ρr
1GL = 2.40483/(kθr
1) ≃
4.91 cm(λ/1 µm)(M⊙/M1)
1
2 (zr/650 AU)
1
2 , which is very small and needs to be aperture-averaged. Fig. 3 demon-
strates the relevant behavior of the PSF (21) at very short spatial scales of a few cm. To estimate the impact of
the large aperture on the light amplification, we average the result (21) over the aperture of the telescope and, using
approximation for the Bessel functions for large arguments [21], we determine:
hµr
1GLi =
1
π(1
2 d)2
Z 1
2 d
0
Z 2π
0
PSF1GL(x) ρdρdφ = 2πkrg1

J2
0 kθ1
1
2 d

+ J2
1 kθ1
1
2 d

≃
≃
4
p
2rg1 zr
dr
= 3.03 × 109
M1
M⊙
1
2
zr
650 AU
1
2
1 m
dr

, (23)
where we recognize the well-known result obtained in [2]. Compared to (18), the aperture-averaging given by (23)
leads to a reduction in the light amplification by a factor of ≃ 38.46, thus resulting only in 2.6% of the maximal value
suggested by (18). Nevertheless, the overall result is still rather impressive.
B. Two lens transmission
Normalized
PSF
-0.4 -0.2 0.0 0.2 0.4
0.0
0.2
0.4
0.6
0.8
1.0
Distance from the optical axis, ρ, [m]
FIG. 3: Normalized PSFs: showing PSF of a single
lens in the receiving lens scenario (21), with M1 =
M⊙, and that for a double-lens transmission (34), with
M2 = M⊙. Clearly, both of these PSFs are identical.
We consider two lenses, positioned on the same line – the
optical axis – and separated by the distance of z12 from each.
The first lens is with the parameters used in Sec. III A, while
lens 2 is given by the mass, M2, Schwarzschild radius rg2 =
2GM2/c2
, and radius R2. To describe power transmission via
a two-lens system, one can develop a scheme similar to one
in Fig. 1. Next, we extended the expression for the optical
path (4) by modeling the contribution of the second lens to the
overall optical path, including that from its gravitational field.
As a result, we treat the EM wave that already passed by
lens 1 to be the source of light incident on lens 2. For an
observer at the lens 1 plane, the angle subtended by the physical
radius of lens 2 is small R2/z12 ≃ θ1, where θ1 is the angle that
determines the diffraction pattern of the incident field (16).
This angle is explicitly evaluated on the lens 2 plane, so that
distance zr in (16) is replaced by z12, yielding the expression
that is relevant for the transmission scenario discussed here
θ1 =
p
2rg1 z̃1
z12
≃ 2.46 × 10−9
M1
M⊙
1
2
zt
650 AU
1
2
10 pc
z12

rad. (24)
Therefore, as R2/z12 ≃ θ1, this lensing geometry corresponds to the strong interference regime of diffraction on lens
1 (see [2] for description). The analytical description of this process was already addressed in Sec. III A.

7
In the case with two-lens transmission, to describe the complex amplitude of an EM wave that is reaching the
lens plane of the lens 2, we use result (16). Specifically, remembering the structure of the EM field (9), we take the
amplitude of the EM at the second lens to be that given by the EM field after it passed lens 1, namely in (9) we
substitute A0(b) → A1GL(b2), which was obtained from (16) by replacing x → b2, we also bring back the x-dependent
term from (7), so that the entire procedure would yield the relevant one-lens amplification factor:
F1GL(b2) =
p
2πkrg1 exp
h
i
kb2
2
2(zt + z12)
i
J0 kθ1b2

. (25)
As a result, the amplitude of the EM field at the observer plane after lens 2 at zr ≪ z12 from it, following the
structure of (6), has the form similar to that of (6) and is given as below (see (B2)–(B4) for complete sturcture)
A2GL(x) = A02(x)F2GL(x), (26)
where the wave amplitude at the observer, A02(x), is given as
A02(x) =
E0
zt + z12 + zr
eikS02
, S02 = zt + z12 + ϕ̂1, (27)
where ϕ̂1 = krg1 ln[2e(zt + z12)/rg1 ] − 1
4 π, where we collected all the constants phase terms, including the relevant
φG1 term in (8) due to lens 1. The amplification factor F2GL(x) in (26) is given as follows
F2GL(x) =
zt + z12 + zr
(zt + z12)zr
k
2πi
ZZ
d2
b2 F1GL(b2) exp
h
ik
b2
2
2(zt + z12)
+
1
2zr
(b2 − x)2
− 2rg2 ln kb2
i
. (28)
See also Appendix B for a path integral derivation of (28). Next, defining the useful notation
1
z̃2
=
1
zt + z12
+
1
zr
⇒ z̃2 =
(zt + z12)zr
zt + z12 + zr
, (29)
we use expression for F1GL(b2) from (25), remove the spherical wave (as in (13)), we present (28) as below
F2GL(x) =
p
2πkrg1
k
iz̃2
1
2π
Z ∞
0
b2db2 J0 kθ1b2

exp
h
ik
b2
2
2z̃2
− 2rg2 ln kb2
i Z 2π
0
dφξ2 exp
h
− ik
b2ρ
zr
cos[φξ2 − φ]
i
=
=
p
2πkrg1
k
iz̃2
Z ∞
0
b2db2 J0 kθ1b2

J0

k
b2ρ
zr

exp
h
ik
b2
2
2z̃2
− 2rg2 ln kb2
i
. (30)
Similarly to (13), we take the integral over b2 by the method of stationary phase, to determine b2 =
p
2rg2 z̃2 and,
thus, the amplitude of the EM field on the image plane at the distance of zr from the second lens is now given as
F2GL(x) =
p
2πkrg1
p
2πkrg2 eiϕ̂2
J0

k
p
2rg1 z̃1
p
2rg2 z̃2
z12

J0

k
p
2rg2 z̃2
zr
ρ

, (31)
with ϕ̂2 = krg2 ln[2e(zt + z12 + zr)/rg2] − 1
4 π, where we collected all the constant phase terms due to the presence of
lens 2 on the eikonal, including the relevant φG2 term in (8). (See Appendix A for alternative derivation of (30).)
Next, we use solution (31), to determine the PSF of the two-thin-lens system
PSF2GL(x) = (2πkrg1 ) (2πkrg2 )J2
0

k
p
2rg1 z̃1
p
2rg2 z̃2
z12

J2
0

k
p
2rg2 z̃2
zr
ρ

, (32)
where we remind that in the two-lens case the distance to the receiver zr from one-lens case (12) is replaced with z12,
so that the effective distance z̃1 has the form z̃1 = ztz12/(zt + z12).
We observe that the argument of the first Bessel function in this expression is very large and is evaluated to be
k
p
2rg1 z̃1
p
2rg2 z̃2/z12 ≥ 9.86 × 106
(1 µm/λ)(M1/M⊙)
1
2 (M2/M⊙)
1
2 (zt/650 AU)
1
2 (zr/650 AU)
1
2 (10 pc/z12). In this
case, the first Bessel function, J2
0 (kb1b2/z12), can be approximated by using its expression for large arguments [21]:
J2
0

k
p
2rg1 z̃1
p
2rg2 z̃2
z12

=
1
p
2πkrg1
p
2πkrg2
z12
√
z̃1z̃2

1 + sin

φ(z̃1, z̃2)

, with δϕ = 2k
p
2rg1 z̃1
p
2rg2 z̃2
z12
. (33)

8
Considering here the term containing sin[φ(z̃1, z̃2)], we note that at optical wavelengths, this term is a rapidly oscil-
lating function of z̃z and z̃2 that averages to 0. Therefore, the last term in the form of J2
0 (x) expansion in (A2) may
be neglected, allowing us to present the averaged PSF from (32) as below:
hPSF2GL(x)i =
p
2πkrg1
p
2πkrg2
z12
√
z̃1z̃2
J2
0

k
p
2rg2 z̃2
zr
ρ

. (34)
This is our main result for the PSF for a two-lens axially-arranged system of monopole gravitational lenses. Note
that result (34) contains similar mass contributions from each of the lenses at both ends of the transmission link,
rg1 , rg2 . The amplification scales with the factor z12/
√
z̃1z̃2 involving the distance between the lenses, z12, as well as
the distances of both transmitter and receiver with respect to the transmitting and receiving lenses, z̃1, z̃2.
This result yields the maximum amplification factor for the two-lens system:
µ2GL = hPSF2GL(0)i =
p
2πkrg1
p
2πkrg2
z12
√
z̃1z̃2
≃
≃ 3.70 × 1014
M1
M⊙
1
2
M2
M⊙
1
2
1 µm
λ
z12
10 pc
650 AU
zz̃1
1
2
650 AU
zz̃2
1
2
. (35)
Clearly, for lensing architectures involving lenses with similar masses rg2 ≃ rg1 , the overall gain behaves as that of a
single lens (18) scaled with the geometric factor of z12/
√
z̃1z̃2, which provides additional gain for the lensing pair.
Based on (34), an observer in the focal region of lens 2, positioned on the optical axis at the distance of zr ≥ R2
2/2rg2
from it, will see one Einstein ring around the lens with the radius θ2 given as:
θ2 =
p
2rg2 z̃2
zr
≃
r
2rg2
zr
≃ 7.80 × 10−6
M2
M⊙
1
2
650 AU
zr
1
2
rad, (36)
which has the same structure as in the receiving lens case when a single lens is a part of a receiver (22).
However, we note that the estimate (35) may be misleading as it pertains only to the case when the receiver
aperture is smaller than the first zero of the Bessel function J0 kθ2ρ

present in (34), namely ρ2GL = 2.40483/(kθ2) ≃
4.91 × 10−2
(λ/1 µm)(M⊙/M2)
1
2 (zr/650 AU)
1
2 m, as shown in Fig. 3, which for optical wavelengths is not practical
and will be averaged by the aperture, as in (23). Therefore, for a realistic telescope, in context of a two-lens system,
similarly to (23), we develop an aperture-averaged value of the PSF, yielding the appropriate light amplification factor
hµ2GLi =
1
π(1
2 dr)2
Z 1
2 dr
0
Z 2π
0
PSF2GL(x) ρdρdφ =
p
2πkrg1
p
2πkrg2
z12
√
z̃1z̃2

J2
0 kθ2
1
2 dr

+ J2
1 kθ2
1
2t dr

≃
≃
4
p
2rg1 z̃1
dr
z12zr
z̃1z̃2
≃
r
2rg1
zt
4z12
dr
= hµ1GLi
z12
zt
= 9.62 × 1012
M1
M⊙
1
2
650 AU
zt
1
2
1 m
dr
z12
10 pc

. (37)
Note that after averaging, the dependence on the mass of the second lens is absent in (37). The reason behind this
is that the small scale of the diffraction pattern in (34) necessitated the aperture averaging, which caused the second
mass rg2 to drop out. (This is analogous to (21)–(23), where the same procedure also led to removing a factor of
√
rg1 .) This is because the telescope’s size, dr, is much larger than the first zero of the diffraction pattern in (34):
dr ≫ ρ2GL = 2.40483/(kθ2) ≃ 0.38λ
r
zr
2rg2
≃ 4.91 × 10−2
λ
1 µm
M⊙
M2
1
2
zr
650 AU
1
2
m. (38)
Note that when condition (38) in not satisfied and the aperture-averaging may not be relevant (i.e., due to a larger
wavelength, larger receiver distance, etc.), one would have to use the entire PSF with the J2
0 (x) factors included, as
in the case (19). In these cases, the relevant PSFs are (19), (21) for one lens, and either (34) or (A13) for two lenses.
Comparing result (37) with the transmission scenarios involving a single lens (18) and (23), we see that the two-
lens system provides significant additional light amplification captured by the factor (z12/zt). It is 82.5 times more
effective compared to the transmitting lens case (19) and is 3.17 × 103
times more effective than the receiving lens
scenario (23). Clearly, transmission via a pair of lenses benefits from gravitational amplification at both ends of the
transmission link, thus enabling interstellar power transmission with modern-day optical instrumentation.
IV. POWER TRANSMISSION
To assess the effectiveness of power transmission using gravitational lensing, we consider three transmission scenarios
that involve lensing with either a single lens or double lenses. These scenarios differ not only in the number of lenses
used but also in the positions of the transmitter and receiver relative to the lenses.

9
A. Single lens: transmission from its focal region
First, we consider transmission via a single lens with the transmitter positioned in its focal region at a distance
of zt ≥ R2
1/2rg1 from the lens, while the receiver is at interstellar distance of zr ≫ zt. This is a transmitting lens
scenario is the case iii) in Sec. II B and was addressed by (19). We consider transmitter to be a point source.
We assume that transmission is characterized by the beam divergence set by the telescope’s aperture dt yielding
angular resolution of θ0 ≃ λ/dt = 1.00 × 10−6
(λ/1 µm)(1 m/dt) rad. When the signal reaches the receiver at the
distance of (zt +zr) from the transmitter, the beam is expanded to a large spot with the radius of ρ∗ = (zt +zr)(λ/dt).
In addition, while passing by the lens, the light is amplified according to (19). As a result, a telescope with the aperture
dr receives a fraction of the transmit power that is evaluated to be:
Pt
1GL = P0 PSFt
1GL(x)
π(1
2 dr)2
πρ2
∗
= P0
π(1
2 dr)2
π(zt + zr)2
dt
λ
2
2πkrg1 J2
0

k
p
2rg1 zt
zr
ρ

≃
≃ 3.06 × 10−13
P0
1 W
1 µm
λ
2 dt
1 m
2 dr
1 m
210 pc
zr
2 M1
M⊙

W. (39)
For the same transmitting scenario, a free space laser power transmission in the vacuum is described as
Pfree =
P0π(1
2 dr)2
π(zt + zr)2
dt
λ
2
≃ 2.62 × 10−24
P0
1 W
1 µm
λ
2 dt
1 m
2 dr
1 m
210 pc
zr
2
W. (40)
Comparing results (39) and (40), we observe that signal transmission relying on a single gravitational lens amplifies
the received power by Pt
1GL/Pfree ≃ 1.17 × 1011
, as prescribed by (18). We note that, depending on the transmit-
ter’s performance, the power amplification value (39) would have to be adjusted to account for a realistic system’s
throughput. Below, we consider several transmitter implementation approaches relevant to this scenario.
First, from a practical standpoint, we observe that, a single transmitter with the combination of parameters w0 and
λ chosen in (39), will be able to form a beam only with very short impact parameters of ρmult ≃ zt
1
2 θ0 = zt(λ/2dt) =
0.07R⊙ (λ/1 µm)(1 m/dt)(zt/650 AU)(R1/R⊙). The corresponding EM field will be totally absorbed by the lens.
Therefore, the parameter choice (39) would require a special transmitter design that would rely on multiple laser
transmitter heads. Considering the Einstein ring that is formed with the radius of RER =
p
2rg1 zt, one choice would
be to take nt = 2πRER/2ρmult = 2π(dt/λ)
p
2rg1 /zt ≃ 48.98 (dt/1 m)(1 µm/λ) (M1/M⊙)
1
2 (650 AU/zt)
1
2 laser heads
arranged in a conical shape each with the same offset angle of RER/zt =
p
2rg1 /zt ≃ 1.61′′
(M1/M⊙)
1
2 (650 AU/zt)
1
2
from the mean direction to the lens. In effect, such a transmitter will illuminate the Einstein ring on the circumference
of the lens with the signal of total power ntP0 that will then be proceed toward the receiver.
In the scenario with multiple transmitting heads, the signal from nt telescopes illuminates the Einstein ring at RER.
However, only small fraction of this light will be deposited in the ring. That fraction is proportional to the ratio of
the Einstein ring’s area as seen by the receiver, AER0 = π(1
2 dr)2
PSFt
1GL, where PSF is from (19), to the combined area
of diffraction-limited fields projected by nt transmitting telescopes on the circumference of the ring at the lens plane,
ntπ(zt(λ/dt))2
. As the signal propagates toward the receiver, its deviation from the optical axis is controlled by the
gravitational field of lens 1 and grows as zrθt
1, with θt
1 from (20). Thus, the total signal reaching the receiver must be
scaled again by the ratio of the area subtended by the ring with RER to the area resulting from this growth.
Putting this all together, we have
Pt
1GL = ntP0
AER0
ntπ(zt(λ/dt))2
πR2
ER
π(zrθt
1)2
≃ P0 2πkrg1
dt
λ
2 d2
r
4z2
r
J2
0

k
p
2rg1 zt
zr
ρ

, (41)
which, with zt ≪ zr, is identical to (39). Therefore, if such a multiple-heads transmitter can be built, this approach
retains the flexibility of choosing any desirable wavelength to initiate the transmission.
Next, we note that a single transmitter with a small aperture of dsm
t = 6 cm may also be used to illumi-
nate the regions around the lens with the practically-important impact parameters, namely psing = zt(λ/2dsm
t ) ≃
1.17R⊙ (λ/1 µm)(6 cm/dt)(zt/650 AU)(R1/R⊙). In this case, one would be forced to choose very small transmit
aperture or longer wavelengths, thus limiting the flexibility for interstellar communications.
In addition, if implemented at the SGL, such a transmitter will illuminate the area πp2
sing πR2
⊙ resulting in the
situation when a major part of the emitted light field, (R⊙/psing)2
≃ 85.86 %, will be absorbed by the lens and only
small fraction of it, ∼ 14.15 %, will illuminate the region with the Einstein ring. This will reduce the transmission
throughput to only t1tr ∼ 0.188, thus requiring a source with a higher transmit power compared to the case with
multiple transmitters. In any case, the total transmit power in this scenario would have to be adjusted, yielding in
this case the power amplification value of t1trPt
1GL/Pfree ≃ 4.29 × 108
(1 m/dt)(M1/M⊙)
1
2 (zt/650 AU)
1
2 .

10
Finally, going back to the multi-head transmitter design discussed above, we note that, using a transmitter with
nt ∼ 49 laser heads each pointing in different directions may be a bit too complicated and other designs should be
considered. As an example, one may consider a holographic diffusing element1
that can be used to sculpt an outgoing
beam to practically any shape. Such a diffuser may allow for a uniform illumination of the Einstein ring at a specified
angular separation from the lens, thus offering a plausible approach to a transmitter design. In this case, there will
be no significant loss in the optical throughput and the entire power amplification of (39) will be at work.
We observe that each of the three transmission architectures above are different, yielding different optical trans-
mission throughputs, specific noise contributions, and will be characterized by different SNR performances [3, 4]. In
any case, these types of the transmission scenarios may be used to search for the signals before moving to the focal
region of lens 2 (discussed in Sec. IV C) that would be needed to establish a reliable communication infrastructure.
B. Single lens: reception at its focal region
Another single lens transmission scenario, involves transmission from an interstellar distance into the focal region
of a lens (the case iv). in Sec. II B). In this case, transmitter is now positioned at large distance of zt from the lens,
while the receiver is at the focal region of that lens at zr ≥ R2
1/2rg1 ≪ z12 from it, so that in this case zt ≪ zr. This
case is also covered by (23). The only difference from scenario discussed in Sec. IV A is that the laser beam divergence
in this case will naturally result in a large spot size illuminated by the transmitter at the lens plane, thus reducing
the incident power and overall link performance.
We note that a telescope with aperture dr positioned on the optical axis in the focal area of a lens and looking back at
it would see the Einstein ring formed around the lens. The energy deposited in the ring will be the same as that received
by the observer. This means that the effective collecting area of the receiving telescope Atel = π(1
2 dr)2
hµr
1GLi, with
hµr
1GLi from (23), is equal to the area subtended by the observed Einstein ring AER = 2πRERwER, were RER =
p
2rg1 zr.
Equating Atel = AER, allows us to determine the width the ring wER = 1
2 dr, thus establishing the effective area
subtended by the Einstein ring as seen by the receiving telescope in the case of a single lens transmission
AER1 = πdr
p
2rg1 zr. (42)
Clearly, a telescope with the angular resolution of 1.22(λ/dr) = 1.22 × 10−6
(λ/1 µm)(1 m/dr) ≫ dr/2zr = 5.14 ×
10−15
(dr/1 m)(650 AU/zr), will not be able to resolve the width of the ring; however, the length of its circumference
is resolved with nt = 2π
p
2rg1 /zr/(λ/dr) ≃ 48.98 (M1/M⊙)
1
2 (650 AU/zr)
1
2 (1 µm/λ)(dr/1 m) resolution elements.
This information is useful when considering a transmitter’s design or evaluating a receiver’s performance.
To evaluate the effectiveness of such scenario, consider transmitting telescope that illuminates the Einstein ring
around the lens at the radius of RER =
p
2rg1zr. However, because of the diffraction transmitted energy will spread
over a much larger area with the radius of ρ′
∗ = zt(λ/dt) = 443.54R⊙ (λ/1 µm)(zt/10 pc)(1 m/dt)(R1/R⊙). Thus, not
all the energy will be received at the Einstein ring, some of the energy will be lost. By the time the signal reaches
the receiver, its deviation from the optical axis is controlled by gravity changing by zrθr
1, where θr
1 is from (22). As a
result, we estimate the total received power in this case when the single lens acts a part of a receiver as below
Pr
1GL = P0
AER1
π(zt(λ/dt))2
πR2
ER
π(zrθr
1)2
=
P0π(1
2 dr)2
πz2
t
dt
λ
2 4
p
2rg1zr
dr
=
= 7.96 × 10−15
P0
1 W
1 µm
λ
2 dt
1 m
2 dr
1 m
10 pc
zt
2 M1
M⊙
1
2
zr
650 AU
1
2
W, (43)
which is a factor 38.47 times smaller than that obtained in the scenario with transmission from the focal region (39)
discussed in Sec. IV A. As a result, there is some difference between transmitting from the focal region of a lens to
a receiver at large interstellar distance vs transmitting from a large distance into the focal region of the lens. In
addition, there may be different noise sources involved affecting the SNR performance of a transmission link.
C. Two lenses: transmission and reception from/to the focal regions
Now we consider scenario that involves a two-lens power transmission, where transmitter, is positioned in the focal
region of lens 1 at the distance of zt R2
1/2rg1 from it, sends a signal toward lens 2 that is separated from lens 1 by
1 As shown here: https://www.rpcphotonics.com/engineered-diffusers-information.

11
the interstellar distance z12 ≫ zt. After passing by lens 2, the signal is detected by a receiver positioned in the focal
region of lens 2 at the distance of zr R2
2/2rg2 from it. Analytically, this case is described by solution (A16).
The power transmission, as outlined above, is characterized by (39), where the light amplification is from (A16).
In this case, we follow the same approach that we used to derive (39). We may treat the two-lens system in the same
manner as we dealt with a single-lens transmission (39). For that, we realize that when the signal reaches the receiver
at the distance of (zt +z12 +zr), the beam is expanded to a large spot with the radius of ρ∗∗ = (zt +z12 +zr)(λ/dt) ≃
z12(λ/dt). In addition, while passing by the two-lens system, the light is amplified according to (A16).
As a result, a telescope with the aperture dr receives a fraction of the transmit power that is evaluated to be:
P2GL = P0 hµ2GLi
π(1
2 dr)2
πρ2
∗∗
=
P0π(1
2 dr)2
πz2
12
dt
λ
2
r
2rg1
zt
4z12
dr
≃
≃ 2.53 × 10−11
P0
1 W
1 µm
λ
2 dt
1 m
2 dr
1 m
10 pc
z12
2 M1
M⊙
1
2
650 AU
zt
1
2
W, (44)
which clearly demonstrates the benefits of the gravitational amplification provided by the double-lens systems while
yielding the gain of P2GL/Pfree =
p
2rg1 /zt(4z12/dr) ≃ 9.62 × 1012
(M1/M⊙)
1
2 (650 AU/zt)
1
2 (1 m/dr)(z12/10 pc),
which is ∼ (2/π2
)(λ/dr)(z12/
p
2rg1 zt) ≃ 82.50 (λ/1 µm)(1 m/dr)(M⊙/M1)
1
2 (650 AU/zt)
1
2 (z12/10 pc) times higher
than in the case when a singe lens acts as a part of a transmitter, shown by (39), or z12/zt ≃ 3.17 ×
103
(z12/10 pc)(650 AU/zt) times higher than that of a single lens acting as a part of the receiver as seen by (43).
We may consider an alternative derivation for the two-lens transmission link (44). A telescope with aperture dr
would see the two Einstein rings formed around the lens. They are unresolved, but the energy, deposited in the ring
will be the same as that received by the telescope. This means that the effective collecting area of the receiving
telescope Atel = π(1
2 dr)2
hµr
2GLi, with hµr
2GLi is now from (A16), is equal to the area subtended by the observed
Einstein ring AER = 2πRERwER, were RER =
p
2rg1 zt. Equating Atel = AER, allows us to determine the width the
ring wER = 1
2 dr(z12/zt), thus establishing the effective area subtended by the Einstein ring as seen by the receiving
telescope in the case of a two lens transmission
AER2 = πdrz12
r
2rg1
zr
= AER1
z12
zt
, (45)
where AER1 from (42). Therefore, the amount of light captured by the ring increases with distance as (z12/zt), so as
the amplification factor (A16). This factor is the key source of additional amplification for two-lens transmissions.
Again, a telescope with the resolution of 1.22λ/dr = 1.22 × 10−6
(λ/1 µm)(1 m/dr) ≫ (z12/zt)dr/2zr = 1.6 ×
10−11
(dr/1 m)(650 AU/zr)2
(zr/10 pc), will not be able to resolve the width of the Einstein ring; however, the length
of its circumference is resolved with nt = 2π
p
2rg1 /zr/(λ/dr) ≃ 48.98 resolution elements as in the earlier case (42).
We can do that in the manner similar to the approach we developed while developing (41). For that, there are
important steps to consider, namely
• Consider nt transmitting apertures dt positioned at the distance of zt in the focal region of lens 1 (as in the case
of (41)). Assuming that these telescopes coherently illuminate the Einstein ring with the radius of RER around
lens 1. Because of the diffraction, transmitted energy spreads over the area with radius of ρ′
∗ = z12(λ/dt). Thus,
the fraction of the signal that will be deposited in the ring is AER2/π(z12(λ/dt))2
, were AER2 is from (45); also here
we use the amplification factor hµ2GLi from (A16), thus the width of the ring, in the case, is wER = 1
2 dr(z12/zt).
• As the signal propagates toward lens 2, its deviation from the optical axis grows as z12θt
1, where θt
1 is from (24).
By the time is reaches the receiver behind lens 2, its deviation from the optical axis will be changed again by
the amount of zrθ2, where θ2 is from (A6), again requiring an appropriate scaling by πR2
ER′ /π(zrθ2)2
, where
RER′ =
p
2rg2 zr is the radius of the Einstein ring seen by the averaged PSF (A16) around lens 2.
As a result, using the same approach that was used to derive (41), we estimate the total received power as below
P2GL = ntP0
AER2
ntπ(zt(λ/dt))2
πR2
ER
π(z12θt
1)2
πR2
ER′
π(zrθ2)2
≃
P0π(1
2 dr)2
πz2
12
dt
λ
2
r
2rg1
zt
4z12
dr
, (46)
which is identical to (44). As a result, in the case of the two-lens transmission, the total power received is higher than
that for a single-lens transmission. In other words, the second lens adds more power by focusing light, as expected.

12
V. DETECTION SENSITIVITY
Now that we have established the power estimates for the transmission links characteristic for various lensing
architectures involving one and two lenses, we need to consider practical aspects of such transmission links. For this,
can evaluate the contribution of the brightness of the stellar atmospheres to the overall detection sensitivity.
A. Relevant signal estimates
First of all, based on the results for power transmission for various configurations considered, namely (39), for a
lens near the transmitter, (43), for a lens on a receiving end, and (44) for the pair of lenses,, we have the following
estimates for photon flux, Qt
1GL, Qr
1GL, Q2GL, received by the telescope in all three of these cases
Qt
1GL =
λ
hc
Pt
1GL ≃ 1.54 × 106
P0
1 W
1 µm
λ
dt
1 m
2 dr
1 m
210 pc
zr
2 M1
M⊙

phot/s, (47)
Qr
1GL =
λ
hc
Pr
1GL ≃ 4.01 × 104
P0
1 W
1 µm
λ
dt
1 m
2 dr
1 m
10 pc
zt
2 M1
M⊙
1
2
zr
650 AU
1
2
phot/s, (48)
Q2GL =
λ
hc
P2GL ≃ 1.27 × 108
P0
1 W
1 µm
λ
dt
1 m
2 dr
1 m
10 pc
z12
2 M1
M⊙
1
2
650 AU
zt
1
2
phot/s. (49)
Therefore, based on the estimates above, we have significant photon fluxes in each of the three configurations
considered. However, each of the three configurations will have different contributions from the brightness of the
stellar atmospheres along the relevant transmissions links. Below, we will provide estimates for the relevant noise.
B. Dominant noise sources
The Einstein rings that form around the stellar lenses appear on the bright background of the stellar atmospheres
(coronas.) To assess this noise contribution from the stellar atmospheres, we need to estimate their contribution to
the signal received by the receiver. Such signals are different, thus establishing the practically relevant constraints.
Similar to case of imaging with the SGL, we know that the most doming noise sources are those that come for the
lenses themselves, namely the light emitted by them and the brightness of their coronas in the areas where Einstein
rings will appear. Not all that light can be blocked by chronographs and must be accounted for in the sensitivity
analysis. Clearly, there are other sources of noise, especially at small angular separations from the lenses, and they
must be treated which considering realistic transmission scenarios.
Below, we will focus on the main anticipated sources of noise – those from lenses themselves and from their coronas.
1. Single lens: transmission from its focal region
In the case, when the lens is at the transmitter node, the situation is quite different from the one that we considered
in Sec. V B 2. In this case the angular extend of the Einstein ring is given by (20). The angular size of the stellar
lens is R1/zt = 2.26 × 10−9
(R1/R⊙)(10 pc/zt) rad, similar to that of the Einstein ring. A conventional telescope with
a practical diameter will not be able to resolve this star, therefore, the use of a coronagraph is out of the question.
Thus, the light emitted by that stellar lens and received by the telescope will be the noise that must be dealt with.
In other words, the light from the Einstein ring will arrive together with the unattenuated light from the star.
We consider the Sun as a stand-alone for the lens 1. In this case, assuming the Sun’s temperature2
to be T⊙ =
5 772 K, we estimate the solar brightness from Planck’s radiation law:
B⊙ = σT 4
⊙ =
Z ∞
0
Bλ(T⊙)dλ =
Z ∞
0
2hc2
λ5 ehc/λkT⊙ − 1
dλ = 2.0034 × 107 W
m2 sr
, (50)
2 See https://en.wikipedia.org/wiki/Sun

13
where we use a blackbody radiation model with σ as the Stefan-Boltzmann constant, kB the Boltzmann constant.
This yields the total power output of the Sun of L⊙ = πB⊙4πR2
⊙ = 3.828 × 1026
W, which is one nominal solar
luminosity as defined by the IAU.3
When dealing with laser light propagating in the vicinity of the Sun, we need to be concern with the flux within
some bandwidth ∆λ around the laser wavelength λ, assuming we can filter the light that falls outside ∆λ, then
B⊙(λ, ∆λ) = Bλ(T⊙)∆λ =
2hc2
λ5 ehc/λkBT⊙ − 1
∆λ ≃ 1.07 × 105
1 µm
λ
5 ∆λ
10 nm
W
m2 sr
. (51)
We take (51) to derive the luminosity of the Sun within a narrow bandwidth as follows
L⊙(λ, ∆λ) = πB⊙(λ, ∆λ)4πR2
⊙ =
2hc2
λ5 ehc/λkBT⊙ − 1
∆λ ≃ 2.05 × 1024
1 µm
λ
5 ∆λ
10 nm

W. (52)
As a result, in the case when a lens used as a part of the transmitter, the power of the signal received from the
lensing star received by a telescope and the corresponding photon flux are derived from (52) and are given as
Pt ⋆
1GL = L1(λ, ∆λ)
π(1
2 dr)2
πz2
r
= 5.39 × 10−12
L1
L⊙
1 µm
λ
5 ∆λ
10 nm
dr
1 m
210 pc
zr
2
W, (53)
Qt ⋆
1GL =
λ
hc
Pr,st
1GL = 2.71 × 107
L1
L⊙
1 µm
λ
4 ∆λ
10 nm
dr
1 m
210 pc
zr
2
phot/s. (54)
2. Single lens: reception at its focal region
The case of a single lens positioned nearby receiver is the most straightforward one. In this scenario, the angular size
of the Einstein ring is given by (22), which is quite large. The receiving telescope with the diameter of dr resolves the
angular diameter of the stellar lens in (R1/zr)/1.22(λ/dr) = 5.86 (R1/R⊙)(650 AU/zr)(1 µm/λ)(dr/1 m) resolution
elements. Thus, one may consider using a coronagraph to block the stellar light (in our case, this would be the Sun.)
Most of the relevant modeling was already accomplished in the context of our SGL studies. For that we know that
the transmitted signal is seen by the telescope within the annulus that corresponds to the Einstein ring around a lens
at the distance of b =
p
2rg1 z̃1 and within the thickness of wER = 1
2 dr, formed by the transmitted light. However,
the telescope sees a much larger annulus with the thickness of wtel ≃ zr(λ/dr), which is wtel/wER = 2zr(λ/d2
r) =
9.72 × 107
(zr/650 AU)(λ/1 µm)(1 m/dr)2
times thicker, thus receiving the light from a much larger area.
While working on the SGL, we developed an approach to account for the contribution of the solar corona to imaging
with the SGL. We take our Sun as the reference luminosity. In the region occupied by the Einstein ring in the focal
plane of a diffraction-limited telescope, the corona contribution is given after the coronagraph as:
Pcor(λ, ∆λ) = ǫcor π(1
2 dr)2
Z 2π
0
dφ
Z ∞
θ0
θdθ Bcor(θ, λ, ∆λ), (55)
where θ = ρ/zr, θ0 = R⊙/zr, and ǫcor = 0.36 is the fraction of the encircled energy for the solar corona (see [14]).
The surface brightness of the solar corona Bcor may be taken from the Baumbach model [22, 23], or use a more
recent and a bit more conservative model [24] that is given by:
Bcor(θ, λ, ∆λ) =

10−6
· B⊙(λ, ∆λ)
h
3.670
θ0
θ
18
+ 1.939
θ0
θ
7.8
+ 5.51 × 10−2
θ0
θ
2.5i
, (56)
where B⊙(λ, ∆λ) is from (51). Using (56) in (55) and following the approach used in [3, 14], we obtain the power of
the signal received from the corona and the corresponding photon flux
Pcor(λ, ∆λ) ≃ 6.58 × 10−12
1 µm
λ
5 L1
L⊙
∆λ
10 nm
dr
1 m
2650 AU
zr
2.5
W, (57)
Qcor =
λ
hc
Pcor ≃ 3.31 × 107
1 µm
λ
4 L1
L⊙
∆λ
10 nm
dr
1 m
2650 AU
zr
2.5
phot/s. (58)
Although these estimates may need to be refined by analyzing the spectral composition of the corona of a particular
stellar lens and the relevant signal as was recently done in [4], the obtained results provide good initial estimates.
3 See https://www.iau.org/static/resolutions/IAU2015_English.pdf

14
3. Two lenses: transmission and reception from/to the focal regions
In the case a two-lens transmission, the situation changes again. This time, one would have to deal not only with the
light received from lens 2 that falls within the annulus around the Einstein ring as in the case discussed in Sec. V B 2,
but also the light that arrive into this annulus from lens 1 must also be accounted form. However, in this case, the
light from lens 1 will be amplified by the gravitational field of lens 2, providing additional noise background.
Consdier formation of this background light, we note that this process is very similar to imaging of the exoplanets
with the SGL [14]. Except, here we concerned with the formation of another Einstein ring formed around lens 2 at
the same location as the transmitted signal, but this time another ring is formed from the light received from lens 1.
Similar to the SGL, ;ens 2 will focus light while reducing the size of the image compared to the source by a factor
of zr/z12 ∼ 3.15 × 10−4
(zr/650 AU)(10 pc/z12), see [14]. For a lens with physical radius R1, positioned at a distance
of z12 from lens 2, the image of this target at a distance of zr, will be compressed to a cylinder with radius
ρ⋆
1 =
zr
z12
R1 = 219.24
R1
R⊙
zr
650 AU
10 pc
z12

km. (59)
Using (59), we see that a telescope with the aperture dr will resolve this object with Nd linear resolution elements
Nd =
2ρ⋆
1
dr
=
2R1
dr
zr
z12
= 4.39 × 105
R1
R⊙
zr
650 AU
10 pc
z12
1 m
dr

. (60)
Therefore, to account for the source provided by the amplified stellar light, we can use the expressions that we
developed for the SGL to describe the total power collected from a resolve exoplanet. The relevant expression in
this case is Eq. (2) in [3]. Replacing in this expression the surface brightness of the exoplanet Bs with Bcor(θ, λ, ∆λ)
describing the solar surface brightness within the bandwidth of ∆λ around the central wavelength λ from (51), we
have At the same time, the power at the Einstein ring at the detector placed in the focal plane of an optical telescope
is dominated by the blur and is given as
P⋆
2GL(ρ) = ǫblurπB⊙(λ, ∆λ)π(1
2 dr)2 2R1
z12
r
2rg2
zr
µ(ρ), with µ(r) =
2
π
E
h ρ
ρ⋆
1
i
, 0 ≤ ρ/ρ⋆
1 ≤ 1, (61)
where ǫblur = 0.69 is the encircled energy fraction for the light received from the entire resolved lens 1 [14] and E[x]
is the elliptic integral [21]. To evaluate the worst case, below we can take ρ = 0 and thus, µ(0) = 1.
Taking this into account and using B1(λ, ∆λ) from (51) and considering the worst case, when ρ = 0 and thus,
µ(0) = 1, from (61), we have the following estimate for the power at the receiver from the amplified light from lens 1
P⋆
2GL ≃ 6.42 × 10−9
L1
L⊙
1 µm
λ
5 ∆λ
10 nm
dr
1 m
2 R1
R⊙
10 pc
z12
M2
M⊙
1
2
650 AU
zr
1
2
W. (62)
This power corresponds to the enormous photon flux received at the telescope
Q⋆
2GL =
λ
hc
P⋆
2GL = 3.23 × 1010
L1
L⊙
1 µm
λ
4 ∆λ
10 nm
dr
1 m
2 R1
R⊙
10 pc
z12
M2
M⊙
1
2
650 AU
zr
1
2
phot/s. (63)
It is clear that photon flux at such high level will form challenging background for two-lens power transmissions.
C. Interstellar transmission: relevant SNRs
We assume that the contributions of the stellar coronas is removable (e.g., by observing the corona from a slightly
different vantage point) and only stochastic (shot) noise remains, we estimate the resulting SNR as usual
SNR =
QER
√
QER + Qnoise
. (64)
The first case we consider is when transmitter sends a signal directly via its local lens with the receiver being at an
interstellar distance. In this case, we use results (47) and (54), we estimate the relevant SNR as below
SNRt
1GL =
Qt
1GL
p
Qt
1GL + Qt ⋆
1GL
≃ 2.87 × 102
P0
1 W
λ
1 µm
10 nm
∆λ
1
2
dt
1 m
2 dr
1 m
10 pc
zr
M1
M⊙
L⊙
L1
1
2
r
t
1 s
. (65)

15
This represents a very handsome SNR level that shows the feasibility of establishing an interstellar power transmission
link where one would transmit from a focal region behind a lens to a receiver at an interstellar distance.
Next, we consider the case when transmitter sends a signal toward a receiver that uses its local lens to amplify
the signal. In this case, there are two scenarios available: 1) an isolated transmitter is positioned in space with no
significant stellar background, and 2) when transmitter is positioned next to a star, i.e., the situation is similar when
we transmit a signal from a ground-based observatory.
In the first of these two cases, the only background noise to consider is the corona noise from the lensing star next
to the receiver. Using (48) to represent the signal of interest, Qr
1GL, and (58) that for the noise from the stellar corona,
Qcor, we estimate the SNR for the case when a single lens is positioned at the receiving end of the link:
SNR
r [1]
1GL =
Qr
1GL
p
Qr
1GL + Qcor
≃
≃ 6.97
P0
1 W
λ
1 µm
10 nm
∆λ
1
2
dt
1 m
210 pc
zt
2 M1
M⊙
1
2
zr
650 AU
1.75L⊙
L1
1
2
r
t
1 s
. (66)
In the second case, we consider the worst case, when the star is directly behind the transmitter. The light from the
star will also be received and amplified by the lens. In this case, the signal Qr
1GL is still given by (48), but the noise in
this case will be due to corona around the lens Qcor from (58) and the light of the star behind the transmitter that
is amplified by the lens Q⋆
2GL, given by (63). The reslting SNR is given as below
SNR
r [2]
1GL =
Qr
1GL
p
Qr
1GL + Qcor + Q⋆
2GL
≃ 0.22
P0
1 W
λ
1 µm
10 nm
∆λ
1
2
dt
1 m
210 pc
z12
3
2
×
×
M1
M⊙
1
2
M⊙
M2
1
4
650 AU
zt
1
4
L⊙
L1
1
2
R⊙
R1
1
2
r
t
1 s
. (67)
The estimates (66) and (67) also support the feasibility of establishing an interstellar transmission where one would
transmit from an interstellar distance directly into a lens.
Finally, we estimate the SNR for the two-lens transmission. In this case, we use (49) for the signal with the relevant
noise source for the corona around lens 2, Qcor, taken from (58) and the amplified light from lens 1, Q⋆
2GL, given by
(63). As a result we have the following SNR for the two-lens transmission
SNR2GL =
Q2GL
p
Q2GL + Qcor + Q⋆
2GL
≃ 7.05 × 102
P0
1 W
λ
1 µm
10 nm
∆λ
1
2
dt
1 m
210 pc
z12
3
2
×
×
M1
M⊙
1
2
M⊙
M2
1
4
650 AU
zt
1
4
L⊙
L1
1
2
R⊙
R1
1
2
r
t
1 s
, (68)
which is the best among the three scenarios considered. Clearly, the fact that the stellar light from lens 1 is amplified
by lens 2 provides a very strong background. As a result, power transmission with a single lens yields a comparatively
strong SNRs compared to the case of two-lens transmission. One can further increase this SNR by increasing the
transmitted power and/or the size of the transmitting aperture. Nevertheless, analysis presented here demonstrates
the feasibility of using gravitational lensing for establishing practical interstellar power transmission links.
VI. DISCUSSION
We considered the propagation of monochromatic EM waves in the presence of non-interacting monopole gravi-
tational lenses. To do that we examined a scenario in which the source, lenses, and observer align along a single
axis—a situation characterized by axial symmetry. We find that the challenging diffraction integrals can be solved
analytically, offering valuable insights. We observe that the number of Einstein rings formed around each successive
lens doubles, but they are not resolvable from each other. The overall light amplification is determined by the mass
of the first lens, positions of the transmitter/receiver, and is affected by the distances between the lenses. Interesting,
but in the realistic observing scenarios, there is no dependence on the mass of the second lens.
It is remarkable that performance of a transmission link is determined by a single factor z/z0 in front of x′
in (8),
that determines image scaling in different lensing scenarios: 1) Lens acts as a part of a transmitter, when the signal
transmission is done from a distance shorter than that of the receiver, z0 ≪ z. In this case, the image size grows as
z/z0, as in the transmitting lens scenario in Sec. III A 1 or in the transmission part at lens 1 in the two-lens scenario
in Sec. IV C, when the image size grows as z12/zt. 2) Lens acts as a part of a receiver with the transmit/receive

16
positions reversed, z0 ≫ z. In this case, the lens focuses light compressing images by z/z0, as in the receiving lens
scenario of Sec. IV B or in the reception part of the two-lens transmission discussed in Sec. IV C.
This scaling changes the width of the Einstein ring formed at each lens plane and thus the area subtended by the
ring, as seen in (45). We anticipate that including higher order terms ∝ b4
in (4) may improve the fidelity of the
solutions. That can be done following the approach that we have demonstrated here. But until that extension is
developed and evaluated, the solutions that were provided here are sufficient to study various transmission links.
As a result, the two-lens transmission configurations shown by (44), are found to represent the most favorable
transmission architectures, providing access to the largest light amplification of (37) and resulting in the gain of
P2GL/Pfree =
p
2rg1 /zt(4z12/dr) ≃ 9.62 × 1012
(M1/M⊙)
1
2 (650 AU/zt)
1
2 (1 m/dr)(z12/10 pc). Configurations, where a
single lens acts as a part of a transmitter were found to be the second best, providing access to the light amplification
factor of (18), thus yielding the gain of Pt
1GL/Pfree ≃ 2πkrg1 ≃ 1.17 × 1011
(M1/M⊙)(1 µm/λ). Finally, configurations,
where a single lens acts as a part of a receiver, are found to amplify electromagnetic signals in accord to (23), thus
resulting in the gain of Pr
1GL/Pfree = 4
p
2rg1 zr/dr ≃ 3.03 × 109
(M1/M⊙)
1
2 (zr/650 AU)
1
2 (1 m/dr).
Note that although throughout the paper we used λ = 1µm to estimate various values, all our expressions are valid
for shorter wavelengths. In fact, not addressing the technical implementations, for shorter wavelength the gravitational
lensing will be even more robust with higher overall gains. Our models work for longer wavelengths as well, but will
some limits. As we have shown in [25], longer wavelengths, would experience much higher diffraction, so that one
may have to consider using the full diffraction-limited PSFs given by (17) and (A13). Therefore, these wavelengths
were not considered here, as our prior work [25, 26] had shown that such longer wavelengths, say above 30 cm, will
be severely affected by stellar atmospheres to the point of being completely blocked by plasma in their coronas.
Our key results are (34), (37) that present the PSF of a light transmission with a pair of gravitational lenses.
Expression (34) has a similar mass contributions from the lenses located at both the start and end of the transmission
link. The amplification is proportional to the factor z12/
√
z̃1z̃2, where z12 is the distance between the two lenses, while
z̃1, z̃2.being the distances from the transmitter to its respective lens and from the receiver to its lens, respectively.
However, as the size of a realistic telescope will be larger than the diffraction pattern set in (34), the result must be
averaged over the telescope’s aperture. We found that after that averaging, the dependence on the mass of the second
lens is absent in (37). (This is analogous to (21)–(23), where the same procedure also removed a factor of
√
rg1 .)
Conclusively, our analysis underscores the fact that, in terms of light amplification, a multi-lens configuration
outperforms a single-lens system by a significant margin. We observe that all the transmission cases are characterized
by different the PSFs, exhibiting different structures with features from a few cm scales to several 100s of meters.
Notably, in the transmitting lens case the structure within the PSF (19) is large and does not need to be averaged.
However, in the two-lens system the fine structure in the PSF is not going to be directly observed with a reasonable-
size telescope. Instead, this structure is averaged out by a modest, say, a 1-m telescope, yielding the average PSFs as
in the receiving lens scenario (23) and that of the two-lens system (A16).
The feasibility of interstellar transmission became evident when we analyzed the SNRs for various transmission
scenarios (see Sec. V C). Although the two-lens transmission benefits from an impressive combined power amplification
available in the two-lens systems, the two-lens case is severely affected by the much amplified background light coming
from the first lens. Clearly, one would have to include other sources of light in the vicinity of the first lens, but their
presents is not expected to significantly affect the achievable SNR (68), which is quite high. On the other hand, both
single-lens transmission cases, where the lens either close to the transmitter or is near the receiver, also show a very
robust link performance, as evidenced by the corresponding SNR estimates (65) and (66), (67).
Concluding, we would like to mention that the results obtained here, could have a profound effect for applications
aiming at interstellar power transmission. Not only we can look for transmitted signals using modern astronomical
techniques [8–11, 27, 28], we may also transmit such signals with space-based platforms in the focal region of the solar
gravitational lens (SGL) using technologies that are either extant or in active development [29, 30].
Looking ahead, it would be interesting to explore non-axially symmetric setups and also transmission in the presence
of non-spherically symmetric lenses, such as those discussed in [12, 16, 31, 32]. These lenses will have a complex PSF
structure exhibiting contribution of various caustics. Although such lenses will disperse some light toward the cusps of
the caustics [33], with a proper transmission alignment such effects may be reduced. Investigating such configurations
via numerical simulations might shed light on the intricate dynamics in such lensing systems. Nonetheless, our
primary focus should be directed towards deriving a formal solution for the situation where a transmitter, positioned
at a significant yet finite distance from a lens, emits a beam—whether plane, spherical, or Gaussian—towards that
lens. Such a solution may be obtained by following the approach taken in [2] that would put all the related analysis
on a much stronger footing. Work on this matter is currently in progress, and findings will be reported elsewhere.

17
Acknowledgments
We kindly acknowledge discussions with Jason T. Wright who provided us with valuable comments and suggestions
on various topics addressed in this document. This work was performed at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and Space Administration.
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Appendix A: Alternative derivation for a two-lens transmission
Here we provide an alternative derivation for the diffraction integral (30), which is repeated below, for convenience:
F2GL(x) =
p
2πkrg1
k
iz̃2
Z ∞
0
b2db2 J0 kθ1b2

J0

k
b2ρ
zr

exp
h
ik
b2
2
2z̃2
− 2rg2 ln kb2
i
. (A1)
Clearly, for any impact parameters smaller that the radius of the second lensing star, b2 ≤ R2, the light rays will
be absorbed by that lens. Therefore, the integration over db2 in (30) really starts from R2. In this case, the argument
of the first Bessel function in this expression is very large and is evaluated to be kθ1b2 ≃ kb2
p
2rg1 z̃1/z12 ≥ 1.07 ×

18
107
(1 µm/λ)(M1/M⊙)
1
2 (zt/650 AU)
1
2 (10 pc/z12)(b2/R⊙). In this case, this Bessel function, J0(x) can approximated
by using its expression for large arguments (see [34])
J0 kθ1b2

=
1
√
2πkθ1b2

ei(kθ1b2− π
4 )
+ e−i(kθ1b2− π
4 )

. (A2)
Note that, as we are concerned with the regions on the optical axis after lens 2, where ρ ≈ 0, we do not need to use
a similar approximation for the second Bessel function in (30).
As a result of using approximation (A2), expression (30) takes the form:
F2GL(x) =
p
2πkrg1
k
iz̃2
Z ∞
R2
√
b2db2
√
2πkθ1
J0

k
b2ρ
zr

ei kθ1b2− π
4

+ e−i kθ1b2− π
4

e
ik
b2
2
2z̃2
−2rg2 ln kb2

. (A3)
We evaluate this integral using the method of stationary phase (as was done in [17]). To do that, we see that the
relevant b2-dependent part of the phase in (A3) is of the form
ϕ±(b2) = k
b2
2
2z̃2
± b2θ1 − 2rg2 ln kb2

∓ π
4 . (A4)
The phase is stationary when dϕ±/db2 = 0, which implies
b2
z̃2
−
2rg2
b2
± θ1 = 0. (A5)
As impact parameter is a positive quantity, b2 ≥ 0, this quadratic equation yields two solutions:
b±
2 = z̃2
1
2
q
θ2
1 + 4θ2
2 ∓ θ1

, where θ2 =
r
2rg2
z̃2
≃
r
2rg2
zr
≃ 7.79 × 10−6
M2
M⊙
1
2
650 AU
zr
1
2
rad, (A6)
where θ2 is the radius of the Einstein ring to be formed around the second lens. The impact parameters b±
2 are two
families of impact parameters describing incident and scattered EM waves, corresponding to light rays passing by the
near side and the far side of lens 2, correspondingly (see [17] for details).
Note that, as opposed to the case with one lens (see (16) and (17)), the two solutions b±
2 from (A6) suggest that
there are now two Einstein rings formed around the second lens with the radii θ±
2 = b±
2 /z̃2 given as below:
θ±
2 = 1
2
q
θ2
1 + 4θ2
2 ∓ θ1

. (A7)
Note that, given the fact that nominally θt
1 ≪ θ2 (see (24)) and (A6)), both rings are very close being only
δθ2 = θ−
2 −θ+
2 ≃ θ1 ≃ 2.46×10−9
rad, which is the angle that unresolvable by a conventional telescope. So, nominally
such a telescope would see both of these ring as one, receiving the signal from both of them. Although these rings
may not be resolved from each other, but yet they carry the relevant energy to the receiver.
Following the approach presented in [17], we again use the method of stationary phase (A4) for both solutions in
(A7). Thus, the amplitude of the EM field on the image plane at the distance of zr from the second lens is given as
F2GL(x) =
p
2πkrg1 eiϕ̂2

a+eiϕ+
J0 kθ+
2 ρ

+ a−eiϕ−
J0 kθ−
2 ρ

, (A8)
where ϕ̂2 is the same as in (31). Also, b±
2 are from (A6), with a± and ϕ± as below:
a2
±(θ1, θ2) =

1
2
p
1 + 4θ2
2/θ2
1 ∓ 1
2
p
1 + 4θ2
2/θ2
1
, (A9)
ϕ±(θ1, θ2) = k

rg2 ± 1
4 z̃2θ2
1
q
1 + 4θ2
2/θ2
1 ∓ 1

− 2rg2 ln 1
2 kz̃2θ1
q
1 + 4θ2
2/θ2
1 ∓ 1

. (A10)
Similarly to (17), we use solution (A8), to determine the PSF of the two-thin-lens system
PSF2GL(x) = 2πkrg1

a2
+J2
0 kθ+
2 ρ

+ a2
−J2
0 kθ−
2 ρ

+ 2a+a−J0 kθ+
2 ρ

J0 kθ−
2 ρ

sin[δϕ±]

. (A11)

19
Considering the mixed term in (A11), that contains sin[δϕ±] with δϕ± is given by
δϕ± = ϕ+ − ϕ− = kz̃2

1
2 θ2
1
q
1 + 4θ2
2/θ2
1 − θ2
2 ln
p
1 + 4θ2
2/θ2
1 − 1
p
1 + 4θ2
2/θ2
1 + 1

, (A12)
we note that at optical wavelengths, this term is a rapidly oscillating function of z̃2 that averages to 0. Therefore, the
last term in (A11) may be neglected, allowing us to present the averaged PSF as below:
hPSF2GL(x)i = 2πkrg1

a2
+J2
0 kθ+
2 ρ

+ a2
−J2
0 kθ−
2 ρ

. (A13)
This result yields the maximum amplification factor for the two-lens system which is obtained by setting ρ = 0:
µ2GL = hPSF2GL(0)i = 2πkrg1

a2
+ + a2
−

. (A14)
Using expressions for a+ and a− from (A9) and remembering the single-lens light amplification factor µ1GL = 2πkrg1
from (18) as well as Einstein angles θ1 and θ2 from (A2) and (A6), correspondingly, we present result (A14) as below
µ2GL = µ1GL
1
u
u2
+ 2
√
u2 + 4
, where u =
θ1
θ2
≡
r
M1
M2
√
z̃1z̃2
z12
. (A15)
Considering the case of two lenses of similar masses, we have rg1 ≃ rg2 , z01 ≃ z02 = 650 AU ≪ z12 = 10 pc, and,
thus, u = θ1/θ2 ≃ 3.15 × 10−4
. This result suggests that (u2
+ 2)/(u
√
u2 + 4) ≃ 3.17 × 103
, which implies that light
amplification of symmetric two similar-lens system a factor of 3.17 × 103
larger than that for a single lens (18).
We observe from (A15) that in the case of lenses with uneven masses, light amplification may be larger. Thus,
for exotic cases when θ1 ≫ θ2, the additional amplification factor scales as (u2
+ 2)/(u
√
u2 + 4) ≃ 1 + (θ2/θ1)4
+
O((θ2/θ1)6
), thus, offering very little additional amplification. For another limiting case, for θ1 ≪ θ2, the scaling is
(u2
+ 2)/(u
√
u2 + 4) ≃ θ2/θ1 + 3
8 (θ1/θ2) + O((θ1/θ2)3
), thus, offering a noticeable additional application.
To put this in context, we observe that the masses of stars in the solar neighborhood within 30 pc are within the
range4
of 0.6–2.7 M⊙. Taking the first lens to be our Sun and using (A15), this range of realistic stellar masses implies
additional light amplification factors of 4.10×103
and 1.93×103
, correspondingly. Also, we observe that the direction
of light propagation for the system with uneven masses may be somewhat important.
The estimates above are a bit misleading as they pertain only to the case the apertures that are smaller than
the first zero of the Bessel functions J0 kθ±
2 ρ

present in (A13), namely ρ+
2GL = 2.40483/(kθ+
2 ) ≃ 4.911 cm and
ρ−
2GL = 2.40483/(kθ−
2 ) ≃ 4.909 cm, which is not practical. Given the fact that both of these values are practically the
same as in the receiving lens case, the PSF behavior (21) and (A13) will be the nearly the identical (shown in Fig. 3),
except the latter expression will be θ2/θt
1 = 1/u ≃ 3.17 × 103
times larger.
Therefore, for realistic telescopes, in the context of a two-lens system, similarly to (23) and (37), we need an
aperture-averaged of the PSF, yielding the relevant light amplification factor. After evaluating the requisite integrals
and applying the large argument approximation to the involved Bessel functions, using (A13), we obtain:
hµ2GLi =
1
π(1
2 d)2
Z 1
2 d
0
Z 2π
0
PSF2GL(x) ρdρdφ =
= 2πkrg1
n
a2
+

J2
0 kθ+
2
1
2 d

+ J2
1 kθ+
2
1
2 d

+ a2
−

J2
0 kθ−
2
1
2 d

+ J2
1 kθ−
2
1
2 d
o
≃
≃
8rg1
d
a2
+
θ+
2
+
a2
−
θ−
2

=
r
2rg1
zt
4z12
dr
= hµ1GLi
z12
zt
= 9.62 × 1012
M1
M⊙
1
2
650 AU
zt
1
2
1 m
dr
z12
10 pc

, (A16)
where θ±
2 and a2
± are given by (A7) and (A9), correspondingly. Clearly, this result is identical to (37).
We see again that after averaging, the PSF (A16) does not depend on the mass of lens 2. This is because the size of
the telescope is larger than the first zero of diffraction pattern in (A13), thus satisfying the condition (38). Note that
other parameter choices, especially in the scenarios where the lens is a part of a transmitter, which is characterized by
a much larger scale of the diffraction pattern imprinted in the PSF, like in (19)–(20), does not require the averaging.
4 http://www.solstation.com/stars3/100-gs.htm

20
FIG. 4: A two-lens geometry for power transmission via gravitational lensing showing the transmitter, two lenses, and the
receiver (compare to Fig. 1 for one-lens geometry). Also shown is the distance from the lens 1 to the transmitter plane, z01,
distance between the lenses, as well as that from the lens to the receiver plane, z02.
Appendix B: EM propagation in the presence of two lenses
In Sec. II A we discussed diffraction of light in the presence of one lens. Here we show how the same approach can
be extend to two and more lenses. Clearly, the amplitude of the EM field is given by the same equation (3) where for
each lens i = 1, 2 we will have a different impact parameter bi and the gravitational phase shift, ψ(bi).
In case of two lenses, the effective path length (eikonal) S(x′
, x, b1, b2) (see (4) for one lens) along a path from the
source position (−z0, x′
) to the observer position (z, x) via points (0, b1) and (0, b2) on the lens planes, to the order
of O(b4
1, b4
2)) (similarly to (4)), has the form (Fig. 4 shows overall geometry of the gravitational lensing system):
S(x′
, b1, b2, x) =
q
(b1 − x′)2 + z2
01 +
q
(b1 − b2)2 + z2
12 +
q
(b2 − x)2 + z2
02 − ψ(b1) − ψ(b2) =
= z01 + z12 + z02 +
(b2 − x′
)2
2(z01 + z12)
+
z01 + z12
2z01z12

b1 −
z01
z01 + z12
b2 −
z12
z01 + z12
x′
2
+
(b2 − x)2
2z02
− ψ(b1) − ψ(b2). (B1)
As a result, the wave amplitude on the observer plane is written as below
A(x′
, x) = A0(x′
, x)F2GL(x′
, x), (B2)
where A0(x, x) is the wave amplitude at the observer in the absence of the gravitational potentials U1, U2:
A0(x′
, x) =
E0
(z01 + z12 + z02)
exp
h
ik

z01 + z12 + z02
i
. (B3)
In the case of a pair of isolated monopole gravitational lenses, the amplification factor F2GL(x, x) is given by the
following nested diffraction integral
F2GL(x′
, x) =
(z01 + z12 + z02)
(z01 + z12)z02
keiφG2
2πi
ZZ
d2
b2 exp
h
ik
(b2 − x′
)2
2(z01 + z12)
+
(b2 − x)2
2z02
− 2rg ln kb2
i
×
×
(z01 + z12)
z01z12
keiφG1
2πi
ZZ
d2
b1 exp
h
ik
z01 + z12
2z01z12

b1 −
z01
z01 + z12

b2 +
z12
z01
x′
2
− 2rg ln kb1
i
, (B4)
where the gravitational phase factors are given as φG1 = krg ln 4k2
z01z12 and φG2 = krg ln 4k2
z12z02.
Result (B4) summarizes the logic of the procedure used in this paper. It has the non-vanishing position of the
transmitter x′
6= 0. The same structure will work for extended lenses, i.e., with higher-order multipole moments. We
take these integrals one after another, each time removing the appropriate spherical wave from the integrand. The
integral of (B4) represents the structure of the lensed EM wave amplitude obtained using the scalar theory of light
using the Fresnel–Kirchhoff diffraction formula [15] and has the form of a path integral [35] (see [12] for details.)

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