Microsoft Word - stancil_text+figs_prDR.doc
 Long Distance Signaling Using Axion-like Particles 
Daniel D. Stancil, Department of Electrical and Computer Engineering 
Carnegie Mellon University, Pittsburgh, PA 15213 
Abstract 
The possible existence of axion-like particles could lead to a new type of long distance 
communication. In this work, basic antenna concepts are defined and a Friis-like equation 
is derived to facilitate long-distance link calculations. An example calculation is 
presented showing that communication over distances of 1000 km or more may be 
possible for 3.5am <  meV and 
8 15 10  GeVag γγ
− −> × .  
PACS: 14.80.Mz, 84.40.Ba, 84.40.Ua, 42.79.Sz 
The axion has been proposed as a solution to the strong-CP problem [1], and is also a 
candidate for the galactic dark matter [2]. Interest in axions has increased recently owing 
to the report by the PVLAS collaboration of optical rotation induced by a magnetic field 
in a vacuum [3], since the creation of axions or other similar particles was one possible 
explanation for this rotation. Subsequently a number of groups began experimental 
searches using axion generation and detection schemes that would more definitively point 
to these new particles as the explanation [4]. The PVLAS result was surprising because it 
suggested coupling between the axion and electromagnetic fields that was much larger 
than thought possible based on solar axion observations by the CAST collaboration [5]. 
Although mechanisms have been proposed to reconcile the reports [6], the PVLAS 
collaboration recently retracted the results [7] and an independent group has reported a 
negative result from a photon regeneration experiment that excludes the PVLAS result 
[8]. 
There does not now appear to be any experimental evidence of a coupling 
strength inconsistent with CAST observations. However, since recent work has suggested 
mechanisms whereby such strong coupling may be possible, I believe it remains 
interesting to consider the implications of stronger-than-expected axion-photon coupling.  
In particular, I would like to call attention to the observation that a new type of long-
distance signaling and communication may be possible. It may be possible to construct a 
communication system that cannot be blocked—even communicating directly through 
the diameter of the earth. This would make reliable worldwide signaling possible without 
the use of either satellites or the ionosphere, and would enable communication to 
locations previously inaccessible, such as submarines at the bottom of the sea and mines 
deep beneath the earth. The signal would also be very difficult to intercept since the axion 
beam would be essentially as narrow as a laser beam used to create it, and most of the 
path would be underground. With advances in power and sensitivity, it may also be 
possible to use axion signaling in space communications. For example, using such a 
system, communication with points on the far side of the moon may be possible without 
the use of lunar satellites. 
Communication systems using neutrinos have also been proposed [9], and would 
have many of the same characteristics as the proposed axion system. However, the 
generation and detection of neutrinos requires massive particle accelerators and 
scintillation detectors [10]. Also, full deflection over 4π steradians would not be 
practical, though limited beam steering could be achieved using a magnetic field to 
deflect the precursor pion beam. Finally, it would be difficult to consider modulation 
techniques more sophisticated than simple amplitude modulation. In contrast, using axion 
mass and coupling values not yet experimentally explored, it appears that worldwide 
communication would be possible with a fully steerable axion system about the size of a 
medium-size telescope. Further, since the signals at the input and output would be 
electromagnetic waves, any existing modulation technology could be used. 
However, as with neutrinos, the lack of strong interactions with matter presents 
challenges with respect to the generation and detection of axions. Sikivie proposed an 
experimental approach for detecting axions via their coupling to the electromagnetic field 
[11]. The coupling was obtained by considering the Lagrangian density 
1 1 1 1
4 4 2 2a a
L F F g aF F a a m aμν μν μμν γγ μν μ= − − + ∂ ∂ − , (1) 
where F A Aμν μ ν ν μ= ∂ − ∂  is the electromagnetic field tensor, ( ),A V Aα = −  where V is the 
electric scalar potential and A  is the magnetic vector potential, a  is the axion field, ag γγ  
is the coupling constant between the electromagnetic and axion fields, and the 
electromagnetic dual tensor is given by 
F F γδαβ αβγδε= . In these equations we have 
taken 1h c= = .  
As an example, consider the coupling between plane waves propagating along the 
z direction caused by a strong static magnetic field parallel to the polarization of the 
incident electromagnetic wave. If the time dependence is exp( )i tω− , where ω is the 
frequency of the incident linearly polarized electromagnetic wave, the equations of 
motion obtained from (1) reduce to the coupled equations 
2 2 2
0 02 2,         .     a a a
m a i g B A A i g B a
z zγγ γγ
ω ω ω ω
+ − = − + =
 (2) 
Thus the static magnetic field B0 couples the photon and axion fields. An apparatus for 
the generation and detection of axions based on this coupling is shown in Figure 1(a). 
This is sometimes referred to as an “invisible light through walls” experiment [4,12,13]. 
An electromagnetic wave with amplitude (0)A  enters a region of magnetic field of 
strength BOT extending over a distance LT. If the coupling is sufficiently weak that the 
change in A  over the transmit conversion region and the change in a over the receive 
conversion region are negligible, then the conversion loss through the system will be [12] 
 0 / in R TP P p p= , (3) 
where inP  is the optical input power, 0P  is the optical output power, Tp , Rp are the 
probabilities of photon-axion (and axion-photon) conversion in the transmitter and 
receiver, respectively, and the conversion probability is [11,12,14] 
1 sin / 2
2 / 2aa
p g B L
k qLγγ
ω ⎡ ⎤
= ⎢ ⎥
. (4) 
Here aq k kγ= −  indicates the phase mismatch between the photon and axion fields.  
The efficiency of generating axions and regenerating photons can be greatly 
increased by adding electromagnetic resonators, as shown in Figure 1(b) [4,15]. In this 
figure a laser is used to generate the incident photons, and mirrors are used to cause the 
light to pass through the magnetic field multiple times, increasing the conversion 
probability by the factor 2 /TF π , where FT is the finesse of the resonator in the axion 
generator (transmitter). This factor can be interpreted as the effective number of photon 
passes in the resonator. Axions are also emitted in the backward direction owing to the 
counter-propagating light in the resonator, resulting in half the particles traveling in an 
unwanted direction. Consequently, the probability of conversion in a given direction is 
increased by the factor /TF π . As also shown in Figure 1(b), a resonator on the photon 
regenerator (receiver) likewise increases the axion-photon conversion probability [15]. 
Since regenerated photons will be emitted in both directions, detectors are placed on both 
ends of the receiving optical resonator, enhancing the photon regeneration probability by 
the factor 2 /RF π , where FR is the finesse of the receiving resonator. (If the power from a 
single end is collected, the factor would be /RF π , as with the transmitter.) Finally, to 
turn this into a communication system, we add appropriate electromagnetic wave 
modulators and detectors as shown in Figure 1(b). 
The conversion loss equation (3) is valid when the transmitter and receiver are 
sufficiently close together that beam diffraction can be neglected, and when the 
transmitter and receiver have equal cross-sectional areas. For signaling over long 
distances, neither assumption will be valid in general. To treat the long-distance case, we 
first calculate the radiated axion field, then calculate the regenerated photons resulting 
when the radiated field reaches the receiver. 
The general solution to Eq. (2) is given by [12] 
 3 0( ) ( ) ( )4
aik r r
a r i g d r A r B r
r rγγ
′ ′ ′=
′−∫ i . (5) 
If the observation point r  is very far away from all points in the source volume V, then 
we obtain the far-field approximation for the axion field 
 3 0( ) ( ) ( )4
a r i g d r e A r B r
′−′ ′ ′≈ ∫ i i . (6) 
Consider the case where the source 0( ) ( )A r B r′ ′i  is only nonzero inside a cylinder of 
radius R and length L as shown in Figure 2. Further, we assume that within this cylinder 
0 0ˆ TB xB=  and 0ˆ / exp( )TA x F A ik zγπ= , where 0A is the amplitude of the incident 
electromagnetic wave, and the cylinder is contained in a resonant cavity with finesse TF . 
Using Eq. (6), the far-field potential is found to be 
sin / 2
2 / 2
aik r az T a TT
a T T T
a Taz T
k k L J k RFe
a r i g A B L s
r k Rk k L
⎡ ⎤ ⎡ ⎤−⎣ ⎦≈ ⎢ ⎥
− ⎢ ⎥⎣ ⎦
, (7) 
where sin ,   cosa a az ak k k kρ θ θ= = , and
T Ts Rπ=  is the cross-sectional area of the source 
region at the transmitter. This expression has its maximum when 0θ = , for which 
,  0az a ak k k ρ= = . The factor containing the Bessel function in (7) has the limit 
( )10lim / 1/ 2a a T a Tk J k R k Rρ ρ ρ→ ⎡ ⎤ =⎣ ⎦ . Consequently, the axion field on axis is 
sin / 2
4 / 2
aik r
a T T T
a r i g A B L s
r qLγγ
≈ . (8) 
The time averaged transmitted power density is 
( ) ( )
a a T T in
S r k a r s p P
= = , (9) 
where (0)in TP S sγ= , and ( )
(0) 0
S k Aγ γω= . If this axion power density is 
incident upon a photon regenerator at distance r that is perfectly aligned with the 
transmitter, then the received power is 
 0 (2 / ) ( )R R a RP F p S r sπ= . (10) 
Substituting Eq. (9) for the power flux ( )aS r  gives 
in aR T
R R T T
P kF F
P p s p s
rπ π π π
= . (11) 
This expression can be understood in terms of antenna theory for electromagnetic waves. 
In this context, we refer to the apparatus consisting of the resonator and the structure 
creating the magnetic field as an axion antenna. In analogy with conventional antenna 
theory, we define the directivity as 
/(4 )
= , (12) 
where Prad is the total power radiated by the transmitting antenna. To obtain the total 
power radiated, we could integrate the power flux (9) over a sphere enclosing the 
antenna. However, it is easier to do the calculation in the near field using the axion field 
at the aperture of the transmitting antenna. Using ( ) ( / ) (0)a T T TS L F p Sγπ= , we have 
 2 ( ) (2 / )rad a T T T T inP S L s F p Pπ= = . (13) 
Substituting (13) for the total radiated power and (9) for the power flux, the directivity 
simplifies to 
 2 2(2 / ) (4 / )( / 2)T a T a TD s sπ λ π λ= = . (14) 
The relation between the directivity and physical area (14) is ½ that found in 
conventional antenna theory, or equivalently, the area appears to be half the physical 
area. This results from the bi-directional radiation properties of the resonator. Defining an 
efficiency as /rad inP Pη = , we also define the antenna gain as 
 2(2 / ) (2 / )T T T T T a TG D F p sη π π λ= = , (15) 
where (2 / )T T TF pη π= . 
Next suppose that at some distant location this transmitted field is incident upon a 
receive antenna with length LR, radius RR, and finesse FR. From (10) and assuming the 
photons emitted from both ends of the receive antenna are collected, the total power 
collected will be 0 , ( )e R aP s S r= , where we have defined the effective area of the 
receiving antenna as 
 , ( / ) ,e R R c R Rs s n F pπ=  (16) 
R Rs Rπ=   is the physical cross-sectional area, and cn  is the number of ends from which 
photons are collected (i.e., 1, 2cn = ). 
As with electromagnetic antennas, the ratio of effective area to gain is found to be 
independent of the details of the antenna, other than whether or not photons are collected 
from both ends of the antenna when used to receive: 
 2, ,/ / /(4 ).e R R e T T c as G s G n λ π= =  (17) 
If photons were collected only from one end of the receive antenna ( 1cn = ), then Eq. (17) 
would be identical to conventional antenna theory.  
With these definitions, the expression for the received power (11) can be 
interpreted as a Friis-like equation: 
 ( ) ( )( )2 2 20 , , ,/ / 4 /( ) /in c T R a e T c T a e RP P n G G r s n s rλ π λ= = . (18) 
Here ,c Tn  is the value used to compute ,e Ts  according to Eq. (16). The ratio , ,/e T c Ts n  is 
independent of the choice of ,c Tn , as it should be since the number of detectors that might 
be used on receive is independent of the transmit properties of the antenna.  
It is also useful to note that if the magnetic field is uniform (i.e., wigglers, or 
quasi-phase matching, are not used [12]), then there is an optimum length for the 
conversion region of an antenna. This occurs when ( )sin / 2 1qL = , or qL π= . The 
optimum length is found to be 
 2( / )opt aL mγλ ω= . (19) 
In obtaining this expression, we have used 2 /(2 )aq m ω≈ , which is valid for am ω .  
The diffraction-limited power pattern of the radiated axion field is determined by 
the aperture size in wavelengths through the Bessel function term in (7): 
 ( ) ( ) 214 sin /( sin )d a aP J k R k Rθ θ θ= ⎡ ⎤⎣ ⎦ . (20) 
The diffraction beam width between first nulls is determined by the first zero of the Airy 
disc, and for small angles is given by the well-known expression 
 1.22 /dFWFN TRγθ λ≈ . (21) 
Similarly, the diffraction beam width at half maximum is determined by the roots of 
( ) 1/ 2dP θ = , or 
 (1.616 / ) / 0.514 /dFWHM a T TR Rγθ π λ λ= ≈ . (22) 
In contrast, the conversion-limited power pattern is given by 
sin cos / 2
2 cos / 2
k k L
k k L
⎡ ⎤⎡ ⎤−⎣ ⎦⎢ ⎥=
, (23) 
and depends on both the length in wavelengths and the velocity mismatch. For the 
optimum length L given by (19), the conversion beam widths are approximately given by 
 2( / ),            1.06( / ) c cFWFN a FWHM am mθ ω θ ω≈ ≈ . (24) 
For quantum-limited detection, the channel capacity is [16] 
 2log (1 / )d RC Nν η ν= Δ + Δ , (25) 
were dη  is the quantum efficiency of the optical detectors, νΔ  is the bandwidth of the 
resonator /(2 )c LFνΔ =  [17], and c is the velocity of light. Equation (25) can be 
combined with (11) or (18) to find the axion parameters that would permit a particular 
channel capacity at a given distance for a particular experimental apparatus. 
As an example, consider transmitters and receivers with 1064 nm (1.17 eV)γλ = , 
10 WinP = , 
20.01 ms = , 0 10 TB = , 3L =  m, 
53.1 10F = × , 0.5dη = , and a minimum 
information capacity of 1 bps. Figure 3 shows the inverse coupling strength curves 
( ) 1/a aM m g γγ=  for distances of 1000 km and the diameter of the earth. Also shown is 
the curve for communication between the earth and the far side of the moon using the 
“4+4” experimental apparatus proposed in [15]. The shaded regions are excluded by the 
results from the BFRT collaboration [13] and Robilliard et al. [8]. The dots represent 
extensions of the curves if quasi-phase matching (QPM) is used by periodically reversing 
the magnetic field [12]. The minimum period of the reversal is taken to be twice the beam 
diameter for the 3 m system example, and 28.6 m for the 4+4 system. From the figure, 
communication over distances in excess of 1000 km should be possible for 3.5am <  meV 
and 72 10  GeVM < ×  ( 8 15 10  GeVag γγ
− −> × ). Of particular interest, we note that the 
range 6 72 10  GeV 2 10  GeVM× < < ×  has not yet been excluded by photon regeneration 
experiments. 
For an area of 0.01 m2, the radius is 0.01/ 0.0564R π= =  m. The half-power 
diffraction beam width for the example antennas is therefore 
6 49.7 10 rad 5.56 10 deg 2dFWHMθ
− −= × ≈ × ≈  arc-sec, while the half-power conversion beam 
width for 0.7am ≈ meV (optimum for L=3 m) is 
4 26.3 10 rad 3.63 10 deg 131cFWHMθ
− −= × ≈ × ≈  arc-sec. Consequently the total beam width is 
determined by the diffraction beam width in this example. While pointing with an 
accuracy of 2 arc-sec would be challenging, it is somewhat less stringent than that 
required for optical deep space communications [18]. The size of such axion transmitters 
and receivers would be roughly that of a medium size professional telescope. It is 
interesting to note that the size of the beam at a distance of the earth’s diameter is about 
124 m. Consequently coordinates obtained with differential GPS at both sites should 
enable the computation of pointing directions to sufficient accuracy. 
Note that the receive resonator must be tuned to the same frequency as the 
transmit resonator to within a fraction of the resonator line width which is 161 Hz in this 
example. This will present a significant challenge, especially since the resonators are 
remote from one another. A possible approach would be to use atomic clocks to stabilize 
both the transmit frequency and a local reference frequency at the receiver. The receive 
resonator would then be locked to the reference to get as close as possible to the correct 
frequency, then slowly tuned until the signal is located. A feedback loop could then be 
closed to lock the receive resonator to the signal. 
In summary, for 3.5am <  meV and 
72 10  GeVM < ×  ( 8 15 10  GeVag γγ
− −> × ), it 
may be possible to realize a new type of wireless signaling that cannot be blocked or 
shielded. An example calculation shows that communication between points located 
diametrically opposite on the earth should be possible. This could enable world-wide 
communication without the use of satellites or the ionosphere. However, with present 
knowledge, the signaling will be limited to low data rates, perhaps on the order of a few 
bits per second for terrestrial links. This estimate assumes 3 m long 
generation/regeneration regions to allow fully steerable instruments. Though not easily 
steerable, the apparatus in the “4+4” experiment proposed by Sikivie et al., [15] may 
enable communication to the far side of the moon for 0.3am <  meV and 
66 10  GeVM < × . 
I would like to acknowledge helpful discussions with Jim Lesh, Rich Holman, 
Jeff Peterson, Pierre Sikivie, and David Tanner during the development of these ideas.  
Electronic address: stancil@cmu.edu 
References 
1. R. D. Peccei  and H. R. Quinn, Phys. Rev. Lett., 38, 1440, (1977); S. Weinberg, 
Phys. Rev. Lett., 40, 223 (1978); F. Wilczek, Phys. Rev. Lett., 40, 279 (1978). 
2. R. Bradley, et al., Rev. Mod. Phys. 75, 777 (2003). 
3. E. Zavattini, et al. (PVLAS Collaboration), Phys. Rev. Lett., 96, 110406 (2006).  
4. See, for example, R. Rabadan, A. Ringwald, and Kris Sigurdson, Phys. Rev. Lett., 
96, 110407 (2006); and J. Jaeckel, E. Masso, J. Redondo, A. Ringwald, and F. 
Takahashi, arXiv:hep-ph/0605313. 
5. S. Andriamonje et al. (CAST collaboration), J. Cosmol. Astropart. Phys. 04, 010 
(2007). 
6. See, for example, R.N. Mohapatra and S. Nasri, Phys. Rev. Lett., 98, 050402 
(2007). 
7. E. Zavattini et al., arXiv:0706.3419 [hep-ex]. 
8. C. Robilliard, et al., arXiv:0707.1296v3 [hep-ex]. 
9. See, for example, J. W. Eerkens, US Patent # 4,205,268, May 27, 1980; J. M. 
Pasachoff and M. L. Kutner, Cosmic Search Vol. 1 No. 3 p. 2 (1979), 
http://www.bigear.org/CSMO/PDF/CS03/cs03p02.pdf ; H. Xie, J. Gao, C. Liu, 
and Q. Zhai, 2006 IET Int. Conf. on Wireless Mobile and Multimedia Networks 
Proc., Hangzhou, China, 6-9 Nov., (2006).  
10. D. G. Michael, et al., Phys. Rev. Lett., 97, 191801 (2006). 
11. P. Sikivie, Phys. Rev. Lett., 51, 1415 (1983); Phys. Rev. D, 32, 2988 (1985). 
12. K. Van Bibber, N.R. Dagdeviren, S.E. Koonin, A.K. Kerman, and H.N. Nelson, 
Phys. Rev. Lett., 59, 759 (1987). 
13. R. Cameron, et al., Phys. Rev. D, 47 3707 (1993). 
14. G. Raffelt and L. Stodolsky, Phys. Rev. D, 37, 1237 (1988). 
15. P. Sikivie, D.B. Turner, and Karl van Bibber, Phys. Rev. Lett. 98, 172002 (2007). 
16. C-C. Chen, in Deep Space Optical Communications, edited by H. Hemmati (John 
Wiley & Sons, Hoboken, NJ, 2006) p. 92. 
17. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 
6th Edition (Oxford Univ. Press, New York, 2007), pp. 171. 
18. C-C. Chen, op cit., p. 106. 
Figure Captions 
Figure 1. (a) Basic system for the generation and detection of axions. (b) An axion communication 
system, with the axion generator and photon regenerator located remote from one another.   
Figure 2. A source with uniform amplitude over a cylindrical region. 
Figure 3. (Color Online) The ranges of inverse coupling parameter M and pseudoscalar mass ma that 
would permit communication at information bandwidths of at least 1 bps using the example system, 
and the 4+4 system proposed in [15]. The shaded regions are ruled out by [8] and [13]. 
LT LR
Mod Laser 
Data In  
Data Out  
 B0T   
Optical 
Detectors 
   B0T      B0R 
Photon 
input 
Photon 
output 
Axions penetrate barrier 
opaque to photons 
LT LR
Photons 
Axions 
Cavity mirrors  
(b)  
Figure 1, Stancil, Phys. Rev. D. 
Figure 2, Stancil, Phys. Rev. D. 
-LT /2 LT /2 
ma (eV)
earth diameter
1000 km
moon, 4+4
Robilliard, et al.
QPM extensions
Figure 3, Stancil, Phys. Rev. D.

  The possible existence of axion-like particles could lead to a new type of
long distance communication. In this work, basic antenna concepts are defined
and a Friis-like equation is derived to facilitate long-distance link
calculations. An example calculation is presented showing that communication
over distances of 1000 km or more may be possible for $m_{a}< 3.5$ meV and
$g_{a\gamma \gamma} > 5 \times 10^{- 8} {\text{GeV}}^{- 1}$.

Microsoft Word - stancil_text+figs_prDR.doc
 Long Distance Signaling Using Axion-like Particles 
Daniel D. Stancil, Department of Electrical and Computer Engineering 
Carnegie Mellon University, Pittsburgh, PA 15213 
Abstract 
The possible existence of axion-like particles could lead to a new type of long distance 
communication. In this work, basic antenna concepts are defined and a Friis-like equation 
is derived to facilitate long-distance link calculations. An example calculation is 
presented showing that communication over distances of 1000 km or more may be 
possible for 3.5am <  meV and 
8 15 10  GeVag γγ
− −> × .  
PACS: 14.80.Mz, 84.40.Ba, 84.40.Ua, 42.79.Sz 
The axion has been proposed as a solution to the strong-CP problem [1], and is also a 
candidate for the galactic dark matter [2]. Interest in axions has increased recently owing 
to the report by the PVLAS collaboration of optical rotation induced by a magnetic field 
in a vacuum [3], since the creation of axions or other similar particles was one possible 
explanation for this rotation. Subsequently a number of groups began experimental 
searches using axion generation and detection schemes that would more definitively point 
to these new particles as the explanation [4]. The PVLAS result was surprising because it 
suggested coupling between the axion and electromagnetic fields that was much larger 
than thought possible based on solar axion observations by the CAST collaboration [5]. 
Although mechanisms have been proposed to reconcile the reports [6], the PVLAS 
collaboration recently retracted the results [7] and an independent group has reported a 
negative result from a photon regeneration experiment that excludes the PVLAS result 
[8]. 
There does not now appear to be any experimental evidence of a coupling 
strength inconsistent with CAST observations. However, since recent work has suggested 
mechanisms whereby such strong coupling may be possible, I believe it remains 
interesting to consider the implications of stronger-than-expected axion-photon coupling.  
In particular, I would like to call attention to the observation that a new type of long-
distance signaling and communication may be possible. It may be possible to construct a 
communication system that cannot be blocked—even communicating directly through 
the diameter of the earth. This would make reliable worldwide signaling possible without 
the use of either satellites or the ionosphere, and would enable communication to 
locations previously inaccessible, such as submarines at the bottom of the sea and mines 
deep beneath the earth. The signal would also be very difficult to intercept since the axion 
beam would be essentially as narrow as a laser beam used to create it, and most of the 
path would be underground. With advances in power and sensitivity, it may also be 
possible to use axion signaling in space communications. For example, using such a 
system, communication with points on the far side of the moon may be possible without 
the use of lunar satellites. 
Communication systems using neutrinos have also been proposed [9], and would 
have many of the same characteristics as the proposed axion system. However, the 
generation and detection of neutrinos requires massive particle accelerators and 
scintillation detectors [10]. Also, full deflection over 4π steradians would not be 
practical, though limited beam steering could be achieved using a magnetic field to 
deflect the precursor pion beam. Finally, it would be difficult to consider modulation 
techniques more sophisticated than simple amplitude modulation. In contrast, using axion 
mass and coupling values not yet experimentally explored, it appears that worldwide 
communication would be possible with a fully steerable axion system about the size of a 
medium-size telescope. Further, since the signals at the input and output would be 
electromagnetic waves, any existing modulation technology could be used. 
However, as with neutrinos, the lack of strong interactions with matter presents 
challenges with respect to the generation and detection of axions. Sikivie proposed an 
experimental approach for detecting axions via their coupling to the electromagnetic field 
[11]. The coupling was obtained by considering the Lagrangian density 
1 1 1 1
4 4 2 2a a
L F F g aF F a a m aμν μν μμν γγ μν μ= − − + ∂ ∂ − , (1) 
where F A Aμν μ ν ν μ= ∂ − ∂  is the electromagnetic field tensor, ( ),A V Aα = −  where V is the 
electric scalar potential and A  is the magnetic vector potential, a  is the axion field, ag γγ  
is the coupling constant between the electromagnetic and axion fields, and the 
electromagnetic dual tensor is given by 
F F γδαβ αβγδε= . In these equations we have 
taken 1h c= = .  
As an example, consider the coupling between plane waves propagating along the 
z direction caused by a strong static magnetic field parallel to the polarization of the 
incident electromagnetic wave. If the time dependence is exp( )i tω− , where ω is the 
frequency of the incident linearly polarized electromagnetic wave, the equations of 
motion obtained from (1) reduce to the coupled equations 
2 2 2
0 02 2,         .     a a a
m a i g B A A i g B a
z zγγ γγ
ω ω ω ω
+ − = − + =
 (2) 
Thus the static magnetic field B0 couples the photon and axion fields. An apparatus for 
the generation and detection of axions based on this coupling is shown in Figure 1(a). 
This is sometimes referred to as an “invisible light through walls” experiment [4,12,13]. 
An electromagnetic wave with amplitude (0)A  enters a region of magnetic field of 
strength BOT extending over a distance LT. If the coupling is sufficiently weak that the 
change in A  over the transmit conversion region and the change in a over the receive 
conversion region are negligible, then the conversion loss through the system will be [12] 
 0 / in R TP P p p= , (3) 
where inP  is the optical input power, 0P  is the optical output power, Tp , Rp are the 
probabilities of photon-axion (and axion-photon) conversion in the transmitter and 
receiver, respectively, and the conversion probability is [11,12,14] 
1 sin / 2
2 / 2aa
p g B L
k qLγγ
ω ⎡ ⎤
= ⎢ ⎥
. (4) 
Here aq k kγ= −  indicates the phase mismatch between the photon and axion fields.  
The efficiency of generating axions and regenerating photons can be greatly 
increased by adding electromagnetic resonators, as shown in Figure 1(b) [4,15]. In this 
figure a laser is used to generate the incident photons, and mirrors are used to cause the 
light to pass through the magnetic field multiple times, increasing the conversion 
probability by the factor 2 /TF π , where FT is the finesse of the resonator in the axion 
generator (transmitter). This factor can be interpreted as the effective number of photon 
passes in the resonator. Axions are also emitted in the backward direction owing to the 
counter-propagating light in the resonator, resulting in half the particles traveling in an 
unwanted direction. Consequently, the probability of conversion in a given direction is 
increased by the factor /TF π . As also shown in Figure 1(b), a resonator on the photon 
regenerator (receiver) likewise increases the axion-photon conversion probability [15]. 
Since regenerated photons will be emitted in both directions, detectors are placed on both 
ends of the receiving optical resonator, enhancing the photon regeneration probability by 
the factor 2 /RF π , where FR is the finesse of the receiving resonator. (If the power from a 
single end is collected, the factor would be /RF π , as with the transmitter.) Finally, to 
turn this into a communication system, we add appropriate electromagnetic wave 
modulators and detectors as shown in Figure 1(b). 
The conversion loss equation (3) is valid when the transmitter and receiver are 
sufficiently close together that beam diffraction can be neglected, and when the 
transmitter and receiver have equal cross-sectional areas. For signaling over long 
distances, neither assumption will be valid in general. To treat the long-distance case, we 
first calculate the radiated axion field, then calculate the regenerated photons resulting 
when the radiated field reaches the receiver. 
The general solution to Eq. (2) is given by [12] 
 3 0( ) ( ) ( )4
aik r r
a r i g d r A r B r
r rγγ
′ ′ ′=
′−∫ i . (5) 
If the observation point r  is very far away from all points in the source volume V, then 
we obtain the far-field approximation for the axion field 
 3 0( ) ( ) ( )4
a r i g d r e A r B r
′−′ ′ ′≈ ∫ i i . (6) 
Consider the case where the source 0( ) ( )A r B r′ ′i  is only nonzero inside a cylinder of 
radius R and length L as shown in Figure 2. Further, we assume that within this cylinder 
0 0ˆ TB xB=  and 0ˆ / exp( )TA x F A ik zγπ= , where 0A is the amplitude of the incident 
electromagnetic wave, and the cylinder is contained in a resonant cavity with finesse TF . 
Using Eq. (6), the far-field potential is found to be 
sin / 2
2 / 2
aik r az T a TT
a T T T
a Taz T
k k L J k RFe
a r i g A B L s
r k Rk k L
⎡ ⎤ ⎡ ⎤−⎣ ⎦≈ ⎢ ⎥
− ⎢ ⎥⎣ ⎦
, (7) 
where sin ,   cosa a az ak k k kρ θ θ= = , and
T Ts Rπ=  is the cross-sectional area of the source 
region at the transmitter. This expression has its maximum when 0θ = , for which 
,  0az a ak k k ρ= = . The factor containing the Bessel function in (7) has the limit 
( )10lim / 1/ 2a a T a Tk J k R k Rρ ρ ρ→ ⎡ ⎤ =⎣ ⎦ . Consequently, the axion field on axis is 
sin / 2
4 / 2
aik r
a T T T
a r i g A B L s
r qLγγ
≈ . (8) 
The time averaged transmitted power density is 
( ) ( )
a a T T in
S r k a r s p P
= = , (9) 
where (0)in TP S sγ= , and ( )
(0) 0
S k Aγ γω= . If this axion power density is 
incident upon a photon regenerator at distance r that is perfectly aligned with the 
transmitter, then the received power is 
 0 (2 / ) ( )R R a RP F p S r sπ= . (10) 
Substituting Eq. (9) for the power flux ( )aS r  gives 
in aR T
R R T T
P kF F
P p s p s
rπ π π π
= . (11) 
This expression can be understood in terms of antenna theory for electromagnetic waves. 
In this context, we refer to the apparatus consisting of the resonator and the structure 
creating the magnetic field as an axion antenna. In analogy with conventional antenna 
theory, we define the directivity as 
/(4 )
= , (12) 
where Prad is the total power radiated by the transmitting antenna. To obtain the total 
power radiated, we could integrate the power flux (9) over a sphere enclosing the 
antenna. However, it is easier to do the calculation in the near field using the axion field 
at the aperture of the transmitting antenna. Using ( ) ( / ) (0)a T T TS L F p Sγπ= , we have 
 2 ( ) (2 / )rad a T T T T inP S L s F p Pπ= = . (13) 
Substituting (13) for the total radiated power and (9) for the power flux, the directivity 
simplifies to 
 2 2(2 / ) (4 / )( / 2)T a T a TD s sπ λ π λ= = . (14) 
The relation between the directivity and physical area (14) is ½ that found in 
conventional antenna theory, or equivalently, the area appears to be half the physical 
area. This results from the bi-directional radiation properties of the resonator. Defining an 
efficiency as /rad inP Pη = , we also define the antenna gain as 
 2(2 / ) (2 / )T T T T T a TG D F p sη π π λ= = , (15) 
where (2 / )T T TF pη π= . 
Next suppose that at some distant location this transmitted field is incident upon a 
receive antenna with length LR, radius RR, and finesse FR. From (10) and assuming the 
photons emitted from both ends of the receive antenna are collected, the total power 
collected will be 0 , ( )e R aP s S r= , where we have defined the effective area of the 
receiving antenna as 
 , ( / ) ,e R R c R Rs s n F pπ=  (16) 
R Rs Rπ=   is the physical cross-sectional area, and cn  is the number of ends from which 
photons are collected (i.e., 1, 2cn = ). 
As with electromagnetic antennas, the ratio of effective area to gain is found to be 
independent of the details of the antenna, other than whether or not photons are collected 
from both ends of the antenna when used to receive: 
 2, ,/ / /(4 ).e R R e T T c as G s G n λ π= =  (17) 
If photons were collected only from one end of the receive antenna ( 1cn = ), then Eq. (17) 
would be identical to conventional antenna theory.  
With these definitions, the expression for the received power (11) can be 
interpreted as a Friis-like equation: 
 ( ) ( )( )2 2 20 , , ,/ / 4 /( ) /in c T R a e T c T a e RP P n G G r s n s rλ π λ= = . (18) 
Here ,c Tn  is the value used to compute ,e Ts  according to Eq. (16). The ratio , ,/e T c Ts n  is 
independent of the choice of ,c Tn , as it should be since the number of detectors that might 
be used on receive is independent of the transmit properties of the antenna.  
It is also useful to note that if the magnetic field is uniform (i.e., wigglers, or 
quasi-phase matching, are not used [12]), then there is an optimum length for the 
conversion region of an antenna. This occurs when ( )sin / 2 1qL = , or qL π= . The 
optimum length is found to be 
 2( / )opt aL mγλ ω= . (19) 
In obtaining this expression, we have used 2 /(2 )aq m ω≈ , which is valid for am ω .  
The diffraction-limited power pattern of the radiated axion field is determined by 
the aperture size in wavelengths through the Bessel function term in (7): 
 ( ) ( ) 214 sin /( sin )d a aP J k R k Rθ θ θ= ⎡ ⎤⎣ ⎦ . (20) 
The diffraction beam width between first nulls is determined by the first zero of the Airy 
disc, and for small angles is given by the well-known expression 
 1.22 /dFWFN TRγθ λ≈ . (21) 
Similarly, the diffraction beam width at half maximum is determined by the roots of 
( ) 1/ 2dP θ = , or 
 (1.616 / ) / 0.514 /dFWHM a T TR Rγθ π λ λ= ≈ . (22) 
In contrast, the conversion-limited power pattern is given by 
sin cos / 2
2 cos / 2
k k L
k k L
⎡ ⎤⎡ ⎤−⎣ ⎦⎢ ⎥=
, (23) 
and depends on both the length in wavelengths and the velocity mismatch. For the 
optimum length L given by (19), the conversion beam widths are approximately given by 
 2( / ),            1.06( / ) c cFWFN a FWHM am mθ ω θ ω≈ ≈ . (24) 
For quantum-limited detection, the channel capacity is [16] 
 2log (1 / )d RC Nν η ν= Δ + Δ , (25) 
were dη  is the quantum efficiency of the optical detectors, νΔ  is the bandwidth of the 
resonator /(2 )c LFνΔ =  [17], and c is the velocity of light. Equation (25) can be 
combined with (11) or (18) to find the axion parameters that would permit a particular 
channel capacity at a given distance for a particular experimental apparatus. 
As an example, consider transmitters and receivers with 1064 nm (1.17 eV)γλ = , 
10 WinP = , 
20.01 ms = , 0 10 TB = , 3L =  m, 
53.1 10F = × , 0.5dη = , and a minimum 
information capacity of 1 bps. Figure 3 shows the inverse coupling strength curves 
( ) 1/a aM m g γγ=  for distances of 1000 km and the diameter of the earth. Also shown is 
the curve for communication between the earth and the far side of the moon using the 
“4+4” experimental apparatus proposed in [15]. The shaded regions are excluded by the 
results from the BFRT collaboration [13] and Robilliard et al. [8]. The dots represent 
extensions of the curves if quasi-phase matching (QPM) is used by periodically reversing 
the magnetic field [12]. The minimum period of the reversal is taken to be twice the beam 
diameter for the 3 m system example, and 28.6 m for the 4+4 system. From the figure, 
communication over distances in excess of 1000 km should be possible for 3.5am <  meV 
and 72 10  GeVM < ×  ( 8 15 10  GeVag γγ
− −> × ). Of particular interest, we note that the 
range 6 72 10  GeV 2 10  GeVM× < < ×  has not yet been excluded by photon regeneration 
experiments. 
For an area of 0.01 m2, the radius is 0.01/ 0.0564R π= =  m. The half-power 
diffraction beam width for the example antennas is therefore 
6 49.7 10 rad 5.56 10 deg 2dFWHMθ
− −= × ≈ × ≈  arc-sec, while the half-power conversion beam 
width for 0.7am ≈ meV (optimum for L=3 m) is 
4 26.3 10 rad 3.63 10 deg 131cFWHMθ
− −= × ≈ × ≈  arc-sec. Consequently the total beam width is 
determined by the diffraction beam width in this example. While pointing with an 
accuracy of 2 arc-sec would be challenging, it is somewhat less stringent than that 
required for optical deep space communications [18]. The size of such axion transmitters 
and receivers would be roughly that of a medium size professional telescope. It is 
interesting to note that the size of the beam at a distance of the earth’s diameter is about 
124 m. Consequently coordinates obtained with differential GPS at both sites should 
enable the computation of pointing directions to sufficient accuracy. 
Note that the receive resonator must be tuned to the same frequency as the 
transmit resonator to within a fraction of the resonator line width which is 161 Hz in this 
example. This will present a significant challenge, especially since the resonators are 
remote from one another. A possible approach would be to use atomic clocks to stabilize 
both the transmit frequency and a local reference frequency at the receiver. The receive 
resonator would then be locked to the reference to get as close as possible to the correct 
frequency, then slowly tuned until the signal is located. A feedback loop could then be 
closed to lock the receive resonator to the signal. 
In summary, for 3.5am <  meV and 
72 10  GeVM < ×  ( 8 15 10  GeVag γγ
− −> × ), it 
may be possible to realize a new type of wireless signaling that cannot be blocked or 
shielded. An example calculation shows that communication between points located 
diametrically opposite on the earth should be possible. This could enable world-wide 
communication without the use of satellites or the ionosphere. However, with present 
knowledge, the signaling will be limited to low data rates, perhaps on the order of a few 
bits per second for terrestrial links. This estimate assumes 3 m long 
generation/regeneration regions to allow fully steerable instruments. Though not easily 
steerable, the apparatus in the “4+4” experiment proposed by Sikivie et al., [15] may 
enable communication to the far side of the moon for 0.3am <  meV and 
66 10  GeVM < × . 
I would like to acknowledge helpful discussions with Jim Lesh, Rich Holman, 
Jeff Peterson, Pierre Sikivie, and David Tanner during the development of these ideas.  
Electronic address: stancil@cmu.edu 
References 
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8. C. Robilliard, et al., arXiv:0707.1296v3 [hep-ex]. 
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18. C-C. Chen, op cit., p. 106. 
Figure Captions 
Figure 1. (a) Basic system for the generation and detection of axions. (b) An axion communication 
system, with the axion generator and photon regenerator located remote from one another.   
Figure 2. A source with uniform amplitude over a cylindrical region. 
Figure 3. (Color Online) The ranges of inverse coupling parameter M and pseudoscalar mass ma that 
would permit communication at information bandwidths of at least 1 bps using the example system, 
and the 4+4 system proposed in [15]. The shaded regions are ruled out by [8] and [13]. 
LT LR
Mod Laser 
Data In  
Data Out  
 B0T   
Optical 
Detectors 
   B0T      B0R 
Photon 
input 
Photon 
output 
Axions penetrate barrier 
opaque to photons 
LT LR
Photons 
Axions 
Cavity mirrors  
(b)  
Figure 1, Stancil, Phys. Rev. D. 
Figure 2, Stancil, Phys. Rev. D. 
-LT /2 LT /2 
ma (eV)
earth diameter
1000 km
moon, 4+4
Robilliard, et al.
QPM extensions
Figure 3, Stancil, Phys. Rev. D.
