Received 16 January 2016; accepted 27 March 2016; published 31 March 2016
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1. Introduction
A gravitational wave (GW) could be detected indirectly by its interaction with the light emitted by astronomical objects. Thus, for instance, the passage of a GW produces a time delay in the signal received from distant sources (Estabrook and Wahlquist [1] ). Similarly, the presence of a stochastic background of GWs can be inferred from a statistical analysis of pulsar timing (Hellings and Downs [2] ). GWs can also interact with the polarization of electromagnetic waves (Hacyan [3] [4] ).
In this paper, we study the effect of GWs on the interferometry of stellar light. Two basic types of interfero- metric devices used in astronomy are considered: the Michelson (see, e.g., [5] ) and the Hanbury Brown-Twiss [6] interferometers. The former uses the interference between two signals, and the latter uses the interference between intensities of light. An intensity interferometer has, in general, some advantages over a Michelson interferometer. It will be shown in the following that the passage of a GW could be more easily detected by intensity interferometry.
Section 2 of the present paper is devoted to the analysis of an electromagnetic wave in the presence of a plane fronted GW. The analysis is based on previous works (Hacyan [3] [4] ) in which the form of the electromagnetic field is deduced using a short-wave length approximation. A general formula for the correlation of electric fields is obtained and the result is applied to interferometric analysis in Section 3; particular cases are worked out.
2. The Electromagnetic Field
The metric of a plane GW in the weak field limit is
(1)
where the two degrees of polarization of the GW are given by the potentials
and
, which are functions of u only. The relation with Minkowski coordinates t and z is
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In the following, quadratic and higher order terms in f and g are neglected, and we set
.
The direction of a light ray in the absence of a GW is k, with
, the frequency of the (monochromatic) wave. We set
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thus defining the angles
and
. In the following, it will be convenient to define the functions
(2)
(3)
In the short-wave length approximation, the electromagnetic potential is taken as
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where S is the eikonal function satisfying the equation
. Then,
is a null-vector defining the direction of propagation of the electromagnetic wave, and
is a four-vector such that
.
The electromagnetic vector is [4]
(4)
where
is a time-like four-vector and
is the frequency measured by a detector with
tangent to its world-line. Choosing
, it follows that
(5)
and the eikonal function is
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As in Ref. [4] , for a plane wave we use a gauge such that
, which is equivalent to
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where
is the unit vector in the direction of propagation of the GW.
The four vector
depends on the coordinate u through the functions
and
. With the gauge
, a particular solution is [4]
(6)
where
and
are constants defining an electromagnetic plane wave in the absence of GWs.
Let us use a tetrad
such that
, where
is the Minkowski matrix. Then, if
, the tetrad is defined by
(7)
Accordingly the tetrad components of
and
are
(8)
and
(9)
Notice in particular that
, and
, as it should be.
The electric field in tetrad components is
(10)
and of course
.
Correlations
For an electromagnetic plane wave with wave vector
, we find after some lengthy but straightforward algebra (keeping only terms of first order)
(11)
where
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are Stokes parameters (
for linear and
for circular polarizations).
3. Interferometry
Consider two detectors with space-time coordinates
and
, each receiving two plane electromagnetic waves with wave-vectors
and
, and use the shorthand notation
(12)
where
(13)
(14)
the subindexes a, b and j refer to the labels 1 and 2 of x and k.
A Michelson interferometer permits to measure the average intensity
(15)
where the second term is the interference term.
A Hanbury Brown-Twiss interferometer permits to measure the interference between intensities:
(16)
where the second term is the interference between the two intensities.
Define
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With this notation, we have for a Michelson interferometer:
(17)
and for a Hanbury Brown-Twiss interferometer:
(18)
Define also the complex functions
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and
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Then
(19)
and
(20)
In the absence of GWs,
, and
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implying that
is time independent. It thus follows that the time variation of
is due entirely to the presence of a GW. This time dependence can be made explicit setting
![]()
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where
,
,
, and
are small terms due to the GW. This implies that the terms
and
are of first order in the potentials f and g of the GW.
It should be noticed that the field correlation
contains terms such as
, which are
highly oscillatory and hinder a precise measurement with a Michelson interferometer. On the other hand, such terms do not appear in the correlation of the intensities:
(21)
The time dependence is included only in the terms
and
, which are entirely due to the passage of the GW. The term with
is not present in this last formula.
3.1. Temporal Coherence
As a particular application of the above formulas, we can calculate the temporal coherence of a single signal in the presence of a GW. This can be obtained setting
,
, and
. Then
and accordingly
(22)
and
(23)
Explicitly, in this particular case,
(24)
which is the only relevant term for the time correlation of the intensity correlation, and is entirely due to the GW.
3.2. Sinusoidal Waves and Pulses
In the particular case of a sinusoidal monochromatic GW of frequency
, we can set
(25)
where
is a complex constant and
a constant phase.
As for a pulse of GW, it can be approximated by a delta function:
. In this case, only
is changed after
. We have
(26)
where
is a function such that
if
and
otherwise. Thus, a pulse of gravitational wave would produce a change both in
and
.
4. Conclusion
The main conclusion from the present results is that the passage of a GW produces a time-dependent perturbation in the intensity interference of a distant light sources, an interference which would otherwise have a
static pattern. Thus, a time variation of
will denote the passage of a gravitational wave. A similar effect would be more difficult to observe with
, a direct signal interferometer, due to the presence of highly oscillating terms, as shown above.