1. Introduction
The DECi-hertz Interferometer Gravitational wave Observatory (DECIGO) is designed to detect gravitational waves at frequencies between 0.1 and 10 Hz. In this frequency band, one of the most important science targets is primordial gravitational waves [
1]. Observation of primordial gravitational waves is expected to provide crucial evidence for cosmic inflation theory. While observation of the primordial gravitational waves by ground-based detectors is challenging due to the ground vibration noise, pendulum thermal noise, etc., existing at low frequencies, and limited interferometer arm lengths, space interferometers enable observations by removing these obstacles. DECIGO also plans to use optical cavities between spacecraft to increase its sensitivity further.
The target sensitivity of DECIGO was established more than ten years ago to detect primordial gravitational waves. However, the recent observation of the cosmic microwave background (CMB) by the Planck satellite and other electromagnetic observations reduced the upper limit for primordial gravitational waves significantly [
2,
3]. This reduction of the upper limit requires further improvement of the target sensitivity of DECIGO [
4]. Therefore, we have been trying to improve the sensitivity by optimizing various parameters of DECIGO, such as the arm length, the laser power and the diameter, reflectivity, and mass of the mirrors.
For this optimization, we have to treat the diffraction loss of light in a Fabry-Perot (FP) cavity properly; we should treat the diffraction loss differently from other optical loss-related quantities such as absorption and transmission. For example, the part of the light that passes outside a mirror due to diffraction obviously does not cause radiation pressure noise. In contrast, light that is absorbed by the mirror causes radiation pressure noise. As for the light coming back to the input mirror from the end mirror of a FP cavity, the part of the light that misses the input mirror due to diffraction does not reach a photodetector positioned to sample light returning from the cavity. In contrast, return light that transmits through the input mirror is detected by the photodetector, and thus contributes to the shot noise at the photodetector. In previous investigations, the impact of the diffraction was not considered. However, it is important to consider the diffraction loss to more correctly design the sensitivity of an interferometer. For these reasons, it is essential to correctly calculate the quantum noise of an interferometer with the diffraction loss. The higher-order mode in a FP cavity is treated as loss in this paper on the condition that the beam cut by the diffraction in a FP cavity is small enough so that the finesse is sufficiently higher than 1. The diffraction of the laser beam in the FP cavity is investigated in other paper such as [
5]. In this paper, the treatment of the diffraction loss in the FP cavity in [
5] is further developed for the calculation of quantum noise. In this paper, we will provide the proper treatment of diffraction loss in terms of quantum noise. It should be noted that this method is applicable to any FP cavities with a relatively small beam cut and the finesse sufficiently higher than 1. It should also be noted that noise sources other than quantum noise are not discussed here. In this paper, we discuss diffraction loss in a FP cavity at
Section 2, calculation of quantum noise including the effect of diffraction loss at
Section 3, result of quantum noise by using the DECIGO parameters in
Section 4, and summary of this paper in
Section 5.
2. Treatment of Diffraction Loss in a FP Cavity
Diffraction influences the amplitude of laser light when the mirrors of FP cavity reflect or transmit it. Because in principle the laser beam extends to infinity in a plane perpendicular to the laser axis, the laser beam suffers a loss when transmitting through or reflecting from a mirror.
Figure 1 illustrates a laser with the wavelength of
entering a FP cavity via the input mirror; it is a distance
l away from the beam waist of the cavity and has radius
R and the mirror has the amplitude transmissivity
t. Assuming that the laser is a Gaussian beam with the Rayleigh length
, the normalized absolute amplitude of the laser through the input mirror is given by
When the resonant mode of the cavity is a
mode, we can treat only this fundamental mode under the condition that the FP cavity has relatively high finesse and the higher-order modes do not resonate in the FP cavity. The higher-order modes can be ignored in the FP cavity with high finesse because there is much more fundamental mode than the higher-order modes due to the substantial amplification of the
mode by the high finesse. Assuming that the contribution of the laser through the outside of the mirror is negligible, the normalized absolute amplitude of the fundamental mode is given by [
6]
With Equations (1) and (3), the amplitude of the fundamental mode after transmitting through the input mirror is given by
Therefore, the laser power after transmitting through the input mirror,
P, is given by
The effective transmissivity
, which is the transmissivity influenced by the diffraction, is given by
We define a diffraction loss factor,
, for each mirror (
) shown in
Figure 2, given by
In the FP cavity, there are two kinds of effects of diffraction loss: leakage loss outside the mirror as expressed by Equation (
1) and higher-order mode loss as expressed by Equation (
3). The leakage loss has to be taken into account when the laser light is reflected or transmitted. Only inside the cavity, higher-order mode loss must be considered with leakage loss when the laser light is reflected by a mirror or transmits through a mirror. While we should treat only
as the leakage loss when we calculate the electric field outside the FP cavity, we have to treat
as the leakage loss coupled with the higher-order mode loss due to the cavity mode. The effective reflectivity
and transmissivity
shown schematically in
Figure 2, are defined by
We calculate each electric field at each point of the cavity, as shown in
Figure 3, in preparation for the calculation of the quantum noise. With the round-trip phase change of the laser field defined as
,
is given by
,
, and
are given by
is multiplied by the negative reflectivity because of reflection from the High Reflection (HR) surface in this case coming from the higher index side instead of the condition where the reflection is coming from the air or vacuum side of the HR coating. In Equations (12) and (13), we should treat only the leakage loss as the diffraction loss because the electric field detected at the photodetector includes the higher-order mode. For this reason, this electric field is multiplied by the coefficient
once. With Equations (12) and (13), the electric field of interference light
is given by
Then the power
of the interference light is given by
With the coefficient to simplify the formula,
, defined as
Equation (
15) can be written as
Here we derive the finesse of the FP cavity, including the effect of diffraction loss. We define the effective finesse as
. The finesse is
;
is free spectral range and
is the cavity bandwidth. First, we derive these terms, including diffraction loss. Using Equation (
10), the laser power
inside the FP cavity can be written as
The length of FP cavity is
. When the frequency of the laser is defined as
, the round-trip phase change
is given by
The half of the maximum laser power is equal to a laser power derived from substituting Equation (
19) and
for Equation (
18), which is represented as
Assuming that
, and using Equation (
20), the cavity bandwidth,
is given by
And
is written as Equation (
22) by substituting Equation (
19) for
and
:
As a result,
is written as Equation (
23) with Equations (21) and (22):
This result shows that the effective finesse is equal to the finesse except for the difference between the reflectivity and the effective reflectivity.
3. Quantum Noise Including Diffraction Loss
We derive the frequency response to gravitational waves in FP interferometers as another preparation for the calculation of the quantum noise. The time
is defined as the round trip time between the input and end mirrors, multiplied by
n. When gravitational waves,
, arrive at FP interferometers, the time, which takes for the laser light round trip, is given by
With Equation (
24),
is given by [
7,
8]
The electric field of the interferometer light is given by a series like Equation (
10), with
,
Using Equation (
25), this can be rewritten as
Here
A, the coefficient to simplify the formula, is given by
Here
, the coefficient to simplify the formula, is given by
Also, we assume this condition given by
Using Equations (29) and (30),
can be written as
2
Next we derive
, which is the strain sensitivity of the shot noise in FP interferometer. First, we calculate
for one arm of a Fabry-Perot Michelson Interferometer (FPMI). Then we calculate the quadrature sum of
in both arms of the interferometer. The shot noise can be regarded as the statistical fluctuations of the photon number at the photodetector. The minimum phase change
when the laser light is detected at the photodetector [
7] is given by
quantum efficiency of the photodetector is
, and the laser power at the photodetector is
. The angular frequency of the laser,
, is given by
Now we calculate
, which is equivalent to
. Assuming that the phase of the light in the interferometer shifts by
when the gravitational wave reaches the interferometer, the electric field
is given by Equation (
31). The phase shift
is then given by
where
, the transfer function between the strain and the phase, is given by [
7]
With the coefficient,
, given by Equation (
16), the absolute value of
is given by
Then Equation (
36) is rewritten as Equation (
37) on the assumption of
.
with
defined by
The gravitational wave strain
, which is equivalent to the phase change
at a certain frequency
f, is given by [
7]
Using Equation (
14),
is given by
Using Equations (37)–(40),
can be written as
Then, because the shot noise from each arm is uncorrelated, the total shot noise,
, can be written as the quadrature sum of the shot noise from each arm, which is given by
Here
is converted to
, which is the total laser power of FPMI. The pre-conceptual design of DECIGO is shown as a reference in
Figure 4. DECIGO uses a differential FP interferometer, whose quantum noise is in principle the same as that of the FPMI.
The radiation pressure noise of a FP cavity is derived from fluctuations of the mirror positions due to fluctuations of the laser power. The fluctuations of the laser power can be attributed to statistical fluctuations of the number of photons. The mirror is subject to the force from the laser radiation pressure. When the laser power is
P, its force is represented as
. The position
x of the mirror follows the equation of motion, which is represented as
The relationship between the fluctuation of the mirror position
and the laser power, which is derived from the Fourier expansion of Equation (
43), is given by
The energy per photon is
, so the laser power
P can be written in terms of the number of photons,
N, as
The fluctuation of
N is proportional to the square root of
N, that is
Then the power fluctuation
is given by
Using Equations (44) and (47), the fluctuation of the mirror position
is given by
In terms of the FP cavity, whose arm length is
L, the response from the gravitational wave with the amplitude of
is equal to the one from
.
is the fluctuation of the mirror position in the FP cavity [
7]. For this reason,
, which corresponds to the phase change by
, is represented as
P in Equation (
49) has contributions from two sources: the light reflected at the input mirror and the laser light circulating inside the FP cavity. As a result, the total radiation pressure noise of an arm cavity in FPMI is derived from two sources. The laser power reflected at the input mirror is negligible because this power is much less than the laser power inside the FP cavity under the condition that the FP cavity has relatively high finesse. The laser power reflected at the end mirror is defined as
, and that at the input mirror is defined as
. Using Equation (
10), electric fields,
and
, are given by
In Equations (50) and (51), we treat only the leakage loss as the diffraction loss because the radiation pressure noise is caused by the laser power, which is just after the reflection. For this reason,
is
multiplied by the reflectivity
of the end mirror and the coefficient
of the leakage loss. Also,
is
multiplied by the effective reflectivity
of the end mirror, the reflectivity
of the input mirror, and the coefficient
of the leakage loss. Using Equations (50) and (51),
and
can be written as
Here we define
and
, which is given by
With Equations (54) and (55), Equations (52) and (53) can be written as
Substituting
and
into Equation (
47), the fluctuation of each conponent of laser power is given by
The term in the square root represents the noise caused by a single reflection. This term is multiplied by the terms related to the finesse,
and
, to represent the fluctuation of the laser power inside the FP cavity. Thus, the radiation pressure noise
of one arm FP cavity is given by
When
is substituted into Equation (
58), assuming that
, it can be rewritten as
Finally, the total radiation pressure noise in a FPMI,
, with no correlation between the noises in the two arms is given by
Assuming that the diffraction is negligible, and the reflectivity
is equal to 1, Equations (42) and (62) can be written as the calculation results,
and
, which are written by
Assuming that
,
and Equations (63) and (64) are rewritten as
These calculation results are consistent with [
9].