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Interferometer performance in the presence of losses
Volume
57, number
1
OPTICS
INTERFEROMETER
COMMUNICATIONS
PERFORMANCE
1 February
IN THE PRESENCE
1986
OF LOSSES
Alex ABRAMOVICI
Received
19 August
1985
The excess noise generated by losses in a Michelson interferometer interferometer strain sensitivity is not significantly altered.
Encouraging progress has been made during the fast few years in improving the performance of broad band gravitational wave detectors, consisting of suspended test masses coupled to the mirrors of a Michelson interferometer [l-4]. The arms of these interferometers contain either optical delay lines [ 1,3,5] or optical resonators [2,4]. The light beams thus cover the arm length many times, which results in increased strain sensitivity as a consequence of the increased effective interferometer base line. Since the multi-pass technique leads to a substantial loss of optical power in the interferometer even when low loss mirrors are used, fluctuations in the amplitude and phase of the optical field are expected, according to the fluctuation dissipation theorem [6] (FDT). For a monochromatic signal, the ensuing reduction in the signal to noise ratio can be compensated for by appropriately increasing the observation time or, in other words, by narrowing the system band width. This does not apply, however, to the pulsed (and thus wide band) signals which gravitational wave detectors are meant to detect. It is the aim of this note to assess to what extent the shot noise limit to the ability of a Michelson interferometer to detect wise band signals is altered by fluctuations due to losses. It follows from the FDT that an optical attenuator which transmits a fraction 0 of the incoming power generates fluctuations of the field amplitude a, with 2 = Ao(1 - 0)/2, where w is the spectral density 6afdt circular frequency of the light. The spectral density of the corresponding phase fluctuations is &j& = Ao(l O)/2a2. The well known spectral density of noise gen-
0 030-4018/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
is evaluated.
It is found
that the shot noise Iunit to
erated by a microwave attenuator (see e.g. ref. [7]) is obtained from the above formulae by replacing Aw -+ 2k, T, as required by the FDT for k, T S hw . To asses the impact of the fluctuations connected to losses on interferometer sensitivity, it should be noted that all interferometers built for gravitational radiation detection are operated on a dark fringe and employ phase modulation of the light beam, as originally proposed by Weiss [5]. Thus, the photodetected current is: i = (@e/2fiw)[
1 - cos(A sin w,t
+ @,)I,
(1)
where n is the quantum efficiency of the photodetector, P is the total power of the light leaving the interferometer, e is the electron charge, w, is the modulation frequency and A is the modulation index. r#+is the phase difference between the light beams in the two arms, proportional to the differential change in arm length, which is the signal to be detected. For Gs
(2)
Eq. (2) emphasizes the fact that the signal @sis shifted to the high modulation frequency w, . This helps in eliminating e.g. the low frequency fluctuations in the output power of the laser employed to illuminate the interferometer. The unavoidable noise present in the photo current is shot noise arising from the DC term in eq. (2): 6i2sh =2ei DC = qPe2A2/4Fiw.
(3) 1
Volume 5 7. number 1
OPTICS COMM~I&ATIONS
According to eq. (2) the equivalent spectral density:
phase noise has
6;c;, = 2hwlnP.
(4)
Shot noise has a white spectrum and therefore has a component also at wm . ~plitude ~uc~atio~s due to losses will not result in excess noise at w, , because of the relatively long storage time of the field in the interferometer arms. However, the phase fluctuations $dt add to the signal I$,. Therefore, the amplitude of the noise affecting the phase shift is larger than just shot noise by the factor: F = KS&
+
2 l/2 = [I ~~;&4shl
+ :q(l
- 0)]1’?(5)
Since the phase shift is proportional to changes in the arm length, it results that the equivalent displacement noise (measured in cm/&) is increased by the same factor F. For example, if r) = 0.5 and half the power is lost,i.e.O=OS,thenF=1.06. In conclusion, little noise is contributed by losses within the interferometer arms, in excess to shot noise. This is in agreement with noise measurements carried out with the Garching interferometer [8]. I am indebted
to Professor Zeev Vager for useful
1 February 1986
suggestions and for his interest in this work. Very stimulating discussions with the Garching group during a visit at the Garching Gravitational Radiation Detector Laboratory are acknowledged.
References [ 1 ] R. Schiliing, L. Schnupp, D.H. Shoemaker, W. Wmkler, K. Maischberger and A. Ruediger, Max PIanck Inst. for Quantum Optics report MPQ 88 (1984). [2] R. Drever. The Caltech laser interferometer. Proc. Fourth Marcel Grossman Meeting (North-Hohand, Amsterdam) in press. 131 J. Livas, The MIT interferometer project, Proc. Fourth Marcel Grossman Meeting (Norm-Hoed, Amsterdam) in press. r41 H. Ward, J. Hough, G.P. Newton, BJ. Meers, N.A. Robertson, S. Hoggan, G.A. Kerr, J.B. Mangan and R.W.P. Drever, IEEE Trans. Instr. and Measurement IM-34 (1985) 261. 151 R. Weiss, Quart. Prog. Report Res. Lab. Electronics MIT 105 (1972) 54. 161 H.B. Caften and T.A. Welton, Phys. Rev. 83 (1951) 34. [71 W.B. Davenport and W.L. Root, An ~troduction to the theory of random signals and noise (M~raw-Hi, New York, 1958). P-A K. Maischberger and D.H. Shoemaker, private communication (1985).
57, number
1
OPTICS
INTERFEROMETER
COMMUNICATIONS
PERFORMANCE
1 February
IN THE PRESENCE
1986
OF LOSSES
Alex ABRAMOVICI
Received
19 August
1985
The excess noise generated by losses in a Michelson interferometer interferometer strain sensitivity is not significantly altered.
Encouraging progress has been made during the fast few years in improving the performance of broad band gravitational wave detectors, consisting of suspended test masses coupled to the mirrors of a Michelson interferometer [l-4]. The arms of these interferometers contain either optical delay lines [ 1,3,5] or optical resonators [2,4]. The light beams thus cover the arm length many times, which results in increased strain sensitivity as a consequence of the increased effective interferometer base line. Since the multi-pass technique leads to a substantial loss of optical power in the interferometer even when low loss mirrors are used, fluctuations in the amplitude and phase of the optical field are expected, according to the fluctuation dissipation theorem [6] (FDT). For a monochromatic signal, the ensuing reduction in the signal to noise ratio can be compensated for by appropriately increasing the observation time or, in other words, by narrowing the system band width. This does not apply, however, to the pulsed (and thus wide band) signals which gravitational wave detectors are meant to detect. It is the aim of this note to assess to what extent the shot noise limit to the ability of a Michelson interferometer to detect wise band signals is altered by fluctuations due to losses. It follows from the FDT that an optical attenuator which transmits a fraction 0 of the incoming power generates fluctuations of the field amplitude a, with 2 = Ao(1 - 0)/2, where w is the spectral density 6afdt circular frequency of the light. The spectral density of the corresponding phase fluctuations is &j& = Ao(l O)/2a2. The well known spectral density of noise gen-
0 030-4018/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
is evaluated.
It is found
that the shot noise Iunit to
erated by a microwave attenuator (see e.g. ref. [7]) is obtained from the above formulae by replacing Aw -+ 2k, T, as required by the FDT for k, T S hw . To asses the impact of the fluctuations connected to losses on interferometer sensitivity, it should be noted that all interferometers built for gravitational radiation detection are operated on a dark fringe and employ phase modulation of the light beam, as originally proposed by Weiss [5]. Thus, the photodetected current is: i = (@e/2fiw)[
1 - cos(A sin w,t
+ @,)I,
(1)
where n is the quantum efficiency of the photodetector, P is the total power of the light leaving the interferometer, e is the electron charge, w, is the modulation frequency and A is the modulation index. r#+is the phase difference between the light beams in the two arms, proportional to the differential change in arm length, which is the signal to be detected. For Gs
(2)
Eq. (2) emphasizes the fact that the signal @sis shifted to the high modulation frequency w, . This helps in eliminating e.g. the low frequency fluctuations in the output power of the laser employed to illuminate the interferometer. The unavoidable noise present in the photo current is shot noise arising from the DC term in eq. (2): 6i2sh =2ei DC = qPe2A2/4Fiw.
(3) 1
Volume 5 7. number 1
OPTICS COMM~I&ATIONS
According to eq. (2) the equivalent spectral density:
phase noise has
6;c;, = 2hwlnP.
(4)
Shot noise has a white spectrum and therefore has a component also at wm . ~plitude ~uc~atio~s due to losses will not result in excess noise at w, , because of the relatively long storage time of the field in the interferometer arms. However, the phase fluctuations $dt add to the signal I$,. Therefore, the amplitude of the noise affecting the phase shift is larger than just shot noise by the factor: F = KS&
+
2 l/2 = [I ~~;&4shl
+ :q(l
- 0)]1’?(5)
Since the phase shift is proportional to changes in the arm length, it results that the equivalent displacement noise (measured in cm/&) is increased by the same factor F. For example, if r) = 0.5 and half the power is lost,i.e.O=OS,thenF=1.06. In conclusion, little noise is contributed by losses within the interferometer arms, in excess to shot noise. This is in agreement with noise measurements carried out with the Garching interferometer [8]. I am indebted
to Professor Zeev Vager for useful
1 February 1986
suggestions and for his interest in this work. Very stimulating discussions with the Garching group during a visit at the Garching Gravitational Radiation Detector Laboratory are acknowledged.
References [ 1 ] R. Schiliing, L. Schnupp, D.H. Shoemaker, W. Wmkler, K. Maischberger and A. Ruediger, Max PIanck Inst. for Quantum Optics report MPQ 88 (1984). [2] R. Drever. The Caltech laser interferometer. Proc. Fourth Marcel Grossman Meeting (North-Hohand, Amsterdam) in press. 131 J. Livas, The MIT interferometer project, Proc. Fourth Marcel Grossman Meeting (Norm-Hoed, Amsterdam) in press. r41 H. Ward, J. Hough, G.P. Newton, BJ. Meers, N.A. Robertson, S. Hoggan, G.A. Kerr, J.B. Mangan and R.W.P. Drever, IEEE Trans. Instr. and Measurement IM-34 (1985) 261. 151 R. Weiss, Quart. Prog. Report Res. Lab. Electronics MIT 105 (1972) 54. 161 H.B. Caften and T.A. Welton, Phys. Rev. 83 (1951) 34. [71 W.B. Davenport and W.L. Root, An ~troduction to the theory of random signals and noise (M~raw-Hi, New York, 1958). P-A K. Maischberger and D.H. Shoemaker, private communication (1985).