Application of these techniques

5.4 Application of these techniques

Using appropriate optical configurations that employ power and signal recyling as described above in Sections 5.1 and 5.2, the required laser power may thus be reduced to a level (in the range of 10 to 100 W) where laser sources are or are becoming available; however stringent requirements on technical noise must be satisfied.

5.4.1 Technical noise requirements

Power fluctuations
As described above in Section 5.1 gravitational wave interferometers are typically designed to operate with the length of the interferometer arms set such that the output of the interferometer is close to a minimum in the output light. If the interferometer were operated exactly at the null point in the fringe pattern, then in principle it would be insensitive to power fluctuations in the input laser light. However in practice there will be small offsets from the null position causing some sensitivity to this noise source. In this case it can be shown [44 ] that the required power stability of the laser in the frequency range of interest for gravitational wave detection may be estimated to be

where $δP ∕P$ are the relative power fluctuations of the laser, and $δL$ is the offset from the null fringe position for an interferometer of arm length $L$ . From calculations of the effects of low frequency seismic noise for the initial designs of long baseline detector [44 ] it can be estimated that the rms motion will be of the order of 10^–13 m when the system is operating. Taking strains of around 2 × 10^–24 (Hz)^–1/2 at 300 Hz, as shown in the lower curve in Figure 3 for LIGO 2, as typical of that desired for upgraded detectors, requires power fluctuations of the laser not to exceed

To achieve this level of power fluctuations typically requires the use of active stabilisation techniques of the type developed for argon ion lasers [76 ].
Frequency fluctuations
For a simple Michelson interferometer it can be shown that a change $δx$ in the differential path length, $x$ , of the interferometer arms causes a phase change $δϕ$ in the light at the interferometer output given by $δϕ = (2π ∕c)(ν δx + xδν)$ where $δν$ is a change in the laser frequency $ν$ and $c$ is the speed of light. It follows From that if the lengths of the interferometer arms are exactly equal (i.e. $x = 0$ ), the interferometer output is insensitive to fluctuations in the frequency of the input laser light, provided that, in the case of Fabry–Pérot cavities in the arms, the fluctuations are not so great that the cavities cannot remain on resonance. In practice however, differences in the optical properties of the interferometer mirrors result in slightly different effective arm lengths, a difference of perhaps a few tens of metres. Then the relationship between thelimit to detectable gravitational wave amplitude and the fluctuations $dν$ of the laser frequency $ν$ is given by [44 ]

Hence to achieve the target sensitivity used in the above calculation using a detector with arms of length 4 km, maximal fractional frequency fluctuations of

are required.This level of frequency noise may be achieved by the use of appropriate laser frequency stabilisation systems involving high finesse reference cavities [44 ].
Although the calculation here is for a simple Michelson interferometer, similar arguments apply to the more sophisticated systems with arm cavities, power recycling and signal recycling discussed earlier and lead to the same conclusions.
Beam geometry fluctuations
Fluctuations in the lateral or angular position of the input laser beam, along with changes in its size and variations in its phasefront curvature may all couple into the output signal of an interferometer and reduce its sensitivity. These fluctuations may be due to intrinsic laser mechanical noise (from water cooling for example) or from seismic motion of the laser with respect to the isolated test masses. As an example of their importance, fluctuations in the lateral position of the beam may couple into interferometer measurements through a misalignment of the beamsplitter with respect to the interferometer mirrors. A lateral movement $δz$ of the beam incident on the beamsplitter, coupled with an angular misalignment of the beamsplitter of $α ∕2$ results in a phase mismatch $δϕ$ of the interfering beams, such that [82 ]

A typical beamsplitter misalignment of $≃$ 10^–7 radians means that to achieve sensitivities of the level described above using a detector with 3 or 4 km arms, and 50 bounces of the light in each arm, a level of beam geometry fluctuations at the beamsplitter of close to 10^–12 m(Hz)^–1/2 at 300 Hz is required.
Typically, and ignoring possible ameliorating effects of the power recycling cavity on beam geometry fluctuations, this will mean that the beam positional fluctuations of the laser need to be suppressed by several orders of magnitude. The two main methods of reducing beam geometry fluctuations are 1) passing the input beam through a single mode optical fibre [63 ] and 2) using a resonant cavity as a modecleaner [82 , 91 , 109 , 3 ].

Passing the beam through a single mode optical fibre helps to eliminate beam geometry fluctuations, as deviations of the beam from a Gaussian TEM00 mode are equivalent to higher order spatial modes, which are thus attenuated by the optical fibre. However there are limitations to the use of optical fibres mainly due to the limited power handling capacity of the fibres; care must also be taken to avoid introducing extra beam geometry fluctuations from movements of the fibre itself.

A cavity may be used to reduce beam geometry fluctuations if it is adjusted to be resonant only for the TEM00 mode of the input light. Any higher order modes should thus be suppressed [82 ]. The use of a resonant cavity should allow the handling of higher laser powers and has the additional benefits of acting as a filter for fast fluctuations in laser frequency and power [91 , 109 ]. This latter property is extremely useful for the conditioning of the light from some laser sources as will be discussed below.

5.4.2 Laser design

From Equation (5 ) it can be seen that the photon-noise limited sensitivity of an interferometer is proportional to $P −1∕2$ where $P$ is the laser power incident on the interferometer, and $λ1∕2$ where $λ$ is the wavelength of the laser light. Thus single frequency lasers of high output power and short wavelength are desirable. With these constraints in mind, laser development has concentrated on argon-ion lasers and Nd:YAG lasers. Argon-ion lasers emitting light at 514 nm have been used to illuminate several interferometric gravitational wave detector prototypes, see for example [90 , 75 ]. They have an output power in the required single spatial (TEM00q) mode of operation typically of around several Watts, sufficient for this type of laser to have been proposed as the initial laser for a full-scale interferometric detector [103 ]. For advanced detectors higher laser powers would be desirable and it has been demonstrated that the output of several argon-ion lasers could be coherently added for this purpose [53 ]. However the disadvantages of argon-ion lasers include the increased optical absorption and damage, and more pronounced effects due to scattering, for light of this shorter wavelength. In addition argon-ion lasers are relatively inefficient.

Nd:YAG lasers, emitting at 1064 nm or frequency doubled to 532 nm, present an alternative. The longer wavelength is less desirable than the 514 nm of the argon-laser, as more laser power is needed to obtain the same sensitivity; in addition, the resulting increase in beam diameter leads to a need for larger optical components. For example in an optical cavity the diameter of the beam at any point is proportional to the square root of the wavelength [54 ] and to keep diffractive losses at each test mass below 1 × 10^–6 it can be shown that the diameter of each test mass must be greater than 2.6 times the beam diameter at the test mass. Thus the test masses for gravitational wave detectors have to be 1.4 times larger in diameter for infrared than for green light. Nd:YAG sources do however have some compelling advantages, and in particular the potential for scaling Nd:YAG laser designs up to levels of 100 W or more [89 ] combined with their superior efficiency, has led all the long baseline interferometer projects to choose some form of Nd:YAG light source.

Compact sources of lower powers of Nd:YAG light have been available for several years in the form of monolithic diode-pumped ring lasers [52 ]. Investigations have shown that the technical noise associated with these lasers may be well controlled and reduced to levels comparable to those needed for gravitational wave interferometer sources [51 , 32 , 14 , 77 , 38 ]. Different approaches to obtaining high powers of low-noise Nd:YAG light have been studied. They all have in common the use of a stable lower power laser as a master oscillator.

One approach is to use a lower power Nd:YAG master oscillator to injection lock a higher power Nd:YAG slave laser, with the length of the slave laser cavity being locked to the frequency of the light from the master oscillator [18 , 68 , 36 ]. Up to 20 W of single frequency laser light have been obtained using this method [89 ], which has the desirable feature that the higher power slave laser light has noise properties which are for the most part dominated by those of the master laser light [29 ]. This is desirable since it is typically easier to apply active noise reduction techniques to stabilise the lower power master lasers. Injection locked systems of this type are being developed for use by the VIRGO, TAMA 300 and GEO 600 projects, each of which requires $≃$ 10 W of laser light for initial operation.

However to adapt this technique for producing still higher powers from the slave laser requires care, since the light power needed from the master oscillator also increases. To meet this requirement systems have been proposed in which a series of lasers are successively injection locked.

An alternative scheme has been developed for use by the LIGO project [106 ]. Light from a master laser is passed through diode-pumped Nd:YAG amplification stages in a master oscillator/power amplifier (MOPA) configuration. This approach has the advantage of allowing a very high continous light power to be obtained using multiple amplification stages, without the need for multiple cavity locking schemes. However the effects of this design configuration on the noise properties of the amplified light must be addressed.

In particular, to obtain high performance from the modulation techniques discussed in section 5.1 it is necessary that at the modulation frequency, the power fluctuations of the laser light used must be shot noise limited in the amount of light detected at the interferometer output (typically up to a few Watts).

Previous studies of the noise properties of optical amplifiers have shown that in a given output power of light from an optical amplifier, power fluctuations exist which are in excess of those obtained from a shot noise limited laser of the same output power [39 ]. This gain dependent excess noise arises from the beating of the spontaneous emission from the amplifier with the light being amplified. Measurements of this excess noise at rf modulation frequencies have been made using a free space Nd:YAG linear optical amplifier system [100 ]. For this type of light source to be suitable for use in an interferometric gravitational wave detector, it is necessary to reduce these high frequency power fluctuations; a suitable technique is to pass the light through a resonant cavity similar to that used to spatially filter the input laser light as described in section 5.4.1 [109 ]. Above the corner frequency $fc$ of the cavity, power and frequency fluctuations of the laser light are reduced by a factor $f ∕f c$ where $f$ is the frequency at which the fluctuation occurs, and

Thus the excess power noise introduced by the amplification process may be reduced to an appropriate level. The noise properties of saturated free space Nd:YAG optical amplifiers remain to be experimentally evaluated.

As mentioned earlier, a light source with the potential to combine the increased efficiency of solid state lasers with the advantage of using shorter wavelength light is a frequency doubled Nd:YAG laser. While single frequency powers in excess of 10 W are obtainable, sources of frequency-controllable doubled light of an acceptable power level have still to be proven in terms of long term reliability, but are likely to become available in the future.