Tolerance of dual recycling laser interferometric gravitational wave detectors to mirror tilt and curvature errors

PHYSICAL REVIEW D VOLUME 48, NUMBER 12 15 DECEMBER 1993

ARTICLES

Tolerance of dual recycling laser interferometric gravitational wave detectors to mirror tilt and curvature errors

D. E. McClelland, C. M. Savage, and A. J. Tridgell Department of Physics and Theoretical Physics, Australian National University, Canberra, 0200, Australia

R. Mavaddat Department of Electrical and Electronic Engineering, University of Western Australia, Nedlands, 6009, Australia

(Received 1 March 1993)

Various configurations of dual recycling laser interferometers are tested, numerically, for their tolerance to wave-front distortions induced by mirror tilts and curvature mismatch. We verify that, in the presence of such geometric imperfections, dual recycling designs experience significantly less power loss than straight power recycling devices. Importantly, we show that the power loss reduction from the detection port is accompanied by maintenance of circulating power in the instrument and hence maintenance of signal response. We confirm predictions that when a cavity is placed at the output port of the interferometer, tolerance to geometric imperfections is further improved by typically an order of magnitude.

PACS numberk): 04.80. +z, 07.60.Ly, 95.75.Kk

I. INTRODUCTION

The need to develop ultrasensitive instruments to detect gravitational waves [I] has led to a number of in- novative ideas [2] for optimizing the performance of laser based Michelson interferometers, depicted in Fig. 1. When subjected to gravitational radiation, opposite length changes are induced in each arm with the sign of the displacement reversing every half period of the radiation. The change in length imposes phase changes on the optical field which are then read out at the detection port. Each arm may contain either a delay line (DL) or a Fabry-Perot (FP) cavity to optimize the storage time of the light in regard to the time during which the phase of the external excitation is one signed. Operation of the interferometer close to a ''null fringe" at the detection port of the main beam splitter (BS1 in Fig. 1) achieves maximum signal-to-noise ratio (SNR) with common mode re- jection minimizing the classical noise on the laser beam [3,4]. In such a configuration, the signal, appearing as sidebands induced on the optical field symmetrically dis- placed about the carrier frequency, is ejected at the main beam splitter toward the detector while the input (carrier) light is directed back toward the laser. Drever [5] proposed reusing this "waste" light by placing a mirror (Mo in Fig. 1) to recycle the power back into the interferometer. In this way Mo forms a split optical cavity, known as the power recycling cavity (PRC), with the "reflecting" elements in the interferometer arms. The power gain achieved translates directly into an increase in the SNR 151. In addition, as the cavity is nonconfocal, the trans- verse spatial modes are nondegenerate in frequency;

hence, the cavity helps to filter frequency and amplitude noise off the input light. The benefits of power recycling have been demonstrated in "in principle" experiments [6-81.

A number of schemes for recycling have since been

Input light -

Mo Power recycling mmor

Signal recycling rmrror T M3

FIG. 1. Optical arrangement for dual recycling. M , is the power recycling mirror; M , is the signal recycling mirror. The power recycling cavity is then formed by mirrors M , with mirrors M I and M 2 ; the signal recycling cavity is formed by mirror M 3 , with mirrors M I and M,.

5475 @ 1993 The American Physical Society

5476 McCLELLAND, SAVAGE, TRIDGELL, AND MAVADDAT 48

proposed, including detuned recycling [9] possible when Fabry-Perot cavities are used in the arms, synchronous (or resonant) recycling [5], and schemes known collective- ly as dual recycling [10,11]. Comprehensive analysis for ideal instruments indicates that the achievable sensitivi- ties are similar [9,10]. However, for practical implementation, dual recycling would appear to have significant advantages [lo].

Dual recycling refers to the recycling of both power and signal. The signal recycling (split) cavity (SRC) is formed by the reflecting elements in the interferometer arms (mirrors M I and M 2 in the simple case of no FP, or a single pass DL) with M 3 as shown in Fig. 1. When the SRC is tuned to the carrier frequency, both gravity wave induced sidebands can be resonantly stored provided they fall within the bandwidth of the cavity. This restricts the choice of reflectivities of the signal recycling mirror. This mode of operation is known as broadband dual recycling. Tuned dual recycling refers to the case where the SRC is tuned to one of the sideband frequencies. Large signal enhancement is predicted at the expense of reduced interferometer bandwidth. In an "in principle" experi- ment, Strain and Meers [12] observed a sevenfold signal enhancement. Such an arrangement is not optimal, however, as the signal occurring at other sideband frequencies is lost.

To overcome this problem Meers and Drever [13] proposed a technique known as doubly resonant recycling. This configuration relies on the coupling between the SRC and a cavity installed in place of mirror M 3 (see Fig.

Input light

0 Power recycling mirror

' M3' Compound cavity mirror

I

f Output light

FIG. 2. Optical arrangement for compound or doubly resonant recycling, featuring a compound cavity "mirror" on the output port of the main beam splitter. R, and R , are amplitude reflectivities.

2 ) to svlit the resonances of the SRC. Meers and Drever [13] show that by appropriate choice of the mirror transmissions the frequencies of the double resonance can be overlapped with the signal sideband frequencies. With this arrangement the optimum response frequency f,,, depends on the storage time in both the SRC and the compound cavity. The required coupling imposes the constraint of a physically long "compound cavity. When the compound cavity is equal in length to the SRC, as would be most easily configured, the bandwidth is the saFe with the gain in response increased by a factor of g2 over the tuned dual recycling case. The optimal response frequency depends on the storage times in both the SRC and the compound cavity.

One of the major benefits predicted for dual recycling is its tolerance to wave-front distortions. Meers and Strain [15] argue that a highly reflecting signal recycling mirror will reduce distortion induced power loss from the interferometer. This should result in the maintenance of the circulating power in the PRC and hence keep the response high. This may be of great importance in enabling gravity wave interferometers to achieve their design sensitivity. However, it is not possible to fully analyze these claims analytically.

Here we will use a numerical model of a dual recycling delay line Michelson laser interferometer, described in detail in Ref. [14], to investigate the response of various dual recycling configurations to imperfections. We use, as a generic distortion, tilt on one mirror of the interferometer. However, we found that the trends reported are also seen when a mirror curvature mismatch is im- posed and hence relevant to other types of geometric distortions. Our results confirm and expand on the experimental demonstration by Meers and Strain [15] and extend the tests to the doubly resonant case.

Work on numerically modeling laser interferometers has to date been directed toward simulating nonideal optical cavities [16] and examining thermally induced distortions in such resonators [17]. Though the model we use is similar to that of Vinet et al. [16], they have not examined recycling laser interferometers, in particular, dual recycling devices.

In the next section we will describe the basis for the predicted tolerance to distortion. In Sec. 111, the numerical model is described and tests of its accuracy are given. We then present numerical results demonstrating the performance and resilience to mirror tilt, of the dual recycling configurations and end with a discussion of the im- plications of such results.

Though developments in recycling have been driven by the gravitational wave quest, the principles are applicable to interferometry generally. For example, such methods would be useful for improving the sensitivity of interferometers responsive to oscillating electric or magnetic fields [4].

11. DISTORTIONS IN DUAL RECYCLING

A major benefit of cavity recycling systems is their predicted tolerance to wave-front distortions. Meers and Strain [15] discuss in detail this issue for the case of dual

48 - TOLERANCE OF DUAL RECYCLING LASER . . . 5477

recycling. Using a modal decomposition analysis they set up a formalism for investigating the action of various imperfections, such as mirror tilt or curvature mismatch.

With an interferometer set on a null fringe, light at the carrier frequency in the fundamental Gaussian mode TEMOO, is reflected at the main beam splitter into the PRC, while light in any higher-order modes is ejected at the beam splitter toward the photodetection system. Now, a simple analysis shows [15,16] that the primary effect of a slight tilt on one mirror is to excite the first- order Gauss-Hermite modes TEMO1,10, which are then ejected out of the PRC. This is demonstrated in Fig. 3, which shows the spatial composition of the beam exiting the interferometer. Note that when the tilt becomes too large, significant amounts of the fundamental are also ejected. Curvature mismatch, on the other hand, can be shown to primarily lead to the excitation of the even order modes TEM02,20 [15,16], which are subsequently ejected from the PRC.

In the absence of signal recycling, the distorted light is lost from the interferometer, reducing the power buildup and hence the response. However, with the introduction of mirror M 3 , much of the distorted light should be reflected back into the interferometer, with the reduction in transmission being related to the transmittivity T 3 of mirror M 3 and the resonance condition of this light in the SRC. The reflected light can then reenter the PRC, contributing to the formation of a new normal mode for the system. However, for the interferometer to benefit from the reduction in loss, good coupling of the input laser power to the new mode will need to be achieved. Similarly, good coupling out of the signal must also be re- tained. This should then translate directly into preserva- tion of the response, even in the presence of some distortion.

In the broadband case, the bandwidth condition requires a relatively low reflectivity (high transmission) mirror. In addition, as the SRC is tuned to the carrier, any light in the fundamental mode leaking into the SRC will be resonantly enhanced by about the same factor as the signal. Thus the effectiveness will most likely be re- stricted to low distortion levels. In the tuned case, however, not only can a mirror of high reflectivity be used but the SRC does not resonate any leaked fundamental mode. A major reduction in loss is therefore anticipated,

FIG. 3. The composition of the field being ejected into the signal recycling cavity for a tilt distortion of 2X lo-' radians. Parameters as listed in Table I. A , and A ,, refer to the amplitudes of the first two Hermite-Gauss modes.

even at quite a large distortion. Meers and Strain [15] report experimental evidence in support of this scenario.

With the introduction of the compound cavity mirror (Fig. 2), even greater reduction in loss can be expected since transmission through the output "mirror" now depends on the transmittivity of two mirrors and the resonance condition of the modes in both the SRC and the compound cavity. A reduction in the loss of an order of magnitude better than for dual recycling is predicted, revealing even greater distortion tolerance.

In Sec. IV we will use the numerical code outlined next, to examine the behavior of dual recycling laser interferometers in the presence of distortion induced by mirror tilts. We will show that the predicted reduction in losses are observed and that this does lead to increased tolerance of the response to such distortion.

111. THE NUMERICAL MODEL

In this section we outline the numerical code developed for dual recycling in a delay-line-type Michelson interferometer. Our approach is similar to that of Ref. [16] and models as closely as possible the physical processes that occur in a real interferometer. However, we allow for time-dependent perturbations, for example, gravitational waves, on the device enabling the response to be extracted directly, without having to infer it from the circulating power. A full account including technical details can be found in Ref. [14].

A. Basic steps

The fundamental objects that are manipulated in the numerical model are light field grids (LFGs). These LFGs, composed of an N X N point array, are snapshots of the complex electric field, in a plane perpendicular to the direction of propagation of the light. Propagation is modeled using the second-order paraxial propagation technique [18] giving the field @(x,y,z) after propagation in terms of the field @ O ( ~ O , y O ) at z = O as

where T, and TF-' represent the Fourier transform and its inverse, respectively, and k , is given by (k : + k j k, and k , being the components of the wave number k in the x and y directions. The technique basically applies plane wave propagation to the LFG, then multiplies the result by a phase sheet in Fourier space to compensate for the off axis components of the wave number. The propagation technique is numerically implemented using fast Fourier transforms.

For reflection, provided the total propagation distance involved in the reflection of the LFG is reasonably small [14], each grid square can then be propagated indepen- dently, just like plane wave propagation. Each grid square will receive a phase change proportional to the height of the reflecting surface at that point. Thus, if we define the reflecting surface by its height H ( x , y ) perpendicular to the direction of propagation of the light, we

5478 McCLELLAND, SAVAGE, TRIDGELL, AND MAVADDAT - 48

can then calculate the reflected LFG using

with i = ( - 1)'12 and R the amplitude reflectivity. The sign of the exponential is positive because a large value for H ( x , y ) means that a grid square will propagate a shorter distance.

The application of the reflection technique to curved surfaces such as spherical mirrors is straightforward. A mirror with radius of curvature C has a height

To evaluate this numerically requires a Taylor expansion to avoid cancellation errors. Finally, because real mirrors are circular and the reflection technique uses a square grid we must truncate any part of the LFG which extends outside the radius of the mirror.

Refraction can be treated similarly [16] writing

where T is the amplitude transmittivity and n is the re- fractive index of the material.

Interference between two LFGs can be simulated numerically simply by the addition of the complex amplitudes at each point on the grids. In the case of an interferometer tuned to a dark fringe the amplitudes of the two LFGs arriving at the beam splitter will be very close and in an ideal situation they will differ only by a small phase difference proportional to the path length difference in the two arms. If we take the complex amplitude of the two LFGs as being represented by A (i, j ) and

before interference takes place, it is then clear that the addition of the LFGs will produce a very small number for small values of 6. Thus it is the accuracy with which addition can be performed that limits the size of S which can be observed.

The model of the beam splitter was idealized [14], with none of the problems of a real device. The beam splitter was assumed to interfere two LFGs arriving at adjacent ports and produce two resultant LFGs at the other two ports.

The model operates by continually cycling through a series of operations, each of which involves several propagation, reflection, refraction (if needed) and beam split- ting steps. The fields are stored in four LFGs one for each of the interferometer arms. At each step in the cy- cle the light propagation time is added to the real time of the system. This means that when dynamic interactions such as external excitation are introduced the timing of the movements can be synchronized with the movement of the LFGs.

B. Locking techniques

In any real interferometer the positions of the mirrors must be locked to optimal positions. In a dual recycling interferometer (without Fabry-Perot cavities in the arms)

there are three separate locks that must be made for suc- cessful operation. The first is that the position of the input mirror M o must maximize the power coupled into the interferometer. The second is that the positioning of the mirrors M I and M , must be locked relative to each other so that a minimum amount of power appears on the dark fringe. The third is that the position of the mirror M 3 must be adjusted so that a particular optical frequency will be resonant in the signal recycling cavity. With the addition of the compound mirror yet another locking condition must be maintained.

To illustrate how locking is performed a simple two mirror cavity will be examined. A LFG representing the incoming laser beam enters the cavity through mirror 1. This grid propagates through the cavity, is reflected from mirror 2, and returns to mirror 1. When the grid reflects off mirror 1 it will interfere with the next grid entering from the laser. If the interference is constructive then the power in the cavity will increase with time. The maximum power will be achieved if the two interfering LFGs have zero relative phase difference. By changing the length of the cavity the relative phase of the returning grid can be adjusted so that the maximum coupling of power into the cavity can be achieved.

To find this length numerically we first create two identical grids and @,. One of these grids (@,I is propagated along the cavity, reflected from the far mirror, propagated back along the cavity and reflected by the first mirror. The relative phase of @, with respect to @, must then be found giving some value 4. The length of the cavity can then be adjusted by an amount

This changes the phase of @, by -4 and gives maximum power buildup.

The only problem with the above procedure is defining the relative phases of two LFGs especially when the grids are different mixtures of several modes, as would be the case in an imperfect interferometer. Vinet et al. [16] used a modal decomposition technique, valid at low distortion levels, to calculate the phase optimizing the over- lap between the incoming wave and the cavity resonant field. We use a maximization routine where successive values of q5 are tried until the optimal one is found [14]. The advantage in the latter technique is that it is not re- stricted to low distortion levels in the cavity, with the disadvantage being in the relative speeds of the two methods.

Using this technique, locking of the power coupled into a cavity and dark fringe locking can be achieved. Lock- ing to the correct frequency in the signal recycling cavity is marginally more complex. The first step is identical to that of locking the power recycling cavity to couple maximum power from the laser. A LFG is propagated from the signal recycling mirror through the interferometer and back to the signal recycling mirror. The value of 4 which maximizes the power in the interference between @ , and @,ei+ is then found as in the above procedure but the distance 61 to move the mirror is now set to

48 - TOLERANCE OF DUAL RECYCLING LASER. . .

where 8, is defined by

and f ppt is the gravitational wave frequency that is being optimized for, with T the round trip time in the SRC. A similar technique is used for the compound cavity mirror.

The locking to a dark fringe must be applied first as it does not rely on the positioning of the two recycling mirrors. The signal recycling mirror system is then locked followed by locking of the power recycling mirror.

C. Multipass arms

From the operational point of view the main effect of including multipass arms is to change the range of frequencies for which the device is sensitive. The only additional complication is the calculation of the appropriate shape of the reflective surface for each reflection. The LFGs will on each step reflect from a different off axis portion of the spherical mirror. This means that the height profile of the reflective surface H ( x , y ) will vary depending on which bounce the reflection takes place. Additionally, the presence of tilts in the mirror will com- plicate the calculation of H (x , y ).

D. External excitation

Implementing the effect of a time-dependent external excitation on the device involves the addition of the appropriate phase change S$(t) to the LFGs as they enter the beam splitter after returning from the delay line arms. For optimally polarized gravitational wave excitation, 8$( t ) is given by [19]

where h is the amplitude (or strain) of the gravitational wave. This phase change is applied by multiplying each of the points in one of the grids by exp[ib$(t)] and each of the points in the other by exp[-i8$(t)], where we have assumed optimum orientation of the interferometer with respect to the wave.

E. The signal extraction scheme

The raw output from the code is a series of LFGs pro- duced through mirror (or cavity) M 3 . In general, the output will consist of several components, some of which are valuable as real signals and some of which are leaked power that would be present in the absence of signal. A signal extraction scheme needs to be invoked to recover signal information only. The latest proposals for long baseline interferometers suggest using an external modu- lation technique [7,20], deriving the local oscillator from the rear surface of the main beam splitter. We employ a numerical equivalent of this method, ensuring that comparison of the signals generated from the different interferometer configurations genuinely reflect their performance. The local oscillator is derived by extracting a fraction (1%) of the field from one arm of the Michelson interferometer. Hence, the magnitude will vary with the square root of the power in the PRC. Full details of the scheme are given in Ref. [14]. Note that we do not in- clude noise (classical or quantum) in this analysis; that is, we compute the signal only, and not the signal-to-noise ratio.

The signal extraction scheme generates, from the raw output data, a response function which is proportional to the product of the magnitude of the local oscillator field with the sum of the magnitudes of the induced sideband fields [14]. The actual value of this response function has little meaning as an absolute number. What is important is how the response of an advanced system compares to a standard arrangement. The results published here are therefore given in terms of the relative response S,,,, defined as the response normalized to the response of a nonrecycled perfect single pass interferometer receiving a 4 kHz gravity wave signal of strain (chosen to reduce computational time) with an input laser power of 10 W. The choice of 4 kHz for the gravity wave frequency is a reasonable value for a nonrecycling single pass long base line instrument. The length of the main arms was chosen to be 3000 m and the radius of curvature for all mirrors was set to 3500 m. All fiducial parameters are given in Table I. The choice of this particular configuration is a matter of computational convenience and is not a particularly significant standard except for its

TABLE I. The configuration used for the tilt results.

Main arm lengths Mirror curvature (all mirrors) Delay line Mirror radii Mirror losses Beam radius at input Laser wavelength Laser power Gravity wave frequency Gravity wave strain

3000 m 3500 m Single pass 10 cm

in power 2.26 cm

Amplitude reflectivity of power recycling mirror 0.95 unless specified

fop, for tuned dual recycling 4000 Hz f otherwise 0


simplicity. As discussed later, however, care must be tak- en in the choice of cavity length to mirror curvature ratio to ensure that all unwanted cavity modes are non- resonant.

F. Including distortions

The two types of distortions that were considered in the model were both geometric distortions. The first was a tilt on one of the mirrors of the interferometer and the second was a curvature mismatch between the mirrors. The implementation of these types of wave front distortion was very simple, as it required only a minor modification to the reflection technique described earlier. In the case of tilts an additional phase wedge was added to the reflection process, and in the curvature mismatch case different mirror radii of curvature were used.

Although these two types of wave front distortion do - - - not match the far more complex surfaces of real mirrors, we expect that the qualitative effect of geometric distortions on the operation of the interferometer would not vary greatly between different distortion types. The inclusion of two distortion types provided a simple check that this was the case.

G. Model verification

With any numerical model it is important to verify the correct operation of the model. Tests were carried out to determine that power build up in the cavities agreed with analytic formulas [14]. We chose a 135 X 135 point array for the light field grid. A convergence criterion of less than a 0.001% change in response over 25 iterations was employed, typically requiring on the order of 100 cycles of the code to achieve convergence. Linearity of the model as a function of applied gravity wave strain was verified over more than ten decades. Unfortunately numerical accuracy problems prevented the program operating at strains below In the case of very small strains the program also became very slow because of the amount of convergence required before the signal could be seen. In practice, a strain of 10-l5 was chosen as a compromise.

Comparisons with the analytic models of the frequency response in the absence of distortions [10,13] were made for the following optical configurations: power recycling; broadband and tuned dual recycling; and doubly resonant recycling. In all cases we found excellent agreement between the numerical and analytic models. This being the case we can now confidently extend the use of the numerical model to areas where the analytic results are not as clear.

IV. NUMERICAL EVALUATION OF WAVE-FRONT DISTORTION

The first tests of the effects of wave-front distortion on the operation of the interferometer were carried out using tilts on mirror M Z . Computational runs were performed to find the observed signal and power losses that occur in power recycled, broadband dual recycled, tuned dual recycled and doubly resonant interferometers. The results

shown below are for a configuration corresponding to that shown in Table I. First, we will consider tuned dual recycling, followed by the broadband case, and then doubly resonant recycling. Note that all quoted mirror reflectivities are amplitude reflectivities.

A. Tuned dual recycling

The discussion in Sec. I1 focused on power loss from the interferometer. A convenient way to display the behavior is to plot the power loss reduction factor (PLRF), defined as the ratio of the power loss with only power recycling to that with dual recycling at the same tilt, as a function of normalized tilt. We define normalized tilt to be the tilt in units of the divergence angle Bc , where Bc is defined as

where w, is the beam (amplitude) waist. For our fiducial interferometer w, = 1.7 1 cm (beam radius at input = 2.26 cm) giving a Bc of 9.9 X rad.

Figure 4 shows the computed PLRF through mirror M 3 as a function of normalized tilt, as signal recycling is progressively introduced (via R3) . Recall that in this case the SRC is tuned to a signal sideband. The reduction in loss in comparison with no signal recycling is evident with losses in a moderately high finesse SRC (R, =0.9) reduced by a factor between 10 and 100, depending on the size of the tilt. The similarity between Figs. 4 (7) of this paper and 5 (4) of Ref. [15], supports both the validi- ty of the approximate analytic model used in Ref. [15] and of the numerical model used here. Figure 5 (of this paper) demonstrates that the reduction in loss converts directly into maintenance of the total power in the recycling cavity, with only a 5% reduction in a high finesse dual recycling instrument, even at relatively large tilts.

Yorrnalized tilt

FIG. 4. Power loss reduction factor (PLRF), the ratio of power loss with power recycling to power loss with dual recycling, as a function of normalized tilt (tilt 0 in units of the divergence Oc), for the tuned dual recycling configuration. For our fiducial interferometer, Bc=9.9X lop6 radians. The different curves arise from varying the amplitude reflectivity R 3 of the signal recycling mirror. Curve a: R3=0.1; b: R3=0.3; c: R3=0.5; d: R3 ~ 0 . 7 ; and e: R3=0.9. The actual power loss for power recycling only at a normalized tilt of 0.15 is 9 W .

TOLERANCE OF DUAL RECYCLING LASER . . . 548 1

Rorrnalized til t

FIG. 5. Power, in units of circulating power in the power recycling cavity in the absence of tilt, as a function of normalized tilt, for the tuned dual recycling configuration. Parameters as for Fig. 4. The circulating power at 0 tilt is 390 W. Curve a: Rj=O.l;b: R3=0.3;c: R3=0.5;d: R3=0.7;ande: R3=0.9.

Clearly, good coupling of input laser power into the power recycling cavity can still be achieved.

Figure 6 shows the response of the interferometer Srei for the same optical configurations and tilt ranges. Note that at zero tilt and no signal recycling, SrCi is equal to 39 (since Ro=0.95) , that is 39 times better than with no power recycling. This value r e f l ~ t s a 65 gain in signal due to power recycling and a 6 3 9 increase in size of the local oscillator over the no recycling case (see Sec. I11 E). This graph displays the two primary features of a tuned dual recycling system. The first feature is the handling of wave-front distortions with the retention of a good response to large mirror tilts. The second is the increased signal due to narrow banding of the frequency response.

B. Broadband dual recycling

Power loss reduction factors as a function of normalized tilt for the broadband case (SRC tuned to the carrier) are shown in Fig. 7. Again the trend of reduced loss in comparison with no signal recycling is evident. An interesting feature of these results however is the behavior of the PLRF at large tilts when the reflectivity of the sig-

Normalized til t

FIG. 6 . Response S,,, as a function of normalized tilt for tuned dual recycling with parameters as for Fig. 4. Curve a: Rj=O.l; b: R3=0.3; c: R3=0.5;d: R3=0.7;ande: R3=0.9.

Normalized tll t

FIG. 7. Power loss reduction factor as a function of normalized tilt for the broadband dual recycling configuration. Curve a: R,=O.l; b: R,=O.3; c: R3=0.5; d: R3=0.7; and e: R3=0.9. Fiducial parameters as listed in Table I. Power loss from a straight power recycling interferometer at a normalized tilt of 0.15 is 9 W.

nal recycling mirror becomes too large. This regime sees significant amounts of the carrier light leaking into the SRC and being resonantly enhanced therefore leading to the enhanced power loss [15]. This is not observed in the tuned case since the leaked carrier is no longer resonant in the SRC.

C. Pointing accuracy vs response

A useful parameter when characterizing the effect of wave-front distortions due to tilts is the pointing accuracy. This is defined as the amount of tilt (normalized to B c ) that is required to produce a specified percentage reduction in signal. A percentage level of 5% was chosen for the following results.

An interesting and concise view of the performance of different optical configurations can be made by consider- ing a graph of pointing accuracy vs response as shown in Fig. 8. Three curves are shown, one each for power, broadband dual and tuned dual recycling. In the case of the power recycling curve the results shown correspond

FIG. 8. Pointing accuracy as a function of response for power recycling, broadband dual and tuned dual recycling, for the fiducial interferometer.

5482 McCLELLAND, SAVAGE, TRIDGELL, AND MAVADDAT 48

to values of the reflectivity of the power recycling mirror ranging from 0.0 to 0.9 in steps of 0.1. The same steps are used for the other two curves except the mirror reiiectivity being changed is for the signal recycling mirror. A value of 0.95 for the power recycling mirror reflectivity was used for both the broadband and tuned dual recycling curves.

In the case of the power recycling curve, the results show that there is a trade-off between pointing accuracy and response so that a high signal level may only be achieved at the expense of increased susceptibility to wave-front distortion. This is equivalent to saying that a high finesse cavity is more difficult to maintain.

The curve for the tuned dual recycling interferometer demonstrates that the addition of the signal recycling mirror increases both the magnitude of the observed signals and the ability of the system to withstand mirror tilts. This illustrates one of the great attractions of dual recycling: its ability to attain large signal levels without sacrificing stability. The trade-off, of course, comes from the significantly reduced bandwidth of the system when a high reflectivity signal recycling mirror is used.

The situation is a little more complex in the case of broadband dual recycling. For small values of the signal recycling mirror reflectivity, the properties of the system are similar to those of tuned dual recycling. The difference comes when the reflectivity has been sufficiently increased to reduce the bandwidth of the system so that the signal sidebands can no longer resonate in the signal recycling cavity. At this stage a roll-off occurs where the response of the system can no longer increase and the pointing accuracy remains relatively steady. The roll-off point will occur at different levels of R , depending on the frequency of the induced sidebands. For low frequencies the roll-off will not occur until R 3 has reached a fairly high level. For higher frequencies the roll-off will occur more quickly, thus reducing the useful- ness of broadband dual recycling for higher signal frequencies.

E. Doubly resonant recycling

In Sec. I1 we discussed the case where the mirror M3 is replaced by a cavity. The length chosen for the compound cavity depended on the requirements. Computa- tionally, the choice of equal lengths for the SRC and the compound cavity allows LFGs cycling in both cavities to be synchronized, significantly reducing the overheads.

The arrangement we consider is that of a symmetric compound mirror cavity with mirror reflectivities R,. Figure 9 shows the P L R F from the interferometer as a function of normalized tilt for varying R ,. Recall that as R , (and therefore T 3 ) changes the frequency of optimum response also changes, from 5000 Hz at R =0.8 down to 2500 H z when R 3 =0.95. Note the change in the vertical scale, 10-1000 in this figure, in comparison with the figures for dual recycling (Figs. 4 and 7), 1-100. This significant increase in the P L R F confirms the speculation that losses emerging from the compound cavity recycling configuration will be much less than in the dual recycling case. With mirror reflectivities in the compound cavity mirror of 0.95, power loss is reduced by up to a factor of 400 with respect to power recycling only.

We can compare the broken line curve of Fig. 9 with the top curve in Fig. 4, for the case of tuning to 4 kHz. At a normalized tilt of 0.03, the doubly resonant interferometer shows about a fourfold improvement in the loss reduction factor in comparison with the dual recycling case.

Further demonstration of the robustness of doubly resonant recycling is given in Fig. 10. This diagram shows P L R F versus tilt as the compound mirror is progressively introduced. The compound cavity now has reflectivity R , for the mirror nearest the beam splitter and R , (set to 0.875) for the output mirror. For the lowest curve, R , =O, corresponding to broadband dual recycling with peak response at 0 Hz. As the compound cavity mirror is

D. Curvature sensitivity

A further set of runs was carried out with the wave front distortion due to curvature mismatch between mirrors M I and M,. However, with the fiducial curvature radii, 3500 m, we initially observed reduced tolerance to distortion. Analysis of the resonance condition of the TEM20,02 modes, the main higher-order modes excited by this type of distortion, showed that these modes were close enough to being resonant (though not resonant) in the SRC to experience substantial gain, hence the increased power loss. This was easily overcome by changing the average curvature radius to 4500 m, rendering these modes off resonant. The trends in power loss and signal maintenance observed with mirror tilt were then recovered. This was, in fact, an interesting lesson in the need for careful optical design of a full dual recycling sys-

X O I I I I ~ I I I / I ~ I t i l t

FIG. 9. Power loss reduction factor (PLRF) as a function of normalized tilt for doubly resonant recycling using a symmetric compound cavity "mirror" with amplitude reflectivities R,. Curve a: R3=0.8; b: R , =0.85; c: R3=0.875 (dashed); d: R , =0.9; and e: R 3 =0.95; with corresponding optimum response frequencies 5.0, 4.335, 4.0, 3.45, and 2.5 kHz, respectively. All other parameters are given in Table I. The actual power loss at a normalized tilt of 0.15 and R , =0.8 is only 0.8

TOLERANCE OF DUAL RECYCLING LASER . . . 5483

FIG. 10. Power loss reduction factor as a function of normalized tilt for doubly resonant recycling with an asymmetric compound cavity mirror. The reflectivity of one mirror of the compound cavity is set to 0.875, while the reflectivity R 4 of the other mirror is varied. Curve a: R 4 = 0 . 0 ; b: R , = O . 125; c: R4=0.375; d: R4=0.625; and e: R4=0.875 (f , , ,=4.0 kHz, dashed curve). All other parameters are given in Table I. The power loss at R,=O.O is 6 W at a normalized tilt of 0.15 (equivalent to broadband dual recycling).

introduced, losses sharply reduce until at R , =0.875 (dashed curve) we reach the loss curve for the 4 kHz tuning, revealing a 20-fold reduction in loss (at higher tilts), when compared to the broadband case.

Results for curvature mismatch show similar improve- ments over dual recycling.

V. DISCUSSION AND CONCLUSION

We have presented a numerical study of the behavior of dual recycling laser interferometers, allowing for the presence of geometric distortions (mirror tilt and curvature mismatch). Our results agree with analytic calcula- tions which predict that dual recycling can lead to significant enhancement of response, traded off against detector bandwidth. The study confirms earlier predictions and experimental work regarding the increased tolerance of dual recycling devices to geometric distortion in comparison with standard power recycling devices.

In the presence of mirror tilt we have shown the following, for a single pass delay line interferometer:

(i) In the case of broadband dual recycling, losses from the interferometer at a tilt of 1.5 X radians (normalized tilt =0.15) can be reduced by a factor of 2 with this factor being much higher at smaller tilts (> 15 times).

(ii) For tuned dual recycling a t small tilts and with only moderate values of the signal recycling mirror reflectivity, reductions in loss on the order of a factor of 20 are possible in comparison with power recycling only. At 1.5 X radian of tilt we find a loss reduction by a t least a factor of 10 is still maintained. This is accom-

panied by narrowbanding of the frequency response. (iii) For doubly resonant recycling (and therefore

"compound" recycling [15]) we have confirmed speculation that compound cavity systems are even less suscepti- ble to geometric distortion with typically about an order of magnitude improvement over the corresponding dual recycling case.

We demonstrated that loss reduction is accompanied by retention of power in the power recycling cavity show- ing that good coupling of the input laser mode to the PRC can be maintained. The result is greater tolerance of the response to mirror tilt. For example, we found that with tuned dual recycling ( R =0.9), the response remains within 5% of the ideal response up to a tilt of 1.3 X l o p 6 radians, a factor of 4 improvement over power recycling. Similar trends were observed with curvature mismatch error.

Though our model ignores many imperfections, we can compare the importance of tilt with absorption in the mirrors. The aim is to ensure that losses due to tilt are much less than the power loss due to absorption. For the fiducial interferometer with an (intensity) absorption of l o p 4 per reflection and no signal recycling, this necessi- tates control of alignment to about 2 X radians (0.0050c), an extremely stringent condition requiring an automatic alignment system. Dual recycling with R 3 = 0 . 9 relaxes this condition by a factor of 5 in the tuned case and 2.5 in the broadband mode.

Optimization of a standard power recycling delay line Michelson interferometer for low-frequency operation requires a many pass delay line to match the storage time to about half the gravity wave period [9]. For example, optimization for 250 Hz is achieved with about 100 passes (-200 reflections). Such a system requires very large mirrors. In addition, the resulting absorption loss, at 10F4 per reflection, reduces the power build up in the PRC by 30% (when Ro=0 .95) and hence the signal by the same factor (SNR reduced by 16%). This relaxes the condition on alignment control, to 2 X lo-' radians. Fur- thermore, it should be noted that maximum power gain, achieved by choosing R o to impedance match the input beam [lo], is inversely proportional to the number of delay line passes.

Alternatively, operation at low frequency can be achieved by simply changing the position of the signal recycling mirror, without the need for a multipass system. This arrangement could however lead to other problems such as increased potential for thermally induced distortions due to extra power loading on the main beam splitter.

We did not consider thermally induced distortions in. this work. The inclusion of heating effects in both the mirrors and the beam splitter will be an important step towards attaining a more realistic model of the optical system. However, we believe that the mode shaping properties of optical cavities should also act with these types of wave-front distortions and hence anticipate better performance from systems incorporating dual re-


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Tolerance of dual recycling laser interferometric gravitational wave detectors to mirror tilt and curvature errors