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gravitational wave astronomy inherently relies on high power resonant optical systems .
the power of the probe light circulating in a cavity is the ultimate lever utilized to increase the sensitivity of a position measurement of a gravitational wave detector test masses carrying information about gravitational wave signals .
the projected continuous wave ( cw ) light power pushes 0.8  mw value in the second generation of gravitational wave detectors , such as advanced ligo ( now in operation ) , advanced virgo and kagra , planned to become operational in the next few months @xcite . while this power value is by far lower
if compared with the optical damage limit of the cavity mirrors , it is high enough to initiate various nonlinear processes resulting in depletion of the probe light and in generation of optical harmonics adding noise to the recorded signal and hindering the desirable sensitivity increase . technical solution allowing suppressing the nonlinear interactions are needed to push the limits of gravitational wave astronomy and to widen the horizon of observable events associated with gravitational wave emission .
resonant opto - mechanical oscillations are expected to have the lowest power threshold if compared with the other nonlinear processes in the cavities , it may cause undesirable parametric instability ( pi ) @xcite .
the pi occurs due to interaction of optical cavity modes and mechanical modes of the cavity mirrors .
the photons of the probe light confined in a selected , usually fundamental , cavity mode pumped at frequency @xmath3 are parametrically converted to mechanical phonons of the cavity mirrors ( having frequency @xmath4 ) as well as lower frequency , or stokes , photons emitted into high order optical modes having frequency @xmath5 .
the power threshold of pi is inversely proportional to the product of quality factors of the optical and mechanical modes participating in the process , so desirable reduction of the optical as well as mechanical attenuation results in undesirable reduction of the pi threshold .
the phenomenon of pi was studied and validated experimentally in a table top fabry - perot resonator @xcite as well as in whispering gallery mode resonators @xcite .
recently pi was observed in full scaled advanced ligo interferometer @xcite at relatively small circulating power @xmath6  kw as compared with @xmath7  mw planned in a advanced ligo .
efficiency of pi depends on phase matching , comprising nonzero overlap integral and energy conservation , of the optical and mechanical modes .
there is a significant probability that these conditions are always fulfilled in long - base gravitational wave detectors because of dense spectrum of optical modes of large cavities and dense spectrum of mechanical modes of large area cavity mirrors . since the mirrors involved into the system
are not identical , they have slightly different associated mechanical frequencies that can lead to pi .
several techniques of reducing pi impact have been studied recently .
they involve either braking the phase matching of the nonlinear process by changing frequency spectra of the modes participating in the ip process , or reducing pi efficiency by damping nonessential modes .
for instance , one can move the opto - mechanical system out of resonance by controlling surface temperature of the mirrors @xcite .
this is possible since the optical ( @xmath3 and @xmath8 ) and mechanical ( @xmath4 ) eigenfrequencies of the system depend on the mirror temperature @xmath9 in different ways , so the pi favorable condition @xmath10 , ultimately breaks .
the drawback of this technique is related to its lack of selectivity .
all the modes of the optical cavity move at nearly the same pace , and while one pair of stokes and mechanical modes comes out of the resonance , another pair comes in . this drawback can be partially suppressed by modulation of the temperature of the mirror surface .
alternative stabilization method involves damping of mechanical modes either in a passive or an active way .
it was found that introducing an annular strip at the rims of cavity mirrors reduces quality ( q- ) factors of elastic modes @xcite .
however , this strip reduces q - factor of the modes within the whole spectrum , including reception band of the antenna ( @xmath11  hz ) .
this is undesirable , since low mechanical attenuation at these frequencies is essential for achieving the desirable detection sensitivity .
active electro - mechanical feedback allows reducing q - factor of several particular elastic modes @xcite .
the method is too selective to suppress all high - frequency modes in the entire bandwidth of interest ( @xmath12  khz ) and , hence , does not solve the problem of instability of highly overmoded opto - mechanical system .
therefore , a universal method of pi suppression is still needed .
we here propose a solution based on optimization of the shape of the cavity mirrors leading to increase of the diffraction losses of all high - order optical modes of the optical cavity and subsequent increase of the pi threshold .
note that diffraction losses a cavity modes can be increased rather significantly by properly shaping mirrors of the cavity @xcite , however , with loss increase of main mode which is inappropriate for laser gravitational detector detector .
we propose the method allows realizing an optical cavity containing only one family of low loss bounded modes .
this is achievable in the case of large area optical mirrors . in a realistic gravity wave detector ,
though , the size of the mirrors is limited and suppression of the high - order modes is associated with loss increase of the fundamental mode family .
we study this practically interesting case using numerical simulations and show that it is feasible to increase the pi threshold at least by an order of magnitude by proper shaping mirrors of a ligo interferometer keeping diffraction loss of main mode at acceptable low level .
we show that the stability of the modified interferometer with respect to the mirror tilt and shape variations is acceptable .
finally , we found that the optimized cavity can still be interrogated using conventional gaussian beams .
the lowest pi intracavity threshold power evaluated for a fabry - perot ( fp ) resonator can be found from expression @xcite @xmath13 where @xmath14 is the round trip optical attenuation coefficient of the stokes mode , @xmath15 is the mass of the mirror , or test mass , @xmath16 is the quality factor of the elastic mode , @xmath17 is the speed of light in the vacuum , @xmath18 is a mismatching factor , @xmath19 is volume of the mirror , @xmath20 is the mechanical mode displacement , @xmath21 is the same normal displacement on the mirror surface , and @xmath22 is distribution of main and stokes optical modes on mirror surface . the integration is performed over mirror volume ( @xmath23 ) and mirror surface ( @xmath24 ) .    equation ( [ p ] ) is obtained for the all - resonant case : @xmath25 . substituting to eq .
( [ p ] ) parameters of ligo system , presented in table  ( [ param ] ) , and assuming full overlapping ( @xmath26 ) , we find that the pi threshold power , @xmath27 , is more than two orders of magnitude smaller if compared with the envisioned power level @xmath28 @xcite . to increase the threshold towards the desirable value we propose to increase @xmath14 to @xmath29 by inducing leakage of the stokes light out of the cavity due to enhanced diffraction of the high order optical modes .
this increase results in a small practically acceptable increase of the attenuation of the fundamental mode @xmath30 .
.parameters of ligo used in calculations [ cols="<,^",options="header " , ]      round trip loss ( red curve ) not exceeds 5 ppm , @xmath31 round trip loss ( blue curve ) is approximately @xmath32 ppm and ratio of these losses ( green curve ) reach a local maximum .
this figure corresponds to the @xmath33 parameters set in table  ( [ table1 ] ) .
, scaledwidth=45.0% ]     and gaussian one.,scaledwidth=45.0% ]    the tolerance requirements to approach the parameters described in table  ( [ table1 ] ) are reasonable .
for example , for variant 3 in table  [ table1 ] the parameters @xmath34 , @xmath35 , and @xmath36 have to hold with accuracy about @xmath37 , @xmath38 , and @xmath38 respectively to keep the value of the loss within 3  db of the predicted values .
it means that shape of the mirrors has to be manufactured within tolerance @xmath39 , which is practically feasible .
another important factor is related to the dynamic stability of the modified fp cavity , which is discussed in the next section .
the optimization of the mirrors shape results in reduction of the radiative loss of the higher order fp modes . as the consequence ,
the resonator sensitivity to the tilt of the mirrors increases if compared with the sensitivity of a conventional fp resonator .
the tilt lifts orthogonality and results in linear coupling among the optical modes .
the coupling is largest for the axial symmetric and the dipole modes .
it is reasonable to expect that the angle sensitivity of the cavity attenuation is approximately proportional to the square root of the clipping loss value of the dipole mode .
there is no known way of accurate analytical evaluation of the loss increase due to mirror tilt .
moreover , the numerical simulations become rather involved since the tilt breaks the symmetry of the system . to evaluate this effect
we use method of successive approximations that is based on fusion of the both numerical and analytical methods . according to this method , the round trip loss depends on small tilt angle @xmath40 of one of the mirrors of the fp cavity as @xmath41,\\ \nonumber      u_{00,00 } & \equiv    \int |\psi_{00}(x)|^2\ , x^3\ , dx ,      \quad   \int |\psi_{j } ( x)|^2x\ , dx=1\,,\\      u_{j,00 } & \equiv    \int \psi^*_{j } ( x)\psi_{00 } ( x)\ , x^2\ , dx\,.\end{aligned}\ ] ] where @xmath42 is the wave number , @xmath43 and @xmath44 are the calculated numerically forward trip eigenvalue and eigenvector of the unperturbed problem ( no tilt )
, @xmath45 is a permissible angle to characterize tilt stability .
numeric calculations for parameters sets 1,2,3 listed in table  [ table1 ] give the following permissible tilt angles : @xmath46 to figure out if this value is large , we calculate similar number for current ligo interferometer ( table  [ param ] ) and find @xmath47rad .
in other words , the dynamic range of the mirror angle deviation of conventional interferometer is an order of magnitude better than one of the interferometer with modified mirrors .
this is expected as the loss parameter of the first dipole mode is approximately 1,000  ppm ( 10  ppm ) for the single mode ( conventional ) resonator .
the considered here model of a fp resonator is a simplification of the realistic ligo system with one exception .
the last one includes two fp resonators and a recycling mirror that allows increasing the effective finesse of the multi - resonator system .
it is possible to show that the ligo resonator effective loss is proportional to @xmath48 , where @xmath49 and @xmath50 are power transmission coefficients of the input and recycling mirrors , respectively , and @xmath51 is the attenuation per round trip in the fp resonator .
in advanced ligo interferometer @xmath52 and @xmath53 @xcite , so the effective transmission coefficient is about @xmath54  ppm , whereas diffraction loss @xmath55  ppm .
hence , increase of diffraction loss of the fundamental mode by about an order of magnitude ( to @xmath0  ppm ) seems to be acceptable to use squeezing about @xmath56 .
the increase of the attenuation of the dipole mode beyond 100  ppm results in reduction of pi in accordance with eq .
we have shown that one can reduce impact of parametric instability of an interferometric gravity wave detector by optimizing geometrical shape of its mirrors .
the improvement stems from the dependence of the threshold of the instability on the losses of the optical modes involved in the process .
modification of the mirror shape increases the diffraction loss of the higher order optical modes resulting in the instability threshold increase , occurring at the cost of scrutinizing the mirror tilt stability requirements . to explain the effect , we have created a semi - analytical model of the diffraction loss of a fabry - perot cavity having an arbitrary mirror shape and found that engineered increase of diffraction loss of the high order modes with necessity leads to larger sensitivity of the fundamental mode loss to the mirror tilt .
optimizing the ratio of the losses of the cavity modes it is possible to achieve a significant suppression of the opto - mechanic instability and also keep acceptable tolerances of the system implementation .
we validated results of our predictions with numerical simulations .
the authors acknowledge fruitful discussions with william kells .
mikhail v. poplavskiy and sergey p. vyatchanin acknowledge support from the russian foundation for basic research ( grant no .
14 - 02 - 00399a and grant no .
13 - 02 - 92441 in frame of program aspera ) , and national science foundation ( grant no .
phy-130586 ) .
fundamental sensitivity of an optical interferometric gravitational wave detector increases with increase of the optical power which , in turn , limited because of the opto - mechanical parametric instabilities of the interferometer . 
 we propose to optimize geometrical shape of the mirrors of the detector to reduce the diffraction - limited finesse of unessential optical modes of the interferometer resulting in increase of the threshold of the opto - mechanical instabilities and subsequent increase of the measurement sensitivity . 
 utilizing parameters of the ligo interferometer we found that the proposed technique allows constructing a fabry - perot interferometer with round trip diffraction loss of the fundamental mode not exceeding @xmath0  ppm , whereas the loss of the first dipole as well as the other high order modes exceed @xmath1  ppm and @xmath2  ppm , respectively . 
 the optimization comes at the price of tighter tolerances on the mirror tilt stability , but does not result in a significant modification of the optical beam profile and does not require changes in the the gravity detector read - out system . 
 the cavity with proposed mirrors is also stable with respect to the slight modification of the mirror shape .