Frequency characteristic of a uniformly rotating non-dithered laser gyro in exact and polynomial forms

This manuscript is about frequency characteristic of a uniformly rotating laser gyro

Abstract-In some inertial navigation systems of a carousel type, laser gyros operate in regime of uniform rotation (their monoblocks do not dither).For development of such systems and computer simulation of their work, one needs to have the "exact" (in wide range of angular velocities  ) analytical expression for counterpropagating waves beat frequency ) (  beat beat   of uniformly rotating device.This expression may be obtained by solving the well-known system of laser gyro dynamic equations with accuracy to second order in parameters of counterpropagating waves linear coupling.But before use of such non-dithered laser gyros in the named inertial system, their metrological parameters (scale factors and null shifts) must be preliminary calibrated.For synthesis and qualitative analysis of methodical errors of the procedure of such calibration on a quickly rotating platform of a single-axis stand, one needs also to have the approximate analytical expression for counterpropagating waves beat frequency in the form of polynomial  are not complete and, therefore, must be modified.In the paper, the result of such modification is presented.

Fragment of output characteristic of a uniformly rotating non-dithered laser gyro (LG)
Index Terms-ring laser gyroscope, ring gas laser, frequency characteristic.
Frequency characteristic of a uniformly rotating non-dithered laser gyro in exact and polynomial forms realized by means of a constant discharge current with use a symmetric scheme: one cathodetwo anodes [1]- [5].
According to relations (6.45)-(6.47)from [6], the system of equations describing the dynamics of dimensionless intensities ) and phase difference  of counterpropagating waves of such laser gyro (under condition of equal currents in its discharge legs) may be presented in the form In deriving these equations it was taken into account that the electromagnetic wave with 1  j propagates in the direction of the laser gyro rotation.
In system (1): j  ,  ,  ,  ,  are the Lamb coefficients which characterize the properties of the active medium; is the laser gyro scale multiplier which is determined mainly by its geometrical component ( L -axis contour perimeter, A -area enclosed by axis contour) but which takes into account also the properties of the active medium ( [7]) by means of very small parameter a K ( a K < 0, a K <<< 1);  is the angular velocity with which the laser gyro rotates in the inertial space; In some inertial navigation systems of a carousel type (see paragraphs 3.7.5 and 3.8.7 in [5]), laser gyros operate in regime of uniform rotation (their monoblocks do not dither).For development of such systems and computer simulation of their work, one needs to have the "exact" (in wide range of  ) analytical expression for counterpropagating waves beat frequency beat  of uniformly rotating device.Such expression may be obtained as a result of solving system (1) with accuracy to second order in parameters j r of Manuscript received September 12, 2021.The author is with the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine (e-mail: ea_bndrk@ukr.net).
counterpropagating waves linear coupling.If relation is known, then the number of information pulses N  , accumulated on the laser gyro output during time t  , can be found from the differential equation where dt dN is the pulse repetition rate on the gyro output, and f k is the "frequency multiplication coefficient" (as a rule, ).In the literature, equation ( 2) is called "frequency characteristic of a uniformly rotating laser gyro".
But before use of such non-dithered laser gyros in the named inertial system, their metrological parameters (scale factors and null shifts) must be preliminary calibrated (see item b of point 12.9.3.1 in [8]).For synthesis and qualitative analysis of methodical errors of the procedure of such calibration on a quickly rotating platform of a single-axis stand, one needs also to have the approximate analytical expression for counterpropagating waves beat frequency in asymptotic limit of high values of  .This relation may be taken in the form of polynomial The first term in the right-hand side of ( 4) is well-known (see, for example, [1], [6], [9], [10]).The second term is known from [10] (see formulas (6.31), (6.32), (6.7) therein).And the third term is known from [11] (see expression (23) therein).
As one can see, the second and the third terms in (4) describe only the reversible (with respect to  ) components of beat  .There are not the nonreversible ones in (4).NOTE:  In the literature, in works [11], [12], [13], in addition to (4), there are correspondingly three qualitatively different versions of expressions for the nonreversible components of beat  .It is important to note that these relations are obtained with accuracy to the fourth order in parameters j r .Their common feature is that they are proportional to the combination    ) of counterpropagating waves amplification due to nonreciprocal resonator losses.This more important factor will manifest itself already to second order in parameters j r .


In (4), the following notations are used:   M  is the laser gyro counterpropagating waves frequencies splitting caused by its rotation in the inertial space with angular velocity  (in ideal device is the sum of arguments of complex integral coefficients } exp{ j j r  of counterpropagating waves linear coupling; is the small dimensionless parameter which characterizes the degree of inequality of counterpropagating waves amplification caused by difference of the laser gyro resonator losses.In ideal device, the losses for both waves are equal, and 0 is the small dimensionless parameter which characterizes the degree of laser gyro resonator frequency detuning from the center of the active medium emission line.This detuning is caused by a small (with respect to 0  ) systematic error L  of the perimeter control extremum system (its periodical search steps are not considered).In ideal case of accurate laser gyro resonator frequency tuning to the line center, 0  L , 0     , and is the so-called "null shift" of the laser gyro frequency characteristic.In other words, it is the counterpropagating waves frequencies splitting (even when 0   ) caused by a multiplicative interaction of the factor of unequal waves amplification and the factor of resonator frequency detuning.If quantity 0  is known, then the laser gyro null shift 0  , which has dimension of angular velocity, may be calculated as which is confirmed theoretically and experimentally in [14]. Known expressions for coefficients of polynomial P beat  As analysis of the literature shows, the known relations for

K and
) 4 ( K .In the literature, there are not expressions for these coefficients calculated to second order in parameters j r .So the corresponding relations for these quantities must be found.NOTE:  In the literature ( [11], [12], [13]), there are qualitatively different expressions for these coefficients but calculated to the fourth order in parameters j r .

B. Principal goals of this paper
As analysis of the literature shows, the known exact expression (4) for counterpropagating waves beat frequency


According to the author's report [18], the modified exact expression for beat  may be written in the form Expression ( 9) is derived on the base of system (1) with the help of the procedure which is generalization (for the case 0  T ) of the method developed earlier by the author in [19] for the case 0  T .The first, the second, and the third terms in must be much less than unity.In modern devices, operating with sufficiently high level of pumping (see paragraph 3.3.2 in [5]), the named condition, as a rule, is satisfied.NOTE:  For the laser gyro (with a four-mirror square resonator) operating total pressures the Ne He  mixture from 1 to 5-6 Torr, a set of engineer formulas for calculation of the parameters ) of system (1) is proposed in [20].A set of relations for estimating the parameters  may be obtained on the base of relation (9)   for beat  with the help of the following approximate formulas (which are valid for  Taking into account   M  , after substituting (11) into (9) and collecting the corresponding terms, we obtain ( [23]): As one can see, expressions ( 12)-( 14) and ( 5)- (7) for coefficients  16), (17) exact expression for beat  .As analysis of the literature shows, the known expression for beat  and the known relations for some of coefficients of polynomial P beat , and transmission of radiation on the mirrors.
of analysis of the literature Known exact expression for beat  As analysis of the literature shows, the known expression for beat  (calculated to second order in parameters j r of counterpropagating waves linear coupling) has the form which describes the influence of the factor of asymmetry ( 2 1 r r  ) of counterpropagating A waves linear coupling.As it will be shown below, such factor is less significant than another onethe factor of inequality ( 2 1


which has dimension of angular frequency and characterizes the halfwidth of synchronization zone of the laser gyro counterpropagating waves frequencies in approximation 0 , which has dimension of angular velocity, may be calculated in such approximation as


counterpropagating waves amplification due to nonreciprocal resonator losses.Moreover, the sought for (to second order in parameters are still unknown.So the principal goals of this paper are: 1) to propose the modified exact expression for beat 

( 9 )
confirm the known expression (4) for beat  .But the fourth and the fifth terms in (9) are substantially new.They describe two nonreversible with respect to  components of beat  caused by the factor of inequality of counterpropagating waves amplification due to nonreciprocal resonator losses.Expression (9) may be used in the range [ is valid, if the condition of weakness of counterpropagating waves linear coupling is fulfilled.It implies that for all given possible values of laser gyro total discharge current, the values of ratios . Formulas for simulating the dynamics of the parameters j r , j  during the device operation in the self-heating regime are proposed in[22].III.MODIFIED EXPRESSIONS FOR COEFFICIENTSOF POLYNOMIAL