## Abstract

By utilizing the generalized ray matrix for spherical mirror reflection, two new sensitivity factors are introduced considering the perturbation of the optical-axis caused by the radial and axial displacements of a spherical mirror in a nonplanar ring resonator. Based on this, a novel way for finding the location of the singular points of the sensitivity factors is presented. It is found that some nonplanar ring resonators with the effective modes may have the singular points of the sensitivity factors. The unsuitable regions for nonplanar ring resonators are also obtained from the perspective of the sensitivity factors.

©2011 Optical Society of America

## 1. Introduction

Nonplanar ring resonators have shown the rapid progress for applications such as ring laser gyroscopes and laser-Compton X-ray source [1–5]. For the practical application of the ring laser gyroscopes, besides the variations of the beam path geometry caused by environmental influences (temperature, mechanical stress, and so on), the manufacturing tolerance of the resonator block and the inaccuracies during the assembling procedure should also be considered, because they may influent the beam path position and the lock-in threshold [6]. Therefore, the study on the optical-axis perturbation resulting from the mirror misalignment is important for the cavity design, improvement and alignment of nonplanar ring resonators [7]. The misalignment and the optical-axis stability of the nonplanar rings have been analyzed by some researchers [8–10]. Sheng and Yuan *et al*. [7, 11] gave the definition of the sensitivity factors regarding the effect of misalignment angles of the spherical mirror on the movement of optical-axis and also analyzed the optical-axis perturbation singularity in nonplanar ring resonators in which case the optical-axis movements diverge. In this paper, we introduce two new sensitivity factors with which the effects of the spherical mirror’s radial and axial displacements on the movement of the optical-axis can be taken into consideration and then the generalized sensitivity factors can be formed. Based on this, we find that some resonators with effective modes may have the singular points of the sensitivity factors, named as unsuitable regions. And the unsuitable regions of the nonplanar ring resonators can be also obtained from the perspective of sensitivity factors.

## 2. Analysis method and generalized sensitivity factors

As shown in Fig. 1
, we take the four-equal-sided nonplanar ring resonator as an example and assume that the resonator contains four segments. Each of the segments has a free-space propagation *L _{j}* (

*j*= 1, 2, 3, 4), a reflection on one spherical mirror with radius of curvature

*R*(infinite for the plane mirror) and incident angle

_{j}*A*, and a coordinate rotation angle

_{j}*φ*.

_{j}Firstly, we analyze the round-trip propagation matrix including the perturbation to any of the optical elements. Following the definition in Refs [12–14], we define the matrix for translation along the optical-axis by *L _{j}* as T(

*L*) and

_{j}*R*(

*φ*), respectively. The generalized ray matrix for spherical mirror reflection can be written as [15]

_{j}*θ*and

_{jx}*θ*are the misalignment angles of the spherical mirror P

_{jy}*in its local tangential and sagittal planes, respectively;*

_{j}*δ*and

_{jx}*δ*are the radial displacements, and

_{jy}*δ*is the axial displacement [16][REMOVED HYPERLINK FIELD]. The ray matrix for each segment of the resonator is defined as

_{jz}Then the round-trip matrix can be written as

The 5 × 5 matrix *M* operates on a five-component ray vector *V* given by

*r*,

_{x}*r*are the optical-axis heights from the reference axis along the

_{y}*x*,

*y*axes, and

*r*

^{'}

*,*

_{x}*r*

^{'}

*are the angles between the optical-axis and the reference axis in the*

_{y}*x*,

*y*planes, respectively. On the basis that the optical-axis has to reappear itself in the ring resonator,

*V*can be obtained by solving the following equation

Matrix Eq. (5) can be solved by producing a new set of linear equations as following

We rename the two sensitivity factors SD and ST defined in Ref [7]. as SD1 and ST1, respectively. And they have the forms

To make all the sensitivity factors dimensionless, the definition of ST1 here is a little different from Ref [7]. where a coefficient 1*/L* (*L* is the total length of the resonator) before *∂r*'*/∂θ* is contained. SD1 and ST1 present the axis decentration and tilt sensitivity of the optical-axis on the *i*th mirror as the result of the misalignment angles, respectively.

Considering that each of the five independent perturbation sources from the misaligned *j*th mirror (*θ _{jx}*,

*θ*,

_{jy}*δ*,

_{jx}*δ*,

_{jy}*δ*) possibly exist, it is needed to analyze the effect of each source has on the movement of the optical-axis. That is, the effects of the radial and axial displacements of the

_{jz}*j*th mirror on the movement of the mirror optical-axis in a nonplanar ring resonator should be considered by introducing two new sensitivity factors defined as follows

*j*th mirror, respectively. SD1, ST1, SD2 and ST2 constitute the generalized sensitivity factors with which one can take all the perturbation sources of a mirror into consideration. Considering that SD1 and ST1 have been studied in Refs [7, 11], we focus on the sensitivity factors SD2 and ST2 in the following.

Assuming that P_{1}, P_{2} are spherical mirrors with the identical radius of curvature *R*, while P_{3}, P_{4} are planar mirrors (Fig. 1), the numerical results presented here are for the case that the incident angles on all four mirrors are identical [11], i.e. *A*
_{1}=*A*
_{2}=*A*
_{3}=*A*
_{4}=*A*. The beam travels along the direction P_{1}→P_{2}→P_{3}→P_{4}→P_{1} and the perturbation originates from the misalignment of mirror P_{1}. Figure 2
shows SD2 and ST2 versus *L/R* with *A* = 43.87° (corresponding image rotation angle *ρ*=90°).

It can be seen that both the SD2 and ST2 characterizing the movement of the optical-axis on mirror P_{1} have two common singular points at *L/R*=3.98 and *L/R*=7.26, where the optical-axis movements diverge. The values of the sensitivity factors approach infinite at these singular points and the absolute value of the sensitivity factors increase sharply when *L/R* approaches the singular points. We have also studied the behavior of the optical-axis on other mirrors and found that the SD2 and ST2 have the same singular points as discussed above. Similarly, beginning from the definition of the sensitivity factors, we can identify that the singular points locate at *L/R=*4.51 and *L/R=*6.54 when *A*=40.06° (*ρ*=180°), while the singular points locate at *L/R=*5.07 and *L/R=*5.73 when *A=*31.74° (*ρ*=270°).

Defining the 4×4 matrix on the left side of Eq. (6) as *M'*, we can obtain the determinant of *M'* (det *M'*) versus *L/R* as shown in Fig. 3
. Comparing with the singular points mentioned above, it can be seen that the location of the singular points overlap with the zero value point of det *M'*. As we know that det *M'* can be expressed by *L/R* and *A*, so we can solve the equation of det *M'*=0 and *L/R* can be expressed in terms of *A* (here the expression is named as *f*(*A*) and *f*(*A*) is the function of the zero points in det *M'*). By expressing the sensitivity factors in terms of *L/R* and *A*, it is found that the left (right) limit of the sensitivity factors approach plus (minus) or minus (plus) infinity when *L/R* are close to *f*(*A*). So that the zero points in det *M'* is just the singular points of the sensitivity factors. It is an easier way to find the accurate location of the singular points by calculating the determinant of *M'* rather than the traditional way.

## 3. Analysis of the generalized sensitivity factors

Bearing in mind that a stable nonplanar ring resonator (of which the beam size is finite) has effective modes, we obtain the stability map of the resonator as shown in Fig. 4 by using the matrix method of generalized Gaussian beams [17][REMOVED HYPERLINK FIELD].

It has been pointed out [7] that the resonator does not have effective modes under the singular points when the image rotation angles *ρ* are 270°, 90° or 0°. However, the cases for other image rotation angles haven’t been studied yet. We have numerically calculated all the singular points of SD1, ST1, SD2 and ST2 when *ρ* varies from 0° to 360° (0°<*A*<45°). As we know that there is only one singular point for some sensitivity factors in a given resonator while there are two singular points for other sensitivity factors in the same resonator [7], we ignore the details and show the locations of all the singular points in Fig. 4.

The track of the singular points almost overlap with the right side of the 2nd stable area when *A*<10°. However, it split into two branches with the augment of *A*. Although the branch II spreads to the unstable area, the branch I crosses the 2nd stable area when 5.095<*L/R*<5.335, which means the corresponding resonators with effective modes have the singular points of the sensitivity factors.

As we know that the optical-axis perturbation should be suppressed at a certain level for a ring laser gyroscope designed with the certain output accuracy. We must consider not only the singular points but also other points with large value of sensitivity factors, with which the resonators can be regarded as an unsuitable ones. It can be seen from Fig. 2 that these points locate in the vicinity of the singular points analyzed above. Figure 5 shows the quantitatively defined unsuitable regions of a nonplanar ring resonator, where the sensitivity factors of the resonator are larger than 100, the unsuitable regions can be obtained (Fig. 5). The movement of the optical-axis responds more violently to small misalignment of the mirrors in the unsuitable regions than in other regions, and one must avoid adopting the parameters in the unsuitable regions for designing the resonators.

## 4. Conclusion

We have presented the definition of the sensitivity factors SD2 and ST2 for the first time as we know considering the radial and axial displacement of the spherical mirror. All the four parameters SD1, ST1, SD2 and ST2 form the generalized sensitivity factors and the influence of both the radial and axial misalignments have been analyzed. Besides, we have found that the zero points in det *M'* is just the singular points of the sensitivity factors, so we can find the singular points by analyzing the determinant of the coefficient matrix of the linear equations. It provides an easier way to find the accurate location of the singular points. By calculating all the singular points with *A* ranging from 0° to 45°, it can be found that some nonplanar ring resonators with effective modes may suffer the singular points of sensitivity factors. An example of unsuitable areas regarding the large sensitivity factors is also presented. The analysis in this paper could be helpful to avoid the violent movement of the optical-axis to small misalignment of the mirrors in nonplanar ring resonators.

## Acknowledgements

This work is supported by the Science Foundation of Aeronautics of China under Grants No 20090853014.

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