## Abstract

By utilizing the novel coordinate system for Gaussian beam reflection and the generalized ray matrix for spherical mirror reflection, the generalized sensitivity factors SD1, ST1, SD2 and ST2 influenced by both the radial and axial displacements of a spherical mirror in a nonplanar ring resonator have been obtained. Besides, the singular points of different kinds of non-planar ring resonators under the conditions of incident angle *A* ranging from 0° to 45° or total coordinate rotation angle *ρ* ranging from 0°to 360°have also been obtained through the analysis of the determinant of the coefficient matrix of the linear equations. The analysis in this paper is important to the cavity design of non-planar ring resonators and it could be helpful to avoid the violent movement of the optical-axis to small misalignment of the mirrors in non-planar ring resonators.

©2013 Optical Society of America

## 1. Introduction

There have been some kinds of planar ring resonators which are widely used for laser gyroscopes [1–3]. Non-planar ring resonators (NPRO) are also widely used for high precision ring laser gyroscopes including Zero-Lock Laser Gyroscopes [1–9]. The ray matrix technique is a fast way to gain an understanding of how a ray of light propagates through a series of optics [3], and augmented ray matrix method has been widely used for optical-axis perturbation analyzing in planar or non-planar ring resonators [3–20].

The coordinate system of Gaussian beam reflection is important because it is the bridge between the theoretical analysis and experimental research. For example, one can find out the perturbation direction of optical-axis in optical-axis analysis by referring to the detailed coordinate system. Traditional ray matrices should be based on suitable coordinate systems and the ray matrices should be consistent with related coordinate systems. In another word, before deducing the ray matrices, the suitable coordinate system should be established. The traditional ABCD ray matrix of Gaussian beam reflection is not consistent with traditional coordinate system for Gaussian beam reflection (TCS) because incorrect position of beam will be obtained in the numerical analyses by utilizing TCS. To solve this inconsistency, novel coordinate system for Gaussian beam reflection (NCS) has been proposed in Ref. [9] and NCS is consistent with traditional ABCD ray matrix of Gaussian beam reflection [9]. The generalized ray matrix of spherical mirror reflection in Ref. [10] is obtained based on NCS. By utilizing NCS, the optical-axis perturbation rules of square ring resonators have been obtained and the validity of NCS has been approved by related optical-axis experiments [9,10]. There exist problems in many analytical results of several other related articles because those analyses are based on TCS. Those related articles have been listed and discussed in detail in ref 0.9 [4–8,17–20].

The numerical analysis results of sensitivity factors for optical-axis perturbation in NPRO have been obtained in Refs. [6–8], and those results are incorrect because those analysis are based on TCS [6–8]. By utilizing NCS, generalized sensitivity factors influenced by both the radial and axial displacements of a spherical mirror in NPRO have been obtained in this article. Besides, the singular points of different kinds of NPRO under the conditions of incident angle *A* ranging from 0° to 45° or total coordinate rotation angle *ρ* ranging from 0°to 360°have also been obtained. These analyses are important to the cavity design of NPRO and it could be helpful to avoid the violent movement of the optical-axis to small misalignment of the mirrors in NPRO.

## 2. Analysis method

As shown in Fig. 1, let’s take the four-equal-sided non-planar ring resonator (NPRO) as an example and assume that the resonator contains four segments. Each of the segments has a free-space propagation *Lj* (*j* = 1, 2, 3, 4), a reflection on one spherical mirror *m _{j}* with radius of curvature

*R*(infinite for the plane mirror), an incident angle

_{j}*Aj*and a coordinate rotation angle

*φj*.

*β*is the folded angle.

The matrix for spherical mirror reflection and matrix for such a single segment as shown in Fig. 1:

*θ*and

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

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

*δjx*and

*δjy*are the radial displacements, and

*δjz*is the axial displacement.

*T(L*and

_{j})*R(φ*represent the matrices for free space propagation and coordinate rotation respectively.

_{j})*M(R _{j}, A_{j})* is dependent on the detailed coordinate system and

*φ*should be illustrated in the detailed coordinate system. The form of

_{j}*M(R*shown in Eq. (1) is not independent and it is dependent on the detailed coordinate system. A positive or negative sign may be added to

_{j}, A_{j})*δjz*in the standard ray-matrix elements

*M(R*(1, 5) and

_{j},A_{j})*M(R*(2, 5) for different mirrors with the consideration of the detailed coordinate system [10]. It would be better for

_{j},A_{j})*M(R*to be replaced by

_{j},A_{j})*M(m*which represents the matrix of mirror

_{j})*m*. and then Eq. (2) became

_{j}*φ*and the

_{j}*R(φ*used in Refs. [7] and [8] are incorrect and

_{j})*M(m*of some mirrors are incorrect too. Therefore, the results of Refs. [7] and [8] are incorrect.

_{j})NPRO for numerical analysis is shown in Fig. 2 with detailed coordinate axes, where *m _{1}*,

*m*are spherical mirrors with the identical radius of curvature

_{2}*R*. The incident angles on all four mirrors are identical [7,8], i.e.

*A*=

_{1}*A*=

_{2}*A*=

_{3}*A*=

_{4}*A*. The cavity lengths of all four sides are equal and the total cavity length is

*L*.

The positive orientations of *δ _{jz}(j = 1,2,3,4)*,

*δ*and

_{jx}(j = 1,2)*δ*are shown in Fig. 2 and these orientations are their translational axes respectively. The definitions of

_{jy}(j = 1,2)*δ*,

_{jz}(j = 1,2,3,4)*δ*and

_{jx}(j = 1,2)*δ*are similar to those in Fig. 1 of Ref. [10]. The sign of

_{jy}(j = 1,2)*δ*in

_{jz}(j = 1,2,3,4)*M(m*is dependent on the detailed coordinate system and a reflection on mirrors

_{j})*m*,

_{1}*m*,

_{2}*m*and

_{3}*m*can be written as [10]

_{4}Coordinate rotation matrix can be found in textbook on lasers [3] and *R(φ _{j})* is

As shown in Fig. 2, the detailed coordinate systems (based on TCS and NCS) and related coordinate rotations in NPRO have been illustrated. For every optical reflecting element (such as *m _{2}* and

*m*), its coordinate systems are composed of the coordinate system of incident beam and the coordinate system of reflected beam. That is to say, every optical reflecting element has two coordinate systems for the incident beam and the reflected beam respectively, and it is not to say that there exists only one coordinate system on every side of the beam path. When a beam propagates from one optical element to another, a coordinate transformation is needed to make the coordinate system of the reflected beam (after being reflected from previous optical element) consistent with the coordinate system of the incident beam (before being reflected from the next optical element).

_{3}The beam propagates along each leg in the clockwise direction as *P _{1}*→

*P*→

_{4}*P*→

_{3}*P*→

_{2}*P*. A coordinate transformation is needed when the beam propagates from one optical element to another. Before the incident beam is reflected from mirror

_{1}*m*, the initial coordinate systems

_{1}*(x*based on TCS and

_{t2r}, y_{2r}, z_{2r})*(x*based on NCS should be rotated into

_{2r}, y_{2r}, z_{2r})*(x*with the angle of

_{t1}, y_{1}, z_{1})*φ*<0 and into

_{t1}*(x*with the angle of

_{1}, y_{1}, z_{1})*φ*<0 respectively, and it is the first coordinate rotation. Then the beam is reflected from

_{1}*m*and meanwhile, the coordinate systems have become

_{1}*(x*and

_{t1r}, y_{1r}, z_{1r})*(x*which are based on TCS and NCS separately. The beam is followed by free-space propagation

_{1r}, y_{1r}, z_{1r})*L*. Before the incident beam is reflected from mirror

_{1}*m*, the coordinate systems

_{4}*(x*based on TCS and

_{t1r}, y_{1r}, z_{1r})*(x*based on NCS should be rotated into

_{1r}, y_{1r}, z_{1r})*(x*with the angle of

_{t4}, y_{4}, z_{4})*φ*>0 and into

_{t4}*(x*with the angle of

_{4}, y_{4}, z_{4})*φ*<0 respectively, and it is the second coordinate rotation. Then the beam is reflected from

_{4}*m*and meanwhile, the coordinate systems have become

_{4}*(x*based on TCS and

_{t4r}, y_{4r}, z_{4r})*(x*based on NCS separately. The beam is followed by free-space propagation

_{4r}, y_{4r}, z_{4r})*L*. Before the incident beam is reflected from mirror

_{4}*m*, the coordinate systems

_{3}*(x*based on TCS and

_{t4r}, y_{4r}, z_{4r})*(x*based on NCS should be rotated into

_{4r}, y_{4r}, z_{4r})*(x*with the angle of

_{t3}, y_{3}, z_{3})*φ*<0 and into

_{t3}*(x*with the angle of

_{3}, y_{3}, z_{3})*φ*<0 respectively, and it is the third coordinate rotation. Then the beam is reflected from

_{3}*m*and meanwhile, the coordinate systems have become

_{3}*(x*based on TCS and

_{t3r}, y_{3r}, z_{3r})*(x*based on NCS separately. The beam is followed by free-space propagation

_{3r}, y_{3r}, z_{3r})*L*. Finally, before the incident beam is reflected from mirror

_{3}*m*, the coordinate systems

_{2}*(x*based on TCS and

_{t3r}, y_{3r}, z_{3r})*(x*based on NCS should be rotated into

_{3r}, y_{3r}, z_{3r})*(x*with the angle of

_{t2}, y_{2}, z_{2})*φ*>0 and into

_{t2}*(x*with the angle of

_{2}, y_{2}, z_{2})*φ*<0 respectively, and it is the fourth coordinate rotation. Then the beam is reflected from

_{2}*m*and meanwhile, the coordinate systems have become

_{2}*(x*—the initial coordinate system based on TCS

_{t2r}, y_{2r}, z_{2r}*)*and

*(x*—the initial coordinate system based on NCS

_{2r}, y_{2r}, z_{2r}*)*separately, and then the beam is followed by free-space propagation

*L*. The coordinate rotation angles have the following relationships

_{2}*φ*is the absolute value of

*φ*. The problem existed in utilzing TCS has been pointed out in Ref. [9]. Coordinate rotation of

_{j}(j = 1,2,3,4)*φ*and Eq. (9) is incorrect because it is based on incorrect TCS.

_{t j}(j = 1,2,3,4)The relationship between *A* and *φ* is

The relation between incident angle *A* and the coordinate rotation angle *φ* is shown in Eq. (11) and it is illustrated in Fig. 3. It is easy to find out that the larger the coordinate rotation angle *φ*, the smaller the incident angle *A*. By utilizing Eq. (11), the corresponding total image rotation angle *ρ* varies from 0° to 360° and the incident angle varies from 45° to 0° when the coordinate rotation angle varies from 0° to 90°.

It would be better that image rotation in Ref. [5] is called beam rotation and its value is equal to the value of total coordinate rotation angle *ρ*. *ρ* can be written as

The initial position is point *P _{1}* and the round trip matrix can be written as

The resonator optical-axis that is invariant under the round-trip propagation coincides with the eigenvector of M with eigenvalue 1:

_{x}and r

_{y}are the ray heights from the reference axis along the x and y axes respectively and they are called optical-axis decentration in this article. r

_{x}′ and r

_{y}′ are the angles that the ray make with the reference axes in the x and y plane(which are vertical to axis

*y*and axis

*x*separately) respectively and they are called optical-axis tilt. The impact of the perturbation sources

*δ*,

_{jz}(j = 1,2,3,4)*δ*and

_{jx}(j = 1,2)*δ*on optical-axis perturbation can be obtained by solving Eq. (14).

_{jy}(j = 1,2)## 3. Analysis results of the generalized sensitivity factors

With reference to the definitions of SD1, ST1, SD2 and ST2 in Ref. [8], Fig. 4 and Fig. 5 shows the results of (SD1, ST1) and (SD2, ST2) respectively (versus *L/R* with *A* = 43.866° (*ρ* = 90°). *φ _{1}* =

*φ*=

_{2}*φ*= 22.5° and

_{3}= φ_{4}= φ*ρ*=

*φ*+

_{1}*φ*= 90°). It can be seen that both the SD1 and ST1 caused by the angular misalignments of mirror

_{2}+ φ_{3}+ φ_{4}*m*have four common singular points at

_{1}*L/R*= 0.403,

*L/R*= 0.774,

*L/R*= 3.447 and

*L/R*= 6.619, where the optical-axis movements diverge. SD2 and ST2 caused by the translational displacements of mirror

*m*have the same four common singular points as mentioned above. The values of the sensitivity factors approach infinite at these singular points and the absolute value of the sensitivity factors increase sharply when

_{1}*L/R*approaches the singular points. SD1, ST1, SD2 and ST2 caused by the misalignments of other mirrors have also been studied and it is found that they have the same singular points as discussed above. Similarly, it has been identified that the singular points are located at

*L/R*= 1.075,

*L/R*= 1.531,

*L/R*= 3.484 and

*L/R*= 4.960, when

*A*= 40.060° (

*ρ*= 180°), while the singular points are located at

*L/R*= 0.543,

*L/R*= 0.556,

*L/R*= 4.799 and

*L/R*= 4.909, when

*A =*31.742° (

*ρ*= 270°). It has also been identified that the singular points are located at

*L/R*= 0,

*L/R*= 3.771 and

*L/R*= 7.543, when

*A*= 45° (

*ρ*= 0°) which is corresponding to a square planar ring resonator.

Defining the 4 × 4 matrix on the left side of Eq. (6) in Ref. [8] as *M'*, we can obtain the determinant of *M'* (det *M'*) versus *L/R* as shown in Fig. 6. Compared 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, det *M'* can be expressed by *L/R* and *A*, therefore, the equation of det *M'* = 0 can be solved 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'*). After the sensitivity factors are expressed in terms of *L/R* and *A*, it is found that the left (right) limit of the sensitivity factors approaches plus (minus) or minus (plus) infinity when *L/R* is close to *f*(*A*), and the zero points in det *M'* are 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.

A stable nonplanar ring resonator (of which the beam size is finite) has effective modes. By utilizing the matrix method of generalized Gaussian beams, we obtain the stability map of the resonator as shown in Fig. 7. The detailed stability map with *ρ* ranging from 0°to 360°is shown in Fig. 7(a) and Fig. 7(b). The detailed stability map with *A* ranging from 0°to 45°is shown in Fig. 7(c) and Fig. 7(d). In region 0, the resonator is stable, with (in general) elliptical isophotes. In region 1, the major axis of the isophotes ellipse is infinite, and in region 2 both axes are infinite. As mentioned above, *L/R* can be expressed in terms of *A* (or *φ*) after the equation of det *M'* = 0 is solved. Then the accurate location of the singular points can be found out in NPRO with different incident angle *A*. The unsuitable tracks of the singular points of a NPRO are shown in Fig. 7(a), Fig. 7(b), Fig. 7(c) and Fig. 7(d) with red marked lines.

## 4. Conclusion

In summary, by utilizing the novel coordinate system for Gaussian beam reflection, the generalized sensitivity factors SD1, ST1, SD2 and ST2 influenced by both the radial and axial displacements of a spherical mirror in a nonplanar ring resonator have been obtained. Besides, the singular points of different kinds of NPRO under the conditions of incident angle *A* ranging from 0° to 45° or total coordinate rotation angle *ρ* ranging from 0°to 360°have also been obtained through the analysis of the determinant of the coefficient matrix of the linear equations. The analysis in this paper is important to the cavity design of NPRO and it could be helpful to avoid the violent movement of the optical-axis to small misalignment of the mirrors in NPRO.

## Acknowledgments

This work was supported by the National Science Foundation of China under grant 61078017.

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