1. Introduction

Beam position (transverse sections of the beam) needs to be controlled in some applications such as nanometric optical tweezers, optical trapping systems, ring laser gyroscopes and other 3D optical systems [1–6]. There have been some kinds of planar ring resonators which are widely used for laser gyroscopes [6–8]. Non-planar ring resonators (NPRO) are also widely used for high precision ring laser gyroscopes including Zero-Lock Laser Gyroscopes [6,9–15]. The ray matrix technique is a fast way to gain an understanding of how a ray of light propagates through a series of optics [8], and augmented ray matrix method has been widely used for optical-axis perturbation analyzing in planar or non-planar ring resonators [8–20]. However, the perturbation sources in most of the previous articles are angular misalignments or translational movements of the mirrors in the ring resonator. As far as non-planar ring resonators, Faraday-wedge is a typical optical-wedge in NPRO [15]. The detailed generalized ray matrix of paraxial optical-wedge transmission has not been analyzed before. Optical-wedge induced optical-axis perturbation and the beam position controlling method to overcome Optical-wedge induced optical-axis perturbation have not been analyzed either. A theoretical beam position controlling method for 3D optical system will be proposed here and NPRO (a typical 3D optical system) will be chosen to show its application.

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]. As shown in Fig. 1(a) of Ref. [9], the only difference between TCS and NCS is that the coordinate systems of the incident beam and reflected beam are all right-handed coordinates in TCS. However, those systems in NCS cannot be all right-handed coordinates simultaneously. The problem existing in TCS has been pointed out [9]. There exist problems in many analytical results of several other related articles because those analyses are based on TCS [13,14,21]. Those related articles have been listed and discussed in detail in Ref. [9] [11,13,14,20–25]. A simple 3D optical system and related coordinate system are shown in Fig. 1. The coordinate systems and related coordinate rotation are based on NCS. As shown in Fig. 1, for every optical reflecting element (such as m₂ and m₃), 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). In Fig. 1, when a beam propagates from m₂ to m₃, the coordinate rotation (with the angle of φ₃) of the coordinate transformation is needed. Coordinate rotation and beam rotation based on NCS have been discussed in detail in Ref. [9]. When a ray propagates through such a 3D optical system, the output ray and the input ray has the following relationship

 

Fig. 1 Schematic diagram of the beam position controlling method for a simple 3D optical system. P_j(j = 1,2,3,4): the reflection points, n_j(j = 2,3): the binormals at points P_j(j = 2,3), m_j(j = 2,3): reflecting mirrors, F_j(j = 2,3,4): facular (transverse section) of the incident beam before being reflected from points P_j(j = 2,3,4), (x_j, y_j, z_j)(j = 2,3): coordinate systems for the incident beam before being reflected from points P_j(j = 2,3), (x_jr, y_jr, z_jr)(j = 2,3): coordinate systems for the reflected beam after being reflected from points P_j(j = 2,3), φ₃: coordinate rotation angle. (Note: The initial nonideal optical-axes and the ideal optical-axes after special perturbations are represented by black solid lines and red dashed lines respectively; reflecting mirror’s positions after axial displacements are illustrated with red dashed lines; the positive directions of y_j and y_jr(j = 2,3) are along the directions of n_j(j = 2,3); the positive directions of z_j and z_jr(j = 2,3) are along the direction of beam propagation; (x₂, x_2r) and (x₃, x_3r) are located at the incident planes of P₁P₂P₃ and P₂P₃P₄ respectively.)

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By utilizing augmented ray matrix method, the affection of perturbation sources on the optical-axis perturbation can be analyzed and it has been explained in some articles [8, 10, 16, 18–20]. On the other hand, by choosing the perturbation sources with special values, the beam position (transverse section of the beam) can be controlled to an ideal position as we want. A simple example of beam position controlling is shown in Fig. 1. The initial nonideal optical-axes can be transferred into the ideal optical-axes after reflecting mirrors m₂ and m₃ are given special perturbations of axial displacements. This controlling method is called novel theoretical beam position controlling method for 3D optical system. To the best of our knowledge, this novel theoretical beam position controlling method which is based on augmented ray matrix method is proposed for the first time in this paper.

The first key point of the theoretical beam position controlling method is to utilize NCS because traditional ABCD ray matrices are consistent with NCS. When TCS is utilized, incorrect position of beam will be obtained in the numerical analyses because traditional ABCD ray matrix is inconsistent with TCS [9]. The second key point of the theoretical beam position controlling method is to find out the augmented ray matrix of various kinds of optical elements by utilizing NCS. The generalized ray matrix of spherical mirror reflection (typical example of reflecting optical component) has been obtained based on NCS in Ref. [10] and the validity of NCS has been approved by related optical-axis experiments [9,10]. However, the generalized augmented ray matrices for transmission from an optical component or a cascaded series of optical elements have not been analyzed before.

Non-planar ring resonators (NPRO) is a typical 3D optical system and NPRO will be chosen as an example to show the application of this novel theoretical beam position controlling method in this paper. When a ray propagates inside a resonator, the output ray and the input ray has the following relationship

2. Generalized ray matrices of paraxial dielectric interface transmission and paraxial optical-wedge transmission

A general optical component can be expressed by an augmented 5 × 5 ray matrix with the consideration of perturbation sources such as angular misalignment and translational displacements. The output ray and the input ray has the following relationship

A dielectric interface I_i with normal incidence has been chosen as an example to show the perturbation sources in Fig. 2 . Generally the dielectric interface I_i has three kinds of translational displacements along the axes of T_ix, T_iy and T_iz, and also has three kinds of angular misalignments around the axes of R_ix, R_iy and R_iz.

 

Fig. 2 Angular misalignments of a paraxial dielectric interface I_i with normal incidence. (a) definition of the interface’s misalignment angle θ_ix and (b) angular misalignment of the dielectric interface around rotational axis R_ix. I_i0: the blue solid line which is the initial position of I_i, I_i1: the red dot line which is the position of I_i after θ_ix>0, I_i2: the red solid line which is the position of I_i after θ_ix<0, P_i0: the incident point, x, y and z: the coordinate axes of the incident ray and refractive rays, T_ix, T_iy and T_iz: three translational axes, R_ix, R_iy and R_iz: three rotational axes, θ_ix, θ_iy, θ_iz: angular misalignments around R_ix, R_iy and R_iz separately, L1_i: incident ray, L1_o0 and L1_o1: transmission rays refracted from I_i0 and I_i1, θ_iix: incident angle, θ_oix: refracted angle, n₁: refractive index before transmission, n₂: refractive index after transmission, △θ_oix: the angle between L1_o1 and z axis. (Note: z axis is parallel to the normal direction of dielectric interface; the positive direction of T_ix, T_iy and T_iz are along the directions of x, y and z separately; the positive direction of R_ix, R_iy and R_iz are along the directions of y, -x and z separately.)

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For the paraxial dielectric interface transmission, three kinds of translational displacements can be ignored. The rotational movement θ_iz around R_iz can also be ignored because the dielectric interface has a spherical symmetry. In summary, θ_ix and θ_iy are two rotational movements around R_ix and R_iy respectively, and they are the two kinds of possible perturbation sources for a dielectric interface transmission. θ_ix, θ_iy, and θ_iz are called angular misalignments in this paper.

As shown in Fig. 2(a), the angular misalignment θ_ix has been chosen as an example to show the definitions of θ_ix, θ_iy and θ_iz. Look at the dielectric interface in front of the rotation axis R_ix, when the dielectric interface rotates clockwise with respect to its rotation axis R_ix, the induced misalignment angle of θ_ix is negative and θ_ix<0. When the dielectric interface rotates counterclockwise with respect to its rotation axis R_ix, the induced misalignment angle of θ_ix is positive and θ_ix>0. The misalignment angles of θ_iy and θ_iz are defined similarly.

The angular misalignment θ_ix will be analyzed first. As shown in Fig. 2(b), firstly the decentration terms representing the radial displacement along the x and y axes are analyzed. Without the angular misalignment θ_ix, the refraction ray L1_o0 and the incident ray L1_i do not have any changes. After the angular misalignment θ_ix, the refraction ray has been changed from L1_o0 to L1_o1. The coordinates of the point P_i0 are both $(0,0)$ in the coordinate axes of incidental ray and refraction ray. The augmented 5 × 5 ray matrix of the dielectric interface I_i is represented by M_I_i in this paper, the decentration term which is represented by standard ray-matrix element M_I_i(1, 5) should not be modified. The incidental angle θ_iix is

According to Eq. (4), and considering that the angular misalignment θ_ix is very small, the relationship between the refraction angle θ_oix and the incidental angle θ_iix can be written as

The exit angle of L1_o1 is

In summary, the generalized ray matrix M_I_i for a dielectric interface transmission with all kinds of possible perturbation sources including θ_ix and θ_iy (θ_iz can be ignored) can be written as:

Optical-wedge is a common optical element which is widely used in various resonators. Ray matrix of paraxial optical-wedge transmission will be discussed here. An optical-wedge with wedge angle θ is shown in Fig. 3 .

 

Fig. 3 Schematic diagram of optical-wedge transmission with the consideration of wedge angle θ and angular misalignments θ_x, θ_y and θ_z = 0. I₁, I₂: the two dielectric interfaces of the optical-wedge, θ: wedge angle of the optical-wedge (it is the angle between I₁ and I₂), l: length of the optical-wedge, x, y and z: the coordinate axes of the incident ray and refractive rays, R_x, R_y and R_z: three rotational axes, θ_x, θ_y, θ_z: three kinds of angular misalignments around the axes of R_x, R_y and R_z respectively, I₂₀₀: the black dashed line which is the virtual position of the I₂ without the consideration of θ>0, I₁₀, I₂₀: the blue solid lines which are the initial position of I₁ and I₂ with the consideration of θ>0, I₁₁, I₂₁: the two red dot lines which are the positions of I₁ and I₂ with the consideration of θ_x>0 and θ>0, P₁, P₂: the incident points on I₁ and I₂, L1_i0, L2_i0: incident rays, L1_o0, L1_o1: transmission rays refracted from I₁₀ and I₁₁, L2_o00, L2_o0 and L2_o1: transmission rays refracted from I₂₀₀, I₂₀ and I₂₁, θ_i1x, θ_i2x: incident angles, θ_o1x, θ_o2x: refracted angles, n₁: refractive index of atmosphere. n₂: refractive index of the optical-wedge. (Note: z axis is parallel to the normal direction of dielectric interface I₁; the positive direction of R_x, R_y and R_z are along the directions of y, -x and z separately.)

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The optical-wedge is composed of two dielectric interfaces I₁, I₂ and a free space with length l. θ_x, θ_y and θ_z are three kinds of angular misalignments around the axes of R_x, R_y and R_z separately. The definitions of θ_x, θ_y and θ_z are similar as the definition of θ_ix in Fig. 2(a). The position of the dielectric interface I₁ has been changed from the initial position of I₁₀ to I₁₁ after an angular misalignment θ_x. The virtual dielectric interface I₂₀₀ is parallel to I₁₀ and the angle between I₂₀₀ and I₂₀ is θ. Wedge angle θ can be treated as a constant angular misalignment around the axis of R_x. The position of the dielectric interface I₂ has been changed from I₂₀₀ to I₂₁ after an angular misalignment θ_x + θ. The incidental angles θ_i1x and θ_i2x are

The affections of θ_y can be analyzed similarly. The ray matrix of the optical-wedge is composed of the ray matrices of the two dielectric interfaces I₁, I₂ and a free space with length $l$. The augmented 5 × 5 ray matrices of the dielectric interface I₁ and I₂ are represented by M_I₁ and M_I₂ in this paper. The ray matrices of M_I₁ and M_I₂ can be written as

M_l is the augmented 5 × 5 ray matrix of the free space with length l and it can be written as

The ray matrix of the optical-wedge is M_w and it is composed of the ray matrices of M_I₁, M_I₂ and M_l(Eq. (9), Eq. (11) and Eq. (12)). M_w can be written as:

3. Analysis of non-planar ring resonators

Non-planar ring resonators (NPRO) are widely used in multioscillator ring laser gyroscopes. As shown in Fig. 4 , a most suitable non-planar resonator which is widely used to create multioscillator ring laser gyroscopes has been chosen as an example. The resonator has coordinate rotation of 90°. The Faraday-wedge is a typical optical-wedge and the wedge angle is θ. The optical-axis locations x_e, y_e, x_g and y_g are the optical-axis deviations from the longitudinal axis of the ideal diaphragm and the center of the longest discharge capillary along the x and y axes respectively, and the center of the longest discharge capillary is also the center of the gain medium. The positive orientation of x_e, y_e, x_g and y_g are shown in Fig. 4. For a high accuracy laser gyro, in order to make the total diffraction loss lowest and to improve the performance, it would be much better to make the optical-axis pass through the centers of diaphragm (point P_e) and discharge capillary (point P_g) simultaneously.

 

Fig. 4 Coordinate systems and corresponding coordinate rotations based on novel coordinate system for Gaussian beam reflection (NCS) in four equal-sided non-planar ring resonators (NPRO) with a Faraday-wedge, β: folding angle, m_a and m_b: spherical mirrors with common radius R, m_c and m_d: planar mirrors, A_i: the incident angles on all four mirrors, FW: Faraday-wedge with wedge angle of θ, L_w: length of the Faraday -wedge, L_j(j = 1,2,3,4): four sides of the cavity, P_j(j = a,b,c,d): terminal points of the resonator, P_e: the center of the diaphragm, P_g: the center of the discharge capillary, P_f: the midpoint between P_b and P_c, P_h: the midpoint between P_a and P_d, O₁, O₂: the midpoints of straight lines P_bP_d and P_aP_c respectively, n_j(j = a,b,c,d): the binormals at points P_j(j = a,b,c,d), (x_j, y_j, z_j)(j = e,f,g,h): coordinate systems for the incident beam before being reflected from points P_j(j = b,c,d,a), (x_jr, y_jr, z_jr)(j = e,f,g,h): coordinate systems for the reflected beam after being reflected from points P_j(j = b,c,d,a), (x, y, z): coordinate system for the incident ray and refractive rays of FW, R_x, R_y and R_z: three rotational axes of FW, θ_x, θ_y, θ_z: three kinds of angular misalignments around the axes of R_x, R_y and R_z respectively, φ_j(j = e,f,g,h): coordinate rotation angles based on NCS, δ_jz(j = a,b,c,d): axial displacement of mirrors m_i(i = a,b,c,d), δ_jx,δ_jy(j = a,b): radial displacements of the spherical mirrors m_a and m_b. (Note: The cavity length of all four sides are equal and the total cavity length is L; the positive directions of y_j and y_jr(j = e,f,g,h) are along the directions of n_j(j = b,c,d,a); the positive directions of z_j and z_jr(j = 1,2,3,4,b,c) are along the direction of beam propagation; (x_e, x_er), (x_f, x_fr), (x_g, x_gr) and (x_h, x_hr) are located at the incident planes of P_aP_bP_c, P_bP_cP_d, P_cP_dP_a and P_dP_aP_b separately; the positive directions of x, y and z are parallel to the directions of x_e, y_e and z_e; the positive direction of R_x, R_y and R_z are along the directions of y_e, -x_e and z_e separately; the positive directions of δ_ax, δ_bx, δ_az, δ_bz, δ_cz, and δ_dz are along the directions of straight lines P_bP_d, P_aP_c, P_aO₁, P_bO₂, P_cO₁ and P_dO₂ respectively; the positive direction of δ_jy (j = a,b) is along the direction of n_j(j = a,b).)

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In order to analyze the optical-axis perturbation in NPRO, all the possible perturbation sources need to be considered. The Faraday-wedge’s wedge angle (θ) and three kinds of angular misalignments (θ_x, θ_y and θ_z) which are defined in Fig. 3 need to be considered. Planar mirror’s radial displacements can be ignored because planar mirrors m_c and m_d have a radius of ∞. The perturbation sources of δ_iz(i = a,b,c,d) and δ_ix, δ_iy(i = a,b) should be considered too. The positive orientations of δ_iz(i = a,b,c,d) and δ_ix, δ_iy(i = a,b) are shown in Fig. 4 and these orientations are their translational axes respectively. The definitions of δ_iz(i = a,b,c,d) and δ_ix, δ_iy(i = a,b) are similar to the definitions in Fig. 1 of Ref [10]. The generalized 5 × 5 ray matrix for a mirror M_i with perturbation sources of three kinds of translational displacements including δ_ix, δ_iy and δ_iz is represented by M(m_j). The form of M(m_j) is not independent and it is dependent on the detailed coordinate system, and it has been discussed in detail in Ref. [9]. A positive or negative sign may be added to δ_jz in the standard ray-matrix elements M(m_j)(1, 5) and M(m_j)(2, 5) for different mirrors with the consideration of the detailed coordinate system [12]. The augmented 5 × 5 ray matrix of coordinate rotation has been proposed by some previous articles and textbooks [8, 9, 12].

As shown in Fig. 4, the detailed coordinate systems based on NCS and related coordinate rotations in NPRO have been illustrated. The beam propagates along each leg in the counterclockwise direction as P_e→P_f→P_g→P_h→P_e. A coordinate transformation is needed when the beam propagates from one optical element to another. After free-space propagation of L₁/2, the incident beam is reflected from mirror m_b. The initial coordinate system (x_e, y_e, z_e) has become (x_er, y_er, z_er). One can easily find out that the coordinate system (x_er, y_er, z_er—coordinate system for the reflected beam after being reflected from point P_b) is not consistent with the coordinate system (x_f, y_f, z_f—coordinate system for the incident beam before being reflected from point P_c). Here a coordinate rotation of the coordinate transformation is needed. So, meanwhile, the coordinate system (x_er, y_er, z_er) should be rotated into (x_f, y_f, z_f) with the angle of φ_f<0. In the following propagation, there will sequentially appear the coordinate rotations (with the angle of φ_g, φ_h, and φ_e) of the coordinate transformations. Then the beam is followed by free-space propagation L₂. After being reflected from mirror m_c, the coordinate system (x_f, y_f, z_f) has become (x_fr, y_fr, z_fr) and meanwhile, the coordinate system (x_fr, y_fr, z_fr) should be rotated into (x_g, y_g, z_g) with the angle of φ_g<0. Then the beam is followed by free-space propagation L₃. After being reflected from mirror m_d, the coordinate system (x_g, y_g, z_g) has become (x_gr, y_gr, z_gr) and meanwhile, the coordinate system (x_gr, y_gr, z_gr) should be rotated into (x_h, y_h, z_h) with the angle of φ_h<0. Then the beam is followed by free-space propagation L₄. After being reflected from mirror m_a, the coordinate system (x_h, y_h, z_h) has become (x_hr, y_hr, z_hr) and meanwhile, the coordinate system (x_hr, y_hr, z_hr) should be rotated into (x_e, y_e, z_e—the initial coordinate system) with the angle of φ_e<0. Then the beam is followed by free-space propagation L₁/2-L_w and Faraday-wedge transmission. Finally the beam comes back to P_e—the initial position. The coordinate rotation angles have the following relationship

If the initial position is point P_e, The round trip matrix M_E can be written as

△x_e, △y_e, △x_g and △y_g are the optical-axes perturbations at the center of the diaphragm (point P_e) and the center of the discharge capillary (point P_g) along the x and y axes respectively. △x_e, △y_e, △x_g and △y_g caused by the perturbation sources of θ, θ_x, θ_y, θ_z, δ_iz(i = a,b,c,d) and δ_ix, δ_iy(i = a,b) can be obtained by substituting Eq. (15) or Eq. (16) into Eq. (3) and solving Eq. (3).

The following typical parameters were used for calculation in the following numerical analysis by referring to Ref. [15]: This NPRO has 90° total coordinate rotation and φ_e = φ_f = φ_g = φ_h = −22.5°. All four incident angles are equal and A_i = 43.866°, L = 0.36m, and R = 4m. The material of Faraday-wedge is made of flint glass and refractive index of the Faraday-wedge is 1.83957. L_W = 2mm and θ ranges from −10 arc minutes to 10 arc minutes. θ_x and θ_y ranges from −1000 arc minutes to 1000 arc minutes which are caused by the mounting errors of the Faraday-wedge (ranges from about −15° to 15°). θ_z has been chosen as 0° or 90°. When θ_z = 0°, the optical-wedge is horizontally installed in NPRO. When θ_z = 90°, the optical-wedge is vertically installed in NPRO. δ_iz(i = a,b,c,d) ranges from −1 mm to 1 mm. δ_ax, δ_ay, δ_bx and δ_by ranges from −1 mm to 1 mm which is limited by the finite coating area of the spherical mirrors and the finite size of the terminal surfaces where the spherical mirrors are mounted.

A passive ring cavity alignment experiment has been utilized to confirm the rules of the optical-axis perturbation in NPRO. The passive ring cavity alignment experiment will not be further discussed in this paper because it is similar to the CCD-area based alignment experimental facility which has been described in detail in Ref. [10]. The only difference is that the passive ring cavity used here has been changed from square ring resonators to NPRO [10]. The rules of optical-axis perturbation in NPRO have been obtained and have been confirmed by passive ring cavity alignment experiment. The numerical analysis results and the experimental results are shown in Fig. 5 , Fig. 6 , Fig. 7 , Fig. 8 , Fig. 9 , Fig. 10 and Fig. 11 .

 

Fig. 5 Wedge angle-induced optical-axis perturbations in NPRO with a horizontally installed Faraday-wedge. (a) schematic diagram of experimental results on optical-axis perturbation (Note: The definitions of the symbols in Fig. 5(a) are similar to their definitions in Fig. 4; the ideal optical-axes under the condition of θ = 0 and the real optical axes with the consideration of a non-zero wedge angle are represented by blue solid lines and red dot lines respectively.), (b) optical-axis decentrations △x_e, △y_e, △x_g and △y_g versus wedge angle θ.

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Fig. 6 Wedge angle-induced optical-axis perturbation in NPRO with a vertically installed Faraday-wedge. (a) schematic diagram of experimental results on optical-axis perturbation (Note: The definitions of the symbols in Fig. 6(a) are similar to their definitions in Fig. 4; the ideal optical-axes under the condition of θ = 0 and the real optical axes with the consideration of a non-zero wedge angle are represented by blue solid lines and red dot lines respectively.), (b) optical-axis decentrations △x_e, △y_e, △x_g and △y_g versus wedge angle θ.

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Fig. 7 Optical-axis decentrations induced by Faraday-wedge’s angular misalignments in NPRO. Optical-axis decentrations △x_e, △y_e, △x_g and △y_g (a) versus θ_x with θ_y = 0, (b) versus θ_y with θ_x = 0.

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Fig. 8 Optical-axis decentrations induced by spherical mirror’s radial displacements in NPRO. Optical-axis decentrations △x_e, △y_e, △x_g and △y_g (a) versus δ_ax with δ_ay = δ_bx = δ_by = 0, (b) verus δ_ay with δ_ax = δ_bx = δ_by = 0, (c) versus δ_bx with δ_ax = δ_ay = δ_by = 0, (d) versus δ_by with δ_ax = δ_ay = δ_bx = 0.

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Fig. 9 Optical-axis decentrations induced by mirror’s axial displacements in NPRO. Optical-axis decentrations △x_e, △y_e, △x_g and △y_g (a) versus δ_az with δ_bz = δ_cz = δ_dz = 0, (b) versus δ_bz with δ_az = δ_cz = δ_dz = 0, (c) versus δ_cz with δ_az = δ_bz = δ_dz = 0, (d) versus δ_dz with δ_az = δ_bz = δ_cz = 0.

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Fig. 10 Schematic diagram of the experimental beam position controlling method for the elimination of the wedge angle-induced mismatching errors in NPRO with a horizontally installed Faraday-wedge (θ_z = 0°), (a) optical-axis perturbation of NPRO during the mismatching error eliminating process by utilizing all 4 mirror’s axial displacements (Note: The definitions of the symbols in Fig. 10(a) are similar to their definitions in Fig. 4; the optical-axes with the wedge angle-induced mismatching error and the optical-axes after the beam position controlling process are represented by blue solid line and red dot line respectively; all 4 mirror’s positions after axial displacements are illustrated with red solid circles.), (b) optical-axis decentrations in NPRO: △x_e, △y_e, △x_g and △y_g versus δ_az = δ_bz = δ_cz = δ_dz.

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Fig. 11 Schematic diagram of the experimental beam position controlling method for the elimination of the wedge angle-induced mismatching errors in NPRO with a vertically installed Faraday-wedge (θ_z = 90°), (a) optical-axis perturbation of NPRO during the mismatching error eliminating process by utilizing m_a and m_c’ axial displacements (Note: The definitions of the symbols in Fig. 11(a) are similar to their definitions in Fig. 4; the optical-axes with the wedge angle-induced mismatching error and the optical-axes after the beam position controlling process are represented by blue solid line and red dot line respectively; m_a and m_c’s positions after axial displacements are illustrated with red solid circles.), (b) optical-axis decentrations in NPRO: △x_e, △y_e, △x_g and △y_g versus δ_az = δ_cz.

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The perturbation sources of θ_x, θ_y, θ_z, δ_iz(i = a,b,c,d) and δ_ix, δ_iy(i = a,b) are chosen to be 0 in Fig. 5. θ is the only perturbation source. As shown in Fig. 5(a), wedge angle-induced optical-axis perturbations in NPRO with a horizontally installed Faraday-wedge are shown with diagrammatic representation. The absolute values of wedge angle-induced optical-axis decentrations △x_e and △x_g are bigger than absolute values of △y_e and △y_g. When θ is chosen as a typical value of 6′, the optical-axis decentrations are:

Wedge angle-induced optical-axis perturbations in NPRO with a vertically installed Faraday-wedge are shown in Fig. 6. The absolute values of wedge angle-induced optical-axis decentrations △y_e and △y_g are bigger than absolute values of △x_e and △x_g. When θ is chosen as a typical value of 6 arc minutes, the optical-axis decentrations are:

When θ is a typical value of 6 arc minutes, no matter whether the Faraday-wedge is horizontally installed (θ_z = 0) or vertically installed (θ_z = 90°), the optical-axis decentrations at point P_e exceed the area of diaphragm and this is called wedge angle-induced diaphragm mismatching error. The optical-axis decentrations at points P_g reach the edge area of discharge capillary and this is called wedge angle-induced discharge capillary mismatching error. Both of the above mentioned two kinds of mismatching errors are called wedge angle-induced mismatching errors of NPRO in this paper and they are illustrated in Fig. 5(a) and Fig. 6(a) respectively.

Optical-axis decentrations induced by Faraday-wedge’s angular misalignments in NPRO are shown in Fig. 7. The perturbation sources of θ, θ_z, δ_iz(i = a,b,c,d) and δ_ix, δ_iy(i = a,b) are chosen to be 0. θ_x and θ_y are the only perturbation sources in Fig. 7(a) and Fig. 7(b) respectively. As shown in Fig. 7(a), $\left|\Delta x_g\right|$ is much smaller than $\left|\Delta x_e\right|$, $\left|\Delta y_e\right|$ and $\left|\Delta y_g\right|$ induced by θ_x. When θ_x is chosen as a typical maximum value of 5°, the optical-axis decentrations are:

Optical-axis decentrations induced by spherical mirror’s radial displacements in NPRO are shown in Fig. 8. The perturbation sources of θ, θ_x, θ_y, θ_z and δ_iz(i = a,b,c,d) are chosen to be 0. δ_ax, δ_ay, δ_bx and δ_by are the only perturbation sources in Fig. 8(a), 8(b), 8(c) and 8(d) respectively. No matter whether the perturbation source is δ_ax, δ_ay, δ_bx or δ_by, there exist non-zero △x_e, △y_e, △x_g and △y_g as the result of the coordinate rotation effect in NPRO.

From Fig. 5, Fig. 6, Fig. 7, Eq. (17), Eq. (18), Eq. (19) and Eq. (20), one can easily find out that optical-axis decentrations induced by Faraday-wedge’s angular misalignments is much smaller than wedge angle-induced mismatching errors, and wedge angle-induced mismatching errors is the major errors and need to be eliminated in NPRO.

From Fig. 5, Fig. 6, Fig. 8, Eq. (17) and Eq. (18), one can easily find out that even if the spherical mirrors radial displacements δ_ax, δ_ay, δ_bx and δ_by are given typical maximum values of −1mm or 1 mm, the maximum wedge angle-induced mismatching error △x_e = 0.3271mm (shown in Eq. (17)) or △y_e = 0.2943mm (shown in Eq. (18)) cannot be eliminated, so the typical wedge angle-induced mismatching errors in NPRO which are shown in Eq. (17) or Eq. (18) cannot be eliminated effectively. From Fig. 7 and Fig. 8, one can easily find out that optical-axis decentrations induced by Faraday-wedge’s angular misalignments (θ_x or θ_y) can be eliminated effectively by controlling the spherical mirrors radial displacements.

Optical-axis decentrations induced by all 4 mirror’s axial displacements in NPRO are shown in Fig. 9. The perturbation sources of θ, θ_x, θ_y, θ_z and δ_ix, δ_iy(i = a,b) are chosen to be 0. δ_az, δ_bz, δ_cz and δ_dz are the only perturbation sources in Fig. 9(a), 9(b), 9(c) and 9(d) respectively. No matter whether the perturbation source is δ_az, δ_bz, δ_cz or δ_dz, there exist non-zero $\Delta x_e$, $\Delta y_e$, $\Delta x_g$ and $\Delta y_g$ as the result of the coordinate rotation effect in NPRO. For the perturbation source of δ_az, δ_bz, δ_cz or δ_dz, $\left|\Delta x_e\right|$ and $\left|\Delta x_g\right|$ are smaller than $\left|\Delta y_e\right|$ and $\left|\Delta y_g\right|$.

From Fig. 5, Fig. 6, Fig. 9, Eq. (17) and Eq. (18), one can easily find out that if only one mirror’s axial displacement δ_az, δ_bz, δ_cz or δ_dz is used, the typical wedge angle-induced mismatching errors in NPRO (shown in Eq. (17) or Eq. (18)) cannot be eliminated effectively. For example, if δ_az is used for the elimination of the wedge angle-induced mismatching errors, as shown in Fig. 9(a), a negative δ_az is required for the elimination of the wedge angle-induced positive △x_e in Fig. 6 and Eq. (16), unfortunately, a positive △x_g will be induced by the negative δ_az and it will be added to the wedge angle-induced positive △x_g in Eq. (17).

As shown in Fig. 10, one can easily find out that if all 4 mirror’s axial displacements δ_az, δ_bz, δ_cz and δ_dz are used simultaneously, and the typical wedge angle-induced mismatching errors in NPRO with a horizontally installed Faraday-wedge (θ_z = 0°) (shown in Fig. (5)) can be eliminated effectively. For example, when the Faraday-wedge is horizontally installed and the wedge angle is 6′ (the wedge angle-induced mismatching errors are shown in Eq. (17)), if δ_az = δ_bz = δ_cz = δ_dz = −0.47mm is used for the elimination of the wedge angle-induced mismatching errors, the optical-axis decentrations can be written as:

As shown in Fig. 11, one can easily find out that if m_a and m_b’s axial displacements δ_az and δ_cz are used simultaneously, the typical wedge angle-induced mismatching errors in NPRO with a vertically installed Faraday-wedge (shown in Fig. (6)) can be eliminated effectively. For example, if the Faraday-wedge is vertically installed and the wedge angle is 6′ (the wedge angle-induced mismatching errors are shown in Eq. (18)), if δ_az = δ_cz = −0.14mm is used for the elimination of the wedge angle-induced mismatching errors, the optical-axis decentrations can be written as:

Obviously the experimental beam position controlling method (controlling all 4 mirror’s axial displacements simultaneously) is effective to eliminate the wedge angle-induced mismatching errors in NPRO with a horizontally installed Faraday-wedge (θ_z = 0°). The experimental beam position controlling method (controlling m_a and m_c’s axial displacements simultaneously) is effective to eliminate the wedge angle-induced mismatching errors in NPRO with a vertically installed Faraday-wedge. From this point of view, the Faraday-wedge would be better to be vertically installed in NPRO.

It is worthwhile to note that the mirrors axial displacements cannot be modified during alignment process, even if the path length control device is added to the mirrors after the alignment process, the mirror’s modifying range is still limited. So, the eliminating process will be accomplished by controlling the allowances of the terminal faces of the ring cavity block and δ_iz(i = a,b,c,d) are modified during machined process. The mismatching errors shown in Eq. (17) or Eq. (18) will be reduced effectively in alignment process. It is also worthwhile to note that the parameters used in the above mentioned numerical analysis can be modified as the request of application. The purpose of the numerical analysis example is to describe the rules of the optical-axis perturbation and the eliminating method intuitively.

4. Conclusion

In summary, a novel theoretical beam position controlling method for 3D optical system has been proposed in this paper. Non-planar ring resonators (NPRO) which is a typical 3D optical system has been chosen as an example to show the application of this novel theoretical beam position controlling method.

To the best of our knowledge, the generalized ray matrix, an augmented 5 × 5 ray matrix for paraxial dielectric interface transmission with 2 kinds of angular misalignments has been proposed based on NCS for the first time in this paper. The detailed coordinate system for deducing the ray matrix and the detailed deducing process have been proposed too. The generalized ray matrix of paraxial optical-wedge transmission which includes the perturbation sources of all kinds of angular misalignments and the wedge angle has been proposed too. This is the first time for the perturbations sources of wedge angle and three kinds of angular misalignments to be accurately considered.

By utilizing the novel coordinate system for Gaussian beam reflection and the generalized ray matrix of paraxial optical-wedge transmission, the rules and some novel results of the optical-axis perturbations of NPRO have been obtained. Wedge angle-induced optical-axis decentrations has been found to be much larger than optical-axis decentrations induced by the Faraday-wedge’s angular misalignments. Optical-axis decentrations induced by spherical mirror’s radial displacements and all 4 mirrors’s axial displacements in NPRO have been obtained. Wedge angle-induced diaphragm mismatching errors (wedge angle-induced diaphragm mismatching error and discharge capillary mismatching error) of NPRO have been found out. It has been found that typical wedge angle-induced mismatching errors cannot be effectively eliminated by spherical mirror’s radial displacements or any one of the mirrors’ axial displacement. All those results have been confirmed by related alignment experiments and the experimental results have been described with diagrammatic representation.

An experimental beam position controlling method (controlling all 4 mirror’s axial displacements simultaneously) is effective to eliminate the wedge angle-induced mismatching errors in NPRO with a horizontally installed Faraday-wedge, while another experimental beam position controlling method (controlling m_a and m_b’s axial displacements simultaneously) is effective to eliminate the wedge angle-induced mismatching errors in NPRO with a vertically installed Faraday-wedge. By utilizing these methods, wedge angle-induced diaphragm mismatching error and wedge angle-induced discharge capillary mismatching error in NPRO can be reduced effectively. That is to say, the optical-axes of NPRO can be made to pass through the center of the diaphragm and the center of the discharge capillary simultaneously.

These interesting findings of NPRO are important to the beam control, cavity design, and cavity alignment of high precision non-planar ring laser gyroscopes. The generalized ray matrices and their deducing methods are valuable for ray analysis of various kinds of paraxial optical-elements and resonators. This novel theoretical beam position controlling method for 3D optical system is valuable for the controlling of various kinds of 3D optical systems.