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Optics Express

Optics Express

  • Editor: C. Martijin de Sterke
  • Vol. 19, Iss. 7 — Mar. 28, 2011
  • pp: 6762–6776
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Generalized ray matrix for spherical mirror reflection and its application in square ring resonators and monolithic triaxial ring resonators

Jie Yuan, Xingwu Long, and Meixiong Chen  »View Author Affiliations


Optics Express, Vol. 19, Issue 7, pp. 6762-6776 (2011)
http://dx.doi.org/10.1364/OE.19.006762


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Abstract

To the best of our knowledge, the generalized ray matrix, an augmented 5×5 ray matrix for a spherical mirror reflection with all the possible perturbation sources including three kinds of displacements and its detailed deducing process have been proposed in this paper for the first time. Square ring resonators and monolithic triaxial ring resonators have been chosen as examples to show its application, and some novel results of the optical-axis perturbation have been obtained. A novel method to eliminate the diaphragm mismatching error and the gain capillary mismatching error in monolithic triaxial ring resonators more effectively has also been proposed. Both those results and method have been confirmed by related experiments and the experimental results have been described with diagrammatic representation. This generalized ray matrix is valuable for ray analysis of various kinds of resonators. These results are important for the cavity design, cavity improvement and alignment of high accuracy and super high accuracy ring laser gyroscopes.

© 2011 OSA

1. Introduction

2. Analysis method

The ray matrix of a general optical component with angular misalignment and translational displacements has the form
(roxrox'royroy'1)=(AxBx00ExCxDx00Fx00AyByEy00CyDyFy00001)(rixrix'riyriy'1),
(1)
where rix, riy, rox and roy are the input ray and output ray heights from the reference axis along the x and y axes respectively and we call them optical-axis decentration. rix', riy', rox' and roy' are the angles that the input ray and output ray makes with the reference axis in the x and y plane respectively and we call them optical-axis tilt. Ax, Bx, Cx and Dx are the standard ray-matrix elements in tangential plane, Ay, By, Cy and Dy are the standard ray-matrix elements in sagittal plane, Ex and Ey are the decentration terms which represent radial displacements along x and y axes. Fx and Fy are the tilt terms which represent the angular misalignments.

A spherical mirror Mi with radius Ri has been chosen as an example to show the perturbation sources in Fig. 1
Fig. 1 Translational displacements of a spherical mirror Mi. (a) axial displacement δiz of Mi and (b) radial displacement δix of Mi. Ri: the radius of spherical mirror Mi, Ai: the incident angle, Mi1: the blue solid arc which is the initial position of Mi, Mi2: the red solid arc in (a) which is the position of Mi after axial displacement δiz, Mi3: the red solid arc in (b) which is the position of Mi after radial displacement δix, P1, P2, P3, P4 and P5: the reflecting points, O1 and O2: spherical centers of Mi1 and Mi3, Tix, Tiy and Tiz: three translational axes, Rix, Riy and Riz: three rotational axes, L1i and L2i: two parallel incident rays, L1o1, L1o2, L2o1 and L2o2: four reflection rays, xi and yi: the coordinate axes of the incident ray, xo and yo: the coordinate axes of the reflected ray, θ1: the angle between line O1P1 and line O1P3, θ2: the angle between line O2P4 and line O2P5, Δx1: the distance between P1 and P3, Δx2: the distance between P4 and P5, Δy: the distance between P1 and P4.
and Fig. 2
Fig. 2 Angular misalignments of a spherical mirror Mi. (a) definition of a mirror’s misalignments angle θix and (b) angular misalignment of the spherical mirror Mi. around rotational axis Rix. Ri: the radius of spherical mirror Mi, Ai: the incident angle, Mi4: the red solid line which is the position of Mi after angular misalignment θix>0, Mi5: the red dashed line which is the position of Mi after angular misalignment θix<0, Mi1: the blue solid arc which is the initial position of Mi, Mi6: the red solid arc which is the position of Mi after angular misalignment θix<0, P1: the reflecting points, O1 and O3: spherical centers of Mi1 and Mi6, Rix, Riy and Riz: three rotational axes, L1i: incident ray, L1o1 and L1o3: two reflection rays, xi and yi: the coordinate axes of the incident ray, xo and yo: the coordinate axes of the reflected ray. θ3: the angle between P1O1 and P1O3.
. The incident angle is Ai. As shown in Fig. 1(a), generally the mirror Mi has 3 kinds of translational displacements and 3 kinds of angular misalignments. δix, δiy and δiz are three kinds of translational displacements along the axes of Tix, Tiy and Tiz respectively. θix, θiy and θiz are three kind of angular misalignments around the axes of Rix, Riy and Riz respectively. θiz can be ignored because the mirror has a spherical symmetry.

In summary, θix, θiy, δix, δiy, and δiz are 5 kinds of possible perturbation sources for a spherical mirror. θix and θiy are angular misalignments. In this paper, δix and δiy are called radial displacements, and δiz is called axial displacement.

The axial displacement δiz will be analyzed first. As shown in Fig. 1(a), Mi1 and Mi2 represent the mirror Mi before and after the axial displacement δiz. The reflection point has been changed from point P1 and to point P2. For a linear resonator, the ray is incident vertically and Ai=0. The transversal offset between P1 and P2 is zero and the reflection point has not been changed under the axial displacement δiz. For a ring resonator, the ray is not incident vertically and Ai0. To discuss this effect in detail, two parallel incident rays L1i and L2i are chosen for examples. L1o1 is the reflection ray of L1i via the reflection at the point P1 of Mi1. L2o1 is the reflection ray of L2i via the reflection at the point P3 of Mi1. L2o2 is the reflection ray of L2i via the reflection at the point P2 of Mi2. The coordinate axes of the incident ray are xi and yi, and the coordinate axes of the reflected rays are xo and yo. Firstly the decentration terms which represent the radial displacement along the x and y axes are analyzed. For the incident ray L2i, the coordinates of the point P2 are (δiz×sin(Ai),0) and (δiz×sin(Ai),0) in the coordinate axes of incidental ray and reflection ray respectively, so the standard ray-matrix elements Mi(1, 5) should be modified into 2δiz×sin(Ai). Secondly, the tilt terms which represent the angular misalignments are analyzed. For the reflection ray of the incident ray L2i in the coordinate axis of reflection ray, the exit angle of L2o1 is 2θ12Δx1/R2δiz×tan(Ai)/Ri and the exit angle of L2o2 is 0. This angle is under the direction of x axis and the angle modification under the direction of y axis is 0, so the standard ray-matrix elements Mi(2, 5) should be modified into 2δiz×tan(Ai)/Ri. The case which is shown in Fig. 1 is that the angle between the positive directions of xi and δiz is bigger than 90 degree. If the angle is smaller than 90 degree, the standard ray-matrix elements Mi(1, 5) and Mi(2, 5) should be modified into 2δiz×sin(Ai) and 2δiz×tan(Ai)/Ri respectively.

The spherical mirror’s radial displacements δix and δiy need to be considered too. As shown in Fig. 1(b), radial displacement δix has been chosen as example for analysis. Firstly the decentration terms which represent the radial displacement along the x and y axes are analyzed. For the incident ray L1i, the coordinates of the point P4 are (0,0) and (Δy×sin(Ai),0) in the coordinate axes of incidental ray and reflection ray respectively, so the standard ray-matrix elements Mi(1, 5) should be modified into 2Δy×sin(Ai)2δix2Ri×sin(Ai). This term can be ignored in this paper because δix<<Ri. The exit angle of L1o2 is 2θ22Δx2/Ri2δix×tan(Ai)/Ri and the exit angle of L1o1 is 0. This angle is under the direction of x axis and the angle modification under the direction of y axis is 0, so the standard ray-matrix elements Mi(2, 5) should be modified into 2δix×tan(Ai)/Ri. Similarly, the standard ray-matrix elements Mi(4, 5) should be modified into 2δiy×tan(Ai)/Ri with the consideration of spherical mirror’s radial displacement δiy.

As shown in Fig. 2(a), the angular misalignment θix has been chosen as example to show the definitions of θix, θiy and θiz. We look at the mirror Mi behind the rotation axis Rix. When the mirror rotates clockwise with respect to its rotation axis Rix, the induced misalignment angle of θix is negative and θix<0. When the mirror rotates counterclockwise with respect to its rotation axis Rix, the induced misalignment angle of θix is positive and θix>0. The misalignment angles of θiy and θiz are defined similarly. θiz can be ignored because the mirror has a spherical symmetry.

The angular misalignment θix has been chosen as example for analysis in Fig. 2(b). Firstly the decentration terms which represent the radial displacement along the x and y axes are analyzed. For the incident ray L1i, the coordinates of the point P1 are both (0,0) in the coordinate axes of incidental ray and reflection ray, so the standard ray-matrix elements Mi(1, 5) should not be modified. The exit angle of L1o3 is 2θ3=2θix and the exit angle of L1o1 is 0. This angle is under the direction of x axis and the angle modification under the direction of y axis is 0, so the standard ray-matrix elements Mi(2, 5) should be modified into 2θix. Similarly, the standard ray-matrix elements Mi(4, 5) should be modified into 2θiy with the consideration of mirror’s angular misalignment θiy.

In summary, a generalized ray matrix for a mirror Mi with all kinds of possible perturbation sources including δix, δiy, δiz, θix, and θiy (θiz can be ignored) can be written as:

M(Mi)=[10002δizsin(Ai)2Ri×cos(Ai)1002δiztan(Ai)/Ri+2(θix+δix/Ri)00100002×cos(Ai)Ri12(θiy+δiy/Ri)00001]
(2)

3. Analysis of square ring resonators

SRR has been chosen as an example in this paper. As shown in Fig. 3(a)
Fig. 3 (a) Schematic diagram of square ring resonator and (b) schematic diagram of alignment experiment. Ma and Mb: spherical mirrors, Mc and Md: planar mirrors, the incident angle is 45°; a, b, c and d: terminal points of the resonator, e: the center of the diaphragm, g: the center of the gain capillary, f: the midpoint between b and c, h: the midpoint between a and d, xj,yj(j=e,f,g,h):x and y coordinate axes at points e, f, g and h, δiz(i=a,b,c,d): axial displacement of Mi(i = a,b,c,d), δix,δiy(i=a,b): radial displacements of the spherical mirrors Ma and Mb, HNLP: He-Ne laser with path length control device, RAD1 and RAD2: reflectors with adjusting device, LB: light bulb, CFLAD: CCD area with focusing lens and adjusting device, PCIBS: personal computer with image grabber and image processing software, FI: facular image.
, the optical-axis locations xe, ye, xg and yg are the optical-axis deviations from the longitudinal axis of the ideal diaphragm and the center of the longest gain capillary along the x and y axes respectively, and the center of the longest gain capillary is also the center of the gain medium. The positive orientation of xe, ye, xg and yg are shown in Fig. 3(a). For a high accuracy laser gyro, in order to make the total diffraction loss be lowest and to improve the performance, it would be much better to make the optical-axis pass through the center of the diaphragm (point e) and the center of the gain capillary (point g) simultaneously.

Planar mirror’s radial displacements δix,δiy(i=c,d) can be ignored because planar mirrors Mi(i=c,d) have a radius of ∞. For a ring laser cavity after machining, the angles of the terminal surfaces have been determined. This means that the angular misalignments θix,θiy(i=a,b,c,d) caused by the angles of the terminal surfaces are 0. In summary, the simplified perturbation sources of δiz(i=a,b,c,d)and δix,δiy(i=a,b) should be considered and the perturbation sources of δix,δiy(i=c,d) and θix,θiy(i=a,b,c,d) should not be considered in the following discussion. The positive orientations of δiz(i=a,b,c,d) and δix,δiy(i=a,b)are shown in Fig. 3(a) and these orientations are their translational axes respectively. The definitions of δiz(i=a,b,c,d) and δix,δiy(i=a,b) are similar to the definition in Fig. 1.

The impact of the simplified perturbation sources δiz(i=a,b,c,d)and δix,δiy(i=a,b) on optical-axis perturbation can be obtained by solving for the eigenvector of the total round-trip matrix of the ring resonator. Δxe, Δye, Δxg and Δyg are the optical-axes perturbations at the center of the diaphragm (point e) and the center of the gain capillary (point g) along the x and y axes respectively, and Δxe, Δye, Δxg and Δyg caused by the above mentioned perturbation sources can be written as

Δxe=24(δax+δbx+δaz+δbz),Δye=22(δay+δby);Δxg=24(δax+δbxδazδbz+2δcz+2δdz),Δyg=22(δay+δby).
(3)

Figure 3(b) shows schematic diagram of alignment experiment [17

17. D. Z. Anderson, “Alignment of resonant optical cavities,” Appl. Opt. 23(17), 2944–2949 (1984). [CrossRef] [PubMed]

, 18

18. N. M. Sampas and D. Z. Anderson, “Stabilization of laser beam alignment to an optical resonator by heterodyne detection of off-axis modes,” Appl. Opt. 29(3), 394–403 (1990). [CrossRef] [PubMed]

]. The passive ring cavity is formed by two spherical mirrors (Ma and Mb) and two planar mirrors (Mc and Md). A light bulb (LB) is used to illuminate the diaphragm and the longest gain capillary. He-Ne laser (HNLP) with wavelength λ=0.6328μm is used as laser source. The laser is reflected by two reflectors which are mounted on adjusting device (RAD1 and RAD2), and then incident into the passive ring cavity. CCD area with focusing lens which is also mounted on a adjusting device (CFLAD) is used to pick up the facular image, then the facular image (FI) is captured into the memory of a personal computer with an image grabber (PCIBS), then the image was transformed into a digital image. The center of the facular image (FI) can be obtained by image processing software (PCIBS). The image grabber has a 8 bits analog-to-digital converter, which is the light intensity range where every element of the CCD area can detect and it can be subdivided into 256 parts. The system can sense a very small deviation of the center of the facular image (FI) because the deviation causes a small alteration in the distribution of light intensity which can be sensed by the CCD area and image grabber. Special arithmetic to processing the digital image was previously published [19

19. J. Yuan and X. W. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74(3), 1362–1365 (2003). [CrossRef]

, 20

20. J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76(12), 125106 (2005). [CrossRef]

]. The arithmetic to calculate the center of the facular image here is similar to the arithmetic to calculate the absolute position of the cross reticle image [19

19. J. Yuan and X. W. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74(3), 1362–1365 (2003). [CrossRef]

, 20

20. J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76(12), 125106 (2005). [CrossRef]

].

First step, the light bulb (LB) is switched on and the laser (HNLP) is switched off, the facular image of illuminated diaphragm and illuminated gain capillary can be obtained respectively by adjusting the focusing lens (CFLAD). The centers of the facular image of illuminated diaphragm and illuminated gain capillary can be made both located at the center of the CCD area by adjusting the focusing lens and the adjusting device (CFLAD). The ideal optical-axis of the passive ring cavity passes through the center of the diaphragm and the center of the gain capillary. As a consequence, the optical-axis of the CCD area and focusing lens has been made aligned with the ideal optical-axis of the passive ring cavity.

Second step, spherical mirror Ma is removed from the cavity first, then the light bulb (LB) is switched off and the laser (HNLP) is switched on. The facular image of incidental laser at different locations can be obtained by adjusting the focusing lens (CFLAD). The centers of the facular image of the incidental laser at different locations can be made located at the center of the CCD area by adjusting the focusing lens and the adjusting device of two reflectors (RAD1 and RAD2). As a consequence, the optical-axis of the incidental laser has been made aligned with the optical-axis of the CCD area and focusing lens and the ideal optical-axis of the passive ring cavity too.

Third step, spherical mirror Ma is mounted on the cavity first, then the path length control device of laser (HNLP) is adjusted and the frequency of the input laser is modified, when the input frequency has a resonance with the passive ring cavity, the centers of the beam frequency facular images at point e and g can be obtained by adjusting the focusing lens (CFLAD). The optical-axis of the beam transmitted by the resonator is called the real optical-axis in this paper and it passes through the centers of the beam frequency facular images at point e and g. When the perturbation sources of δiz(i=a,b,c,d)and δix,δiy(i=a,b) is added to the mirrors of passive ring cavity, the real optical axis will modified with respect to the ideal optical-axis of the passive ring cavity accordingly. As a consequence, the rules of optical-axis perturbation have been obtained and the results in Eq. (3) have been confirmed.

The experimental results are shown with diagrammatic representation in Fig. 4
Fig. 4 Schematic diagram of experimental results on optical-axis perturbations in square ring resonator. (a) optical-axis perturbation caused by spherical mirror’s axial displacements δiz(i=a,b), (b) optical-axis perturbation caused by planar mirror’s axial displacements δiz(i=c,d), (c) optical-axis perturbation caused by spherical mirror’s radial displacements δix(i=a,b) and (d) optical-axis perturbations caused by spherical mirror’s axial displacements δiy(i=a,b). The ideal optical-axes and the real optical axes after special perturbations are represented by blue solid lines and red dashed lines respectively, spherical mirror’s positions after axial displacements and radial displacements are illustrated with red solid arcs, and planar mirror’s positions after axial displacements are illustrated with red solid lines.
. Optical-axis perturbations caused by spherical mirror’s axial displacements δiz(i=a,b) and planar mirror’s axial displacements δiz(i=c,d) are illustrated in Fig. 4(a) and Fig. 4(b) respectively. It can be easily found that δiz(i=a,b,c,d) have no contributions to perturbations in sagittal plane such as Δye and Δyg. In addition, the following novel results can be obtained, that spherical mirror’s axial displacements δaz and δbz have contributions on Δxe and Δxg, while at the same time, planar mirror’s axial displacements δcz and have no contributions to Δxe. Optical-axis perturbation in tangential plane and sagittal plane which are caused by radial displacements δix(i=a,b) and δiy(i=a,b) respectively are illustrated in Fig. 4(c) and Fig. 4(d) accordingly.

4. Analysis of monolithic triaxial ring resonators

Based on the above discussion, we can consider the optical-axis perturbation of MTRR which has been proposed in ref [15

15. J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. 47(5), 628–631 (2008). [CrossRef] [PubMed]

]. For a MTRR, all its three planar ring resonators are SRRs which are mutually orthogonal. As shown in Fig. 5
Fig. 5 Schematic diagram of MTRR with all 3 spherical mirror’s radial displacements and all 6 mirror’s axial displacements. M1, M2 and M3: spherical mirrors with radius R, M4, M5 and M6: planar mirrors, Q1, Q2, Q3, Q4, Q5 and Q6: terminal points of the resonator, PA:, PB:, PC, PD, PE and PB: the midpoints of straight lines Q2Q3, Q1Q3, Q1Q2, Q4Q6, Q5Q6 and Q4Q5 respectively, δ1x,δ1y,δ2x,δ2y,δ3x,δ3y: radial displacements of spherical mirrors M1, M2 and M3, δ1z,δ2z,δ3z,δ4z,δ5z,δ6z: axial displacements of mirrors M1, M2, M3, M4, M5 and M6.
, mirrors M1, M2, M3, M4, M5 and M6 are respectively positioned in the center of each cube body face. The cube is machined such that a small diameter bore connects adjacent mirrors. A closed optical cavity is defined between four coplanar mirrors, which are interconnected by bores. There are three mutually orthogonal closed beam paths, each of which is used to detect angular rotation about its normal axis. The planar ring resonator which is defined by the optical cavity between the mirrors M2, M3, M4 and M6 is called cavity I, the resonator defined by M1, M3, M5 and M6 is called cavity II, and the resonator defined by M1, M2, M5 and M4 is called cavity III.

As shown in Fig. 5, δ1x, δ1y, δ2x, δ2y, δ3x and δ3y represent the radial displacements of spherical mirror Mi(i=1,2,3) from time of 0 to t respectively. δ1z, δ2z, δ3z, δ4z, δ5z and δ6z represent the axial displacement of mirror Mi(i=1,2,3,4,5,6) from time of 0 to t respectively. The positive orientations ofxj,yj(j=A,B,C,D,E,F), are similar to the definitions of Fig. 3 and ref [15

15. J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. 47(5), 628–631 (2008). [CrossRef] [PubMed]

]. The optical-axis perturbations at the points PA, PB, PC, PD, PE and PF, caused by all the 3 spherical mirror’s translations displacements and all the 6 mirror’s axial displacements can be written as
ΔxA=24[δ2yδ3x+δ2z+δ3z],ΔyA=22(δ2xδ3y)ΔxB=24[δ3yδ1x+δ1z+δ3z],ΔyB=22(δ3xδ1y),ΔxC=24[δ1yδ2x+δ1z+δ2z],ΔyC=22(δ1xδ2y)
(6)
and

ΔxD=24[δ2yδ3xδ2zδ3z+2δ4z+2δ6z],ΔyD=22(δ2xδ3y)ΔxE=24[δ3yδ1xδ1zδ3z+2δ5z+2δ6z],ΔyE=22(δ3xδ1y)ΔxF=24[δ1yδ2xδ1zδ2z+2δ4z+2δ5z],ΔyF=22(δ1xδ2y)
(7)

Following the expression in ref [15

15. J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. 47(5), 628–631 (2008). [CrossRef] [PubMed]

, 16

16. X. W. Long and J. Yuan, “Method for eliminating mismatching error in monolithic triaxial ring resonators,” Chin. Opt. Lett. 8(12), 1135–1138 (2010). [CrossRef]

], we can obtain the following equations

ΔxA+ΔyA/2+ΔxB+ΔyB/2+ΔxC+ΔyC/2=22(δ1z+δ2z+δ3z)ΔxC+ΔyD/2+ΔxE+ΔyE/2+ΔxF+ΔyF/2=22(2δ4z+2δ5z+2δ6zδ1zδ2zδ3z)
(8)

That is different from the results of optical-axis perturbation caused by the mirror’s angular misalignments where it is 0 in ref [15

15. J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. 47(5), 628–631 (2008). [CrossRef] [PubMed]

]. We use the functions of xj(t),yj(t)(j=A,B,C,D,E,F) to represent the optical-axis locations those 6 points at the time of t. xj(0),yj(0)(j=A,B,C,D,E,F) is used to represent the optical-axis locations at the time of t=0. The optical-axis perturbations during the period of time from 0 to t can be written as:

Δxj(0t)=xj(t)xj(0)Δyj(0t)=yj(t)yj(0),j=A,B,C,D,E,F
(9)

According to Eq. (8),

j=A,B,CΔxj(0t)+Δyj(0t)/2=22(δ1z+δ2z+δ3z)j=D,E,FΔxj(0t)+Δyj(0t)/2=22(2δ4z+2δ5z+2δ6zδ1zδ2zδ3z)
(10)

By using the following definition of mismatching error C and C2, and utilizing Eq. (9) and Eq. (10), we can obtain

C(t)=j=A,B,Cxj(t)+yj(t)/2=j=A,B,Cxj(0)+yj(0)/2+22(δ1z+δ2z+δ3z)=C(0)+22(δ1z+δ2z+δ3z)C2(t)=j=D,E,Fxj(t)+yj(t)/2=j=D,E,Fxj(0)+yj(0)/2+22(2δ4z+2δ5z+2δ6zδ1zδ2zδ3z)=C2(0)+22(2δ4z+2δ5z+2δ6zδ1zδ2zδ3z)
(11)

C(t) is the total spatial displacements between the ideal optical-axes and the real optical-axes of the cavities I, II and III at the locations of their diaphragms. C2(t) is the total spatial displacements between the ideal optical-axes and the real optical-axes of the cavities I, II and III at the locations of their gain capillaries. C(t) is called the diaphragm mismatching error and C2(t) is called the gain capillary mismatching error in this paper. The distances between the optical axis and the center of the diaphragm at any point of PA, PB, PC and their symmetrical point PD, PE and PF at the time of t can be written as DA(t), DB(t), DC(t), DD(t), DE(t) and DF(t) [15

15. J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. 47(5), 628–631 (2008). [CrossRef] [PubMed]

],

Dj(t)=xj(t)2+yj(t)2(j=A,B,C,D,E,F)
(12)

Obviously, whenC(0)0, the three monoaxial ring resonators cannot be aligned to the best condition of DA(0)=DB(0)=DC(0)=DD(0)=DE(0)=DF(0)=0. In order to make the total diffraction loss of the monolithic triaxial ring resonator be lowest, the values of DA(0), DB(0), DC(0), DD(0), DE(0) and DF(0) should be made smallest. That is to say that the diaphragm mismatching error C should be shared equally [15

15. J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. 47(5), 628–631 (2008). [CrossRef] [PubMed]

]. So the best case should be

xA(0)=xD(0)=xB(0)=xE(0)=xC(0)=xF(0)=C/3yA(0)=yD(0)=yB(0)=yE(0)=yC(0)=yF(0)=0DA(0)=DB(0)=DC(0)=DD(0)=DE(0)=DF(0)=|C|3
(13)

Now let us look back to the Eq. (11), the result in ref [15

15. J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. 47(5), 628–631 (2008). [CrossRef] [PubMed]

] is not valid now. The diaphragm mismatching error C(t) and gain capillary mismatching error C2(t) of the MTRR is not invariant and it can be variable because of spherical and planar mirror’s axial displacements. Whatever the mismatch errors C and C2of the MTRR at the time of t=0 is big or small, C and C2 at the time of t can be decreased and even eliminated to 0 by choosing appropriate δ1z, δ2z, δ3z, δ4z, δ5z and δ6z. If the following conditions have been satisfied:
δ1z+δ2z+δ3z=2×C(0)δ4z+δ5z+δ6z=2×C(0)
(14)
And the simplest case is
δ1z=δ2z=δ3z=2×C(0)/3δ4z=δ5z=δ6z=2×C(0)/3,
(15)
then the following ideal condition can be obtained:

C(t)=0,C2(t)=0xA(t)=xB(t)=xC(t)=xD(t)=xE(t)=xF(t)=0yA(t)=yB(t)=yC(t)=yD(t)=yE(t)=yF(t)=0DA(t)=DB(t)=DC(t)=DD(t)=DE(t)=DF(t)=0
(16)

DA(t), DB(t), DC(t), the distances between the optical axis and the center of the diaphragm at any point of PA, PB, PC have been made smallest. At the same time, DD(t), DE(t) and DF(t), the distances between the optical axis and the center of the gain capillary at any point of PD, PE, PF have been made smallest too. The total diffraction loss of the monolithic triaxial ring resonator has been made lowest.

Cavities I, II and III of MTRR are SRRs, so the alignment experimental setup for MTRR is similar to the setup for SRR. Three sets of experimental setup as shown in Fig. 3(b) are used to align the three SRRs of MTRR simultaneously.

As a consequence, the rules of optical-axes perturbations have been obtained and the results in above theoretical analysis have been confirmed. Experimental results on the method for sharing and eliminating the mismatching errors C and C2 in SRR of MTRR is illustrated in Fig. (6)
Fig. 6 Schematic diagram of experimental results on the method for sharing and eliminating the mismatching errors C and C2 in SRR of MTRR. (a) ideal optical-axis of SRR, (b) optical-axis perturbation of SRR during mismatching error sharing process, (c) optical-axis perturbation of SRR during the mismatching error eliminating process by utilizing spherical mirror’s axial displacements δiz(i=a,b), and (d) optical-axis perturbation of SRR during the eliminating process by utilizing both spherical and planar mirror’s axial displacements δiz(i=a,b,c,d). The ideal optical-axis, the optical-axis after the mismatching error sharing process and the optical axis after the mismatching error eliminating process are represented by blue solid line, green solid line and red dashed line respectively, spherical mirror’s positions after axial displacements are illustrated with red solid arc, planar mirror’s positions after axial displacements are illustrated with red solid line.
. Cavity I is SRR of MTRR and it has been chosen as example to describe its optical-axis perturbation during the mismatching error sharing process and mismatching errors eliminating process. The ideal optical-axis of cavity I without any perturbation sources is illustrated in Fig. 6(a). Without any elimination method, every single monoaxial ring resonator of the MTRR have to share the diaphragm mismatching error C in the three specific directions of xA, xB and xC equally described as Eq. (13). Optical-axis perturbation of the cavity I during this sharing process has been illustrated in Fig. 6(b).

It is worthwhile to note that the mirrors axial displacement 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, in our experiment, the diaphragm mismatching error C(0) and gain capillary mismatching error C2(0) of MTRR will be measured first in the first alignment process, then the ring cavity block will be sent back to be machined. The eliminating process will been accomplished by controlling the allowances of the terminal faces of the ring cavity block and δiz(i=1,2,3,4,5,6) are modified as described in Eq. (14, 15). At last, the diaphragm mismatching error C(t) and gain capillary mismatching error C2(t) of MTRR will be reduced to 0 in the second alignment process.

5. Conclusion

In summary, to the best of our knowledge, for the first time in this paper, the generalized ray matrix has been proposed, which is an augmented 5×5 ray matrix for spherical mirror reflection with all the possible perturbation sources including two kinds of angular misalignments and three kinds of translational displacements. A detailed coordinate system for deducing the ray matrix has also been proposed here. SRR and MTRR have been chosen as examples to show its application. Based on the augmented 5×5 ray matrix method by considering all four mirror’s axial displacements and two spherical mirror’s radial displacements in SRR, some novel results of the optical-axis perturbation have been obtained. By applying those results in three SRRs of a MTRR, the diaphragm mismatching error C of the MTRR has been found out that it does not has any relation with the planar mirror’s axial displacement and it has the relation with the spherical mirror’s axial displacement. The gain capillary mismatching error C2 of the MTRR has been defined in this paper. A novel method to eliminate the diaphragm mismatching error C and the gain capillary mismatching error C2 more effectively and simultaneously by controlling the spherical and planar mirror’s axial displacements has been proposed. By utilizing this method, the diaphragm mismatching error C and gain capillary mismatching error C2 can be reduced to 0. That is to say, the optical-axes of all the three monoaxial ring resonators of MTRR can be made to pass through the center of their diaphragms and the center of their gain capillaries simultaneously. All these novel results in SRR, MTRR and the eliminating method for MTRR have been confirmed by our alignment experiment. The results have been described in detail with diagrammatic representation. This generalized ray matrix is valuable for ray analysis of various kinds of resonators. These results are important for the cavity design, cavity improvement and alignment of high accuracy and super high accuracy SRR and MTRR laser gyroscopes.

Acknowledgments

This work was supported by the National Science Foundation of China under grant 61078017 and 60608002.

References and links

1.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–86 (1985). [CrossRef]

2.

M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt. 19(3), 101–115 (1988). [CrossRef]

3.

A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1389–1399 (2000). [CrossRef]

4.

J. A. Arnaud, “Degenerate Optical Cavities. II: Effect of Misalignments,” Appl. Opt. 8(9), 1909–1917 (1969). [CrossRef] [PubMed]

5.

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, “Analysis of misalignment sensitivity of ring laser resonators,” Sov. J. Quantum Electron. 7, 857–859 (1977). [CrossRef]

6.

H. R. Bilger and G. E. Stedman, “Stability of planar ring lasers with mirror misalignment,” Appl. Opt. 26(17), 3710–3716 (1987). [CrossRef] [PubMed]

7.

R. Rodloff, “A laser gyro with optimized resonator geometry,” IEEE J. Quantum Electron. 23(4), 438–445 (1987). [CrossRef]

8.

J. Yuan, X. W. Long, B. Zhang, F. Wang, and H. C. Zhao, “Optical axis perturbation in folded planar ring resonators,” Appl. Opt. 46(25), 6314–6322 (2007). [CrossRef] [PubMed]

9.

I. W. Smith, “Optical resonator axis stability and instability from first principles,” Proc. SPIE 412, 203–206 (1983).

10.

A. L. Levkit and V. M. Ovchinnikov, “Stability of a ring resonator with a nonplane axial contour,” J. Appl. Spectrosc. (USSR) 40(6), 657–660 (1984). [CrossRef]

11.

S.-C. Sheng, “Optical-axis perturbation singularity in an out-of-plane ring resonator,” Opt. Lett. 19(10), 683–685 (1994). [CrossRef] [PubMed]

12.

A. H. Paxton and W. P. Latham Jr., “Unstable resonators with 90° beam rotation,” Appl. Opt. 25(17), 2939–2946 (1986). [CrossRef] [PubMed]

13.

B. E. Currie, G. E. Stedman, and R. W. Dunn, “Laser stability and beam steering in a nonregular polygonal cavity,” Appl. Opt. 41(9), 1689–1697 (2002). [CrossRef] [PubMed]

14.

J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun. 281(5), 1204–1210 (2008). [CrossRef]

15.

J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. 47(5), 628–631 (2008). [CrossRef] [PubMed]

16.

X. W. Long and J. Yuan, “Method for eliminating mismatching error in monolithic triaxial ring resonators,” Chin. Opt. Lett. 8(12), 1135–1138 (2010). [CrossRef]

17.

D. Z. Anderson, “Alignment of resonant optical cavities,” Appl. Opt. 23(17), 2944–2949 (1984). [CrossRef] [PubMed]

18.

N. M. Sampas and D. Z. Anderson, “Stabilization of laser beam alignment to an optical resonator by heterodyne detection of off-axis modes,” Appl. Opt. 29(3), 394–403 (1990). [CrossRef] [PubMed]

19.

J. Yuan and X. W. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74(3), 1362–1365 (2003). [CrossRef]

20.

J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76(12), 125106 (2005). [CrossRef]

OCIS Codes
(140.3370) Lasers and laser optics : Laser gyroscopes
(140.3410) Lasers and laser optics : Laser resonators
(140.3560) Lasers and laser optics : Lasers, ring
(140.4780) Lasers and laser optics : Optical resonators

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: January 26, 2011
Revised Manuscript: March 13, 2011
Manuscript Accepted: March 15, 2011
Published: March 24, 2011

Citation
Jie Yuan, Xingwu Long, and Meixiong Chen, "Generalized ray matrix for spherical mirror reflection and its application in square ring resonators and monolithic triaxial ring resonators," Opt. Express 19, 6762-6776 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6762


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References

  1. W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–86 (1985). [CrossRef]
  2. M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt. 19(3), 101–115 (1988). [CrossRef]
  3. A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1389–1399 (2000). [CrossRef]
  4. J. A. Arnaud, “Degenerate Optical Cavities. II: Effect of Misalignments,” Appl. Opt. 8(9), 1909–1917 (1969). [CrossRef] [PubMed]
  5. G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, “Analysis of misalignment sensitivity of ring laser resonators,” Sov. J. Quantum Electron. 7, 857–859 (1977). [CrossRef]
  6. H. R. Bilger and G. E. Stedman, “Stability of planar ring lasers with mirror misalignment,” Appl. Opt. 26(17), 3710–3716 (1987). [CrossRef] [PubMed]
  7. R. Rodloff, “A laser gyro with optimized resonator geometry,” IEEE J. Quantum Electron. 23(4), 438–445 (1987). [CrossRef]
  8. J. Yuan, X. W. Long, B. Zhang, F. Wang, and H. C. Zhao, “Optical axis perturbation in folded planar ring resonators,” Appl. Opt. 46(25), 6314–6322 (2007). [CrossRef] [PubMed]
  9. I. W. Smith, “Optical resonator axis stability and instability from first principles,” Proc. SPIE 412, 203–206 (1983).
  10. A. L. Levkit and V. M. Ovchinnikov, “Stability of a ring resonator with a nonplane axial contour,” J. Appl. Spectrosc. (USSR) 40(6), 657–660 (1984). [CrossRef]
  11. S.-C. Sheng, “Optical-axis perturbation singularity in an out-of-plane ring resonator,” Opt. Lett. 19(10), 683–685 (1994). [CrossRef] [PubMed]
  12. A. H. Paxton and W. P. Latham., “Unstable resonators with 90° beam rotation,” Appl. Opt. 25(17), 2939–2946 (1986). [CrossRef] [PubMed]
  13. B. E. Currie, G. E. Stedman, and R. W. Dunn, “Laser stability and beam steering in a nonregular polygonal cavity,” Appl. Opt. 41(9), 1689–1697 (2002). [CrossRef] [PubMed]
  14. J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun. 281(5), 1204–1210 (2008). [CrossRef]
  15. J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt. 47(5), 628–631 (2008). [CrossRef] [PubMed]
  16. X. W. Long and J. Yuan, “Method for eliminating mismatching error in monolithic triaxial ring resonators,” Chin. Opt. Lett. 8(12), 1135–1138 (2010). [CrossRef]
  17. D. Z. Anderson, “Alignment of resonant optical cavities,” Appl. Opt. 23(17), 2944–2949 (1984). [CrossRef] [PubMed]
  18. N. M. Sampas and D. Z. Anderson, “Stabilization of laser beam alignment to an optical resonator by heterodyne detection of off-axis modes,” Appl. Opt. 29(3), 394–403 (1990). [CrossRef] [PubMed]
  19. J. Yuan and X. W. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74(3), 1362–1365 (2003). [CrossRef]
  20. J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76(12), 125106 (2005). [CrossRef]

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