Beam position controlling method for 3D optical system and its application in non-planar ring resonators

Jie Yuan, Meixiong Chen, Xingwu Long, Yanyang Tan, Zhenglong Kang, and Yingying Li

Author Affiliations

Jie Yuan,^{1,}^{*} Meixiong Chen,^{1} Xingwu Long,^{1} Yanyang Tan,^{2} Zhenglong Kang,^{1} and Yingying Li^{1}

^{1}Department of Optoelectronic Engineering, College of Opto-electric Science and Engineering, National University of Defense Technology, Changsha Hunan 410073, China

^{2}Department of International Insurance, Insurance Professional College, Changsha Hunan 410114, China

Jie Yuan, Meixiong Chen, Xingwu Long, Yanyang Tan, Zhenglong Kang, and Yingying Li, "Beam position controlling method for 3D optical system and its application in non-planar ring resonators," Opt. Express 20, 19563-19579 (2012)

A novel theoretical beam position controlling method for 3D optical system has been proposed in this paper. Non-planar ring resonator, which is a typical 3D optical system, has been chosen as an example to show its application. To the best of our knowledge, the generalized ray matrices, augmented 5 × 5 ray matrices for paraxial dielectric interface transmission and paraxial optical-wedge transmission, and their detailed deducing process have been proposed in this paper for the first time. 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 non-planar ring resonators have been obtained. Wedge angle-induced mismatching errors of non-planar ring resonators have been found out and two experimental beam position controlling methods to effectively eliminate the wedge angle-induced mismatching errors have been proposed. All those results have been confirmed by related alignment experiments and the experimental results have been described with diagrammatic representation. These findings are important to the beam control, cavity design, and cavity alignment of high precision non-planar ring laser gyroscopes. Those 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.

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[Crossref]

P.-D. Lin and C.-C. Hsueh, “6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems,” Appl. Phys. B 97(1), 135–143 (2009).
[Crossref]

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), 61–104 (1985).
[Crossref]

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

J. Yuan, X. W. Long, M. X. Chen Z. L. Kang, and Y. Y. Li, “Comment on “Generalized sensitivity factors for optical-axis perturbation in nonplanar ring resonators,” To be published on Opt. Express, 168035 (2012).

H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, “The multioscillator ring laser gyroscope,” in Laser Handbook, M. I. Stitch, and M. Bass, eds. Vol. 4 (North-Holland, 1985) Chap. 3, pp. 229–327.

P.-D. Lin and C.-C. Hsueh, “6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems,” Appl. Phys. B 97(1), 135–143 (2009).
[Crossref]

2008 (3)

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

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[Crossref]

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), 61–104 (1985).
[Crossref]

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), 61–104 (1985).
[Crossref]

Dickinson, M. R.

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[Crossref]

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), 61–104 (1985).
[Crossref]

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[Crossref]

Hsueh, C.-C.

P.-D. Lin and C.-C. Hsueh, “6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems,” Appl. Phys. B 97(1), 135–143 (2009).
[Crossref]

P.-D. Lin and C.-C. Hsueh, “6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems,” Appl. Phys. B 97(1), 135–143 (2009).
[Crossref]

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), 61–104 (1985).
[Crossref]

Roberts, N. W.

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[Crossref]

Sanders, V. E.

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), 61–104 (1985).
[Crossref]

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), 61–104 (1985).
[Crossref]

Scully, M. O.

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), 61–104 (1985).
[Crossref]

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[Crossref]

P.-D. Lin and C.-C. Hsueh, “6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems,” Appl. Phys. B 97(1), 135–143 (2009).
[Crossref]

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[Crossref]

Opt. Commun. (1)

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

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), 61–104 (1985).
[Crossref]

Other (4)

A. E. Siegman, Lasers (University Science, 1986).

J. Yuan, X. W. Long, M. X. Chen Z. L. Kang, and Y. Y. Li, “Comment on “Generalized sensitivity factors for optical-axis perturbation in nonplanar ring resonators,” To be published on Opt. Express, 168035 (2012).

H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, “The multioscillator ring laser gyroscope,” in Laser Handbook, M. I. Stitch, and M. Bass, eds. Vol. 4 (North-Holland, 1985) Chap. 3, pp. 229–327.

G. J. Martin, “Multioscillator ring laser gyro using compensated optical wedge,” U.S. Patent 5,907,402 (25 May 1999).

Cited By

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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), φ_{3}: 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; (x_{2}, x_{2r}) and (x_{3}, x_{3r}) are located at the incident planes of P_{1}P_{2}P_{3} and P_{2}P_{3}P_{4} respectively.)

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, n_{1}: refractive index before transmission, n_{2}: 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.)

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. I_{1}, I_{2}: the two dielectric interfaces of the optical-wedge, θ: wedge angle of the optical-wedge (it is the angle between I_{1} and I_{2}), 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, I_{200}: the black dashed line which is the virtual position of the I_{2} without the consideration of θ>0, I_{10}, I_{20}: the blue solid lines which are the initial position of I_{1} and I_{2} with the consideration of θ>0, I_{11}, I_{21}: the two red dot lines which are the positions of I_{1} and I_{2} with the consideration of θ_{x}>0 and θ>0, P_{1}, P_{2}: the incident points on I_{1} and I_{2}, L1_{i0}, L2_{i0}: incident rays, L1_{o0}, L1_{o1}: transmission rays refracted from I_{10} and I_{11}, L2_{o00}, L2_{o0} and L2_{o1}: transmission rays refracted from I_{200}, I_{20} and I_{21}, θ_{i1x}, θ_{i2x}: incident angles, θ_{o1x}, θ_{o2x}: refracted angles, n_{1}: refractive index of atmosphere. n_{2}: refractive index of the optical-wedge. (Note: z axis is parallel to the normal direction of dielectric interface I_{1}; the positive direction of R_{x}, R_{y} and R_{z} are along the directions of y, -x and z separately.)

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}, O_{1}, O_{2}: the midpoints of straight lines P_{b}P_{d} and P_{a}P_{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_{a}P_{b}P_{c}, P_{b}P_{c}P_{d}, P_{c}P_{d}P_{a} and P_{d}P_{a}P_{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_{b}P_{d}, P_{a}P_{c}, P_{a}O_{1}, P_{b}O_{2}, P_{c}O_{1} and P_{d}O_{2} respectively; the positive direction of δ_{jy} (j = a,b) is along the direction of n_{j}(j = a,b).)

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.

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.

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.

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}.

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}.