Abstract

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.

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References

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2012 (1)

2011 (2)

2010 (3)

2009 (1)

P.-D. Lin and C.-C. Hsueh, “6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems,” Appl. Phys. B97(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]

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]

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics2(6), 365–370 (2008).
[CrossRef]

2007 (2)

2005 (2)

2000 (1)

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

1997 (1)

1994 (1)

1991 (1)

1986 (1)

1985 (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), 61–104 (1985).
[CrossRef]

1969 (1)

Bañas, A.

Chen, M.

Chen, M. X.

Chen, Z.

Chow, W. W.

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. Photonics2(6), 365–370 (2008).
[CrossRef]

Gangopadhyay, S.

Gea-Banacloche, J.

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]

Glückstad, J.

Grigorenko, A. N.

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics2(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. B97(1), 135–143 (2009).
[CrossRef]

Hu, Y.

Huang, S.

Huang, S. L.

Huisken, J.

Kang, Z.

Latham, W. P.

Li, D.

Liang, L. M.

Lin, P.-D.

P.-D. Lin and C.-C. Hsueh, “6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems,” Appl. Phys. B97(1), 135–143 (2009).
[CrossRef]

Liu, H. Z.

Liu, L. R.

Long, X.

Long, X. W.

Lou, C.

Luan, Z.

Massey, G. A.

Palima, D.

Paxton, A. H.

Pedrotti, L. M.

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. Photonics2(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]

Sarkar, S.

Sceats, M. G.

Schleich, W.

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]

Sheng, S.-C.

Siegman, A. E.

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

G. A. Massey and A. E. Siegman, “Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces,” Appl. Opt.8(5), 975–978 (1969).
[CrossRef] [PubMed]

Stelzer, E. H. K.

Stokes, A. D.

Swoger, J.

Tauro, S.

Tuan, H. T.

Wang, F.

Wen, D. D.

Xu, J.

Xu, R. W.

Yuan, J.

Zhang, B.

Zhang, P.

Zhang, Y.

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics2(6), 365–370 (2008).
[CrossRef]

Zhao, H. C.

Zhao, J. L.

Zhao, Y. X.

Appl. Opt. (7)

Appl. Phys. B (1)

P.-D. Lin and C.-C. Hsueh, “6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems,” Appl. Phys. B97(1), 135–143 (2009).
[CrossRef]

Chin. Opt. Lett. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

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

J. Opt. Soc. Am. A (1)

Nat. Photonics (1)

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics2(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]

Opt. Express (4)

Opt. Lett. (3)

Rev. Mod. Phys. (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), 61–104 (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).

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of the beam position controlling method for a simple 3D optical system. Pj(j = 1,2,3,4): the reflection points, nj(j = 2,3): the binormals at points Pj(j = 2,3), mj(j = 2,3): reflecting mirrors, Fj(j = 2,3,4): facular (transverse section) of the incident beam before being reflected from points Pj(j = 2,3,4), (xj, yj, zj)(j = 2,3): coordinate systems for the incident beam before being reflected from points Pj(j = 2,3), (xjr, yjr, zjr)(j = 2,3): coordinate systems for the reflected beam after being reflected from points Pj(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 yj and yjr(j = 2,3) are along the directions of nj(j = 2,3); the positive directions of zj and zjr(j = 2,3) are along the direction of beam propagation; (x2, x2r) and (x3, x3r) are located at the incident planes of P1P2P3 and P2P3P4 respectively.)

Fig. 2
Fig. 2

Angular misalignments of a paraxial dielectric interface Ii with normal incidence. (a) definition of the interface’s misalignment angle θix and (b) angular misalignment of the dielectric interface around rotational axis Rix. Ii0: the blue solid line which is the initial position of Ii, Ii1: the red dot line which is the position of Ii after θix>0, Ii2: the red solid line which is the position of Ii after θix<0, Pi0: the incident point, x, y and z: the coordinate axes of the incident ray and refractive rays, Tix, Tiy and Tiz: three translational axes, Rix, Riy and Riz: three rotational axes, θix, θiy, θiz: angular misalignments around Rix, Riy and Riz separately, L1i: incident ray, L1o0 and L1o1: transmission rays refracted from Ii0 and Ii1, θiix: incident angle, θoix: refracted angle, n1: refractive index before transmission, n2: refractive index after transmission, △θoix: the angle between L1o1 and z axis. (Note: z axis is parallel to the normal direction of dielectric interface; the positive direction of Tix, Tiy and Tiz are along the directions of x, y and z separately; the positive direction of Rix, Riy and Riz are along the directions of y, -x and z separately.)

Fig. 3
Fig. 3

Schematic diagram of optical-wedge transmission with the consideration of wedge angle θ and angular misalignments θx, θy and θz = 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, Rx, Ry and Rz: three rotational axes, θx, θy, θz: three kinds of angular misalignments around the axes of Rx, Ry and Rz 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 θx>0 and θ>0, P1, P2: the incident points on I1 and I2, L1i0, L2i0: incident rays, L1o0, L1o1: transmission rays refracted from I10 and I11, L2o00, L2o0 and L2o1: transmission rays refracted from I200, I20 and I21, θi1x, θi2x: incident angles, θo1x, θo2x: 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 Rx, Ry and Rz are along the directions of y, -x and z separately.)

Fig. 4
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, ma and mb: spherical mirrors with common radius R, mc and md: planar mirrors, Ai: the incident angles on all four mirrors, FW: Faraday-wedge with wedge angle of θ, Lw: length of the Faraday -wedge, Lj(j = 1,2,3,4): four sides of the cavity, Pj(j = a,b,c,d): terminal points of the resonator, Pe: the center of the diaphragm, Pg: the center of the discharge capillary, Pf: the midpoint between Pb and Pc, Ph: the midpoint between Pa and Pd, O1, O2: the midpoints of straight lines PbPd and PaPc respectively, nj(j = a,b,c,d): the binormals at points Pj(j = a,b,c,d), (xj, yj, zj)(j = e,f,g,h): coordinate systems for the incident beam before being reflected from points Pj(j = b,c,d,a), (xjr, yjr, zjr)(j = e,f,g,h): coordinate systems for the reflected beam after being reflected from points Pj(j = b,c,d,a), (x, y, z): coordinate system for the incident ray and refractive rays of FW, Rx, Ry and Rz: three rotational axes of FW, θx, θy, θz: three kinds of angular misalignments around the axes of Rx, Ry and Rz respectively, φj(j = e,f,g,h): coordinate rotation angles based on NCS, δjz(j = a,b,c,d): axial displacement of mirrors mi(i = a,b,c,d), δjxjy(j = a,b): radial displacements of the spherical mirrors ma and mb. (Note: The cavity length of all four sides are equal and the total cavity length is L; the positive directions of yj and yjr(j = e,f,g,h) are along the directions of nj(j = b,c,d,a); the positive directions of zj and zjr(j = 1,2,3,4,b,c) are along the direction of beam propagation; (xe, xer), (xf, xfr), (xg, xgr) and (xh, xhr) are located at the incident planes of PaPbPc, PbPcPd, PcPdPa and PdPaPb separately; the positive directions of x, y and z are parallel to the directions of xe, ye and ze; the positive direction of Rx, Ry and Rz are along the directions of ye, -xe and ze separately; the positive directions of δax, δbx, δaz, δbz, δcz, and δdz are along the directions of straight lines PbPd, PaPc, PaO1, PbO2, PcO1 and PdO2 respectively; the positive direction of δjy (j = a,b) is along the direction of nj(j = a,b).)

Fig. 5
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 △xe, △ye, △xg and △yg versus wedge angle θ.

Fig. 6
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 △xe, △ye, △xg and △yg versus wedge angle θ.

Fig. 7
Fig. 7

Optical-axis decentrations induced by Faraday-wedge’s angular misalignments in NPRO. Optical-axis decentrations △xe, △ye, △xg and △yg (a) versus θx with θy = 0, (b) versus θy with θx = 0.

Fig. 8
Fig. 8

Optical-axis decentrations induced by spherical mirror’s radial displacements in NPRO. Optical-axis decentrations △xe, △ye, △xg and △yg (a) versus δax with δay = δbx = δby = 0, (b) verus δay with δax = δbx = δby = 0, (c) versus δbx with δax = δay = δby = 0, (d) versus δby with δax = δay = δbx = 0.

Fig. 9
Fig. 9

Optical-axis decentrations induced by mirror’s axial displacements in NPRO. Optical-axis decentrations △xe, △ye, △xg and △yg (a) versus δaz with δbz = δcz = δdz = 0, (b) versus δbz with δaz = δcz = δdz = 0, (c) versus δcz with δaz = δbz = δdz = 0, (d) versus δdz with δaz = δbz = δcz = 0.

Fig. 10
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 (θz = 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: △xe, △ye, △xg and △yg versus δaz = δbz = δcz = δdz.

Fig. 11
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 (θz = 90°), (a) optical-axis perturbation of NPRO during the mismatching error eliminating process by utilizing ma and mc’ 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; ma and mc’s positions after axial displacements are illustrated with red solid circles.), (b) optical-axis decentrations in NPRO: △xe, △ye, △xg and △yg versus δaz = δcz.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

( r ox r ox ' r oy r oy ' 1 ) T = M C ( r ix r ix ' r iy r iy ' 1 ) T ,
( r x r x ' r y r y ' 1 ) T = M R ( r x r x ' r y r y ' 1 ) T ,
( r ox r ox ' r oy r oy ' 1 )=( A x B x 0 0 E x C x D x 0 0 F x 0 0 A y B y E y 0 0 C y D y F y 0 0 0 0 1 )( r ix r ix ' r iy r iy ' 1 ),
θ iix = θ ix .
n 1 sin( θ iix )= n 1 sin( θ ix )= n 2 sin( θ oix ), n 1 θ ix n 2 θ oix , θ oix n 1 θ ix n 2 .
Δ θ oix = θ iix θ oix = θ ix ( n 1 / n 2 ) θ ix = θ ix (1 n 1 / n 2 ),
M_ I i =( 1 0 0 0 0 0 n 1 n 2 0 0 (1 n 1 n 2 ) θ ix 0 0 1 0 0 0 0 0 n 1 n 2 (1 n 1 n 2 ) θ iy 0 0 0 0 1 ).
θ i1x = θ x , θ i2x = θ x +θ.
M_ I 1 =( 1 0 0 0 0 0 n 1 n 2 0 0 (1 n 1 n 2 ) θ x 0 0 1 0 0 0 0 0 n 1 n 2 (1 n 1 n 2 ) θ y 0 0 0 0 1 ),
M_ I 2 =( 1 0 0 0 0 0 n 2 n 1 0 0 (1 n 2 n 1 )( θ x +θ)) 0 0 1 0 0 0 0 0 n 2 n 1 (1 n 2 n 1 ) θ y 0 0 0 0 1 ).
M_ I 2 =( 1 0 0 0 0 0 n 2 n 1 0 0 (1 n 2 n 1 )( θ x +θcos( θ z )) 0 0 1 0 0 0 0 0 n 2 n 1 (1 n 2 n 1 )( θ y +θsin( θ z )) 0 0 0 0 1 ).
M l =( 1 l n 2 0 0 0 0 1 0 0 0 0 0 1 l n 2 0 0 0 0 1 0 0 0 0 0 1 ).
M w =M_ I 2 M l M_ I 1 =( 1 l n 1 n 2 2 0 0 l( n 2 n 1 ) θ x / n 2 2 0 1 0 0 ( n 1 n 2 )θcos( θ z )/ n 1 0 0 1 l n 1 n 2 2 l( n 2 n 1 ) θ y / n 2 2 0 0 0 1 ( n 1 n 2 )θsin( θ z )/ n 1 0 0 0 0 1 ).
φ e = φ f = φ g = φ h =φ.
M E = M w T( L 1 /2 L w )R( φ e )M( m a )T( L 4 )R( φ h )M( m d ) T( L 3 )R( φ g )M( m c )T( L 2 )R( φ f )M( m b )T( L 1 /2),
M G =T( L 3 /2)R( φ g )M( m c )T( L 2 )R( φ f )M( m b )T( L 1 /2) M w T( L 1 /2 L w )R( φ e )M( m a )T( L 4 )R( φ h )M( m d )T( L 3 /2).
Δ x e =0.3271mm,Δ y e =0.0069mm,Δ x g =0.2386mm,Δ y g =0.0048mm.
Δ x e =0.0069mm,Δ y e =0.2943mm,Δ x g =0.0065mm,Δ y g =0.2115mm.
Δ x e =0.0224mm,Δ y e =0.0249mm,Δ x g =0.0006mm,Δ y g =0.0336mm.
Δ x e =-0.0286mm,Δ y e =0.0209mm,Δ x g =-0.0365mm,Δ y g =-0.0006mm.
Δ x e =-0.3262mm,Δ y e =-0.0649mm,Δ x g =-0.3255mm,Δ y g =-0.0647mm.
Δ x e =0.0009mm,Δ y e =0.058mm,Δ x g =0.0869mm,Δ y g =0.0599mm.
Δ x e =0.0062mm,Δ y e =-0.2844mm,Δ x g =0.0051mm,Δ y g =-0.2789mm.
Δ x e =0.0131mm,Δ y e =0.0099mm,Δ x g =0.0116mm,Δ y g =0.0674mm.

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