Abstract

A novel coordinate system for Gaussian beam reflection has been proposed in this Letter. Reflection from a spherical mirror is used to describe the novel coordinate system. One single segment of a general resonator is chosen to describe coordinate rotation in detail. Nonplanar ring resonators are chosen to show the application of the novel coordinate system. This novel coordinate system has been proved by two simple experiments and the problem existing in using the traditional coordinate system has been pointed out. This novel coordinate system is valuable for not only the designing of laser resonators but also Gaussian beam propagation analysis.

© 2012 Optical Society of America

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References

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2011 (2)

2008 (1)

J. Yuan and X. W. Long, Opt. Commun. 281, 1204 (2008).
[CrossRef]

2007 (1)

2005 (2)

2000 (1)

A. E. Siegman, IEEE J. Sel. Top. Quantum Electron. 6, 1389 (2000).
[CrossRef]

1997 (1)

1994 (1)

1991 (1)

1985 (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

1969 (1)

Chen, M. X.

Chow, W. W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

Dorschner, T. A.

H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, Laser HandbookM. L. Stitch and M. Bass, eds. (North-Holland, 1985), Vol. 4, Chap. 3.

Gangopadhyay, S.

Gea-Banacloche, J.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

Holtz, M.

H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, Laser HandbookM. L. Stitch and M. Bass, eds. (North-Holland, 1985), Vol. 4, Chap. 3.

Huang, S. L.

Li, D.

Liang, L. M.

Liu, H. Z.

Liu, L. R.

Long, X. W.

Luan, Z.

Massey, G. A.

Pedrotti, L. M.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

Sanders, V. E.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61 (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, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

Scully, M. O.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

Sheng, S.-C.

Siegman, A. E.

A. E. Siegman, IEEE J. Sel. Top. Quantum Electron. 6, 1389 (2000).
[CrossRef]

G. A. Massey and A. E. Siegman, Appl. Opt. 8, 975 (1969).
[CrossRef]

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

Smith, I. W.

H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, Laser HandbookM. L. Stitch and M. Bass, eds. (North-Holland, 1985), Vol. 4, Chap. 3.

Statz, H.

H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, Laser HandbookM. L. Stitch and M. Bass, eds. (North-Holland, 1985), Vol. 4, Chap. 3.

Stokes, A. D.

Tuan, H. T.

Wang, F.

Wen, D. D.

Xu, R. W.

Yuan, J.

Zhang, B.

Zhao, H. C.

Zhao, J. L.

Zhao, Y. X.

Appl. Opt. (5)

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

A. E. Siegman, IEEE J. Sel. Top. Quantum Electron. 6, 1389 (2000).
[CrossRef]

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

Opt. Commun. (1)

J. Yuan and X. W. Long, Opt. Commun. 281, 1204 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Rev. Mod. Phys. (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

Other (2)

H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, Laser HandbookM. L. Stitch and M. Bass, eds. (North-Holland, 1985), Vol. 4, Chap. 3.

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

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Figures (3)

Fig. 1.
Fig. 1.

(a) Coordinate systems for Gaussian beam reflection from a spherical mirror Mi, Li: incident beam, Lr: reflected beam; (b) a simple experiment for verifying novel coordinate system for Gaussian beam reflection (NCS), M0: planar mirror, Li0: an arbitrarily chosen incident ray, Li0r: the reflected ray of Li0 obtained by experiment, Li0rT and Li0rN: the positions of the reflected ray of Li0 by referring to TCS for Gaussian beam reflection (TCS) and NCS respectively; Pj(j=i,0): reflected points, nj(j=i,0): the binormals at points Pj(j=i,0), Aj(j=i,0): incident angles, (xtj,yj,zj) and (xj,yj,zj)(j=i,0): coordinate systems of the incident beam based on TCS and NCS, respectively, (xtjr,yjr,zjr) and (xjr,yjr,zjr)(j=i,0): coordinate systems of the reflected beam based on TCS and NCS, respectively.

Fig. 2.
Fig. 2.

(a) Coordinate systems and corresponding coordinate rotations based on TCS and NCS in four equal-sided nonplanar ring resonators (NPRO), β: folding angle, the cavity length of all four sides are equal and the total cavity length is L, M1 and M2: spherical mirrors with radius of R1 and R2, M3 and M4: planar mirrors, Aj(j=1,2,3,4): incident angles on four mirrors, Pj(j=1,2,3,4): terminal points of the resonator, Pe, Pf, Pg, Ph, O1, O2: the midpoints of straight lines P1P2, P2P3, P3P4, P4P1, P1P3, and P2P4 separately, φtj(j=1,2,3,4) and φj(j=1,2,3,4): coordinate rotation angles based on TCS and NCS respectively; (b) one single segment of a general resonator, Pj(j=a,b,c,d): the reflection points, φtc and φc: coordinate rotation angles based on TCS and NCS respectively, Fj(j=b,c,d): facular (transverse section) of the incident beam before being reflected from points Pj(j=b,c,d); (c) schematic diagram of experimental result on optical-axis perturbation caused by spherical mirror’s radial displacements in square ring resonators (SRR), SRR: a special case of NPRO in the case of β=0° and A1=A2=A3=A4=A=45°, δjx(j=1,2): radial displacements of Mj(j=1,2) with the same value of δ>0, PjT(j=f,h) and PjN(j=f,h): the positions of points Pj(j=f,h) after perturbations of δjx=δ>0(j=1,2) by referring to TCS and NCS, respectively. 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 radial displacements, are illustrated with red solid arcs. nj(j=1,2,3,4,b,c): the binormals at points Pj(j=1,2,3,4,b,c), (xtj,yj,zj) and (xj,yj,zj) (j=1,2,3,4,b,c): coordinate systems for the incident beam (based on TCS and NCS, respectively) before being reflected from points Pj(j=1,2,3,4,b,c), (xtjr,yjr,zjr) and (xjr,yjr,zjr)(j=1,2,3,4,b,c): coordinate systems for the reflected beam (based on TCS and NCS, respectively) after being reflected from points Pj(j=1,2,3,4,b,c). (Note: The positive directions of yj and yjr(j=1,2,3,4,b,c) are along the directions of nj(j=1,2,3,4,b,c); the positive directions of zj and zjr(j=1,2,3,4,b,c) are along the direction of beam propagation; (xt1,xt1r,x1,x1r), (xt2,xt2r,x2,x2r), (xt3,xt3r,x3,x3r) and (xt4,xt4r,x4,x4r) are located at the incident planes of P4P1P2, P1P2P3, P2P3P4, and P3P4P1 separately).

Fig. 3.
Fig. 3.

Stability map for NPRO1.

Equations (5)

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(roxroxroyroy1)T=M(rixrixriyriy1)T,
φt1=φt2=φt3=φt4=φ,
φ1=φ2=φ3=φ4=φ,
ρ=|φ1|+|φ2|+|φ3|+|φ4|=4φ.
Δxf=Δxh=2/4×(δ1x+δ2x)=2/2×δ>0,Δyf=Δyh=0.

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