Abstract

We report an axial-asymmetric high-Q Fabry–Perot cavity supporting nondegenerate Hermite–Gaussian modes of the same mode order. Axial asymmetry of mirror surface was introduced by mechanically grinding off one side of a cylindrical mirror substrate without degrading the original mirror quality. The bases of the resulting Hermite–Gaussian modes were aligned with respect to the direction of grinding, making it possible to prescribe the mirror principal axes.

© 2012 Optical Society of America

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2011 (1)

P. Zhang, Y. Guo, Z. Li, Y. Zhang, Y. Zhang, J. Du, G. Li, J. Wang, and T. Zhang, Phys. Rev. A 83, 031804 (2011).
[CrossRef]

2010 (2)

W. Seo, H. Hong, M. Lee, Y. Song, Y. Chough, W. Choi, C. Fang-Yen, R. Dasari, M. Feld, J. Lee, and K. An, Phys. Rev. A 81, 053824 (2010).
[CrossRef]

Y. Choi, S. Kang, S. Lim, W. Kim, J. Kim, J. Lee, and K. An, Phys. Rev. Lett. 104, 153601 (2010).
[CrossRef]

2009 (3)

A. Kubanek, M. Koch, C. Sames, A. Ourjoumtsev, P. Pinkse, K. Murr, and G. Rempe, Nature 462, 898 (2009).
[CrossRef]

S. Gopalakrishnan, B. Lev, and P. Goldbart, Nat. Phys. 5, 845 (2009).
[CrossRef]

H.-G. Hong, W. Seo, M. Lee, Y. Song, Y. T. Chough, J.-H. Lee, and K. An, Opt. Express 17, 15455 (2009).
[CrossRef]

2008 (2)

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. Morizur, P. Lam, and H. Bachor, Science 321, 541 (2008).
[CrossRef]

H. Kimble, Nature 453, 1023 (2008).
[CrossRef]

2007 (1)

T. Wilk, S. Webster, A. Kuhn, and G. Rempe, Science 317, 488 (2007).
[CrossRef]

2005 (1)

2004 (1)

T. Puppe, P. Maunz, T. Fischer, P. Pinkse, and G. Rempe, Phys. Scr. T112, 7 (2004).
[CrossRef]

2003 (1)

K. Vahala, Nature 424, 839 (2003).
[CrossRef]

2002 (2)

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[CrossRef]

T. Udem, R. Holzwarth, and T. Hänsch, Nature 416, 233 (2002).
[CrossRef]

2001 (2)

G. Guthöhrlein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, Nature 414, 49 (2001).
[CrossRef]

S. Feng and H. G. Winful, Opt. Lett. 26, 485 (2001).
[CrossRef]

1992 (2)

G. Rempe, R. Thompson, H. Kimble, and R. Lalezari, Opt. Lett. 17, 363 (1992).
[CrossRef]

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

1987 (1)

1969 (1)

D. Hanna, IEEE J. Quantum Electron. 5, 483 (1969).
[CrossRef]

Abramovici, A.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

Althouse, W.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

An, K.

W. Seo, H. Hong, M. Lee, Y. Song, Y. Chough, W. Choi, C. Fang-Yen, R. Dasari, M. Feld, J. Lee, and K. An, Phys. Rev. A 81, 053824 (2010).
[CrossRef]

Y. Choi, S. Kang, S. Lim, W. Kim, J. Kim, J. Lee, and K. An, Phys. Rev. Lett. 104, 153601 (2010).
[CrossRef]

H.-G. Hong, W. Seo, M. Lee, Y. Song, Y. T. Chough, J.-H. Lee, and K. An, Opt. Express 17, 15455 (2009).
[CrossRef]

Bachor, H.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. Morizur, P. Lam, and H. Bachor, Science 321, 541 (2008).
[CrossRef]

Berman, P.

P. Berman, Cavity Quantum Electrodynamics (Academic Press, 1994).

Choi, W.

W. Seo, H. Hong, M. Lee, Y. Song, Y. Chough, W. Choi, C. Fang-Yen, R. Dasari, M. Feld, J. Lee, and K. An, Phys. Rev. A 81, 053824 (2010).
[CrossRef]

Choi, Y.

Y. Choi, S. Kang, S. Lim, W. Kim, J. Kim, J. Lee, and K. An, Phys. Rev. Lett. 104, 153601 (2010).
[CrossRef]

Chough, Y.

W. Seo, H. Hong, M. Lee, Y. Song, Y. Chough, W. Choi, C. Fang-Yen, R. Dasari, M. Feld, J. Lee, and K. An, Phys. Rev. A 81, 053824 (2010).
[CrossRef]

Chough, Y. T.

Chuu, C.

Dasari, R.

W. Seo, H. Hong, M. Lee, Y. Song, Y. Chough, W. Choi, C. Fang-Yen, R. Dasari, M. Feld, J. Lee, and K. An, Phys. Rev. A 81, 053824 (2010).
[CrossRef]

Delaubert, V.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. Morizur, P. Lam, and H. Bachor, Science 321, 541 (2008).
[CrossRef]

Drever, R.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

Du, J.

P. Zhang, Y. Guo, Z. Li, Y. Zhang, Y. Zhang, J. Du, G. Li, J. Wang, and T. Zhang, Phys. Rev. A 83, 031804 (2011).
[CrossRef]

Fang-Yen, C.

W. Seo, H. Hong, M. Lee, Y. Song, Y. Chough, W. Choi, C. Fang-Yen, R. Dasari, M. Feld, J. Lee, and K. An, Phys. Rev. A 81, 053824 (2010).
[CrossRef]

Feld, M.

W. Seo, H. Hong, M. Lee, Y. Song, Y. Chough, W. Choi, C. Fang-Yen, R. Dasari, M. Feld, J. Lee, and K. An, Phys. Rev. A 81, 053824 (2010).
[CrossRef]

Feng, S.

Fischer, T.

T. Puppe, P. Maunz, T. Fischer, P. Pinkse, and G. Rempe, Phys. Scr. T112, 7 (2004).
[CrossRef]

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[CrossRef]

Goldbart, P.

S. Gopalakrishnan, B. Lev, and P. Goldbart, Nat. Phys. 5, 845 (2009).
[CrossRef]

Gopalakrishnan, S.

S. Gopalakrishnan, B. Lev, and P. Goldbart, Nat. Phys. 5, 845 (2009).
[CrossRef]

Guo, Y.

P. Zhang, Y. Guo, Z. Li, Y. Zhang, Y. Zhang, J. Du, G. Li, J. Wang, and T. Zhang, Phys. Rev. A 83, 031804 (2011).
[CrossRef]

Gürsel, Y.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

Guthöhrlein, G.

G. Guthöhrlein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, Nature 414, 49 (2001).
[CrossRef]

Hanna, D.

D. Hanna, IEEE J. Quantum Electron. 5, 483 (1969).
[CrossRef]

Hänsch, T.

T. Udem, R. Holzwarth, and T. Hänsch, Nature 416, 233 (2002).
[CrossRef]

Hanssen, J.

Harb, C.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. Morizur, P. Lam, and H. Bachor, Science 321, 541 (2008).
[CrossRef]

Hayasaka, K.

G. Guthöhrlein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, Nature 414, 49 (2001).
[CrossRef]

Holzwarth, R.

T. Udem, R. Holzwarth, and T. Hänsch, Nature 416, 233 (2002).
[CrossRef]

Hong, H.

W. Seo, H. Hong, M. Lee, Y. Song, Y. Chough, W. Choi, C. Fang-Yen, R. Dasari, M. Feld, J. Lee, and K. An, Phys. Rev. A 81, 053824 (2010).
[CrossRef]

Hong, H.-G.

Horak, P.

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[CrossRef]

Janousek, J.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. Morizur, P. Lam, and H. Bachor, Science 321, 541 (2008).
[CrossRef]

Kang, S.

Y. Choi, S. Kang, S. Lim, W. Kim, J. Kim, J. Lee, and K. An, Phys. Rev. Lett. 104, 153601 (2010).
[CrossRef]

Kawamura, S.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

Keller, M.

G. Guthöhrlein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, Nature 414, 49 (2001).
[CrossRef]

Kim, J.

Y. Choi, S. Kang, S. Lim, W. Kim, J. Kim, J. Lee, and K. An, Phys. Rev. Lett. 104, 153601 (2010).
[CrossRef]

Kim, W.

Y. Choi, S. Kang, S. Lim, W. Kim, J. Kim, J. Lee, and K. An, Phys. Rev. Lett. 104, 153601 (2010).
[CrossRef]

Kimble, H.

Koch, M.

A. Kubanek, M. Koch, C. Sames, A. Ourjoumtsev, P. Pinkse, K. Murr, and G. Rempe, Nature 462, 898 (2009).
[CrossRef]

Kubanek, A.

A. Kubanek, M. Koch, C. Sames, A. Ourjoumtsev, P. Pinkse, K. Murr, and G. Rempe, Nature 462, 898 (2009).
[CrossRef]

Kuhn, A.

T. Wilk, S. Webster, A. Kuhn, and G. Rempe, Science 317, 488 (2007).
[CrossRef]

Lalezari, R.

Lam, P.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. Morizur, P. Lam, and H. Bachor, Science 321, 541 (2008).
[CrossRef]

Lange, W.

G. Guthöhrlein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, Nature 414, 49 (2001).
[CrossRef]

Lee, J.

Y. Choi, S. Kang, S. Lim, W. Kim, J. Kim, J. Lee, and K. An, Phys. Rev. Lett. 104, 153601 (2010).
[CrossRef]

W. Seo, H. Hong, M. Lee, Y. Song, Y. Chough, W. Choi, C. Fang-Yen, R. Dasari, M. Feld, J. Lee, and K. An, Phys. Rev. A 81, 053824 (2010).
[CrossRef]

Lee, J.-H.

Lee, M.

W. Seo, H. Hong, M. Lee, Y. Song, Y. Chough, W. Choi, C. Fang-Yen, R. Dasari, M. Feld, J. Lee, and K. An, Phys. Rev. A 81, 053824 (2010).
[CrossRef]

H.-G. Hong, W. Seo, M. Lee, Y. Song, Y. T. Chough, J.-H. Lee, and K. An, Opt. Express 17, 15455 (2009).
[CrossRef]

Lev, B.

S. Gopalakrishnan, B. Lev, and P. Goldbart, Nat. Phys. 5, 845 (2009).
[CrossRef]

Li, G.

P. Zhang, Y. Guo, Z. Li, Y. Zhang, Y. Zhang, J. Du, G. Li, J. Wang, and T. Zhang, Phys. Rev. A 83, 031804 (2011).
[CrossRef]

Li, Z.

P. Zhang, Y. Guo, Z. Li, Y. Zhang, Y. Zhang, J. Du, G. Li, J. Wang, and T. Zhang, Phys. Rev. A 83, 031804 (2011).
[CrossRef]

Lim, S.

Y. Choi, S. Kang, S. Lim, W. Kim, J. Kim, J. Lee, and K. An, Phys. Rev. Lett. 104, 153601 (2010).
[CrossRef]

Maunz, P.

T. Puppe, P. Maunz, T. Fischer, P. Pinkse, and G. Rempe, Phys. Scr. T112, 7 (2004).
[CrossRef]

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[CrossRef]

Meyrath, T.

Morizur, J.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. Morizur, P. Lam, and H. Bachor, Science 321, 541 (2008).
[CrossRef]

Murr, K.

A. Kubanek, M. Koch, C. Sames, A. Ourjoumtsev, P. Pinkse, K. Murr, and G. Rempe, Nature 462, 898 (2009).
[CrossRef]

Ourjoumtsev, A.

A. Kubanek, M. Koch, C. Sames, A. Ourjoumtsev, P. Pinkse, K. Murr, and G. Rempe, Nature 462, 898 (2009).
[CrossRef]

Pinkse, P.

A. Kubanek, M. Koch, C. Sames, A. Ourjoumtsev, P. Pinkse, K. Murr, and G. Rempe, Nature 462, 898 (2009).
[CrossRef]

T. Puppe, P. Maunz, T. Fischer, P. Pinkse, and G. Rempe, Phys. Scr. T112, 7 (2004).
[CrossRef]

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[CrossRef]

Puppe, T.

T. Puppe, P. Maunz, T. Fischer, P. Pinkse, and G. Rempe, Phys. Scr. T112, 7 (2004).
[CrossRef]

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[CrossRef]

Raab, F.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

Raizen, M.

Rempe, G.

A. Kubanek, M. Koch, C. Sames, A. Ourjoumtsev, P. Pinkse, K. Murr, and G. Rempe, Nature 462, 898 (2009).
[CrossRef]

T. Wilk, S. Webster, A. Kuhn, and G. Rempe, Science 317, 488 (2007).
[CrossRef]

T. Puppe, P. Maunz, T. Fischer, P. Pinkse, and G. Rempe, Phys. Scr. T112, 7 (2004).
[CrossRef]

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[CrossRef]

G. Rempe, R. Thompson, H. Kimble, and R. Lalezari, Opt. Lett. 17, 363 (1992).
[CrossRef]

Ritsch, H.

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[CrossRef]

Sames, C.

A. Kubanek, M. Koch, C. Sames, A. Ourjoumtsev, P. Pinkse, K. Murr, and G. Rempe, Nature 462, 898 (2009).
[CrossRef]

Schreck, F.

Seo, W.

W. Seo, H. Hong, M. Lee, Y. Song, Y. Chough, W. Choi, C. Fang-Yen, R. Dasari, M. Feld, J. Lee, and K. An, Phys. Rev. A 81, 053824 (2010).
[CrossRef]

H.-G. Hong, W. Seo, M. Lee, Y. Song, Y. T. Chough, J.-H. Lee, and K. An, Opt. Express 17, 15455 (2009).
[CrossRef]

Shoemaker, D.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

Sievers, L.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

Simon, R.

Song, Y.

W. Seo, H. Hong, M. Lee, Y. Song, Y. Chough, W. Choi, C. Fang-Yen, R. Dasari, M. Feld, J. Lee, and K. An, Phys. Rev. A 81, 053824 (2010).
[CrossRef]

H.-G. Hong, W. Seo, M. Lee, Y. Song, Y. T. Chough, J.-H. Lee, and K. An, Opt. Express 17, 15455 (2009).
[CrossRef]

Spero, R.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

Thompson, R.

Thorne, K.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

Treps, N.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. Morizur, P. Lam, and H. Bachor, Science 321, 541 (2008).
[CrossRef]

Udem, T.

T. Udem, R. Holzwarth, and T. Hänsch, Nature 416, 233 (2002).
[CrossRef]

Vahala, K.

K. Vahala, Nature 424, 839 (2003).
[CrossRef]

Vogt, R.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

Wagner, K.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. Morizur, P. Lam, and H. Bachor, Science 321, 541 (2008).
[CrossRef]

Walther, H.

G. Guthöhrlein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, Nature 414, 49 (2001).
[CrossRef]

Wang, J.

P. Zhang, Y. Guo, Z. Li, Y. Zhang, Y. Zhang, J. Du, G. Li, J. Wang, and T. Zhang, Phys. Rev. A 83, 031804 (2011).
[CrossRef]

Webster, S.

T. Wilk, S. Webster, A. Kuhn, and G. Rempe, Science 317, 488 (2007).
[CrossRef]

Weiss, R.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

Whitcomb, S.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

Wilk, T.

T. Wilk, S. Webster, A. Kuhn, and G. Rempe, Science 317, 488 (2007).
[CrossRef]

Winful, H. G.

Zhang, P.

P. Zhang, Y. Guo, Z. Li, Y. Zhang, Y. Zhang, J. Du, G. Li, J. Wang, and T. Zhang, Phys. Rev. A 83, 031804 (2011).
[CrossRef]

Zhang, T.

P. Zhang, Y. Guo, Z. Li, Y. Zhang, Y. Zhang, J. Du, G. Li, J. Wang, and T. Zhang, Phys. Rev. A 83, 031804 (2011).
[CrossRef]

Zhang, Y.

P. Zhang, Y. Guo, Z. Li, Y. Zhang, Y. Zhang, J. Du, G. Li, J. Wang, and T. Zhang, Phys. Rev. A 83, 031804 (2011).
[CrossRef]

P. Zhang, Y. Guo, Z. Li, Y. Zhang, Y. Zhang, J. Du, G. Li, J. Wang, and T. Zhang, Phys. Rev. A 83, 031804 (2011).
[CrossRef]

Zou, H.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. Morizur, P. Lam, and H. Bachor, Science 321, 541 (2008).
[CrossRef]

Zucker, M.

A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, Science 256, 325 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

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Figures (3)

Fig. 1.
Fig. 1.

(a) Procedure of mirror modification. The small gray dot in the center of the mirror surface indicates where low-order cavity modes are located. (b) Layout of our FPC.

Fig. 2.
Fig. 2.

(a) CCD images of transverse intensity distribution of cavity modes. (b), (c) Horizontal (along x axis) and vertical (along y axis) slices of transverse intensity distributions, respectively. Filled dots are data and solid curves are fits to Hermite–Gaussian functions.

Fig. 3.
Fig. 3.

(a) Resonance frequency spectrum of high-order transverse modes. It was obtained from cavity transmission of a probe laser as the cavity was scanned. Low peak heights of some high-order modes are due to imperfect coupling of the probe laser into those modes. (b) Deviation of resonance frequencies from those of the axial-symmetric cavity as described by Eq. (2). The solid and dashed curves are linear fits to HGl0 and HGom modes, respectively. The difference of two slopes indicates asymmetry of mirror surface.

Tables (1)

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Table 1. Resonance Frequency Difference between Modes of the Same Mode Order

Equations (4)

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Ul,m(x,y,z)=Ul,m0w0w(z)Hl(2xw(z))Hm(2yw(z))×exp[x2+y2w2(z)],
ωl,m=qνF+(1+l+m)ΔζπνF,
Ul,m(x,y,z)=Ul,m0w0xw0ywx(z)wy(z)Hl(2xwx(z))Hm(2ywy(z))×exp[x2wx2(z)y2wy2(z)],
ωl,m=qνF+(1/2+l)ΔζxπνF+(1/2+m)ΔζyπνF,

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