Abstract

Optical resonators are usually examined wave optically. We consider geometrical imaging in stable canonical resonators. We show that, with important exceptions related to eigenmode degeneracy, stable resonators generally image all transverse planes into each other. This insight leads to an intuitive understanding of important properties of the corresponding eigenmodes, most notably their well-known structural stability, i.e., the property that the eigenmodes retain their shape on propagation.

© 2002 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. J. Courtial, Opt. Commun. 151, 1 (1998).
  3. Ref. 1, Chap. 19.
  4. J. Courtial and M. J. Padgett, Phys. Rev. Lett. 85, 5320 (2000).
  5. J. Morgan, Introduction to Geometrical and Physical Optics (McGraw-Hill, New York, 1953).
  6. Ref. 1, Chap. 15.3.
  7. I. N. Bronstein and K. A. Semendjajew, Taschenbuch der Mathematik, 23rd ed. (Teubner Verlagsgesellschaft, Leipzig, Germany, 1987), p. 240.
  8. The Borel measure of the set of all the irrational numbers in any interval a,b is b-a, whereas that of the rational numbers in the same interval is 0. This is related to the fact that rational numbers are countable.
  9. A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985), Chap. 4.3.
  10. Ref. 1, p. 762.

2000 (1)

J. Courtial and M. J. Padgett, Phys. Rev. Lett. 85, 5320 (2000).

1998 (1)

J. Courtial, Opt. Commun. 151, 1 (1998).

Bronstein, I. N.

I. N. Bronstein and K. A. Semendjajew, Taschenbuch der Mathematik, 23rd ed. (Teubner Verlagsgesellschaft, Leipzig, Germany, 1987), p. 240.

Courtial, J.

J. Courtial and M. J. Padgett, Phys. Rev. Lett. 85, 5320 (2000).

J. Courtial, Opt. Commun. 151, 1 (1998).

Morgan, J.

J. Morgan, Introduction to Geometrical and Physical Optics (McGraw-Hill, New York, 1953).

Padgett, M. J.

J. Courtial and M. J. Padgett, Phys. Rev. Lett. 85, 5320 (2000).

Semendjajew, K. A.

I. N. Bronstein and K. A. Semendjajew, Taschenbuch der Mathematik, 23rd ed. (Teubner Verlagsgesellschaft, Leipzig, Germany, 1987), p. 240.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Yariv, A.

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985), Chap. 4.3.

Opt. Commun. (1)

J. Courtial, Opt. Commun. 151, 1 (1998).

Phys. Rev. Lett. (1)

J. Courtial and M. J. Padgett, Phys. Rev. Lett. 85, 5320 (2000).

Other (8)

J. Morgan, Introduction to Geometrical and Physical Optics (McGraw-Hill, New York, 1953).

Ref. 1, Chap. 15.3.

I. N. Bronstein and K. A. Semendjajew, Taschenbuch der Mathematik, 23rd ed. (Teubner Verlagsgesellschaft, Leipzig, Germany, 1987), p. 240.

The Borel measure of the set of all the irrational numbers in any interval a,b is b-a, whereas that of the rational numbers in the same interval is 0. This is related to the fact that rational numbers are countable.

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985), Chap. 4.3.

Ref. 1, p. 762.

Ref. 1, Chap. 19.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

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Figures (2)

Fig. 1
Fig. 1

Imaging in a canonical resonator with one planar and one thin mirror of focal length f. As far as geometrical imaging is concerned, resonators of this type are representative of all canonical resonators. The positions of the image and object planes are described in terms of their distance in front of the curved mirror.

Fig. 2
Fig. 2

Examples of periodic imaging in terms of (a) Ons and (b) Ωns, represented as angles. The bold numbers indicate the round-trip numbers, n. In the latter representation, each round trip simply advances the angle Ω by δ. The figure is drawn for δ=2/5 2π.

Equations (12)

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f=f1f2f1+f2-l0,    l=l021+ff1+ff2.
1on+12l-on+1=1f.
Ωn=π+2 arctan On-L1-L2,
On=onf-1,    L=lf-1
Ωn+1=Ωn+δmod 2π,
δ=π+2 arctan L1-L2.
δ=pq2π,
Mn=-in/on=-1/On.
ωN,n+r=ωN+s,n,
=rsπ,
=arctan z2zR-arctan z1zR.
δ=4+π2mod π-π2.

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