1. §3.3 Optical Resonators with Spherical Mirrors We will show the field solutions inside the spherical mirror resonator are Gaussian Beams Z=0 00 z R2R2 R1R1 Lecture 5

2. §3.3 Optical Resonators with Spherical Mirrors I. Optical Resonator Algebra R > 0 when convex mirror faces cavity R < 0 when concave mirror faces cavity R > 0R < 0 Z=0 00 z2z2 R2R2 R1R1 z1z1 Knowing: Find: Knowing:Find:

3. §3.3 Optical Resonators with Spherical Mirrors For example: In practice we have R 1 and R 2, as well as cavity length L, find    and mirror location? Z=0 00 z2z2 R2R2 R1R1 z1z1 L and

4. §3.3 Optical Resonators with Spherical Mirrors II. The Symmetrical Resonator The minimum spot size is locate at and For symmetrical confocal resonator

5. §3.3 Optical Resonators with Spherical Mirrors Example: Design a Symmetrical Resonator Find the relation of mirror curvatures and minimum spot size

6. §3.4 Mode Stability Criteria The previous sections show the stability depends on the l, R 1 and R 2. 1) A Symmetrical Resonator l / R  /  conf 1 2 1 0 Plane-parallel Cavity Concentric Cavity Loss high

7. §3.4 Mode Stability Criteria Stability condition for optical resonators

8. §3.5 Modes in a Generalized Resonator – The Self-Consistent Method Self-consistency condition: reproduces itself after one round trip I. Self-Consistent Mode Compare to: General Stability Condition of an Arbitrary Resonator

9. §3.5 Modes in a Generalized Resonator – The Self-Consistant Method The other plane can be obtained by applying the ABCD law to q s

10. II. Stability of the Resonator Modes §3.5 Modes in a Generalized Resonator – The Self-Consistent Method We just find the existence condition and its solution of steady-state resonator modes. Now we need investigate whether the modes are stable. Perturbate the steady-state solution

11. §3.5 Modes in a Generalized Resonator – The Self-Consistent Method At steady state:

12. §3.6 Resonance Frequencies of Optical Resonators We now consider the resonance frequency (longitudinal mode) of a given spatial mode A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern measured in a plane perpendicular (i.e. transverse) to the propagation direction of the wave. A longitudinal mode of a resonant cavity is a particular standing wave pattern formed by waves confined in the cavity.

13. §3.6 Resonance Frequencies of Optical Resonators I. Varying q with a fixed l and m q is some integer Longitudinal phase shift: Intermode frequency spacing (or FSR)

14. §3.6 Resonance Frequencies of Optical Resonators II. Varying l and m with a fixed q All modes with the same value of l + m for a given q are degenerate. For different l + m

15. §3.6 Resonance Frequencies of Optical Resonators 1) For a confocal resonator n l + m = const. q = const. q q +1 q +2 q +3 l+ml+m l + m +1 l + m +3 l + m +5 l + m +7

16. §3.6 Resonance Frequencies of Optical Resonators 2) For a nearly planar resonator n q q +1 123123 Bad for spectral analyzer

17. 3) For the general case §3.6 Resonance Frequencies of Optical Resonators +:When both and are positive -:When both and are negative

18. §3.7 Losses in Optical Resonators An understanding of the mechanisms by which electromagnetic energy is dissipated in optical resonators and the ability to control them are of major importance in understanding and operating a variety of optical devices. Loss per passPhoton lifetimeQuality factor 1) Photon lifetime 2) Loss per pass with cavity length l

19. §3.7 Losses in Optical Resonators For a resonator with mirrors ’ reflectivity R 1 and R 2 and average distributed loss constant a 3) Quality factor

20. §3.7 Losses in Optical Resonators The most common loss mechanisms in optical resonators are: 1. Loss resulting from nonperfect reflection. 2. Absorption and scattering in the laser medium. 3. Diffraction losses.