Passive Optical Resonators

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1 Passive Optical Resonators

2 Optical Cavities and Feedback Back mirror I 0 I 1 Laser medium with gain, G Output mirror I 3 R = 100% R < 100% I 2 Cavities are essential components of the lasers. They provide the required feedback for laser oscillation. Cavities also act as resanators and frequency filters. Hence they drastically reduce the number of modes that can oscillate with low loss. Here, we will consider passive optical resonators, which do not include an active medium within. Practiacal resonator sizes range from micrometers to meters.

3 Closed vs. Open Cavities We used closed cavities when we discussed blackbody radiation. We have seen that the cavity modes correspond to stationary electric field configurations. In lasers we use open cavities, which have much smaller number of modes. (~10 9 vs ~10) The electric field of a mode can be written as: Where τ c is called the «cavity photon decay time». We are going to find stable solutions for u(r).

4 Transverse vs. Longitudinal Modes Self-reproducing solutions of u(r) are the transverse modes of the cavity. They yield the spatial mode distribution of the output beam. Resonant frequencies (or ω) for maximum power output form the longitudinal modes. We will use geometrical optics for qualitative examination and wave optics for rigorous calculation of modes.

5 Plane-Parallel (Fabry-Perot) Resonator This is essentially a Fabry-Perot etalon. The resonant frequencies, or the longitudinal modes of the cavity are given by: This also means that the cavity modes form standing waves, where the amplitude is zero at the mirrors. The frequency difference between consequitive modes is: These frequencies are what we called the «longitudinal modes». n determines the number of half wavelengths of the mode along the cavity.

6 Concentric Resonator This resonator is made of two spherical mirrors of coincident centers. All rays have the same path length. Resonant frequencies are the same as the plane parallel resonator of separation L.

7 Confocal Resonator In this case, the resonant frequencies cannot be obtained from geometrical optics, hence they are different from plane parallel resonator.

8 Generalized Spherical Resonator Resonators can also have cavity separation between concentric and confocal resonators. The mirrors can also be convex. For this case, we cannot find a complete loop with ray tracing. A general resonator has mirrors (convex or concave) with arbitrary radii of curvature, separated by an arbitrary distance. A resonator is called «stable» if the rays tend to stay within the cavity. A resonator is «unstable» if the rays diverge after some bounces. An unstable resonator.

9 Ring Resonator Here, the optical path is arranged in a ring configuration. For resonant frequencies, the total phase shift after one round trip should be an integer multiple of 2π. Ring resonators can also be stable or unstable. Resonance frequencies are given by: Where L p is the length of the perimeter of the ring.

10 Stable Resonators Multiple bounces between the cavity mirrors can also be considered as continuous propagation through periodic lenses. For stable propagation, the field should reproduce itself after one round trip. We will use wave optics to find the longitudinal modes, and ABCD matrix method to find the stability condition.

11 Eigenmodes and Eigenvalues Using the Fresnel-Kirschoff integral we can write the reproduction condition as: K is the propagation kernel, and is determined by the cavity. The reproduction condition requires Where σ is a complex number Real part of σ describes the change in intensity (i.e. number of photons) and its imaginary part describes the change in round trip phase (hence affects longitudinal modes). Self-reproducing field distributions are called eigenmodes and corresponding are called eigenvalues. eigenmodes: eigenvalues: l and m are pair of integers.

12 Eigenvalues describe the losses and resonances Remember the relationship between the intensity and electric field: r I = cε E 1 2 ~ 0 2 Using this, we can define a round trip fractional power loss (also called diffraction loss): In wave optics, the total round trip phase includes imaginary part of σ : Hence, the resonant frequencies become:

13 Photon Lifetime The energy in the cavity decays due to internal losses and mirror transmissions. After one round trip, the intensity becomes is the round-trip time T i is fracional internal loss per pas. After m round trips, the intensity and total number of photons become: For large number of round trips, we can ignore discreteness of t m and set an exponential decay: Using the expression for t m : Hence, we obtain the photon lifetime:

14 Photon Lifetime and the Bandwidth Remember that we also defined logarithmic losses: Therefore, the photon lifetime can also be written as: γ ln γ ln γ ln 1 γ γ γ γ 2 photon lifetime equals the transit time in the laser cavity divided by the cavity loss. The electric field of the output beam will be in the form: By Fourier transform, we find that the corresponding bandwidth is: This bandwidth is essentially the same as Fabry-Perot bandwidth.

15 The Cavity Quality Factor We can find that this is equal to: Or, using the definition of the bandwidth: Hence, the Q-factor is equal to the resonance frequency divided by its bandwidth.

16 Photon Lifetime and Q-Factor: Example Using R 1 = R 2 = 0.98 T i = 0 L = 90 cm We obtain: τ transit = 3 ns τ c = 150 ns Note that τ c >> τ transit Using 633 nm wavelength, we find: Q ~ Very high Q means that very small fraction of the energy lost per one optical cycle.

17 Stability Condition To find the condition for stable oscillation within the cavity, we will use the ABCD matrices. For one round-trip: For two round-trips: For n roundtrips:

18 Stability Condition If the resonator is to be stable, we require that, for any initial point (r 0, r 0 ), output point (r n, r n ), does not diverge as n increases. This means that the matrix must not diverge as n increases. This condition holds even if the cavity contains other elements.

19 Defining: Sylvester s Theorem If θ has an imaginary part, jnθ term becomes real and diverges with n. Therefore, n th power of the matrix does not diverge if θ is real. Then, we obtain the stability condition:

20 Generalized Spherical Resonator By defining dimensionless g parameters: The stability condition becomes:

21 Graphical Representation of Stability R 1 = R 2 A: concentric B: confocal C: Plane Points A, B and C lie on the boundary. Such resonators are called marginally stable.

22 Stable Resonator Modes The Gaussian beam should reproduce itself after one round-trip. Remember the Gaussian beam complex q parameter and its propagation through ABCD matrix: 1 1 λ i q z R z π w z 2 ( ) ( ) ( ) OR: In order to have a real beam width, q has to be complex, or the discriminant has to be negative: Using: (verify that this is the case for all matrices we saw) OR This is the stability condition. This means that Gaussian beam solutions can be found only for stable resonators.

23 Hermite-Gauss modes are stable solutions of cavities

24 ABCD Matrices Have Determinat of 1

25 Eigenmodes By using the ABCD matrix for the cavity and the condition that the beam should reproduce itself after one round trip: Using the round-trip ABCD matrix, we can show that the condition above is equivalent to requiring that the radius of curvature of the beam must match the radius of curvature of the mirrors at mirror position.

26 Spot Sizes Within the Cavity Again, using the q transformation, we can find that the spot sizes at the mirrors and minimum spot size within the cavity are: For a symmetric resonator, R 1 = R 2 = R, g 1 = g 2 = 1 L/R. Hence:

27 Symmetric Resonator

28 Eigenvalues To find the diffraction loss and resonant modes of the cavity, we need to find the cavity eigenvalues. For Gaussian beam modes of a stable resonator, the magnitude of σ is 1. That means, the diffraction loss is zero. The phase of σ determines the resonant modes. Round trip phase shift should be an integer multiple of 2π. We obtain: A 1 and D 1 are for single pass matrix. Indices l and m indicate Hermite polynomial order. In particular, for a two mirror resonator, the resonant frequencies are:

29 For a symmetric confocal resonator, g 1 = g 2 = 0 : Frequency Spectra Spacing between two transverse modes (consequtive l, m) is c/4l. Spacing between two longitudinal modes (consequtive n) is c/2l. For a near-plane resonator, R>>L, g 1 = g 2 = ~ 1-L/R :

30 Unstable Resonators The stable resonators require small spot sizes. Hence the effective gain volume and power amplification will be limited. With unstable resonators, the beam size is not limited. Higher power outputs can be obtained.

31 Unstable Resonator Configurations Unstable resonators do not have Hermite-Gaus solutions. The solution can be obtained from numerical solution of the FK integral. Edges of the output mirror.

32 Advantages of Unstable Resonators - Mode volume is larger and controllable. - Transverse modes are easily discriminated. - All mirrors can be perfectly reflecting. Disadvantages of Unstable Resonators - The output beam is ring-shaped. - The beam has diffraction rings. - The laser is more sensitive to the cavity perturbations.

33 Gaussian - Supergaussian Apertures The reflectivity of a Gaussian (n = 2) and supergaussian (n>2) aperture is: The smooth aperture reduces the diffraction rings.

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