Non-planar and planar ring cavities

The multi-pass ring cavity is constructed by a pair of identical spherical mirrors (two laser mirrors are called input coupler and output coupler). While the reentrant condition is satisfied and all laser beam paths locate on the same plane, it is called planar ring cavity; otherwise, it is non-planar. The set-up is shown in Fig. 2.1. The parameter “d0” is used to show the gain medium displacement from the optical axis of the cavity. The reentrant condition and stability of ring cavity will be discussed in this section.

A series of multi-pass configurations can exist in this ring cavity. In planar ring cavity, parameter “N” is used to specify a particular configuration [1.7]. An N-point planar ring cavity refers to a ring cavity with N beam spots on each coupler while the reentrant condition is satisfied. Take the planar N=2 ring cavity as an example, as shown in Fig. 2.2, the laser beam bounces back and forth between the two cavity mirrors M1 and M2.

Optical axis

I/C Gain medium O/C

Laser Laser diode

Lens

d0

Fiber lens

Fig. 2.1. Multi-pass ring laser set-up.

Fig. 2.2. Beam path in the 2-point planar ring cavity. O2 is the spherical center of M2.

Assuming that the beam is symmetrical about the optical axis, the coordinates for the laser spots on the input and output couplers can be expressed as

where d is the distance between the outermost laser spot on the coupler and the optical axis of the cavity (d is equal to d0 in planar ring cavity), L is the cavity length, and R is the radius of curvature of the coupler. The reentrant conditions for planar N=2 and 3 ring cavities can be expressed using the reflection law as

The ratio of L over R of the N=3 planar cavity is exactly half that of the N=2 planar cavity, and they are plotted in Fig. 2.3. The cavity length slightly decreases when d increases.

Fig. 2.3. Normalized cavity lengths (L/R) for N=2 and N=3 planar ring cavities.

The reentrant conditions in the planar ring cavity become very complicated when N 4≥ . A Fortran program was used to solve the problem. A ray-tracing software package, Beam4 (by Stellar Software), was used to generate the beam paths, which are shown in Table 2.1. The round trip length for the planar ring cavity is approximately equal to N times that of the linear cavity.

In non-planar ring cavity, two parameters, “N” and “M”, are needed to define a particular configuration [1.5]. “N” is the number of beam spots at a coupler (all laser spots at a coupler locate on a circle), and “M” is the number of times the beam circulates in a round-trip, the cavity length, L, can be expressed as when viewing along the optical axis of the cavity.

Table 2.2 depicts how different the planar and non-planar ring cavities are. The yellow and green spots refer to the laser spots on input and output couplers respectively.

Comparing 3-point planar with (3, 1) non-planar ring cavities, although the cavity length and the beam path of side view are quite close, the beam path of the end views are quite different. In planar ring cavity, the beam path of the end view always looks

0.4

like a straight line. In (N, 1) non-planar ring cavity, the beam paths of the end view always forms a regular 2N-edged polygon. Comparing the (3, 1) and (3, 2) non-planar ring cavities, not only the cavity lengths are different, but also the beam paths.

Table 2.1. Beam paths in planar ring cavities with R=80 mm and d=10 mm.

N Side view Round trip length / 2L

2 2.00012

3 3.00025

4 4.00089

5 5.00258

6 6.00641

7 7.01431

Table 2.2. The beam path for N is 3 of planar and non-planar ring cavity.

Figure 2.4 shows the ratio of cavity length to R of the planar and non-planar ring cavities. The L/R ratio of both cavities decrease with increasing N. The cavity length of N-point planar ring cavity is slightly shorter than that of the non-planar (N, 1).

After discussing the reentrant condition of the ring cavities, the ring cavity modes are analyzed. We use ABCD matrix method to analyze the transverse mode stability of planar ring cavity [1.7], because the planar ring cavity is an orthogonal configuration.

Fig. 2.4. Normalized lengths of planar and non-planar ring cavities.

Non-planar N=3 M=1

Planar N=3

side view

end view

M=2 Ring

cavity

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

2 3 4 5 6 7 8

N

L/R

N-point planar ring cavity (N, 1) non-planar ring cavity

i i i i matrix and stability criterion starting from the cavity center for the x- and y-directions are expressed as

The cavity is stable if the following equation is satisfied.

In an N-point planar ring cavity (N≧2, integer), we define %n to be

When N is an even number, there exists %n different incidence angles and %n 1+ different beam pass lengths when the beam propagates in a round-trip of the cavity.

When N is an odd number, there exist %n 1+ different incidence angles and %n 1+ different beam lengths. As shown in Fig. 2.5, the round-trip ABCD matrix can be expressed as Mi02, where

where is the Gauss symbol of .

2 2

p₁

Fig. 2.5. Beam paths in the planar ring cavities when N is (a) even and (b) odd.

The effective radii of curvature in the plane perpendicular to the plane of incidence and in the plane of incidence are R/cosθand Rcosθ, respectively. Figure 2.6(a) shows the stability analysis in the x-direction. The cavity is always critically stable for all N and d. Figure 2.6(b) shows the stability analysis in the y-direction. The cavity is stable when N is an odd number, but otherwise is unstable. The Sy is closer to 1 for smaller d.

When the thickness of the gain medium is taken into consideration in the cavity, Figs. 2.7 and 2.8 show the stabilities with gain media of various effective thicknesses of the 2-point and 3-point planar ring cavities. The effective thickness of the gain medium is defined as

where n and t are the refractive index and thickness of the gain medium. In Fig. 2.7, the gain medium is placed in the center of the planar N=2 cavity so that the cavity still had a symmetrical configuration. The cavity is more stable with a thicker gain medium both in the x- and y-directions. When the gain medium is placed on one side of the beam path, the symmetry of the cavity is destroyed. The cavity is stable in the x-direction, as

z

shown in Fig. 2.7(c), but becomes unstable with the thicker gain medium in the y-direction, as shown in Fig. 2.7(d).

(a)

(b)

Fig. 2.6. Stability analyses in (a) x-direction and (b) y-direction for the empty planar N=2-7 ring cavities with R=80 mm.

0.997 0.998 0.999 1.000 1.001 1.002 1.003 1.004

0 0.05 0.1 0.15 0.2

d/R Sx

N=2N=3 N=4N=5 N=6N=7

0.997 0.998 0.999 1.000 1.001 1.002 1.003 1.004

0 0.05 0.1 0.15 0.2

d/R Sy

N=2N=3 N=4N=5 N=6N=7

Fig. 2.7. Stabilities with gain media of various effective thicknesses of planar N=2 ring cavities. Stability analyses in (a) x-direction and (b) y-direction with the gain medium at the center of the cavity. Stability analyses in (c) x-direction, and (d) y-direction with the gain medium on the side arm of the cavity.

Figure 2.8 shows the stability analysis for the planar N=3 ring cavity with the gain medium on one side arm of the beam path. The cavity is stable in the x-direction, as shown in Fig. 2.8(a), and is stable only when d/R is larger than a critical value in the y-direction, as shown in Fig. 2.8(b).

Table 2.3 conclude the characteristics and advantages of planar and non-planar ring cavities. The non-planar ring cavity with long round-trip lengths has strong potentiality to be applied in a compact mode-locked laser with moderate repetition rate.

Additionally, all beams and laser spots on mirrors of the non-planar ring cavities are identical. Therefore, if chirp mirror is designed on cavity mirrors, the dispersion compensation can be enhanced by 2N fold for mode-locked lasers and no extra

0.99

dispersive component is needed. For planar ring cavities, good linear polarized laser is achieved for cubic gain medium, such as Yb:YAG and Nd:YAG. The ellipticity increases with higher N because the beam hits the mirror 2N times during a round-trip propagation.

Fig. 2.8. Stability analyses in (a) x-direction and (b) y-direction for various effective thicknesses of gain medium placed on one side arm of the beam path in the planar N=3 ring cavity.

Table 2.3. Advantages of planar and non-planar ring cavities.

Non-planar Planar

x-y plane y-z plane x-z plane

Advantages

Compact and less components Enhance dispersion