COMSOL 6.4 - Bow-Tie Laser Cavity

Lasers are ubiquitous in application areas such as cutting, ablation, telecommunication, and spectroscopy, among others. Typically, lasers are produced by a laser cavity or optical cavity containing a set of mirrors, a gain medium, and possibly some other optical components such as prisms or lenses.

Stability analysis of the laser cavity ensures that light remains confined in the cavity, allowing the laser to operate reliably. If the laser cavity is not stable, laser production may abruptly stop as light escapes from the cavity into the surroundings. The stability of the laser cavity can be analyzed by the standard ABCD matrix analysis based on the paraxial approximation, or alternatively by geometrical optics simulation.

This is a model of a symmetric bow-tie laser cavity consisting of two flat mirrors and two spherical mirrors. A single ray is released from the surface of one of the flat mirrors, initially with a very small angle relative to the surface normal. Then the ray is traced for a predefined time period that is sufficiently long for many reflections to occur. Ray tracing continues until the predefined computation time has passed if the laser cavity is stable, whereas the time-dependent study terminates earlier if the ray escapes from the cavity. A demonstrates the effect of cavity length on stability and compares the result with the ABCD matrix theory.

Model Definition

The laser cavity consists of two identical flat end mirrors and two identical spherical mirrors. The spherical mirrors have surface radii of curvature R = 0.5 m. Each flat mirror is a distance L1 = 0.1 m from the symmetry plane. Each spherical mirror is a distance L2 from the symmetry plane; this half-distance is increased from 0.02 m to 0.6 m using a . The end mirror surfaces are also tilted at a small angle θ (SI unit: rad) relative to the symmetry axis. The cavity geometry is illustrated in Figure 1.

A single ray is released from the center of one of the flat mirrors, at a very small angle to the surface normal. The ray is traced for a predefined total computation time, T0 (SI unit: s), which is sufficient for it to be reflected a large number of times, at least several hundred reflections for the largest value of L2.

The feature is used to end the time-dependent study early if the ray gets out of the cavity; in that case, the last computation time, T1 (SI unit: s) is stored. The cavity stability is represented by the ratio T1/T0, with a value of 1 indicating that the ray is still inside the cavity and the configuration is stable.

Figure 1: Optical layout of a laser cavity in a symmetric bow-tie configuration.

ABCD Matrix Theory

The result of the ray tracing analysis can be compared to an analytic solution based on ABCD matrix theory, as long as the paraxial approximation holds. In ABCD matrix theory, while following Hecht’s notation (Ref. 1), a ray is characterized by the ray angle θ (SI unit: rad) and the ray position y (SI unit: m) relative to the optical axis in a 2-by-1 column vector as

Elements of the optical system are represented as 2-by-2 matrices that are multiplied by this vector. Propagation through a distance L is denoted by the matrix

Reflection at a mirror with the radius of curvature R in the air by

After one round trip of propagation, in which the ray is reflected by every mirror once and returns to its initial position, the new angle θ and position y of the ray can be described by the matrix product of each propagation or reflection in the sequence, multiplied by the initial angle θ0 and position y0,

where T is the product of eight 2-by-2 matrices, corresponding to the four reflections and four propagation distances. According to Kogelnik’s stability theory (Ref. 2), the system is stable if the initial angle and position give bounded values when multiplied by an arbitrarily high power of the matrix T; this stability criterion can also be written as

where Tr stands for the trace of a matrix. For the parameter values used in this model, the stable values of L2 satisfy the inequality

(1)

With L2 in meters. The inequality is satisfied for 0 ≤ L2 ≤ 0.15 and for 0.25 ≤ L2 ≤ 0.455.

Results and Discussion

Figure 2 shows the ray propagation when the half-distance between the flat mirrors is L2 = 0.12 m. The ray remains confined inside the cavity for the full duration of the study. If the total computation time is sufficiently long, the result means the cavity is stable for this particular parameter value.

Figure 3 is a 1D plot of the stability versus the mirror half-distance. It shows good agreement with the values 0 ≤ L2 ≤ 0.15 and for 0.25 ≤ L2 ≤ 0.455 which were found to satisfy Equation 1 from the ABCD matrix analysis in the previous section. The slight difference in the results can be attributed to the mirror tilt, which is considered in the ray optics simulation but ignored in the ABCD matrix theory. The ray tracing simulation gives a higher-fidelity result because it does not rely on the paraxial approximation.

Figure 2: Ray tracing result for L = 0.1 m. The ray is confined in the cavity after the total computation time T0 = 1 μs.

Figure 3: Stability plot as a function of the cavity length showing a good agreement with the ABCD matrix theory.

References

1. E. Hecht, Optics, 4th ed., Addison-Wesley, 1998.

2. H. Kogelnik and T. Li, “Laser beams and resonators,” Applied Optics, vol. 5, no. 10, pp. 1550–1567, 1966.

Ray_Optics_Module/Laser_Cavities/bow_tie_laser_cavity

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

Load the global parameters for the laser cavity from a text file.

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file bow_tie_laser_cavity_parameters.txt.

Definitions

Define some functions that will be used to compare the ray tracing results to ABCD matrix theory during postprocessing.

Analytic 1 (an1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.04*(51200*x^4-40960*x^3+8832*x^2-256*x-23).

Locate the section. In the table, enter the following settings:


Plot	Argument	Lower limit	Upper limit	Fixed value	Unit
√	x	0	0.6	0

This analytic function is the expression from the left-hand side of Equation 1. It is not used later in this model, but you can click the button and use the resulting plot to estimate the regions of stability and points of marginal stability.

Rectangle 1 (rect1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 0.

In the text field, type 0.15.

Click to expand the section. In the text field, type 0.001.

Rectangle 2 (rect2)

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.25.

In the text field, type 0.455.

Analytic 2 (an2)

In the toolbar, click

In the window for , locate the section.

In the text field, type rect1(x)+rect2(x).

Locate the section. In the table, enter the following settings:


Plot	Argument	Lower limit	Upper limit	Fixed value	Unit
√	x	0	0.6	0

Geometry 1

Create the laser cavity geometry. The flat mirrors are cylinders. The spherical mirrors are based on a part from the Part Library.

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type D_FM/2.

In the text field, type L_FM.

Locate the section. In the text field, type X_FM.

In the text field, type Y_FM.

In the text field, type Z_FM.

Locate the section. From the list, choose .

In the text field, type 180+th.

Click

Part Libraries

In the toolbar, click

In the window, click .

In the window, select > > > in the tree.

Click

In the dialog, select in the list.

Click .

Geometry 1

Spherical Mirror 3D 1 (pi1)

In the window, under > click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
R	-R_SM	-0.5 m	Radius of curvature (+convex/-concave)
Tc	10[mm]	0.01 m	Center thickness
d0	D_SM	0.0125 m	Mirror full diameter
d1	D_SM	0.0125 m	Mirror surface diameter
d_clear	D_SM	0.0125 m	Clear aperture diameter
d_hole	0	0 m	Center hole diameter
nix	sin(RY_SM)	0.087156	Local optical axis, x-component
niy	0	0	Local optical axis, y-component
niz	cos(RY_SM)	0.99619	Local optical axis, z-component
n_extra_r	0	0	Number of extra radial points
n_extra_a	0	0	Number of extra azimuthal points

Locate the section. Find the subsection. In the text field, type X_SM.

In the text field, type Y_SM.

In the text field, type Z_SM.

Click

Reflect each mirror across the xy-plane to complete the cavity geometry.

Mirror 1 (mir1)

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select both objects.

In the window for , locate the section.

Select the checkbox.

Click

In the window toolbar, click

next to

, then choose . Compare the resulting geometry to Figure 1.

Geometrical Optics (gop)

In the window, under click .

In the window for , locate the section.

Click

. The mirror domains can be excluded because rays never actually pass through them in this model.

Locate the section. In the text field, type 0.

Ray Properties 1

In the window, under > click .

In the window for , locate the section.

In the λ0 text field, type wl.

Mirror 1

In the toolbar, click

and choose .

Select Boundaries 7, 8, 15, and 18 only; that is, select all of the boundaries facing the inside of the cavity.

Add a node to release one ray from the flat mirror surface at the predefined initial ray angle with respect to the normal.

Release from Grid 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the qx,0 text field, type X_FM.

In the qy,0 text field, type Y_FM.

In the qz,0 text field, type Z_FM.

Locate the section. Specify the L0 vector as


sin(th+dth)	x
0	y
cos(th+dth)	z

Ray Termination 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Mesh 1

Adjust the default mesh to improve the resolution of the curved mirror surfaces.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Size

In the window, under > click .

In the window for , click to expand the section.

In the text field, type D_SM/20.

In the text field, type D_SM/40.

Click

Study 1

Step 1: Ray Tracing

Add a to end the simulation when the ray gets out of the cavity.

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,dt,T0).

From the list, choose .

Add a to vary the spherical mirror half distance to see the effect on the stability. Instead of entering uniform length intervals, use smaller increments close to the marginally stable regions predicted by Equation 1.

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
L2 (Spherical mirror half distance)	range(0.02,0.02,0.14) range(0.142,0.002,0.152) range(0.16,0.02,0.24) range(0.242,0.002,0.252) range(0.26,0.02,0.44) range(0.442,0.002,0.452) range(0.46,0.02,0.6)	m

In the toolbar, click

Results

Ray Trajectories (gop)

In the window for , locate the section.

From the list, choose .

Surface 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Ray Trajectories 1

. This ensures that the ray path is rendered correctly even for the smallest distance between the mirrors, when there are several thousand reflections.

In the window, click .

In the toolbar, click

Rotate and zoom in as needed. For values of L2 that satisfy the stability criterion (Equation 1), the plot should look like Figure 2. Otherwise, the ray eventually escapes from the cavity.

1D Plot Group 2

In the toolbar, click