Two-Mirror 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.

In this model, two concave mirrors are placed at opposite ends of a laser cavity. This two-mirror cavity is the most basic configuration for studying laser stability. A single ray is released from a point within the cavity, initially with a very small angle relative to the optical axis. 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 spherical end mirrors with radii of curvature both equal to R = 1 m. The mirrors are separated by distance L (SI unit: m) as shown in Figure 1. A ray is released from the center of one of the mirrors, at a very small angle to the optical axis. 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 L.

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 the laser cavity.

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

and reflection in a mirror with the radius of curvature R in the air by

After the ray is reflected by both mirrors once and returns to its original 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

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. Some arithmetic reduces these inequalities to

(1)

which predicts that the stable range is 0 ≤ L ≤ 2R. Note that the computation may result in unstable results for L = R and L = 2R, where the cavity is marginally stable.

Results and Discussion

Figure 2 shows the ray propagation when the distance between the mirrors is L = 0.2 m, the smallest parameter value used. 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 cavity length, which shows good agreement between the computed results and the ABCD matrix theory. The two results differ when the value of L is slightly outside the region of stability, for example at L = 2.2 m. This is because the stability criterion derived from the ABCD matrix theory holds for an arbitrarily large number of reflections, whereas in the ray optics simulation the maximum number of reflections is finite.

At points of marginal stability, such as L = 2.0 m, the two results disagree because the analytic result from ABCD matrix theory is entered into the model as a smoothed step function.

Figure 2: Ray tracing result for L = 0.2 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/two_mirror_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 two_mirror_laser_cavity_parameters.txt.

Definitions

Create a function. This function will be used during postprocessing to define the theoretical stability criterion.

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 2*R.

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

Geometry 1

Create a laser cavity geometry.

Part Libraries

In the toolbar, click

and choose .

In the window, under click .

In the window, select in the tree.

Click

In the dialog box, 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	-1 m	Radius of curvature (+convex/-concave)
Tc	12[mm]	0.012 m	Center thickness
d0	D	0.025 m	Mirror full diameter
d1	0	0 m	Mirror surface diameter
d_clear	0	0 m	Clear aperture diameter
d_hole	0	0 m	Center hole diameter
nix	0	0	Local optical axis, x-component
niy	0	0	Local optical axis, y-component
niz	-1	-1	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

Click to expand the section. Click to select row number 2 in the table.

Click .

In the dialog box, type Mirrors in the text field.

Click .

Spherical Mirror 3D 2 (pi2)

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
niz	1	1	Local optical axis, z-component

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

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


Name	Keep	Physics	Contribute to
Mirror surface		√	Mirrors

Click

. 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.

Mirror 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose . These are the mirror surfaces that face the inside of the cavity.

Release from Grid 1

In the toolbar, click

and choose .

In the window for , locate the section.

Specify the L0 vector as


sin(th)	x
0	y
cos(th)	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/20.

Click

Study 1

Add a to vary the cavity length to see the effect on the stability.

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

In the table, click to select the cell at row number 1 and column number 2.

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
L (Cavity length)		m

Click

In the dialog box, type 0.2 in the text field.

In the text field, type 0.2.

In the text field, type 2.4.

Click .

Step 1: Ray Tracing

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

In the window, click .

In the window for , locate the section.

In the text field, type 0 T0.

From the list, choose .

In the toolbar, click

Results

Ray Trajectories (gop)

In the window for , locate the section.

From the list, choose .

Click to expand the section. From the list, choose .

In the text area, type Ray trajectory, L=0.2.

Surface 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Ray Trajectories 1

In the window, click .

In the window for , locate the section.

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

Color Expression 1

In the window, expand the node, then click .

In the window for , locate the section.

Clear the check box.

In the toolbar, click

For values of L 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