Laser Cavity with a Thin Lens

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 flat mirrors are placed at opposite ends of a laser cavity and a thin convex lens is placed halfway between them. This particular type of cavity is often used in ABCD matrix analysis to account for the thermal lensing effect, in which thermal lensing is modeled as a thin focusing lens with a corresponding focal length. 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 flat end mirrors and a thin biconvex spherical lens with focal length f = 0.5 m. The mirrors are positioned at a distance L (SI unit: m) to either side of the lens 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 a laser cavity with a thin lens (not drawn to scale) at the center.

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 flat mirror is given by

Transmission through a thin lens with a focal length f is given by

After the ray is reflected by both mirrors once, passes through the thin lens twice, 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, reflection, and refraction in the sequence, multiplied by the initial angle θ0 and position y0,

where T is the product of the matrices for all elements encountered in one round trip through the cavity, of which there are eight altogether.

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 ≤ 2f.

Results and Discussion

Figure 2 shows the ray propagation when the half-distance between the mirrors is L = 0.1 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 = 1.1 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 the point of marginal stability, L = 1.0 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 = 0.1 m. The ray is confined in the cavity after the total computation time T0 = 1000 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/laser_cavity_thin_lens

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

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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 laser_cavity_thin_lens_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*f.

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

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 Equi-Convex Lens 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	0.5 m	Radius of curvature
Tc	2[mm]	0.002 m	Center thickness
d	D	0.025 m	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

Locate the section. Find the subsection. In the text field, type L-1[mm].

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type D/2.

In the text field, type 10[mm].

Locate the section. In the text field, type -10[mm].

Cylinder 2 (cyl2)

Right-click and choose .

In the window for , locate the section.

In the text field, type 2*L.

Click

Click the

button in the toolbar. Compare the resulting geometry to Figure 1.

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	n	1	Refractive index

Geometrical Optics (gop)

In the window, under click .

In the window for , locate the section.

Click

Select Domain 2 only. 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 .

Select Boundaries 4 and 11 only; that is, select the mirror boundaries facing the inside of the cavity.

Material Discontinuity 1

In the window, click .

In the window for , locate the section.

From the list, choose . Assume an ideal antireflective coating on the lens surface so that reflected rays can be neglected.

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 lens 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 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 7 and 8 only.

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type D/10.

Select the check box.

In the associated 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 (Mirror-lens distance)		m

Click

In the dialog box, type 0.1 in the text field.

In the text field, type 0.1.

In the text field, type 1.2.

Click .

Step 1: Ray Tracing

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

In the window, click .

In the window for , locate the section.

In the text field, type range(0,dt,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=0.1.

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