Fabry–Perot Resonator

By properly arranging a set of mirrors, we can create passive optical resonators that are of great use to probe and manipulate light or to build lasers. The textbook example of a free space optical resonator is the combination of two spherical mirrors along the optical axis – a Fabry–Perot resonator. A peculiar property of such resonators is the fact that a light beam of defined shape and frequency can be transmitted with high transmission through the resonator even though it consists of two highly reflecting mirrors. In this model, we will study the transmission properties of a Fabry–Perot resonator and compare this to the analytic solution.

Figure 1: Geometry configuration of a Fabry–Perot resonator. The resonator consists of curved front and rear mirrors. Inside the cavity (located between the mirrors) and outside the cavity, there is air. The incident Gaussian beam propagates from left to right. Thus, to the left of the cavity there is both the incident and the reflected beam. Inside the cavity there is both a right- and left-propagating beam. To the right of the cavity, there is only the transmitted beam.

Two mirrors with radii ρ1 and ρ2, separated a distance L, can create a stable optical resonator, if they fulfill the stability criterion 0 < g1g2 < 1, where g1,2 = 1- L/ρ1,2 are the stability parameters of the two mirrors. A stable optical resonator defines a set of spatial modes, each one with a defined mode shape. If we want to excite a spatial mode of the resonator, we also have to consider that the distance of the two mirrors creates a resonance condition that needs to be fulfilled for a certain spatial mode. While it is possible to have high transmission on resonance, the resonator is highly reflective for off resonant frequencies.

We can define the free spectral range of the resonator ΔνFSR and the finesse F as

and

where c is the velocity of light and R1 and R2 are the respective mirror reflectance. The bandwidth of the resonance is then defined as

Please note that, depending on the cavity configuration, different spatial modes can exhibit different resonance frequencies. Any input field can be decomposed into the spatial modes of the resonator. Therefore, the resonator will act as a spatial as well as a frequency filter. We will focus here on efficiently exciting the fundamental mode of the resonator.

The fundamental mode of the resonator can be described by the waist,

where λ is the wavelength.

For the case of equal radii of curvature for the mirrors, ρ1 = ρ2, the waist is located at the center of the resonator.

Model Definition

To efficiently model the full wave solution of the 50 mm long Fabry–Perot resonator, the Electromagnetic Waves, Beam Envelopes interface in the bi-directional formulation is used. The first and second waves are two counter-propagating plane waves in vacuum. The geometry consists of the cavity separating the two mirrors and two exterior domains, one in front and one behind the cavity.

The mirrors are approximated as highly reflecting thin dielectric layers, modeled using the Transition Boundary Condition. To compute the reflectivity R of the thin dielectric layer (Transition Boundary Condition – TBC), we assume that the layer has a refractive index n and the surrounding layers have the refractive indices n1 and n2, respectively.

Figure 2: A mirror modeled as a thin dielectric layer with refractive index n, surrounded by materials with refractive index n1 and n2, respectively. The reflectances R1 and R2, for each interface, are indicated at the bottom of the picture. Each mirror is modeled using a Transition Boundary Condition feature.

The two material interfaces, depicted in Figure 2, have the amplitude reflectivities

and

respectively.

The corresponding power reflectances are then

The reflectance R of the film due to interference is

where

is the accumulated phase when passing through the layer

Using a thickness d= λ/100 and refractive indices n = 15 and n1 = n2 = 1, both mirrors have a reflectance of 0.973. This corresponds to a finesse of F = 116.2.

A Gaussian beam is launched using the Scattering Boundary Condition. The input beam waist corresponds to the analytical solution of the cavity mode and the waist is located at the center of the cavity.

Results and Discussion

The sweep over one free spectral range shows only one resonance (Figure 3). This indicates that only one spatial mode is excited.

Figure 3: A frequency sweep over one free spectral range.

Figure 4 shows the norm of the electric field for the first and second waves at resonance. The first wave propagates from left to right and the second wave propagates from right to left. It is clear that at resonance, the field in the cavity is much larger than outside the cavity. Furthermore, there is almost no reflected wave at resonance and all power is transmitted.

Figure 4: The norm of the electric field for the first wave (bottom) and the second wave (top) at the resonance frequency. The arrows show the power flow for the two waves.

The second study (see Figure 5) carries out a finer sweep over the cavity resonance. On resonance, all light is transmitted through the cavity as is expected for good mode match.

Figure 5: A frequency sweep over the resonance peak.

Figure 6 shows the electric field norm for the first wave at resonance. It confirms that the field inside the cavity is much higher than outside the cavity at resonance.

Figure 6: The norm of the electric field for the first wave at resonance.

Wave_Optics_Module/Verification_Examples/fabry_perot_resonator

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

Geometry Parameters

In the window, under click .

In the window for , type Geometry Parameters in the text field.

Locate the section. Click

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

Cavity Parameters

In the toolbar, click

and choose .

In the window for , type Cavity Parameters in the text field.

Locate the section. Click

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

Geometry 1

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type rho1.

Locate the section. In the text field, type rho1.

Circle 2 (c2)

In the toolbar, click

In the window for , locate the section.

In the text field, type rho2.

Locate the section. In the text field, type l_cav-rho2.

Click

Click the

button in the toolbar.

Union 1 (uni1)

In the toolbar, click

and choose .

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

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type l_total.

In the text field, type h_cav/2.

Locate the section. In the text field, type -l_in.

Intersection 1 (int1)

In the toolbar, click

and choose .

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

In the window for , click

Click the

button in the toolbar.

Definitions

In the window, expand the node.

Axis

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Click

Click the

button in the toolbar.

Materials

Air

In the window, under right-click and choose .

In the window for , type Air in the text field.

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	n1	1	Refractive index
Refractive index, imaginary part	ki_iso ; kiii = ki_iso, kiij = 0	0	1	Refractive index

Electromagnetic Waves, Beam Envelopes (ewbe)

In the window, under click .

In the window for , locate the section.

From the list, choose . This reduces the degrees of freedom for the model and restricts the solution to out-of-plane TE waves.

Perfect Magnetic Conductor 1

In the toolbar, click

and choose condition to model even symmetry on the optical axis.

Select Boundaries 2, 4, and 7 only.

Transition Boundary Condition 1

Now, add a to model the mirrors. Assign a refractive index that is much larger than the refractive index of air, to create a high reflectivity for the mirrors.

In the toolbar, click

and choose .

Select Boundaries 9 and 10 only.

In the window for , locate the section.

From the n list, choose . In the associated text field, type n.

From the k list, choose . In the d text field, type d.

Scattering Boundary Condition 1

In the toolbar, click

and choose .

Use a to launch an incident Gaussian beam polarized in the z direction. This is compatible with the out-of plane setting for the interface.

Select Boundary 1 only.

In the window for , locate the section.

From the list, choose .

In the w0 text field, type w0.

In the p0 text field, type l_in+l_cav/2, which places the focal plane for the Gaussian beam at the center of the cavity.

Specify the Eg0 vector as


0	x
0	y
1	z

Reference Point 1

In the toolbar, click

and choose .

If no subfeature is added to the , the reference point will appear at the center point of the boundary. Add a subfeature to make sure the reference point appears on the optical axis.

In the window for , locate the section.

Click

Select Point 1 only.

Scattering Boundary Condition 2

In the toolbar, click

and choose . This makes the boundary transparent for the transmitted field.

Select Boundary 8 only.

Definitions

Create integral operators for use in reflectance and transmittance calculation.

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 1 only.

Integration 2 (intop2)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 8 only.

Variables 1

In the window, right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
R	intop1(ewbe.nPoav2)/intop1(-ewbe.nPoav1)		Reflectance
T	intop2(ewbe.nPoav1)/intop1(-ewbe.nPoav1)		Transmittance

Mesh 1

The beam envelopes interface allows us to resolve only the resulting field envelope after demodulation with the prescribed phase function. Here, we adjust the number of elements perpendicular and along the optical axis to resolve this envelope properly.

In the window, under click .

In the window for , locate the section.

In the NT text field, type 60. This will create the mapped mesh with sixty elements over the wavefront.

In the NL text field, type 30. This will create the mapped mesh with thirty elements along the simulation domain. This will be sufficient to study the Fabry-Perot cavity.

Click