Defining the modal basis

Using the correct modal basis (via beam parameters) at each point in the model configuration is important for computing couplings through Scattering matrices and other quantities such as the spatial distribution \(u_{nm}\) used for, e.g., calculating beam profiles. For these reasons, it is necessary to understand how Finesse sets beam parameters at nodes and how you can use this information to model the systems that you are investigating.

Complete details on the beam tracing algorithm of Finesse are included in Tracing the beam, this page will focus on how to use cavities and manual setting of beam parameters.

Using cavity objects

It is important in any analysis involving cavities that HOM couplings are computed using a (propagated) cavity eigenmode as the basis. If the beam parameters at nodes inside a cavity get set to values which are not equal to the corresponding cavity eigenmode then we can get mismatches where there shouldn’t be any and / or matches where there should be mismatches.

A key role of the Cavity object is to prevent this from happening by influencing the beam tracing algorithm when instances of this class are present in a model. The eigenmodes of all cavities in a model are computed upon adding it to the model and these are then used to set beam parameters in the rest of the model via propagating them through the ABCD matrix formalism, see Transformation of the beam parameter for details on this.

Adding cavities to a model

Cavity objects can be added to a model in the standard way via a call to Model.add() using the Python API. The constructor of the Cavity class requires a name for the cavity and the source node (i.e. starting point) of the cavity; with an optional via node if the cavity path should traverse via a specific node.

For simple cavities where the surfaces are not bounded (on the inside of the cavity path) by AR / HR coating surfaces, you only ever need to specify the source node or port. The cavity path will then be determined automatically from just this node. An example of this is shown below, using a simple Fabry-Perot type cavity model:

import finesse
import finesse.components as components

model = finesse.Model()
model.parse("""
l L0 P=1
s s0 L0.p1 ITM.p1

m ITM R=0.99 T=0.01 Rc=-2.5
s sCAV ITM.p2 ETM.p1 L=1
m ETM R=0.99 T=0.01 Rc=2.5

# Add a cavity object with source node as output node
# of the second port of mirror ITM
cav FP ITM.p2
""")

# showing the cavity path
print(model.FP.draw())

o ITM.p2.o
╰──o ETM.p1.i
   ╰──o ETM.p1.o
      ╰──o ITM.p2.i

If you have a model with AR / HR coatings at surfaces then you will need to specify the via node argument in order for the cavity path to include the surface itself. See the example below:

model = finesse.Model()
model.parse("""
l L0 P=1
s l0 L0.p1 PRM.p1 L=1

m PRM T=0.046 L=37.5u

s lpr1 PRM.p2 BS.p1 L=9

bs BS T=0.5 L=37.5u alpha=60

s ly BS.p2 ITMAR.p1 L=100

# Y arm input mirror with AR surface
m ITMAR R=0 L=20u
s ITMsub ITMAR.p2 ITM.p1 L=0.2 nr=1.44963098985906
m ITM T=7000u L=37.5u Rc=-5580

lambda(1550n)
""")

# INCORRECT -- doesn't include ITM
model.add(components.Cavity("PRC_wrong", model.PRM.p2.o))

print(model.PRC_wrong.draw())

o PRM.p2.o
╰──o BS.p1.i
   ╰──o BS.p2.o
      ╰──o ITMAR.p1.i
         ╰──o ITMAR.p1.o
            ╰──o BS.p2.i
               ╰──o BS.p1.o
                  ╰──o PRM.p2.i

You can see from this cavity path that the cavity “end” point is ITMAR, not ITM, as by default the cavity will find the shortest path back to the source node. To correct this, you must use the via node:

# CORRECT -- now we specify that the path must traverse via ITM
model.add(components.Cavity("PRC", model.PRM.p2.o, model.ITM.p1.i))

print(model.PRC.draw())

o PRM.p2.o
╰──o BS.p1.i
   ╰──o BS.p2.o
      ╰──o ITMAR.p1.i
         ╰──o ITMAR.p2.o
            ╰──o ITM.p1.i
               ╰──o ITM.p1.o
                  ╰──o ITMAR.p2.i
                     ╰──o ITMAR.p1.o
                        ╰──o BS.p2.i
                           ╰──o BS.p1.o
                              ╰──o PRM.p2.i

Showing the importance of including Cavity objects

Below is an example highlighting the importance of including Cavity objects in a model for HOM modelling. In this example, the detuning of the input mirror of a Fabry-Perot cavity is scanned whilst the power on transmission is detected.

import finesse

model = finesse.Model()
# define a configuration with no Cavity object present
model.parse("""
l L0 P=1
s s0 L0.p1 ITM.p1

m ITM R=0.99 T=0.01 Rc=-2.5
s sCAV ITM.p2 ETM.p1 L=1
m ETM R=0.99 T=0.01 Rc=2.5

# set a beam parameter with w0 = 1 mm, z-z0 = -0.3 m at the laser
gauss gL0 L0.p1.o w0=1m z=-0.3

# detect the transmitted power
pd trns ETM.p2.o

modes(maxtem=4)
""")

out_no_cav = model.run("xaxis(ETM.phi, lin, -180, 180, 500)")

Now we add a Cavity object to the model and run it again, we also display the mismatches that are now present in the system due to the manually set beam parameter at the laser and the beam parameters automatically set in the cavity based on its eigenmode. See Creating and detecting mode mismatches for details on the Model.mismatches_table() method and more.

# source node is output of second port of ITM
# -> cavity path is then determined automatically from this
model.add(finesse.components.Cavity("FP", model.ITM.p2.o))

print(model.mismatches_table())

out_cav = model.run("xaxis(ETM.phi, lin, -180, 180, 500)")

┌──────────────────────╥──────────────┬──────────────┐
│       Coupling       ║ Mismatch (x) │ Mismatch (y) │
╞══════════════════════╬══════════════╪══════════════╡
│ ITM.p1.i -> ITM.p2.o ║       0.2460 │       0.2460 │
├──────────────────────╫──────────────┼──────────────┤
│ ITM.p1.i -> ITM.p1.o ║       0.5437 │       0.5437 │
├──────────────────────╫──────────────┼──────────────┤
│ ITM.p2.i -> ITM.p1.o ║       0.2460 │       0.2460 │
└──────────────────────╨──────────────┴──────────────┘

And then plot the results of each to show the difference:

import matplotlib.pyplot as plt
finesse.configure(plotting=True)

fig = plt.figure()
plt.semilogy(out_no_cav.x1, out_no_cav["trns"], label="No cavity tracing")
plt.semilogy(out_cav.x1, out_cav["trns"], label="With cavity tracing")
plt.xlabel("ETM phi [deg]")
plt.ylabel("Transmitted power [W]")
plt.legend();

From this plot you can see that with the cavity included we get the expected resonance peaks corresponding to the fundamental (TEM00) mode and the 02/20 and 04/40 modes; the latter arising due to the mismatch present within the system. Without the cavity object present, the transmitted power trace obtained is nonsensical.

Defining manual beam parameters

Before we demonstrated how we can use the cavity command to automatically set the beam parameters in a model. In certain cases you may also need to specify what beam parameter to use manually. For example, if you wanted the beam shape coming out of a laser to be of a particular shape and see what mismatch there is to a cavity.

Here we start with a cavity and defines the mode shape using the cavity command.

import finesse
import finesse.components as components
import matplotlib.pyplot as plt

finesse.init_plotting()

model = finesse.Model()
model.parse("""
l L0 P=1
s s0 L0.p1 ITM.p1

m ITM R=0.99 T=0.01 Rc=-2.5
s sCAV ITM.p2 ETM.p1 L=1
m ETM R=0.99 T=0.01 Rc=2.5
cav FP ITM.p2
# detect the transmitted power
pd trns ETM.p2.o
modes(maxtem=4)
""")

# run the simulation to get the ideal case
out0 = model.run("xaxis(ETM.phi, lin, -180, 180, 500)")

Next we use gauss command to set what shape the beam is at the output of the laser, this will add a new object to the model called g1.

model.parse("gauss g1 L0.p1.o w0=1m z=0")
out1 = model.run("xaxis(ETM.phi, lin, -180, 180, 500)")

We can see what beam parameters are being set by a “gauss” command using:

model.gausses

{<OpticalNode L0.p1.o @ 0x7c8ea41dda90>: <'g1' @ 0x7c8e9df78080 (Gauss)>}

Lastly we can plot the result and see that we have scattering into higher order modes due to the mismatch.

fig = plt.figure()
plt.semilogy(out0.x1, out0["trns"], label="Ideal mode matching")
plt.semilogy(out1.x1, out1["trns"], label="With manual laser shape")
plt.xlabel("ETM phi [deg]")
plt.ylabel("Transmitted power [W]")
plt.legend();

We can also look at the table of mismatches that are ocurring in the ABCD beam tracing calculations. With this we should see siginifcant mismatches around the ITM mirror.

model.mismatches_table()

Coupling	Mismatch (x)	Mismatch (y)
ITM.p1.i -> ITM.p2.o	0.2559	0.2559
ITM.p1.i -> ITM.p1.o	0.5824	0.5824
ITM.p2.i -> ITM.p1.o	0.2559	0.2559

Remove a manual beam parameter or cavity

To remove a manually set beam parameter or cavity command you can use the remove command with the name of the target:

model.remove('g1')

Defining manual beam parameters with the Python API

Above we defined the beam parameter using the KatScript command. To achieve the same result from using the Python API we need to look at the optical nodes q property. We can print these easily:

model = finesse.script.parse("""
l L0 P=1
s s0 L0.p1 ITM.p1

m ITM R=0.99 T=0.01 Rc=-2.5
s sCAV ITM.p2 ETM.p1 L=1
m ETM R=0.99 T=0.01 Rc=2.5
cav FP ITM.p2
# detect the transmitted power
pd trns ETM.p2.o
modes(maxtem=4)
""")
model.beam_trace()
print(model.L0.p1.o.qx)
print(model.L0.p1.o.qy)

BeamParam(w0=581.96 um, z=-500 mm, w=650.66 um, Rc=-2.5 m)
BeamParam(w0=581.96 um, z=-500 mm, w=650.66 um, Rc=-2.5 m)

We can also easily set them as well:

# qx and qy can be set separately or you can set .q to set both at the same time
model.L0.p1.o.q = finesse.BeamParam(w0=1e-3, z=0)
model.beam_trace()

<BeamTraceSolution of beam_trace @ 0x7c8e9df94980 children=0>

Setting the beam parameter this way will not automatically perform a new beam trace through the model. If you want to query the new beam parameters due to your change you must run a model.beam_trace().

Doing the above will have the same effect as using a gaussian beam command as seen before. If you check you will see a Gauss is actually added to the model with an auto-generated name and can be removed if needed.

model.gausses

{<OpticalNode L0.p1.o @ 0x7c8e9deb3800>: <'gL0_p1_o' @ 0x7c8e9dddcec0 (Gauss)>}