Getting started

strapy was designed primarily for modelling displacement measuring optical interferometers. This guide is intended to give a brief introduction to using strapy for modelling a simple Michelson interferometer, shown in figure 1, and should provide a starting point for modelling other more complex optical systems. Python scripts implementing the example described here may be found in the examples folder of the strapy GitHub repository. To run this example code matplotlib will need to be installed (pip install matplotlib). Further examples can be found in the tests folder.

Schematic of Michelson interferometer.

Figure 1: schematic of a simple Michelson interferometer with nodes used for strapy modelling labeled. Beam splitter, BS; reference mirror Mref; moving measurement mirror Mmes; linear polariser Pol; photodiode PD1.

As can be seen in figure 1, strapy describes optical systems as a network of interconnected nodes. The interferometer shown is linked together with 10 nodes, with counter propagating electric field components defined at each node.

Once installed strapy, along with the numpy and matplotlib libraries, can be imported, and an empty model initialised with

import numpy as np
from matplotlib import pyplot as plt
import strapy

model = strapy.Model()

The Model instance defines the structure of the optical network to modelled All models must have at least one source of light, which can be added with the strapy.Model.Model.add_component method

model.add_component(strapy.components.Source, 'laser', 'n0')

When adding a component, a unique name (in the above example laser) and the node, or nodes, to which the compoent is attached must be specified. In this case the source has been connected to a node named n0. For components that connect to multiple nodes a tuple of node names should be specified, for example in the case of the beam splitter component in figure 1

model.add_component(strapy.components.BeamSplitter, 'BS', ('n1', 'n2', 'n3', 'n4'))

Similarly, the other components can be added with

model.add_component(strapy.components.Mirror, 'Mmes', 'n5')
model.add_component(strapy.components.Mirror, 'Mref', 'n6')
model.add_component(strapy.components.Polariser, 'pol', ('n7', 'n8'))

All components in strapy must be linked together with transfer matrix components, in the majority of cases this linking component will be a strapy.components.Stack. For the interferometer in figure 1, the following stacks will link all components

model.add_component(strapy.components.Stack, 's01', ('n0', 'n1'))
model.add_component(strapy.components.Stack, 's35', ('n3', 'n5'))
model.add_component(strapy.components.Stack, 's26', ('n2', 'n6'))
model.add_component(strapy.components.Stack, 's47', ('n4', 'n7'))
model.add_component(strapy.components.Stack, 's89', ('n8', 'n9'))

All free stack ends must be terminated with a beam dump, so that the network matrix equation is solvable. This can be added with

model.add_component(strapy.components.Dump, 'd9', 'n9')

Finally, in order to read out optical properties from the model, a detector must be added. A detector can be added at any node and will not affect the modelled optics; detectors can be added to a beam path without blocking the beam. In this case an amplitude and intensity detector will be added at node 9

model.add_detector('PD1', 'n9', ('amplitude', 'intensity'))

As for the strapy.Model.Model.add_component method, when adding a detector a unique name (PD1), and a node must be specified. In addition, the optical properties to be detected can be specified, by default the electric field ampltide vector with for the four S and P polarised counter propagating components is detected. For the available detector properties, see strapy.Detector.

Once added to the model, the optical properties of individual components can be specified. In general, components should default to sensible properties for an ‘ideal’ component, however explicitly specifying all properties should be preferred. Individual components are accessed through a python dictionary contained in the strapy.Model.Model, for this model properties will be specified as follows

model.components['laser'].amplitude[0] = 0
model.components['laser'].amplitude[1] = 1

model.components['BS'].rP = np.sqrt(0.5)
model.components['BS'].rS = np.sqrt(0.5)
model.components['BS'].tP = np.sqrt(0.5)
model.components['BS'].tS = np.sqrt(0.5)

model.components['Mmes'].rP = 1
model.compoents['Mmes'].rS = 1

model.components['Mref'].rP = 1
model.components['Mref'].rS = 1

model.components['pol'].rotation = 0
model.components['pol'].extinction = 0
model.components['pol'].loss = 0

This defines an S polarised source beam, with an ideal 50:50 non-polarising beam splitter, and an output polariser orientated to transmit S polarised light. Once properties have been defined, for components where the scattering or transfer matrix must be calculated from manually specified properties, the update method must be called for the properties to take effect. In this case this must be done for the pol component

model.components['pol'].update()

The model can then be built, this step creates the network matrix from the specified components, and need only be completed once unless further components or detectors are added to the model.

model.build()

The model is then ready for evaluation. Currently all optical path lengths are set to zero, the default for strapy.components.Stack instances, which should result in constructive interference at the detector. In order to see the interference fringes produced by this model, the length of the stack connecting node 3 to node 5 (leading to the moving measurement mirror) can be varied with the strapy.components.Stack.set_length method, and the results plotted with

xs = np.linspace(0, 1, 100)
ints = np.empty(xs.shape, dtype=float)

for i, x in enumerate(xs):
    model.components['s35'].set_length(x)
    model.evaluate()

    ints[i] = model.detectors['PD1'].intensity

plt.plot(xs, ints)
plt.show()

The resulting output is shown in figure 2. As expected, a sinusoidal intensity is produced with a a period of half the optical wavelength.

Output of example Michelson interferometer.

Figure 2: output of interferometer depicted in figure 1 for a displacement sweep of one wavelength.

This demonstrates the basic usage of strapy, however as there is no possibility of optical cavity formation, the true utility of the software may not be clear. Cavity formation can be demonstrated by adding a partially reflecting mirror in the path between nodes 4 and 7, creating a pair of coupled weak Fabry-Perot cavities through the beam splitter. This can be accomplished by employing the pyctmm module to specify the s47 stack as a multilayer stack of optical materials.

A pyctmm stack consisting of an air path with a 10 nm thick layer of gold in the centre can be created with

import pyctmm

stack = pyctmm.create_stack(3, 633e-9, 0)

pyctmm.set_ind(stack, 0, 1, 0)
pyctmm.set_ind(stack, 1, 0.2, -3)
pyctmm.set_ind(stack, 2, 1, 0)

pyctmm.set_d(stack, 0, 0)
pyctmm.set_d(stack, 1, 10e-9)
pyctmm.set_d(stack, 2, 0)

Stack ‘s47’ can be set to the pyctmm stack with the strapy.components.Stack.set_pyctmm method

model.components['s47'].set_pyctmm(stack)

As the measurement and reference arms of the interferometer should now be coupled, the output will depend on their relative positions. This can be observed by scanning both arms through a wavelength

xs = np.linspace(0, 1, 100)
ys = np.linspace(0, 1, 100)
ints = np.empty((len(xs), len(ys)), dtype=float)

for i in range(len(xs)):
    for j in range(len(ys)):
        model.components['s35'].set_length(xs[i])
        model.components['s26'].set_length(ys[j])
        model.evaluate()

        ints[i, j] = model.detectors['PD1'].intensity

ints = (ints - np.min(ints))/(np.max(ints) - np.min(ints))

plt.imshow(ints, extent=[0,1,0,1])
cbar = plt.colorbar()
cbar.set_label("Normalised intensity")
plt.xlabel('$M_{mes}$ displacement (wavelengths)')
plt.ylabel('$M_{ref}$ displacement (wavelengths)')
plt.show()

The result of this scan is shown in figure 3, with the output intensity dependant on the relative position of the two mirrors as expected.

Output of example Michelson interferometer with partial mirror in output path.

Figure 3: output of interferometer depicted in figure 1 with a partially reflecting mirror in the output path.