# Modeling of the Sagnac effect by the methods of geometric optics

High-precision inertial navigation systems (INS) are one of the most high-tech elements of the modern aerospace and military industry. The task of such systems, built on the basis of accelerometers and optical gyroscopes, is the determination of angular velocities, accelerations of a moving object, and, in general, the orientation of the object in three-dimensional space.

The specific feature of the ANN simulation is that in the modeling domain, both rectilinear and rotational motion often occur, which strongly affects the operation of these systems and, accordingly, requires accounting. In this article we briefly describe what the Sagnac effect and how the device based on it can be numerically investigated in the COMSOL Multiphysics package

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What are the optical gyroscopes

Specificity of accounting for rotation in the simulation of

The Sagnac effect: theoretical foundations

The Sagnac interferometer model in COMSOL Multiphysics

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Conclusion

Optical gyroscopes and the Sagnac effect

Perhaps, it is the classical Sagnac interferometer that best demonstrates the need for high-precision registration of the non-inertial motion of the modeling domain.

The simplest Sagnac interferometer consists of the following components:

Light source

A beam splitter that directs the light of the source along two different trajectories, and then combines them

A set of mirrors (usually consisting of two or three mirrors)

The beam splitter and mirrors form a triangular or rectangular trajectory along which light propagates in both directions. At this time, the navigation system itself (as well as the air or space ship into which it is installed) also rotates at a certain angular velocity. Observing the interference of light rays (in consequence of the Sagnac effect) that propagate along these trajectories, it is possible to determine the angular velocity of rotation of the system with very high accuracy.

Measuring small rotations is vital for determining and controlling the orientation of objects in the modern defense and space industries. Currently, the most widely used ring laser and fiber optic gyroscopes, the principle of which is also based on the Sagnac effect. Note that the ring laser gyroscope is highly accurate, cheap and easy to maintain, because unlike mechanical gyroscopes it does not contain rotating parts.

Modeling of light propagation in rotating optical components

How to calculate the path of light propagation in a rotating system of mirrors, prisms and beam splitters? In order not to go into the theory of relativity, let us assume that the speed of rotation is much less than the speed of light, but it is large enough that we have to take into account the rotation. There are at least two approaches to the solution of this problem:

Rewrite the equations for the propagation of light in a noninertial reference frame

Rotate the construction in real time for the propagation of rays

The difference between these approaches is that in one case the model is in a non-inertial reference system (variant No. 1), connected to a moving interferometer, or in a "laboratory" frame of reference fixed in space (variant No. 2). Since it is much easier to implement the second variant, we will use this approach for modeling the Sagnac interferometer.

Package COMSOL Multiphysics

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Problems with variable geometry of the computational domain [/b]

The work of complex physical and technical systems often involves changing the geometry of objects, their movement or rotation. In addition, a change in geometry may be required in solving optimization problems or in analyzing the sensitivity of the model to geometric dimensions. To correctly model the processes in these cases, it is necessary to take into account the corresponding geometric transformations in the calculation model. COMSOL Multiphysics

^{ ® }allows to solve such problems with the help of moving grids and the change of geometrical model directly in the process of modeling.

In This video review (in Rus.) , we consider examples of tasks in which you need to configure and use a variable geometry, and also on concrete illustrative examples we show the main tools and special interfaces COMSOL Multiphysics

^{ ® }for working with a variable geometry.

Analysis of deformed and moving structures has traditionally been carried out with great care, since it is used in a wide variety of areas: in the analysis of thermal stress, fluid interaction with the structure, multiphase flows, as well as in galvanic engineering, piezoelectric devices, and so on. In fact, to accurately trace the rays in a moving structure it will suffice to indicate the angular velocity of the system, and then run a standard calculation based on technologies of geometric optics .

An example of setting the angular velocity of rotation in the interface COMSOL Multiphysics [/b]

The Sagnac effect: theoretical foundations

Before proceeding to the description of the model implemented in the package, let's briefly examine what the Sagnac effect is.

Imagine that light propagates strictly along a circle (for example, over a fiber optic cable) in two opposite directions, as illustrated in Fig. 1. The point of launching the rays is

. The dashed line shows the direction clockwise, and the solid solid line is the direction counter-clockwise. Light rays in this setting will be opposite to each other, since they propagate along the circumference in opposite directions.

If the ring were stationary, the ray trajectories would intersect twice: first at the opposite point of the circle, and then at the starting point

. Now, imagine that the ring rotates counterclockwise around its center at some angular velocity. If we follow the movement of the point

during the propagation of light, we will see that a ray propagating clockwise will return to it when it is already in a new position,

. When to the point

the beam propagating counterclockwise will return, it will move further and will be in position

.

is located at a greater distance from

, than

because the circle also rotates counterclockwise.

Fig.1. Spread the light clockwise and counterclockwise along the edge of the rotating circle.

Obviously, the illustration in Fig. 1 for clarity is significantly scaled, and in reality the distance between points is 10 billion times smaller. However, even in this case, the difference in the optical path traversed leads to a phase shift and, accordingly, interference.

Without going into the theoretical calculations (but if they are interesting, we recommend the following classical paper : Post, Evert J. "Sagnac effect", Reviews of Modern Physics, 3? no. ? p. 47? 1967 ), between the angular velocity

and the difference in the optical path

can be expressed as:

Where

- the area of the considered circle, and

Is the speed of light.

In general, the Sagnac effect is even more general than the example described above. The trajectory of the propagation of two opposing beams can have any shape, but the delay between them will always be proportional to the size of the region, which is limited by the contour in which the rays propagate. In addition, this effect is also observed in cases when the center of rotation does not coincide with the center of the contour.

Test model of the Sagnac interferometer based on the optical ray tracing

To verify how in COMSOL Multiphysics

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and, consequently, the sensitivity of the device, we consider the test design of the Sagnac interferometer, in which the light propagates not along the circumference, but along the perimeter of the triangle, at the vertices of which there are two mirrors and a beam splitter (Fig. 2).

Fig.2. Scheme of the Sagnac interferometer.

The original beam passes through the beam splitter, resulting in the formation of two beams of equal intensity. At the moment of exit from the beam splitter, they are at the same point and have the same phase. Since the mirror system rotates, then by the time when the rays return to the beam splitter, their optical paths (and, consequently, the phases) are different from each other.

In practice, instead of small values

, systems often detect a frequency shift (or beat frequency)

:

Here

- the effective length of the contour along which the rays propagate, and

- their frequency. Please note that

is determined directly in the calculation.

The process of numerical ray tracing can be easily automated, for example, for parametric analysis. In Fig. 3 shows the results of parametric analysis in a wide range of angular velocity values - from relatively small to very large.

Fig.3. (a) Ray trajectories in a test interferometer. The divergences in the trajectory of the two rays are so insignificant that they are not noticeable even on a close-up. (B) The dependence of the beat frequency on the angular frequency of rotation of the system.

The corresponding beat frequency is in excellent agreement with the theoretical values. By varying the distance between the mirrors, it can be shown that the slope of this line is proportional to the area of the triangular region enclosed between the opposing beams.

A look into the future or practical application of numerical simulation of optical gyroscopes

The above results demonstrate that by conducting ray tracing in a rotating geometry (frame) using the described technique, it is possible to calculate with high accuracy the sensitivity of devices based on the Sagnac effect, if the rotation speed is small compared to the speed of light (i.e., without taking into account relativistic effects). So, thanks to this new model the modeling specialists and engineers working with angular orientation systems will now have a ready-made working pattern for studying the Sagnac effect, which underlies the operation of ring laser gyroscopes.

An attentive reader will probably ask the question of the need for such a numerical simulation, taking into account the fact that the Sagnac effect is described quite accurately by the formula presented above. It should be taken into account that real ANNs are much more complicated than the simplest setup with a beam splitter and two mirrors, considered above. Such systems are installed together with other sensitive devices in a confined space, an additional frame is required, which ensures that the optical components are stationary relative to each other. In addition, often INS work in aggressive environments, and they are subject to mechanical stresses, temperature and electromagnetic fields. These factors affect the behavior and sensitivity of the gyroscope, which requires more detailed and careful consideration, and can not be described by the same simple formula.

Thus, the presented ray tracing in the Sagnac interferometer or ring laser gyroscope will be only the first step in the high-precision and complex multi-physical analysis of large optical systems. COMSOL Multiphysics

^{ ® }allows you to trace rays in the most realistic conditions, in particular taking into account the heating and thermal deformation of optical components, which will open new opportunities for better understanding and assessment of the sensitivity and accuracy of complex inertial navigation systems.

Geometric optics in COMSOL Multiphysics [/b]

The Ray Optics module of COMSOL Multiphysics

^{ ® }provides a wide range of functions for such calculations. In this case, the trajectories of such rays can be calculated at large distances with a minimum cost of computing because there is no need to express the wavelength using a finite element grid. Examples of using COMSOL Multiphysics

^{ ® }in this area include modeling laser resonators , lens systems, optical Bragg filters, interferometers, spectrographs , monochromators, and the like.

In This video review (in Rus.) we will tell about all the key featuresx and the advantages of this approach and module, including the possibility of combining with full wave calculations, solving related thermal and mechanical problems, and advanced post-processing tools, incl. on the analysis of monochromatic aberrations.

# More information

This material is based on the following articles:

C.Boucher. Optics simulation , magazine Laser Focus World, August 2018

Modeling of the Sagnac interferometer and ring laser gyroscope using geometric optics , corporate blog COMSOL

For more detailed acquaintance with the described techniques and examples, you can request a free full-featured demo version of COMSOL Multiphysics

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