Sagnac Interferometer

The Sagnac effect is a phenomenon that arises when light propagates around a closed loop in a rotating frame of reference. Although the effect is fundamentally relativistic in nature, it can still be observed in a pure geometrical optics simulation when the frame of reference rotates slowly. An understanding of the Sagnac effect is essential to modern guidance and navigation systems, an area in which optical gyroscopes (or gyros) often prove to be a cost-effective alternative to mechanical gyros; they benefit from comparatively low maintenance costs because they have no moving parts.

This is a model of a simple Sagnac interferometer consisting of two mirrors and a beam splitter arranged in a triangle. The entire modeling domain rotates; as a result, the rays propagating in opposite directions in the triangle have different optical path lengths due to the Sagnac effect. This can be used to deduce the angular velocity of the system.

Model Definition

The model geometry is shown in Figure 1. The two mirrors and the beam splitter form an equilateral triangle in which light propagates in both directions.

Figure 1: Diagram of the Sagnac interferometer. The entire assembly rotates, and rays propagate through the triangle in both directions.

The entire apparatus rotates at a constant angular velocity Ω (SI unit: rad/s).

In an active ring laser gyro, at least one side of the triangle would typically include a lasing medium, and the light exiting the triangle would typically pass through a prism to combine the outgoing beams. However, in this greatly simplified Sagnac interferometer model, these components are not considered.

For certain numerical considerations in the following sections it is convenient to know some geometry dimensions. Also assume the rays propagate in a vacuum, n = 1.

Table 1: Interferometer Specification.
Name	Expression	Value	Description
λ0	N/A	632.8 nm	Vacuum wavelength
R	N/A	10 cm	Ring radius
b		17.3 cm	Triangle side length
P	P = 3b	52.0 cm	Triangle perimeter
A		130. cm2	Triangle area

The Sagnac Effect

The Sagnac effect is most easily illustrated by two counterpropagating beams of light, each constrained within a ring that is rotating at constant angular velocity Ω. This is shown in Figure 1; the beam propagating in the direction of rotation is shown as a solid line, whereas the beam propagating opposite the direction of rotation is shown as a dashed line. Assume that these diagrams are being observed from an inertial frame of reference.

Figure 2: Diagram demonstrating the Sagnac effect in a rotating frame of reference

Initially both beams are released simultaneously from point P0. Since the ring is rotating, the ray indicated by the dashed line reaches the release point at a new location P1 before it reaches the original location P0. Conversely, the ring indicated by the solid line reaches the release point at a third location P2, having already passed through the original location at P0. Thus the dashed line travels a shorter distance due to the rotation whereas the solid line travels a longer distance.

The counterpropagating beams thus recombine after having propagated for slightly different distances and times. It follows that there will be a phase difference between the beams when they recombine; this could be observed, for example, as a shift of the interference fringes.

In Ref. 1 it is shown that the magnitude of the optical path difference due to the Sagnac effect is not affected by the shape of the path but only its enclosed area. Assuming the axis of rotation is perpendicular to the plane of the interferometer, so that the angular velocity can be treated as a scalar, the difference in transit time for the counterpropagating beams is

where the physical constant c0 = 299,792,458 m/s is the speed of light in a vacuum. The corresponding difference in optical path length is

(1)

Measuring the Optical Path Difference

For the area given in Table 1 and an angular velocity of 1 deg/h (which is not unreasonable for spacecraft), Equation 1 gives a path difference of about 8 × 10-16 m, approximately the radius of a proton. Such a difference is impractically small to measure, so instead of comparing optical paths directly, most devices instead report frequency differences.

For the above parameter values, the frequency difference is about 1 rad/s. This value, called the beat frequency, is much easier to read compared to the path difference or transit time difference.

The ratio of beat frequency to the angular velocity is sometimes called the scale factor S,

The scale factor is a measure of the sensitivity to small rotations.

Numerical Precision

This model uses the Geometrical Optics interface, in which rays are traced through the model geometry while being reflected or refracted at surfaces. Because the ray tracing calculation uses double-precision arithmetic, the smallest relative difference that can be detected between two optical paths is

where ε is sometimes called machine precision or machine epsilon. At smaller values, the difference in optical path for the counterpropagating beams returns zero due to cancellation error. From Equation 1 it is possible to compute the angular velocity corresponding to the smallest measurable optical path difference in double-precision arithmetic,

Solving for Ωmin yields

For the parameter values given in Table 1, Ωmin is approximately 6.66 × 10-7 rad/s or 0.137 deg/h. Thus the number of degrees of precision the result can be no greater than log10(Ω/Ωmin).

Results and Discussion

The ray trajectories are shown in Figure 3, where the color expression indicates the ray index, i ∈ {1, 2, 3, 4}. The rays that hit the obstruction have indices 1 and 4; this is because the incident ray splits once when entering the ring, and each of the two counterpropagating rays splits again while exiting the ring.

Figure 3: Ray propagation in a Sagnac interferometer consisting of two mirrors and a beam splitter. The entire device rotates counterclockwise, resulting in a small phase difference between the rays.

In Figure 4 the beat frequency is given as a function of the angular velocity. The beat frequency is on the order of 1 Hz even for the smallest angular velocity shown, wheres the smallest optical path difference would have been on the femtometer scale.

The slope of this line, the scale factor, is shown in Figure 5, where it is compared to the analytic result from a simple geometric analysis of the interferometer. At lower angular velocity values, the computed scale factor is noisy and inaccurate because of cancellation error; at the lowest value, the path difference can only be known to one digit of precision.

At higher values of Ω, the two lines differ, but for a different, more physical reason. The analytic expression for scale factor has been written assuming that optical path within the triangle is equivalent to distance. However, for a short interval when entering and leaving the beam splitter, the refractive index is not unity. Ref. 1 provides more detailed expressions for the optical path difference and transit time difference when the interferometer contains a co-moving medium. That this is a real, physical effect can also be demonstrated by re-running the model with an extremely thin beam splitter; then the two lines show better agreement at larger values of Ω.

Figure 4: Beat frequency in the Sagnac interferometer as a function of angular velocity. The relationship is linear; the slope is the scale factor of the interferometer.

Figure 5: Scale factor of the Sagnac interferometer as a function of angular velocity.

Reference

1. E.J. Post, “Sagnac effect,” Reviews of Modern Physics, vol. 39, no. 2, pp. 475–493, 1967.

Ray_Optics_Module/Spectrometers_and_Monochromators/sagnac_interferometer

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

Parameters 1

Load the global parameters for the interferometer 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 sagnac_interferometer_parameters.txt.

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose . This is more convenient than the SI unit, given the dimensions of the geometry.

The interferometer geometry is a triangular arrangement of two mirrors and a beam splitter. The geometry also includes an obstruction to absorb the outgoing rays.

Bottom-Right Mirror

In the toolbar, click

In the window for , type Bottom-Right Mirror in the text field.

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

In the text field, type hm.

Locate the section. From the list, choose .

In the text field, type xm1.

In the text field, type ym1.

Locate the section. In the text field, type 30.

Top-Right Mirror

In the toolbar, click

In the window for , type Top-Right Mirror in the text field.

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

In the text field, type hm.

Locate the section. From the list, choose .

In the text field, type xm2.

In the text field, type ym2.

Locate the section. In the text field, type 150.

Beam Splitter

In the toolbar, click

In the window for , type Beam Splitter in the text field.

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

In the text field, type hs.

Locate the section. From the list, choose .

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (cm)
Layer 1	hs/2

The interior boundary splits the incoming ray into reflected and refracted rays. The outer surfaces of the glass will be assigned antireflective coatings.

Obstruction

In the toolbar, click

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

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

In the text field, type ho.

Locate the section. From the list, choose .

In the text field, type xo.

In the text field, type yo.

Locate the section. In the text field, type 60.

Click

Click the

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

Definitions

Rotating Domain 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the ω text field, type Omega.

Locate the section. Specify the rax vector as


xc	X
yc	Y

Load some local variable definitions. These will be used later to interpret the results.

Variables 1

In the window, right-click and choose .

Load the variable definitions from a text file.

In the window for , locate the section.

Click

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

Geometrical Optics (gop)

In the window, under click .

In the window for , locate the section.

In the text field, type 3. The Geometrical Optics interface will apply deterministic ray splitting at the beam splitter boundary. A total of three secondary rays will split off from the released ray.

Locate the section. From the list, choose .

Locate the section. Select the check box. The optical path difference between two of the rays will be used to compute the beat frequency of the output.

Ray Properties 1

In the window, under click .

In the window for , locate the section.

In the λ0 text field, type lam0.

Define some boundary conditions for the beam splitter, mirrors, and obstruction.

ARC

In the toolbar, click

and choose .

Select Boundaries 6 and 9 only.

In the window for , locate the section.

From the list, choose .

In the text field, type ARC.

Beam Splitter

In the toolbar, click

and choose .

Select Boundary 8 only.

In the window for , type Beam Splitter in the text field.

Locate the section. From the list, choose .

In the R text field, type 0.5.

Mirrors

In the toolbar, click

and choose .

Select Boundaries 13 and 15 only.

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

Obstruction

In the toolbar, click

and choose .

Select Boundary 3 only.

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

Release from Grid 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the qx,0 text field, type q0x.

In the qy,0 text field, type q0y.

Locate the section. Specify the L0 vector as


L0x	x
L0y	y

Use the feature to delete any rays that escape from the geometry.

Ray Termination 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Materials

Specify the refractive index in the domains. The surroundings are treated as a vacuum with n = 0.

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	1.5	1	Refractive index

Study 1

The study will include a over different values of the angular velocity.

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 1.

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
Omega (Interferometer angular velocity)	10^{range(0,0.2,5)}	deg/h

Step 1: Ray Tracing

In the window, click .

In the window for , locate the section.

From the list, choose .

In the text field, type 0 1.5*P. Since P is the perimeter of the triangle, this is a sufficiently long optical path length for the rays to reach the obstruction.

In the toolbar, click