Setup 1: Quadrature Interferometery Based on Elsworth and James: "An optical screw with a pitch of one wavelength"

Sources

Based on: Y. Elsworth and J. James, ""An optical screw with a pitch of one wavelength," Journal of Physics E: Scientific Instruments 1973 Volume 6.

Overview

This particular design relies on polarized light and two quarter wave plates (λ/4 wave plates or retarders) oriented at the appropriate angles to create the quadrature signal. A complete diagram is shown above. It differs from setup #2 in that it uses the two quarter wave plates instead of a quarter wave plate and a polarizer.

A condensed diagram of this setup is shown below where HWP is the half wave plate, QWP is the quarter wave plate, NBS is the non-ploarizing beam splitter, PBS is the polarizing beamsplitter, and PD1 and PD2 are the photo-detectors Y and X.

A more detailed description of the apparatus is provided below.

Detailed Description of this Michelson Quadrature Interferometer

The linearly polarized input beam (for example, originating from a HeNe laser) is passed through a half-wave plate (or λ/ 2 retarder, (not shown in the diagram below) to adjust its (linear) polarization to the desired orientation. In the diagram below, the input beam has been rotated so that the input beam is polarized in the vertical plane.

Part 1: Input Beams

The beam in the measurement arm is reflected by its mirror back to the beam splitter. Except for a phase change, it remains polarized in the vertical direction.

Upon its reflection from the mirror at the end of the reference arm, the elliptically polarized beam in the reference arm undergoes a change in rotation direction . Once more , it passes the quarter wave plate (retarder) oriented with its fast axis at 22.5 degrees with respect to the vertical. This second pass through this wave plate converts the elliptically polarized beam back into a linearly polarized beam, albeit one polarized at twice the angle of the quarter wave plate orientation, i.e., one with its orientation at 45 degrees to the vertical.

Half of the the beam from the measurement arm is reflected by the beam splitter while half of the beam from the reference arm is transmitted through it and these two parts combine together and continue on towards the right in the above diagram. (Note: since only half of the signals continue "to the right", the other half continues back towards the origin, i.e., the light source, the laser. However, we are not interested in this part of the beam and will ignore it for the rest of this discussion.)

The combined beam that continues to the right is yet another elliptically polarized beam. However, it might be more useful for this discussion to think of it as two distinct linearly polarized beams, with identical frequency but different phases and orientation: while one beam's orientation (from the measurement arm) remains in the vertical plane, the beam from the reference beam is oriented at 45 degrees to it.

Part 3: Final Stage with Detectors

In the final stage of the quadrature interferometer, shown above, the combined beams are passed through a second quarter wave plate. However, unlike the first quarter wave plate, the orientation of this quarter wave plate's fast axis is at 45 degrees with the vertical.

This orientation has two very different effects on the two incoming beams from the measurement and reference arm: it converts the vertically polarized beam into a circularly polarized beam but it has no effect on the beam from the reference arm since it is parallel to the fast axis of the quarter wave plate. (In other words, the reference beam has no polarization component in the direction of the quarter wave plate's slow axis.)

The resulting beam is a complicated elliptically polarized beam, consisting of a linearly polarized beam at 45 degrees with the vertical and a circularly polarized beam which has a phase offset depending on the measurement beam's arm length.

This beam continues on towards the final polarizing beam splitter. It transmits the horizontally polarized beam to the detector labeled "Detector X" while the vertically polarized component is reflected to the detector labeled "Detector Y."

Note that "real" detectors are not able to measure the electric field shown in the illustrations above. Instead they measure the average of the intensity where the intensity corresponds to the square of the electric field. (The reason for measuring the average intensity is that the electric field changes at a rate of 10^14 Hz, which exceeds today's electronic components response time.)

Theory

The theory for this setup is shown here using Wolfram Mathematica.

Some 3-D visualizations created with Autodesk Inventor can be found here.

Caged Setup

Detailed Parts List

See this link for a detailed parts list for all the components.

Detailed Setup Instructions

For a more in depth setup instruction, see Quadrature Michelson Alignment

Open Setup

For a more in depth setup instruction, see in depth setup for the open system byLuke Molacek.

The half wave plate was placed in between the laser and the first mirror as shown below.

One quarter wave plate was placed in the reference arm, and the other was placed in between the non-polarizing beamsplitter and the polarizing beamsplitter as shown below. Note that in this image the reference arm now has a magnetostriction setup in it, which will be discussed in the next section.

Magnetostriction

In this setup, terfenol-D was fixed to a rod on one end, and a mirror on the other end using epoxy and placed inside a solenoid. The solenoid was connected to a MPA-200 Stereo PA Amplifier, which received input from a function generator driving a sinusoidal signal of varying amplitude and frequency. The rod was fixed with two posts with 90 degree clamps which were adjusted so the mirror was slightly off axis from the input beam. The solenoid was set on two supports such that the solenoid was not in contact with the terfenol-D, rod, or mirror in order to reduce dampening of vibrations.

The results seen using the magnetostriction setup with setup 3 need to be studied further. The ellipse on the oscilloscope tended to vary in size and shape with time. The amount to which this occurred seemed to be dependent on the amplifier settings and the frequency of the sinusoidal signal.