Objective:

In this lab students will build a Twyman-Green interferometer.  They will use the interferometer to test a plane parallel glass plate and a microscope cover slip for refractive index and thickness variations.  They will also use the interferometer to determine the frequency separation of the axial modes of a He-Ne laser.

Background:

A LASER (Light Amplification by Stimulated Emission of Radiation) is an electromagnetic oscillator which combines light amplification with feedback.  The laser uses mirrors to feed the light output from an optical amplifier through a delay (the travel time of the light) back into the amplifier input.

Atoms and molecules can interact with light by three related processes, absorption, spontaneous emission, and stimulated emission.  Stimulated emission is the process which provides optical amplification in most lasers.  An excited atom or molecule can be stimulated to release its excess energy E in response to an incident photon of the right frequency (f = E/h ).  The atom emits an additional photon with the same frequency f, resulting in an amplification factor of exactly two.

An optical resonator constructed with two mirrors that face each other provides feedback and delay.

The mirrors re-circulate, or feed back, the light with efficiency R = R₁R₂, the product of the reflectivities of the two mirrors.  The delay time τ = 2L/c is the time it takes light to complete a round trip between the mirrors.  The net gain for a complete round must be equal to unity to sustain steady oscillation, the gain must exactly compensate for the loss.

There is an additional criterion that after each round trip the light wave crests line up with the crests from the previous round trip.  This results in a standing wave.  The condition on the mirror separation is that one round trip contains an integral number of wavelengths 2L = mλ, where m is an integer.  The corresponding resonant frequencies are f_m = c/λ = mc/(2L).  The separation between resonance frequencies is Δf = c/2L.  

A typical visible wavelength laser resonator might have length L = 30 cm operating at wavelength λ = 600 nm so that m = 1 million.  For an L = 30 cm resonator, the spacing between successive resonance frequencies is Δf = c/2L = 500 MHz while the resonance frequencies are near f_m = c/λ = 500 THz, one million times larger.

The frequency of the laser transition can lie in a range Δf about f = E/h, since the energy of the levels has an uncertainty ΔE.  Several of the resonance frequencies f_m may fall into that range Δf.  The spectrum emitted by a laser is a combination of the resonant frequencies f_m that fall into that range Δf.  A representative laser output spectrum is shown below.  

Modified laser resonators can suppress all but one of the resonant frequencies, permitting more precise control of the frequency, a desirable feature in scientific and engineering applications.

The He-Ne laser in the "Projects in Optics" kit supports three axial modes.  The polarization of neighboring modes is orthogonal.  It is possible to investigate these modes without resorting to high-resolution devices by using the Twyman-Green interferometer.

Assume our source emits waves with wavelength λ₁ and λ₂.  Waves with wavelength λ₁ and λ₂ emitted in phase, will destructively interfere after some optical path length l'_c = λ²/(2Δλ); l'_c is often also called the coherence length.  This definition of the coherence length differs from our previous definition by a factor of π.

The phase of a wave propagating into the x-direction is given by φ = kx - ωt.  Look at the wave pattern in space at some time t.  After some distance l the phase difference between two waves with wave vectors k₁ and k₂  which are in phase at x = 0 becomes Δφ = (k₁ - k₂)l.  When Δφ = π, the two waves interfere destructively.
We therefore have 
π = (k₁ - k₂)l'_c = (2π/λ - 2π/(λ + Δλ))l'_c.
(λ + Δλ - λ)l'_c/(λ(λ + Δλ))  ~ Δλl'_c/λ² = 1/2.
l'_c = λ²/(2Δλ).

Since λf = c, we have Δf/f = Δλ/λ.  We can write
l'_c = λ²/(2Δλ) = λf/(2Δf) = c/2Δf,
If we can measure l'_c, we can calculate Δf = c/2l'_c.

Using the interferometer, we can measure the coherence length by monitoring the contrast of the fringes.  When the interference maxima and minima of neighboring modes with λ₁ and λ₂ overlap, we observe high contrast fringes.  If we move one of the mirrors a distance ΔL = l'_c/2, so that the total path increases by l'_c, then the maxima of one mode will overlap the minima of the neighboring mode and the fringe contrast will be greatly reduced.  We have

Δf = c/2l'_c = c/(4ΔL).

Equipment:

• Laser Assembly
• Beam Steering Assemblies
• Lens Chuck Assemblies
• Target Assemblies
• Lens Kit
• Beam Splitter
• Optical Rail
• Meter Stick
• Index Cards
• Project in Optics Workbook

Open a Microsoft Word document to keep a live journal of your experimental procedures and your results.  Include all deliverables, (data, graphs, analysis, outcome).  Write a 'mini-reflection' immediately after finishing each investigation, experiment or activity, while the logic is fresh in your mind.

Procedure:

Part 1

Follow the instructions on pages 67 - 70 of the Projects in Optics Workbook, but simplify the setup described in steps 1-8 by using the procedure described below.

• Turn on the laser.
• Reflect the laser beam from the middle of mirror 1 back onto the laser output hole.  Use a mirror with a center-of-rotation adaptor so you later can turn it easily, and use a collar to fix the height of the mirror, so the height does not change when you loosen the screw in the post holder to turn the mirror.
• Use lens paper in the beam path to observe the incident as well as the reflected beam and adjust the mirror so that the reflected beam overlaps the incident beam.

  

• To simplify the setup, leave out the beam expander.
• Mount the beam splitter approximately 18 inches from mirror 2 and mount each of the other two mirrors without center-of-rotation adaptors approximately 7 inches from the beam splitter.  Mount the reference mirror onto the optical rail.  You should be able to move the reference mirror at least 30 cm back from the position it has when it has the same distance from the beam splitter as the test mirror.
• Rotate mirror 2 so that one of the laser beams strikes the center of the reference mirror, and then rotate the beam splitter so that the other laser beam strikes the center of the test mirror.
• Cover mirror 3 with an index card and adjust mirror 2, so that the laser beam reflects from mirror 2 back onto the laser just above the output hole.  Use lens paper to help you view the reflected beam.
• Cover mirror 2 with an index card and adjust mirror 3, so that the laser beam reflects from mirror 3 back onto the laser just above the output hole.
• Uncover both mirror and observe fringes on an index card.  Magnify the fringes using a 50 mm concave lens.

• Adjust the test mirror to observe fringes of equal thickness on the observation screen.

  

  Measure the separation between the fringes and record the spacing between two successive fringes.
  By what distance does the path length from the test mirror to the observation screen change from one fringe to the next fringe?

• Observe how the fringe pattern changes when you push lightly against the post holder for one of the mirrors.  Every relative movement of 1/4 wavelength (1/4 *633 nm) will change a bright fringe into a dark fringe and vice versa.
• Insert a glass slide or a piece of plastic into one of the paths.  Observe changes in the fringe pattern.

   

• The fringes, being lines of equal phase difference, are actually a contour map of the distorted wave front. Imperfections of the sample show up in terms of wave front aberrations.

Part 1 Deliverables: (to be included in the your journal)

• Visuals:  A picture of your setup.  A picture of the fringe pattern with and without a transparent object inserted into one of the paths. 
• Analysis:  Discuss your results.  By what distance does the path length from the test mirror to the observation screen change from one fringe to the next fringe?  Did you observe imperfections when inserting a transparent object into one of the paths?  If so, estimate the magnitude of the optical path length variations in the slide.

Part 2

Follow the instructions on pages 71 - 74 of the Projects in Optics Workbook.   You already have completed steps 1 - 10.

• Start your experiment with the reference mirror and the test mirror having approximately the same distance from the beam splitter.  Carefully move the reference mirror about its initial position to find the mirror position that gives the strongest possible fringe contrast.  Record the reference mirrors distance from the beam splitter in a table.

  initial position 0 max contrast	position 1 min contrast	position 2 max contrast	position 3 min contrast	Δf =c/4ΔL	Laser length L	Δf =c/2L

• Complete 11 - 14 as described in the workbook.  Try to find positions of minimum, maximum, and minimum fringe contrast.  Record these positions in the table.
• Complete steps 15 and 16 as described in the workbook.  Calculate Δf from your measurements. 

Optional:

• Add a polarizer to the system and carry out the additional experiments as described on page 74 of the workbook.

Part 2 Deliverables: (to be included in the your journal)

• Tables:  Filled out table.
• Analysis:  Discuss your results.  Does your measurement seem reasonable?  Estimate Δf based on the exterior dimensions of the laser.  Did you have any problems making the measurements?  Do you have suggestions for making better measurements?

Laboratory 7 report:

• Make sure you completed the entire lab and answered all questions.  Make sure you show your work and inserted and properly labeled relevant tables and plots.
• Add a short summary reflection at the end of your report.