Coherence and Lasers

Objective:

In this lab students will build a Twyman-Green interferometer and use it to determine the frequency separation of the axial modes of a He-Ne laser.

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.

In a 4-level laser, an atom is first excited by optical, electrical, or other means through the “pump transition” from starting level “0” to a temporary level or group of levels “3”.  The atom quickly relaxes to  level “2”.  Stimulated emission occurs from level “2” to level “1”.  Level “1” quickly decays back to level “0”, so that absorption from level “1” to level ”2” is unlikely.  At any given time more of the atoms are at level “2” than at level “1”, there is a “population inversion”.  The rate of stimulated emission will exceed the absorption rate resulting in an optical amplifier.

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 t = 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 = ml, where m is an integer.  The corresponding resonant frequencies are f_m = c/λ = mc/(2L).  The separation between resonance frequencies is Df = 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 Df = c/2L = 500 MHz while the resonance frequencies are near f_m = c/λ = 500 THz, one million times larger.

The frequency of transition from level “2” to level “1” can lie in a range Df about f = E/h, since the energy of  the levels has an uncertainty DE.  Several of the resonance frequencies f_m may fall into that range Df.  The spectrum emitted by a laser is a combination of the resonant frequencies f_m that fall into that range Df.  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.

A typical He-Ne laser consists of a sealed tube containing a mixture of helium and neon gasses and an optical resonator.  An electrical discharge in the tube excites the helium atoms, which then collide with the neon atoms, transferring excess energy to pump the neon atoms from level “0” to level “2”.  The neon atoms develop a population inversion between levels 2 and 1, providing gain at one of several different visible and infrared frequencies.  The light circulating between the two mirrors is amplified by the stimulated emission from the excited neon atoms on each round trip.  One mirror, called the back reflector, has high reflectivity (>99 %) while the other mirror, called the output coupler, has lower reflectivity (typically 95-98 %) and transmits a fraction (2-5 %) of the light to the outside–this is the laser output that we observe.

The He-Ne 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 l₁ and l₂.  Waves with wavelength l₁ and l₂ emitted in phase, will destructively interfere after some optical path length l'_c = l²/(2Dl); l'_c is often also called the coherence length.  This definition of the coherence length differs from our previous definition by a factor of p.

[The phase of a wave propagating into the x-direction is given by f = kx - wt.  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 Df = (k₁ - k₂)l.  When Df = p, the two waves interfere destructively.
We therefore have 
p = (k₁ - k₂)l'_c = (2p/l - 2p/(l + Dl))l'_c.
(l + Dl-l)l'_c/(l(l + Dl))  » Dll'_c/l² = 1/2.
l'_c = l²/(2Dl).]

Since lf = c, we have Df/f = Dl/l.  We can write

l'_c = l²/(2Dl) = lf/(2Df) = c/2Df,

If we can measure l'_c, we can calculate Df = 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 l₁ and l₂ overlap, we observe high contrast fringes.  If we move one of the mirrors a distance DL = 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

Df = c/2l'_c = c/(4DL).

Equipment:

Procedure:

Follow the instructions on pages 71 - 74 of the Projects in Optics Workbook.  To assemble the components, refer to pages 36 - 44 in the Projects in Optics Workbook.

Required:

Optional:

Laboratory Report:

Open Microsoft Word and prepare a report using the template shown below.

Name:
E-mail address:

Print out your Word document, and hand it to your instructor, or save your Word document (your name_lab8.doc) and attach it to an e-mail message to your lab instructor.