Modes of a Laser . ...... 1
Bryn Mawr College
Department of Physics
Undergraduate Teaching Laboratories
Longitudinal and Transverse Modes of a Laser
Introduction
Simplified descriptions of lasers emphasize that they can emit electromagnetic radiation of a single frequency (and, therefore, a single wavelength) in a highly directional beam. The light is said to be "temporally coherent" because the electromagnetic waves are all in phase, and the light is said to be "spatially coherent" because the emissions in different transverse positions in the beam have fixed phase relationships. However, these are ideal properties that are rarely true without exceptional technological effort. In this laboratory experiment you will explore what characteristics of a laser determine the frequencies (wavelengths) and intensity patterns emitted by a laser. You will investigate a variety of the transverse patterns that can be found for a laser beam, and what changes in the characteristics of the design of the laser cause changes in those transverse patterns. You will also get to investigate how a laser can be made from discrete pieces: mirrors and a gas of atoms (by a current of electrons in a discharge), which can spontaneously emit and amplify a range of wavelengths of light.
A Few Words of Caution
One of the technological advantages, and practical dangers, of laser radiation arises from the spatial and temporal coherence of the beam. This permits all of the energy of the beam to be focused by a lens to an extremely small spot, with the consequent delivery of all of the power of the beam to a very small area. The HeNe lasers ( = 0.6328 m) you will use in the teaching laboratories range in output powers from 1-5 mW (milliwatts). This is just at the level where discomfort might occur if all of the power is focused to a small spot on your retina. It is below the levels that have been found to cause retinal damage. Despite the "safety" of these weak beams, it is wise practice never to look directly at a laser beam. It is safest to remember that you should not put your head at the level of the laser beam or in line with the reflection of a laser beam from a mirror. The diffuse reflection of a laser beam from the wall, or a piece of paper, loses both intensity and some of the spatial coherence because of the roughness of the surface, and so it is safe to view beams in this way. Have a look at the laser safety Wikipedia entry for a discussion of how lasers are classified and safely used:
Theory: Introduction
A string fixed at both ends can vibrate in many different spatial modes: the first mode (longest wavelength and lowest frequency) has nodes only at the fixed ends, the second mode has a third node in the middle and so on. (See the picture on page 484 of Serway, "Physics for Scientists and Engineers," Third Edition.) These are one-dimensional modes. These modes are analogous to the longitudinal modes we will measure for a laser. Two-dimensional modes are a little more complicated than one-dimensional modes. Serway has nice pictures of two-dimensional modes in a drum where the circumference of the vibrating material is a node. For a laser, there is usually no sharp boundary at the sides, only at the mirrors at the ends. However, the curvature of the mirrors and narrow cross-sectional region where there is an amplifying medium limit the laser in the transverse direction. To satisfy Maxwell's equations for the electric and magnetic fields that make up a laser beam inside a laser there must be a balance between diffraction (spreading of a spatially limited beam) and focusing along with amplification to keep the intense part of the beam at the center.
Theory: Longitudinal Modes
When there are no boundary limitations for electromagnetic waves, they can exist at any frequency f (or wavelength since c = f where c is the speed of light). Electromagnetic waves can be created from the oscillating dipole moments in atoms when the electrons of the atoms are in a mixture to two appropriate atomic states. The frequency of oscillation of the dipole moment is related by Planck's formula to the difference in the energies of the two atomic states: E = hf, where h is Planck's constant, f is the frequency of the oscillation, and E is the energy difference. For many reasons most observations of atomic dipole transitions correspond to emission or absorption over a narrow range of energies, typically of order 10-2000 Megahertz (107-109 Hz) wide. This is to be compared to the transition energy itself E, typically on the order of 1015 Hz, which is the frequency for visible light. Laser emission involves a process of stimulated emission, in which atoms are stimulated by some of the light that is present to emit their energy in phase with the initial radiation. This process amplifies an incident beam of light.
One way to confine the light is with two plane and mutually parallel mirrors. Light that travels back and forth between the mirrors forms a standing wave that must satisfy the boundary conditions. If we take the z-axis perpendicular to the mirrors and place the mirrors at z=0 and z=L, then the spatially varying part of the electric field of the standing electromagnetic wave can be expressed as E(z,t) = E0sin(kz) cos(t), where k is the wave number (k = 2/) and is the angular frequency ( = 2f). If there is to be a node at both ends then an integer number of half wavelengths must fit into the distance L. Therefore at z = L, L=q where q is an integer. Since c = f, the values of f are also restricted to fq = qc/2L where the subscript q is a convenient way to keep track of the possible values of f. Hence the boundary conditions give us a discrete set of allowed frequencies separated by f = c/2L. The frequencies fq and their differences are shown in Figure 1. Figure 1 also shows a schematic curve of the emission of the lasing transition. If the wavelength is 0.6328 µm and the laser cavity length is 30 cm, what is the approximate value of “q”? Most commercial HeNe lasers emit light at several of these "cavity frequencies" or "longitudinal modes" because the range of colors emitted on the red transition of neon is broadened to about 1.5 GHz by Doppler shifts of the neon atoms in the discharge.
Theory: Transverse Modes
The solutions for the transverse modes are relatively complicated. Two families of these modes are the most common ones: the Gauss-Hermite modes and the Gauss-Laguerre modes. These modes are called "TEM" modes since the electric and magnetic fields are both transverse to the direction of propagation, even though this is not necessarily the case for confined EM fields. Generally three indices, m, n and q are used to designate a particular modal pattern using the notation TEMmnq. The "q" specifies the longitudinal number of half wavelengths, as discussed above, and the m and n indices are the integer number of transverse nodal lines in the x- and y- directions respectively, across the beam. Because the spatial variation differs for the different modes, the frequencies of these modes are also different. Generally for each of the longitudinal modes q, of the plane wave resonator, there is an infinite family of transverse modes (m, n), with the frequencies increasing as the complexity of the modes increases. Figure 2 shows a schematic picture of the frequencies of longitudinal and transverse modes in the vicinity of the lasing frequency.
For the Gauss-Hermite modes, the transverse intensity patterns are shown in Fig. 13.9 (pp. 592) of Hecht's Optics and these patterns are given by
,(1)
wherex and y are the transverse coordinates and HmandHn are Hermite polynomials. The first few Hermite polynomials are: H0(x) = 1, H1(x) = 2x, H2(x) = 4x2-2. These are the same functions as those used to describe the quantum mechanical harmonic oscillator. The position z=0 corresponds to the where in the resonator the phase fronts are a plane. This would be at the plane mirror if one of the mirrors is plane, and if the mirrors have equal radii of curvature, it would be halfway between the mirrors. At z=0, w(z) has its minimum value w0, which is called the "beam waist".
We see that the intensity patterns involve the product of the Hermite polynomials and a Gaussian function. The Gaussian falls to 1/e2 of its value when x2+y2 = w2(z). So roughly speaking, the transverse "width" of the beam intensity pattern at longitudinal position z is given by 2w(z) with
.(2)
The radius of curvature of the wavefront, as a function of the distance z, is given by:
.(3)
Returning to the equation for the lowest order mode, we see that
,(4)
wherer2 = x2 + y2. This is a circularly symmetric pattern (a spot), and since the transverse profile is simply a Gaussian function of the distance from the center of the beam, this mode is often called the "Gaussian mode", or fundamental mode. See Figure 13.11 in Hecht, pp. 593.
The second family of modes involves replacing the product of the Hermite polynomials for x and y with the Laguerre polynomials for r and , the polar coordinates in the transverse plane. When the circular (azimuthal) symmetry is nearly perfect, the Gauss-Laguerre modes are more commonly found. When the circular symmetry is broken because of the tilted Brewster Angle window or a tilted mirror, then the Gauss-Hermite modes are more commonly found.
Equipment
There are two lasers on the bench. One is the white Metrologic Neon Laser, which we’ll refer to as the “commercial laser.” The other is a glass tube mounted on the rail at the back on the bench, which we’ll refer to as the “open cavity laser.” The power supply for the open cavity laser is on the shelf above the laser.
There are also two Fabry-Perot interferometers on the table. One is very old and brass. You should look carefully at this device and identify the function of each of the adjustment knobs. The other interferometer is sealed inside a silver tube and mounted in a black disk with two knobs that adjust the orientation of the cylinder. This interferometer is powered by a supply on the shelf and its output can be viewed on the neighboring oscilloscope.
The final instrument that you will use is a ccd camera. This camera is connected to the computer, where you can view its output.
Experiment One: Longitudinal Modes of a Laser
Using the old-style Fabry-Perot interferometer consisting of two closely spaced parallel mirrors, observe the ring pattern that is created in transmission when the mirrors are illuminated by a commercial laser beam that is focused by a lens to illuminate a range of angles. To align this interferometer it is best to begin by sending the laser beam through the two mirrors without the lens. Tape a piece of paper to the wall so that you can see the light that is transmitted through the pair of mirrors. Adjust the orientation of the cavity so that the laser beam is normal to the mirrors. Adjust the adjustable mirror so that the dots you see transmitted through the pair of mirrors all fall on top of each other. Now insert the diverging lens into the beam between the laser and the cavity. Fine-tune the adjustable mirror of the Fabry-Perot interferometer to get circular rings. Note that every time the distance between the Fabry-Perot mirrors increases or decreases by half of the laser wavelength, the rings decrease or increase in size by one ring spacing.
Next, use the scanning Fabry-Perot Spectrum Analyzer made by Coherent to display the optical spectrum of the commercial laser. This analyzer is a commercial version of the same kind of Fabry-Perot interferometer you have just investigated, except that it has a detector placed at the center of the transmitted pattern and has curved mirrors so that the light is almost always limited to the fundamental transverse spatial mode. A linearly ramped voltage supplied by the controller changes the voltage on a piezoelectric crystal to which one of the Fabry-Perot Analyzer mirrors is mounted. The crystal changes its length in proportion to the voltage (about one micrometer for a few hundred volts), thereby scanning the separation between the two mirrors of the interferometer.
Steer the output of the commercial laser beam (by using mirrors if necessary) so that it enters perpendicularly into the aperture of the Coherent Fabry-Perot Spectrum Analyzer. Look at the output of the analyzer on an oscilloscope. Trigger the oscilloscope with the “trig out” signal from the back of the controller. Set the offset voltage to zero and the scanning voltage to ~200 volts. Set the oscilloscope to 2 mV/div and the time scale to 1 msec per/div. Adjust the tilt of the analyzer until you see sharp peaks corresponding to the longitudinal modes of the laser. The Fabry-Perot has a "free spectral range" (FSR) of about 7.5 GHz, so that the transmission pattern is repeated with a spacing equivalent to 7.5 GHz. Using the 7.5 GHz spacing as a calibration, measure the spacing between the longitudinal modes of the commercial laser. From your measured values, estimate the actual distance between the mirrors of the HeNe laser.
Insert a polarizer between the laser and the Fabry-Perot Spectrum Analyzer. By rotating the polarizer determine the polarization of the different longitudinal modes of the lasers. What polarization pattern do you observe in the modes?
Finally, use the scanning Fabry-Perot Spectrum Analyzer to display the optical spectrum of the open cavity HeNe laser. Alignment is more difficult with this laser since the beam is weaker that the commercial laser. Examine and identify the discharge tube and the two mirrors making up the open cavity laser. Note that one mirror can be moved along the rail to give different cavity lengths. Steer the output of the laser beam by using a mirror or two to direct it perpendicularly into the aperture of the Fabry-Perot Spectrum Analyzer. Again, look at the output of the Analyzer on an oscilloscope. Adjust the tilt of the analyzer until you see sharp peaks corresponding to the longitudinal modes of the laser. Using the 7.5 GHz FSR of the Fabry-Perot Spectrum Analyzer as a calibration, measure the spacing between the longitudinal modes of this laser for a given cavity length L. From your measured values, estimate the actual distance between the mirrors of the HeNe laser. How does it compare to the length of the cavity measured with a ruler? Now assume you do not know the FSR of the Analyzer. Can you devise a way to use the open cavity laser to measure it? Do this and determine the uncertainty in your result.
Experiment Two: Transverse Modes of a Laser
For each longitudinal position of the movable mirror of the open cavity HeNe laser, study the different transverse modes you can obtain by adjusting the variable aperture in the cavity and by small adjustments in the tilt of the external mirror. Repeat this for several different longitudinal positions of the external mirror. Shine the different mode patterns onto the CCD camera and record the intensity profiles with the associated computer using the beam analysis software.
Using the camera, make measurements of the separation of bright or dark points in a transverse mode as a function of the distance from the output end mirror of the laser. The TEM10 or TEM20 modes work well. Use mirrors to fold the beam back and forth across the room to get the longer distances. Plot the separation of the mode structure as a function of distance. Use this information to determine the divergence of the laser beam and to estimate the location of the beam waist inside the laser. From this result, estimate the curvature of the fixed end mirror, assuming the output mirror is flat.
Aligning the Open Cavity Laser
If the open cavity laser does not emit light when the discharge is turned on, probably the external output mirror is misaligned. To align this mirror, we use a working commercial laser as a "straight line" reference. First the external output mirror is removed. Second the optical discharge tube is aligned horizontally parallel to the rails of the optical bench at the desired height and at the desired transverse position. Third, by use of two or more steering mirrors, the beam from the commercial laser is directed along the optical bench, through the Brewster Angle window of the discharge tube, through the inside of the discharge tube (with minimal touching of the sidewalls) and onto the sealed end mirror. A small amount of light is transmitted by this mirror and it can be viewed on a card. It will consist of a compact circular spot when the light is aligned properly through the discharge tube. The reflected light from this end mirror travels back through the tube and can be viewed with a card placed around the output of the commercial laser. By fine tuning the alignment of the steering mirrors to make the commercial laser beam exactly perpendicular to the sealed end mirror, both the transmitted and reflected beams form overlapping circular spots. Then the external end mirror can be inserted and centered on the commercial laser beam path. The tilt of this mirror is adjusted to make the reflection from its surface a nice round spot on the card around the exit aperture of the commercial laser. At this point, if the open cavity laser discharge tube has a voltage applied across it, and the commercial laser is blocked, some spontaneously emitted red laser light should be visible from the open cavity laser. Then the alignment can be improved by minor tweaking of the tilt adjustment screws of the external output mirror.