HIGHER ORDER GAUSSIAN LASER BEAMS

HIGHER ORDER GAUSSIAN LASER BEAMS

In the real world, the truly 100-percent, single transverse mode, Gaussian laser beam (also called a pure or fundamental mode beam) described by equations 36.7 and 36.8 is very hard to find. Low-power beams from helium neon lasers can be a close approx- imation, but the higher the power of the laser, and the more com- plex the excitation mechanism (e.g., transverse discharges, flash-lamp pumping), or the higher the order of the mode, the more the beam deviates from the ideal.

To address the issue of higher order Gaussian beams and mixed mode beams, a beam quality factor, M2, has come into general use. A mixed mode is one where several modes are oscillating in the res- onator at the same time. A common example is the mixture of the lowest order single transverse mode with the doughnut mode, before the intracavity mode limiting aperture is critically set to select just the fundamental mode. Because all beams have some wavefront defects, which implies they contain at least a small admixture of some higher order modes, a mixed mode beam is also called a “real” laser beam.

For a theoretical single transverse mode Gaussian beam, the value of the waist radius–divergence product is (from equation 36.11):

 

It is important to note that this product is an invariant for trans- mission of a beam through any normal, high-quality optical system (one that does not add aberrations to the beam wavefront). That is, if a lens focuses the single mode beam to a smaller waist radius, the convergence angle coming into the focus (and the divergence angle emerging from it) will be larger than that of the unfocused beam in the same ratio that the focal spot diameter is smaller: the product is invariant.

For a real laser beam, we have

 

where W0 and V are the 1/e2 intensity waist radius and the far-field half-divergence angle of the real laser beam, respectively. Here we have introduced the convention that upper case symbols are used for the mixed mode and lower case symbols for the fundamental mode beam coming from the same resonator. The mixed-mode beam radius W is M times larger than the fundamental mode radius at all propagation distances. Thus the waist radius is that much larger, contributing the first factor of M in equation 36.16.

The second factor of M comes from the half-angle divergence, which is also M times larger. The waist radius–divergence half-angle product for the mixed mode beam also is an invariant, but is M2 larger. The fundamental mode beam has the smallest divergence allowed by diffraction for a beam of that waist radius. The factor M2 is called the “times-diffraction-limit” number or (inverse) beam quality; a diffraction-limited beam has an M2 of unity.

For a typical helium neon laser operating in TEM00 mode, M2 < 1.05. Ion lasers typically have an M2 factor ranging from 1.1 to 1.7. For high-energy multimode lasers, the M2 factor can be as high as 30 or 40. The M2 factor describes the propagation char- acteristics (spreading rate) of the laser beam. It cannot be neglected in the design of an optical train to be used with the beam. Trunca- tion (aperturing) by an optic, in general, increases the M2 factor of the beam.

The propagation equations (analogous to equations 36.7 and 36.8) for the mixed-mode beam W(z) and R(z) are as follows:

 

and

The Rayleigh range remains the same for a mixed mode laser beam:

 

Now consider the consequences in coupling a high M2 beam into a fiber. Fiber coupling is a task controlled by the product of the focal diameter (2Wf) and the focal convergence angle (vf). In the tight focusing limit, the focal diameter is proportional to the focal length f of the lens, and is inversely proportional to the diameter of the beam at the lens (i.e., 2Wf ∝ f/Dlens).

The lens-to-focus distance is f, and, since f x vf is the beam diameter at distance f in the far field of the focus, Dlens ∝ fvf. Com- bining these proportionalities yields

 

for the fiber-coupling problem as stated above. The diameter- divergence product for the mixed-mode beam is M2 larger than the fundamental mode beam in accordance with equations 36.15 and 36.16.

There is a threefold penalty associated with coupling a beam with a high M2 into a fiber: 1) the focal length of the focusing lens must be a factor of 1/M2 shorter than that used with a fundamen- tal-mode beam to obtain the same focal diameter at the fiber; 2) the numerical aperture (NA) of the focused beam will be higher than that of the fundamental beam (again by a factor of 1/M2) and may exceed the NA of the fiber; and 3) the depth of focus will be smaller by 1/M2 requiring a higher degree of precision and stability in the optical alignment.

CALCULATING A CORRECTING SURFACE

CALCULATING A CORRECTING SURFACE

A laser beam is refracted as it passes through a curved output mirror. If the mirror has a flat second surface, the waist of the refracted beam moves closer to the mirror, and the divergence is increased. To counteract this, laser manufacturers often put a radius on the output coupler’s second surface to collimate the beam by mak- ing a waist at the output coupler. This is illustrated by the case of a typical helium neon laser cavity consisting of a flat high reflector 

and an output mirror with a radius of curvature of 20 cm sepa- rated by 15 cm. If the laser is operating at 633 nm, the beam waist radius, beam radius at the output coupler, and beam half-angle divergence are

respectively; however, with a flat second surface, the divergence nearly doubles to 2.8 mrad. Geometrical optics would give the focal length of the lens formed by the correcting output coupler as 15 cm; a rigorous calculation using Gaussian beam optics shows it should be 15.1 cm. Using the lens-makers formula

  

with the appropriate sign convention and assuming that n = 1.5, we get a convex correcting curvature of approximately 5.5 cm. At this point, the beam waist has been transferred to the output cou- pler, with a radius of 0.26 mm, and the far-field half-angle divergence is reduced to 0.76 mrad, a factor of nearly 4.

Correcting surfaces are used primarily on output couplers whose radius of curvature is a meter or less. For longer radius output cou- plers, the refraction effects are less dramatic, and a correcting sec- ond surface radius is unnecessary.

LOCATING THE BEAM WAIST

LOCATING THE BEAM WAIST

For a Gaussian laser beam, the location (and radius) of the beam waist is determined uniquely by the radius of curvature and optical spacing of the laser cavity mirrors because, at the reflect- ing surfaces of the cavity mirrors, the radius of curvature of the propagating beam is exactly the same as that of the mirrors. Con- sequently, for the flat/curved cavity shown in figure 36.7 (a), the beam waist is located at the surface of the flat mirror. For a sym- metric cavity (b), the beam waist is halfway between the mirrors; for non-symmetric cavities (c and d), the beam waist is located by using the equation

 

and

 

where L is the effective mirror spacing, R1 and R2 are the radii of curvature of the cavity mirrors, and z1 and z2 are the distances from the beam waist of mirrors 1 and 2, respectively. (Note that dis- tances are measured from the beam waist, and that, by conven- tion, mirror curvatures that are concave when viewed from the waist

are considered positive, while those that are convex are considered negative.)

In any case but that of a flat output mirror, the beam waist is refracted as it passes through the mirror substrate. If the output coupler’s second surface is flat, the effective waist of the refracted beam is moved toward the output coupler and is reduced in diam- eter. However, by applying a spherical correction to the second sur- face of the output coupler, the location of the beam waist can be moved to the output coupler itself, increasing the beam waist diam- eter and reducing far-field divergence. (See Calculating a Correct- ing Surface.)

It is useful, particularly when designing laser cavities, to under- stand the effect that mirror spacing has on the beam radius, both at the waist and at the curved mirror. Figure 36.8 plots equations 36.7 and 36.8 as a function of R/z (curved mirror radius divided by the mirror spacing). As the mirror spacing approaches the radius of curvature of the mirror (R/z = 1), the beam waist decreases dra- matically, and the beam radius at the curved mirror becomes very large. On the other hand, as R/z becomes large, the beam radius at the waist and at the curved mirror are approximately the same.

NEAR-FIELD VS. FAR-FIELD DIVERGENCE

NEAR-FIELD VS. FAR-FIELD DIVERGENCE

Unlike conventional light beams, Gaussian beams do not diverge linearly, as can be seen in figure 36.6. Near the laser, the divergence angle is extremely small; far from the laser, the divergence angle

approaches the asymptotic limit described in equation 36.11 above.

The Raleigh range(z ), defined as the distance over which the  beam radius spreads by a factor of √2, is given by

 

The Raleigh range is the dividing line between near-field diver- gence and mid-range divergence. Far-field divergence (the number quoted in laser specifications) must be measured at a point >zR (usually 10zR will suffice). This is a very important distinction because calculations for spot size and other parameters in an opti- cal train will be inaccurate if near- or mid-field divergence values are used. For a tightly focused beam, the distance from the waist (the focal point) to the far field can be a few millimeters or less. For beams coming directly from the laser, the far-field distance can be measured in meters.

BEAM WAIST AND DIVERGENCE

BEAM WAIST AND DIVERGENCE

Diffraction causes light waves to spread transversely as they propagate, and it is therefore impossible to have a perfectly collimated beam. The spreading of a laser beam is in accord with the predic- tions of diffraction theory. Under ordinary circumstances, the beam spreading can be so small it can go unnoticed. The following for- mulas accurately describe beam spreading, making it easy to see the capabilities and limitations of laser beams. The notation is consis- tent with much of the laser literature, particularly with Siegman’s excellent Lasers (University Science Books).

Even if a Gaussian TEM00 laser-beam wavefront were made perfectly flat at some plane, with all rays there moving in precisely parallel directions, it would acquire curvature and begin spread- ing in accordance with

 

and

 

where z is the distance propagated from the plane where the wave- front is flat, l is the wavelength of light, w0 is the radius of the 1/e2 irradiance contour at the plane where the wavefront is flat, w(z) is the radius of the 1/e2 contour after the wave has propagated a dis- tance z, and R(z) is the wavefront radius of curvature after propa- gating a distance z. R(z) is infinite at z = 0, passes through a minimum at some finite z, and rises again toward infinity as z is further increased, asymptotically approaching the value of z itself.

The plane z = 0 marks the location of a beam waist, or a place where the wavefront is flat, and w0 is called the beam waist radius.

The irradiance distribution of the Gaussian TEM00 beam, namely,

 

where w = w(z) and P is the total power in the beam, is the same at all cross sections of the beam. The invariance of the form of the distribution is a special consequence of the presumed Gaussian distribution at z = 0. Simultaneously, as R(z) asymptotically approaches z for large z, w(z) asymptotically approaches the value

 

where z is presumed to be much larger than pw02/l so that the 1/e2 irradiance contours asymptotically approach a cone of angular radius

 

This value is the far-field angular radius (half-angle divergence) of the Gaussian TEM00 beam. The vertex of the cone lies at the center of the waist (see figure 36.6).

It is important to note that, for a given value of l, variations of beam diameter and divergence with distance z are functions of a sin- gle parameter, w0, the beam waist radius.

Transverse Modes

Transverse Modes

The fundamental TEM00 mode is only one of many transverse modes that satisfies the condition that it be replicated each round-trip in the cavity. Figure 36.9 shows examples of the primary lower-order Hermite-Gaussian (rectangular) modes.

Note that the subscripts m and n in the mode designation TEMmn are correlated to the number of nodes in the x and y direc- tions. The propagation equation can also be written in cylindrical form in terms of radius (r) and angle (f). The eigenmodes (Erf) for this equation are a series of axially symmetric modes, which, for stable resonators, are closely approximated by Laguerre-Gaussian functions, denoted by TEMrf. For the lowest-order mode, TEM00, the Hermite-Gaussian and Laguerre-Gaussian functions are iden- tical, but for higher-order modes, they differ significantly, as shown in figure 36.10.

The mode, TEM01*, also known as the “bagel” or “doughnut” mode, is considered to be a superposition of the Hermite- Gaussian TEM10 and TEM01 modes, locked in phase and space quadrature. (See W.W. Rigrod, “Isolation of Axi-Symmetric Optical-Resonator Modes,” Applied Physics Letters, Vol. 2 (1 Feb. ‘63), pages 51–53.)

In real-world lasers, the Hermite-Gaussian modes predominate since strain, slight misalignment, or contamination on the optics tends to drive the system toward rectangular coordinates. Nonethe- less, the Laguerre-Gaussian TEM10 “target” or “bulls-eye” mode is clearly observed in well-aligned gas-ion and helium neon lasers with the appropriate limiting apertures.

Principle of Laser

Lasers are devices that produce intense beams of light which are monochromatic, coherent, and highly collimated. The wavelength (color) of laser light is extremely pure (monochromatic) when compared to other sources of light, and all of the photons (energy) that make up the laser beam have a fixed phase relationship (coherence) with respect to one another. Light from a laser typically has very low divergence. It can travel over great distances or can be focused to a very small spot with a brightness which exceeds that of the sun. Because of these properties, lasers are used in a wide variety of applications in all walks of life. The basic operating principles of the laser were put forth by Charles Townes and Arthur Schalow from the Bell Telephone Laboratories in 1958, and the first actual laser, based on a pink ruby crystal, was demonstrated in 1960 by Theodor Maiman at Hughes Research Laboratories. Since that time, literally thousands of lasers have been invented (including the edible “Jello” laser), but only a much smaller number have found practical applications in scientific, industrial, commercial, and military applications. The helium neon laser (the first continuous-wave laser), the semiconductor diode laser, and air-cooled ion lasers have found broad OEM application. In recent years the use of diode-pumped solid-state (DPSS) lasers in OEM applications has been growing rapidly. The term “laser” is an acronym for (L)ight (A)mplification by (S)timulated (E)mission of (R)adiation. To understand the laser, one needs to understand the meaning of these terms. The term “light” is generally accepted to be electromagnetic radiation ranging from 1 nm to 1000 mm in wavelength. The visible spectrum (what we see) ranges from approximately 400 to 700 nm. The wavelength range from 700 nm to 10 mm is considered the near infrared (NIR), and anything beyond that is the far infrared (FIR). Conversely, 200 to 400 nm is called ultraviolet (UV); below 200 nm is the deep ultraviolet (DUV). To understand stimulated emission, we start with the Bohr atom. THE BOHR ATOM In 1915, Neils Bohr proposed a model of the atom that explained a wide variety of phenomena that were puzzling scientists in the late 19th century. This simple model became the basis for the field of quantum mechanics and, although not fully accurate by today’s understanding, still is useful for demonstrating laser principles. In Bohr’s model, shown in figure 36.1, electrons orbit the nucleus of an atom. Unlike earlier “planetary” models, the Bohr atom has a limited number of fixed orbits that are available to the electrons. Under the right circumstances an electron can go from its ground state (lowest-energy orbit) to a higher (excited) state, or it can decay from a higher state to a lower state, but it cannot remain between these states. The allowed energy states are called “quantum” states and are referred to by the principal “quantum numbers” 1, 2, 3, etc. The quantum states are represented by an energy-level diagram. Basic Laser Principles www.mellesgriot.com Introduction to Laser Technology For an electron to jump to a higher quantum state, the atom must receive energy from the outside world. This can happen through a variety of mechanisms such as inelastic or semielastic collisions with other atoms and absorption of energy in the form of electromagnetic radiation (e.g., light). Likewise, when an electron drops from a higher state to a lower state, the atom must give off energy, either as kinetic activity (nonradiative transitions) or as electromagnetic radiation (radiative transitions). For the remainder of this discussion we will consider only radiative transitions. PHOTONS AND ENERGY In the 1600s and 1700s, early in the modern study of light, there was a great controversy about light’s nature. Some thought that light was made up of particles, while others thought that it was made up of waves. Both concepts explained some of the behavior of light, but not all. It was finally determined that light is made up of particles called “photons” which exhibit both particle-like and wave-like properties. Each photon has an intrinsic energy determined by the equation

E h = n

where n is the frequency of the light and h is Planck’s constant. Since, for a wave, the frequency and wavelength are related by the equation

ln = c

where l is the wavelength of the light and c is the speed of light in a vacuum, equation 36.1 can be rewritten as

E hc = l .

It is evident from this equation that the longer the wavelength of the light, the lower the energy of the photon; consequently, ultraviolet light is much more “energetic” than infrared light. Returning to the Bohr atom: for an atom to absorb light (i.e., for the light energy to cause an electron to move from a lower energy state En to a higher energy state Em), the energy of a single photon must equal, almost exactly, the energy difference between the two states. Too much energy or too little energy and the photon will not be absorbed. Consequently, the wavelength of that photon must be

D D = = − hc E

D = = − hc E EE E where m n.

Likewise, when an electron decays to a lower energy level in a radiative transition, the photon of light given off by the atom must also have an energy equal to the energy difference between the two states.

SPONTANEOUS AND STIMULATED EMISSION

In general, when an electron is in an excited energy state, it must eventually decay to a lower level, giving off a photon of radiation. This event is called “spontaneous emission,” and the photon is emitted in a random direction and a random phase. The average time it takes for the electron to decay is called the time constant for spontaneous emission, and is represented by t. On the other hand, if an electron is in energy state E2, and its decay path is to E1, but, before it has a chance to spontaneously decay, a photon happens to pass by whose energy is approximately

E24E1, there is a probability that the passing photon will cause the electron to decay in such a manner that a photon is emitted at exactly the same wavelength, in exactly the same direction, and with exactly the same phase as the passing photon. This process is called “stimulated emission.” Absorption, spontaneous emission, and stimulated emission are illustrated in figure 36.2.

Now consider the group of atoms shown in figure 36.3: all begin in exactly the same excited state, and most are effectively within the stimulation range of a passing photon. We also will assume that t is very long, and that the probability for stimulated emission is 100 percent. The incoming (stimulating) photon interacts with the first atom, causing stimulated emission of a coherent photon; these two photons then interact with the next two atoms in line, and the result is four coherent photons, on down the line. At the end of the process, we will have eleven coherent photons, all with identical phases and all traveling in the same direction. In other words, the initial photon has been “amplified” by a factor of eleven. Note that the energy to put these atoms in excited states is provided externally by some energy source which is usually referred to as the “pump” source.

Of course, in any real population of atoms, the probability for stimulated emission is quite small. Furthermore, not all of the atoms are usually in an excited state; in fact, the opposite is true. Boltzmann’s principle, a fundamental law of thermodynamics, states that, when a collection of atoms is at thermal equilibrium, the relative population of any two energy levels is given by where N2 and N1 are the populations of the upper and lower energy states, respectively, T is the equilibrium temperature, and k is Boltzmann’s constant. Substituting hn for E24E1 yields For a normal population of atoms, there will always be more atoms in the lower energy levels than in the upper ones. Since the probability for an individual atom to absorb a photon is the same as the probability for an excited atom to emit a photon via stimulated emission, the collection of real atoms will be a net absorber, not a net emitter, and amplification will not be possible. Consequently, to make a laser, we have to create a “population inversion.” POPULATION INVERSION Atomic energy states are much more complex than indicated by the description above. There are many more energy levels, and each one has its own time constants for decay. The four-level energy diagram shown in figure 36.4 is representative of some real lasers. The electron is pumped (excited) into an upper level E4 by some mechanism (for example, a collision with another atom or absorption of high-energy radiation). It then decays to E3, then to E2, and finally to the ground state E1. Let us assume that the time it takes to decay from E2 to E1 is much longer than the time it takes to decay from E2 to E1. In a large population of such atoms, at equilibrium and with a continuous pumping process, a population inversion will occur between the E3 and E2 energy states, and a photon entering the population will be amplified coherently.

THE RESONATOR

Although with a population inversion we have the ability to amplify a signal via stimulated emission, the overall single-pass gain is quite small, and most of the excited atoms in the population emit spontaneously and do not contribute to the overall output. To turn this system into a laser, we need a positive feedback mechanism that will cause the majority of the atoms in the population to contribute to the coherent output. This is the resonator, a system of mirrors that reflects undesirable (off-axis) photons out of the system and reflects the desirable (on-axis) photons back into the excited population where they can continue to be amplified.

Now consider the laser system shown in figure 36.5. The lasing medium is pumped continuously to create a population inversion at the lasing wavelength. As the excited atoms start to decay, they emit photons spontaneously in all directions. Some of the photons travel along the axis of the lasing medium, but most of the photons are directed out the sides. The photons traveling along the axis have an opportunity to stimulate atoms they encounter to emit photons, but the ones radiating out the sides do not. Furthermore, the photons traveling parallel to the axis will be reflected back into the lasing medium and given the opportunity to stimulate more excited atoms. As the on-axis photons are reflected back and forth interacting with more and more atoms, spontaneous emission decreases, stimulated emission along the axis predominates, and we have a laser.

Finally, to get the light out of the system, one of the mirrors is has a partially transmitting coating that couples out a small percentage of the circulating photons. The amount of coupling depends on the characteristics of the laser system and varies from a fraction of a percent for helium neon lasers to 50 percent or more for high-power lasers

Links

http://www.bgu.ac.il/~glevi/website/Guides/Lasers.pdf

https://www.youtube.com/watch?v=1LmcUaWuYao

https://www.youtube.com/watch?v=o8tHfNjiae4

https://www.youtube.com/watch?v=FAc5bXX4qog

https://www.youtube.com/watch?v=2YrKdq8TyFk