Properties of Lasers

Lasers: Basic Characterstics

Laser has certain unique properties, namely, high monochromaticity, coherence and directionality, compared to ordinary sources of light, though both are electromagnetic radiations. These properties are briefly discussed in the following sections.

Monochromaticity

The energy of a photon determines its wavelength through the relationship E = hc/λ, where c is the speed of light, h is Planck's constant, and λ is wavelength. In an ideal case, the laser emits all photons with the same energy, and thus the same wavelength, it is said to be monochromatic. The light from a laser typically comes from one atomic transition with a single precise wavelength. So the laser light has a single spectral color and is almost the purest monochromatic light available.

However, in all practical cases, the laser light is not truly monochromatic. A truly monochromatic wave requires a wave train of infinite duration. The spectral emission line from which it originates does have a finite width, because of the Doppler effect of the moving atoms or molecules from which it comes. Compared to the ordinary sources of light, the range of frequency (line width) of the laser is extremely small. This range is called line width or bandwidth.

Why the laser light is monochromatic? Following are the factors responsible for making the laser beam monochromatic:

• Laser light consists of essentially one wavelength, having its origin in stimulated emission from one set of atomic energy levels. This is possible because laser transition, in principle, involves well-defined energy levels.
• EM wave of frequency n = (E₂ - E₁) only can be amplified, n has a certain range which is called line width. This line width is decided by various broadening factors such as Doppler effect of moving atoms and molecules.
• The generation of laser is such that the laser cavity forms a resonant system and laser oscillation is sustained only at the resonant frequencies of the cavity. This leads to the further narrowing of the laser line width. So laser light is usually very pure in wavelength, we say it has the property of monochromatic.

The lasers, in general, generate light in a very narrow band around a single, central wavelength. The degree of monochromoticity can be quantitatively described in terms of wavelength bandwidth or frequency bandwidth. The narrower is the line width, higher degree of the monochromocity of the laser has. However this depends on the type of laser, and special techniques can be used to improve monochromaticity. Typically, the frequency bandwidth of a commercial He-Ne laser is about 1500MHz (full width at half-maximum, FWHM). In terms of wavelength, it means that at a wavelength of 632.8nm this means a wavelength bandwidth of about 0.01nm. On the other hand, the bandwidth of a typically diode laser with a wavelength of 900nm is about 1nm as compared to LED, which has a bandwidth of approximately 30-60 nm.

Monochromatic output, or high frequency stability, is of great importance for lasers being used in interferometric measurements since the wavelength is the measure of length and distance and must be known with extreme precision, at least one part in a million, and it must remain constant with time. The same holds true for lasers used in chemical and many other scientific analytical applications. Both these techniques are important in quality control and inspection. For these applications, frequency stabilized 632.8 nm HeNe laser (a frequency of approximately 473 THz) with a 1 MHz bandwidth are commercially available.

Another important laser: the Nd:YAG laser used in most laser designators, generates an output beam at 1.064 microns, with a typical bandwidth of 0.00045 microns, an amazingly narrow line width of 0.04 percent of the central wavelength. This spectrally pure output is critical for a multitude of applications, including remote sensing for specific chemical constituents and high signal-to-noise ratio (SNR) communications.

This property of monochromoticity has excellent applications in high-resolution spectroscopy to observe specific transitions in a molecule. A practical application is the separation of isotopes in the nuclear industry where the fissionable isotope of Uranium, 235U, is separated from the non-fissionable one 238U by exploiting the minute difference in their energy levels.

Coherence

When an excited atom, depending on its lifetime at the higher energy level, comes down to lower energy level, a photon is emitted, corresponding to the equation,

where h is the Planks constant, n is the frequency of the emitted photon and E₂ and E₁ correspond to higher and lower energy levels respectively. This type of natural emission occurs in different directions and is called spontaneous emissions. It is characterized by the lifetime of the upper excited state after which it spontaneously returns to lower state and radiates away the energy by emission. Interestingly, apart from spontaneous emission, an excited atom can be induced to emit a photon by another photon of same frequency - i.e. a passing photon can stimulate a transition from a higher level to the lower level, thus resulting in the emission of two photons, which is gain. The two emitted photons are said to be in phase, which means that the crest or the trough of the wave associated with one photon will occur at the same time as on the wave associated with the other photon. An avalanche of similar photons is created and these photons have a fixed phase relationship with each other. This fixed phase relationship between the photons from various atoms in the active medium results in the laser beam generated having the property of coherence. Since the radiation emitted is by the stimulation process, it is referred to as the stimulated emission and the generation of laser is by stimulated emission.

In the case of spontaneous emission, the emission is natural where as in the case of stimulated emission, it is induced or stimulated. Further there is no amplification in the case of spontaneous emission as well as no phase relationship between emitted photons, as it happens in the case of stimulated emission. But one has to remember that under normal conditions, there is far more atoms in the lower level than in the upper level and as such absorption dominates stimulated emission. In order to reverse this trend, there must be much more atoms in the upper level than in the lower level. This specific condition is called population inversion and is essential for stimulated emission to be in a predominant position for generation of laser. In the case of laser, the stimulated emission process is responsible for the emission of photons and amplification. Since the emitted photons have a definite phase relationship with each other, coherent output is produced. i.e. the atoms emit photons in phase with the incoming stimulating photons and emitted waves adds to the incoming waves, generating brighter output. Addition is due to the relative phase relationship. Photons of ordinary light also come from atoms without any phase relationship with each other and are not coherent. Therefore, laser is called a coherent light source where as an ordinary light is called an incoherent light source.

To sum up, the two conditions necessary for laser action are population inversion and stimulated emission. Inside a laser, the stimulated emission occurs in a resonant cavity with mirrors at both ends. Thus by repeating this process of interaction of photon with excited atoms many times, one can produce a highly coherent beam of light. Since a common stimulus triggers the emission events, which provide the amplified light, the emitted photons are "in step" and have a definite phase relation to each other. These emitted photons having a definite phase relation to each other, generates coherent output, i.e. the atoms emit photons in phase with the incoming stimulating photons and emitted waves add to the incoming waves, generating brighter output. Addition is due to the relative phase relationship. Photons of ordinary light also come from atoms, but independent of each other and without any phase relationship with each other and are not coherent. Therefore, laser is called a coherent light source where as an ordinary light is called an incoherent source of light. The concept of coherence can be well understood from the following figure.

Figure (a) depicts a typical beam of light waves from an ordinary source traveling through space. One can see that these waves do not have any fixed relationship with each other. This light is said to be "incoherent", meaning that the light beam has no internal order. Figure (b), on the other hand, illustrates the light waves within a highly collimated laser beam. All of these individual waves are in step, or "in phase", with one another at every point. "Coherence" is the term used to describe such a property of laser light.

There are two types of coherence - spatial and temporal.

Correlation between the waves at one place at different times, or along the path of a beam at a single instant, are effectively the same thing, and are called "temporal coherence". Correlation between different places (but not along the path) is called "spatial coherence".

To understand coherence, let us take two points on a wave front, at time equal to zero. There will be a certain phase difference between these two points and if it remains same even after lapse of a period of time, then the electromagnetic wave (em) has perfect coherence between the two points. In case, the phase difference remains same for any two points anywhere on the wave front, then we say that the electromagnetic wave has perfect spatial coherence, where as if this is true only for a specific area, then the electromagnetic wave is said to have only partial spatial coherence. Spatial coherence is related to directionality and uniphase wave fronts.

Now let us consider a single point on the wave front. There will be a phase difference between time, t = 0 and t = d t of the electromagnetic wave. If this phase difference remains same for any value of d t, then we say that the em wave has perfect temporal coherence. But if this is only for a specific value of d t, then the em wave has partial temporal coherence.

It may be understood that these two types of coherence are independent of each other. i.e. an em wave with partial temporal coherence can have perfect spatial coherence.

Some important points

• Coherence is a property of waves that indicates the ability of the waves to interfere with each other. Two waves that are coherent can be combined to produce an unmoving distribution of constructive and destructive interference (a visible interference pattern) depending on the relative phase of the waves at their meeting point. Waves that are incoherent, when combined, produce rapidly moving areas of constructive and destructive interference and therefore do not produce a visible interference pattern.
• Another way of saying the same thing is that coherence is a measure of the ability of a light source to produce high contrast interference fringes when the light is interfered with itself in an interferometer. High coherence means high fringe visibility with excellent contrast, (i.e., good black and white fringes, or black and whatever color the light is); low coherence means washed-out fringes, and zero coherence means no fringes.
• One of the ways of understanding coherence is to predict it. Suppose one can take a snapshot of the waves, and then take another snapshot at a later time. If the two snapshots look almost identical, even with a long time interval, then we have a high degree of coherence.
• A wave can also be coherent with itself, a property known as temporal coherence. If a wave is combined with a delayed copy of itself, the duration of the delay over which it produces visible interference is known as the coherence time of the wave, Δt_c. From this, a corresponding coherence length l_c, can be estimated as:
  l_c = Δt_c

  where c is the speed of the light wave.

  Coherence time (Δt_c) relates to the finite bandwidth of the source and in general, it is proportional to the bandwidth

  Δt_cα1/Δn
  where Δ n is the frequency bandwidth. The temporal coherence comes from the monochromaticity of the laser beam. The narrower the line width Δλ or Δ n of the light source, the better is its temporal coherence.
• If the laser supports the oscillation of multiple longitudinal modes but no higher-order transverse modes, it means that the laser output has finite temporal coherence but perfect spatial coherence. The longitudinal modes in a laser are equally spaced in frequency by c/2L where c is the velocity of light and L is the effective length of the laser resonator. The number of longitudinal modes determines the coherence length of the laser. The relation between the coherence length l_c, the longitudinal mode spacing c/2L and the number of modes N (N>!) is
  N = [c/2 l_c ] / c/2 L = L/ l_c
• Small coherence lengths are obtained with lasers, which support oscillation over a very wide spectral bandwidth.
• There are certain gas lasers which have very long coherence length of tens of meters, while other lasers, especially Diode lasers have coherence length of the order of millimeters.
• Typically for a commercially available He-Ne laser with 632.8 nm wave-length and 0.01 nm spectral bandwidth, the coherence length is about 4cm or at the most of the order of the length of its resonator because of the presence of many longitudinal modes.
• A frequency stabilized 632.8 nm HeNe laser (a frequency of approximately 473 THz) with a 1 MHz bandwidth would have a coherence length of about 300 meters. In this case, the coherence length is much longer than the length of the cavity because only a single longitudinal is forced to be active in it at any given time.
• An 800 nm laser diode with a 1 nm spectral width would have a coherence length of about 0.64 mm.
• A 600 nm LED with a spectral width of 60 nm would have a coherence length of around 6 μm.
• He-Ne lasers are also much more spatially coherent than LEDs. LEDs generally have a very short spatial coherence length, typically only a couple of wavelengths.
• In holography, the temporal coherence length determines the maximum depth of the object in a reflection hologram, and the spatial coherence length determines the lateral size. Holography, which is based on interference between light beams, long coherence length enables taking holograms of large bodies, which require greater depth of field. Both the light reflected from the near part of the body, and the light reflected from the far part of the body, will still be coherent with the reference beam.
• Spatial coherence refers to how spherical the wave front is. Does every portion of the wave front appear to have exactly the same center of curvature?
• The requirement for high temporal coherence is in coherent, or heterodyne detection. In these systems, energy reflected off the target is mixed with energy from the original laser to create a fringe pattern. The photons are supposed to maintain the fixed phase relationship for the time needed to hit the target and return in order to have proper contrast in the fringe pattern.
• One of the important applications includes Doppler velocity measurements of the target through the measurements of the frequency shift because of the moving targets. The frequency shift from the target-reflected energy is a function of the target velocity. However, if the frequency of the laser itself is shifting (because of poor coherence) during the time of flight, this creates a broadening or an error in the frequency of the returned beam that limits how accurately one can measure the Doppler velocity.
• Spatial coherence is high for sphere waves and plane waves, and is related to the size of the light source. A point source emits spatially coherent light, while the light from a finite source has lower coherence. Spatial coherence can be increased with a spatial filter; a very small pinhole preceded by a condenser lens. The spatial coherence of light will increase as it travels away from the source and becomes more like a sphere or plane wave. Light from distant stars, though far from monochromatic, has extremely high spatial coherence.
• Coherence length is defined as the length over which energy in two separate waves remains constant. With respect to the laser, it is the greatest distance between two arms of an interferometric system for which sufficient interferometric effects can be observed.
• Using Michelson Interferometer, one can estimate the coherence length by measuring the maximum path difference between the two beams, which still show the interference pattern.
• Since the temporal coherence is a measure of the ability of the radiation to perform interference, as a result of differences in path lengths between the two beams, it is thus important in interferometry and holography.
• Ordinary light is not coherent because it comes from independent atoms, which emit on time scales of about 10-8 seconds. There is a degree of coherence in sources like the mercury green line and some other useful spectral sources, but their coherence does not approach that of a laser.

Beam diameter

It is very interesting to note that, the intensity of laser light is not same throughout the cross section of the beam. This is because of the fact that the cavity also controls the trans-verse modes, or intensity cross sections. The ideal beam has a symmetric cross section: The intensity is greater in the middle and tails off at the edges. This is called the Transverse Electromagnetic Mode (TEM ₀₀) output as shown in the figure. The subscripts n and m (0 and 0 in this case) in the TEM _nm are correlated to the number of nodes in the x and y directions. A theoretical TEM ₀₀ beam has a perfect Gaussian profile. Detailed discussion on modes is given in the next section. Lasers can produce many other TEM modes, which would be discussed in later sections. In general, one can say that laser beams have a symmetric intensity profile. i.e. if we run across the beam, the intensity is minimum at the edge and as we move towards the center it increases and is maximum at the center and then it falls in a similar fashion as on the other side, where from we started. In fact, we can start at any point on the rim of the laser beam and the result will be same, as discussed earlier. Beam diameter is defined as the diameter of a circular beam at a certain point where the intensity drops to a certain fraction of its maximum value. The common definitions are half the intensity i.e. full width at half maximum (FWHM), 1/e (0.368) and 1/e² (0.135) of the maximum value. In other words, beam diameter is the diameter of the laser beam cross section between points near the outer edge of the beam where its intensity is only 50 % (FWHM), 63% (1- 1/e) and about 86% (1-1/e²) of the intensity at the beam center.

Directionality and beam divergence

One of the important properties of laser is its high directionality. The mirrors placed at opposite ends of a laser cavity enables the beam to travel back and forth in order to gain intensity by the stimulated emission of more photons at the same wavelength, which results in increased amplification due to the longer path length through the medium. The multiple reflections also produce a well-collimated beam, because only photons traveling parallel to the cavity walls will be reflected from both mirrors. If the light is the slightest bit off axis, it will be lost from the beam. The resonant cavity, thus, makes certain that only electromagnetic waves traveling along the optic axis can be sustained, consequent building of the gain.

The high degree of collimation arises from the fact that the cavity of the laser has very nearly parallel front and back mirrors, which constrain the final laser beam to a path, which is perpendicular to those mirrors. Collimation refers to the degree to which the beam remains parallel with distance. A perfectly collimated beam would have parallel sides and would never expand at all. Its divergence angle would be exactly zero. Diffraction plays an important role in determining the size of laser spot that can be projected at a given distance. The oscillation of the beam in the resonator cavity produces a narrow beam that subsequently diverges at some angle depending on the resonator design, the size of the output aperture, and resulting diffraction effects on the beam. These diffraction effects usually referred as a beam-spreading effect are a result of the light waves passing through a small opening. These diffraction phenomena impose a limit on the minimum diameter of a light point after passing through an optical system. For a laser, the beam emerging from the output mirror can be thought of as the opening or aperture, and the diffraction effects on the beam by the mirror will limit the minimum divergence and spot size of the beam. For beams in TEM ₀₀ mode, diffraction is usually the limiting factor in beam divergence.

In fact one can say that, divergence angle describes the directionality of the laser. For a perfect spatially coherent laser beam, the diffraction limited divergence angle θ is given by,

where λ and D are the wavelength and diameter of the laser beam respectively. K is a constant factor that is usually unity but depends on the wavelength. The relationship clearly demonstrates that beam divergence increases with wavelength, and decreases as beam (or output lens) diameter increases. In other words, a smaller diameter beam will suffer more divergence and greater spread with distance than a larger beam. For a perfect gaussian beam, the divergence θ_o (half angle), is related to beam waist radius w_o as

Using the above equations, and assuming K or 2 x (1 / π) as unity, let us calculate the minimum divergence (full angle) that can be theoretically achievable for the most well known lasers, i.e. Nd:YAG (l = 1.06 mm with 3mm diameter) and He-Ne laser (l = 0.6328 mm with 1mm beam diameter). The divergence angles are 0.353 milli-rad or 0.02014° and 0.6328 mrad or 0.03607° respectively. Compare this with the divergence of the light from a torchlight (20° or more) and the high directionality of laser beams becomes quite obvious. As the spatial coherence becomes partial or the degree of coherence reduces, the divergence increases accordingly and for calculating the divergence, the diameter of the beam D is to be replaced with the coherence area in the above-mentioned equation.

Consider the size (diameter) of a collimated beam as it propagates as shown in the figure. It can be seen that the diameter increases. This increase of the beam size is due to the beam divergence and the same is measured in milliradians (mrad). It is either measured as full angle (measure of increase in diameter) or as half angle (measure of increase in radius).For example, the diameter of a beam of 1mrad full angle divergence, after propagation of 1Km, will be 1m (Physical optics). For small angle, the divergence can be approximated as the ratio of the beam diameter to the distance from the laser aperture.

Brightness

While summing up the discussion on monochromaticity (narrow line width) and directionality (low divergence) of laser, radiance of laser cannot be missed out. It is defined as the power emitted per unit surface area per unit solid angle. The units are watts per square meter per steradian. A steradian is the unit of solid angle, which is three-dimensional analogue of conventional two-dimensional (planar) angle expressed in radians. For small angles the relation between a planar angle and the solid angle of a cone with that planar angle is to a good approximation:

where θ is the planar angle and Ω is the solid angle as shown in the figure.

The radiance of a 1mm He-Ne laser with 1 mm out put diameter and a divergence of 1 milli-radian is 1.6 x 10⁹ Watts/m²-steradian, which can be estimated in the following manner.

The solid angle corresponding to one millirad is:

and the radiance is power divided by the area of the beam and the solid angle. Thus radiance B is

B = 10^-3W/(0.785 x 10^-6)(0.8 x 10^-6) = 1.6 x 10⁹ Watts/m²-steradian

The radiance of a milliwatt helium neon laser is far greater than 10⁶ Watts/m²-steradian, that of the sun which emits more than 10²⁶ W.

This is a unique advantage for many of the laser applications in various areas.

Laser Modes

As we know that part of the laser light in the laser cavity emerges through the output mirror. The optical waves within an optical resonant cavity are characterized by their resonant modes, which are discrete resonant conditions governed by the dimensions of the cavity. The laser beam radiated from the laser cavity is thus not arbitrary. Only the waves oscillating at modes that match the oscillation modes of the laser cavity can be produced. The laser modes governed by the axial dimensions of the resonant cavity are called the longitudinal modes, and the modes determined by the cross-sectional dimensions of the laser cavity are called transverse modes.

Longitudinal Mode

Generally speaking light modes means possible standing EM waves in a system. The number of modes in this meaning is huge. Laser mode means the possible standing waves in laser cavity. We see that stimulated lights are transmitted back and forth between the mirrors and interfere with each other, as a result only light of those frequencies, which create nodes at both mirrors are allowed. In other words, if the round trip distance is integer multiples of the wavelength ?, only then it can result in a standing wave. Thus, the cavity length must be an integer multiplication of half their wavelengths. The result is the condition of resonance: light waves are amplified strongly, if and only if, they satisfy the equation:

where L is the cavity length, n is the refractive index of the laser medium, nL is theoptical path, N is an integer and λ denotes the wavelength.

The integer N cannot be an arbitrary number. It is limited by the fluorescence curve and only the modes for which the gain of laser of the laser medium G (λ) > 1 would be supported.

The above equation can be rewritten as:

and

Where c and f are the velocity and frequency of light.

Assuming a cavity of length 50 cm, it gives us the possible number of modes as 159 x 10⁴ and the separation between two modes as 300 MHz. However, if the laser bandwidth is of the order of 2.5 GHz, it can support only 6 longitudinal modes.

Some important points related to longitudinal modes:

• Modes governed by the axial dimensions of the resonant cavity are called longitudinal modes. The longitudinal modes are formed when the two waves with the same frequency and amplitude are moving against each other.
• The to and fro movement of the electromagnetic radiation is controlled by the laser cavity end mirrors and only the waves with nodes at both ends are sustained or allowed, which means that the cavity length should be an integral multiple of the half wavelengths. Thus the cavity length and the refractive index of the laser medium determine the frequencies that are allowed inside the cavity.
• The other important aspect is that the frequencies are spaced at equal intervals.
• In applications where power is more important as in most high power applications for material processing or medical surgery, multimode lasers can be used. As such the laser is being used as a mean for transferring the energy on to the target. Thus there is not much importance for the longitudinal laser modes. However, for applications involving interference such as holography or interferrometric measurements, and in applications related to spectroscopic and photochemical, where single well-defined wavelength is required, single mode lasers are very critical.
• Increase in the cavity length increases the number of possible laser modes under the fluorescence curve. However, it reduces the frequency gap between the adjacent modes. This leads to that a single mode laser can be made by reducing the length of the cavity, such that only one longitudinal mode will remain under the fluorescence curve with G_L>1.
• The multiple longitudinal mode structure gives rise to a power fluctuation phenomenon termed mode sweeping. All unstabilized helium neon lasers exhibit this effect, which is due to thermal instability causing variation in the cavity length. As the cavity length changes, there is a small change in mode spacing which is typically 10 kHz or less under normal conditions.

Transverse Mode

The configuration of the optical cavity determines the transverse modes of the laser output, which characterizes the intensity distribution of laser beam in the transverse plane that is perpendicular to the direction of propagation. If we intersect the output laser beam and study the transverse beam cross section, we find the light intensity can be of different distributions (patterns). These are called Transverse Electromagnetic Modes (TEM). Two indices are used to indicate the TEM modes - TEM_pq, p and q are integer numbers indicating the number of points of zero illumination (between illuminated regions) along x axis and y - axis respectively.

As explained earlier that the amplitude of a light beam is increased in a laser by multiple passes of coherent light waves through the active medium. The process is accomplished by an active medium placed between a pair of mirrors that act as a feedback mechanism. During each round trip between the mirrors, the light waves are amplified by the active medium and reduced by internal losses and laser output. A number of different combinations of mirrors, such as plane and curved, have been utilized in practical laser. Some of them are shown in the figure.

Most common form of structure is a stable resonator, which concentrate light along the laser axis, extracting energy efficiently from that region, but not from the outer regions far from the axis. This cavity will then have a set of nearly loss less resonant modes, which will have the form of very nearly perfect Hermite-gaussian or Laguerre-gaussian mathematical functions. The lowest-order mode will have an essentially ideal gaussian profile with a certain spot size, which depends only on the spacing and radii of the mirrors and the wavelength of the light and not on the mirror diameter, which is assumed to be very large typically four to five times of the beam size. This spot size, called the "gaussian spot size" and can be estimated by a simple formula in terms of the cavity length L, the end mirror radii r₁ and r₂, and the wavelength. The beam thus it produces has an intensity peak in the center, and a Gaussian drop in intensity with increasing distance from the axis. The fundamental TEM₀₀ mode is only one of many transverse modes that satisfy the round-trip propagation criteria.

For most applications for example like holography, the TEM₀₀ mode is considered most desirable, but multi-mode beams can often deliver more power, though with a poorer beam quality, and may be acceptable in applications where power is the main criterion.

The laser can be forced to lase in a single TEM₀₀ mode by simply putting a pinhole with proper diameter between the two mirrors. The pinhole diameter should be equal to the diameter of the lower mode as this would allow only this mode to pass through the pinhole, and all higher modes will be attenuated. Since radiation inside the optical cavity undergoes multiple passes, only the basic mode will be amplified, and appear in the output.

Beam Quality

Discussion on properties of laser will not be complete without making an assessment of beam quality. Laser beam quality is important since the closer a real laser beam is to diffraction-limited, the more tightly it can be focused, the greater depth of field, and the smaller the diameter of beam-handling optics need to transmit the beam. For applications such as directed energy applications, a better beam quality translates into better delivery of optical power to the target in the far field. For material processing, the more tightly focused the laser beam results in the higher intensities. The design of optical delivery systems for laser systems is highly dependent on the laser's beam quality.

It was thus felt that to recognize, quantify and determine the beam propagation characteristics, a figure of merit would be very necessary and useful. Therefore, the concept of a dimensionless beam propagation parameter, M² was developed in 1970 for all types of lasers. M² is a quantitative measure of the quality of the laser beam and according to ISO standard 11146, it is defined as the beam parameter product (BPP) divided by λ / π. The beam divergence, as discussed earlier, is

where w_0R is the beam radius at the beam waist and θ the wave length.

(Beam parameter product (BPP) is the product of a laser beam's divergence angle and the diameter of the beam at its narrowest point (the "beam waist"). Its units are mm mrad.

M² can also be defined in the following manner:

• The ratio of the BPP of an actual beam to that of an ideal Gaussian beam at the same wavelength. This parameter is a wavelength-independent measure of beam quality.
• It is the ratio of the divergence of the real beam to that of a theoretical diffraction-limited beam of the same waist size with a Gaussian beam profile.

ISO standard 11146 has laid down procedures for the measurement of M² also. This was necessitated by the use of large number of high power lasers for industrial applications like cutting, drilling and welding, with high cost of investment. Here it is necessary to focus the laser beams 'tightly' to produce highest possible radiance with minimum collateral damage. Technically, only high quality and reliable laser beams can ensure this aspect as well as profitable return to the investment. M² beam quality factor limits the degree to which a laser beam can be focused for a given beam divergence, which in turn is limited by the numerical aperture of the focusing lens. A word of caution that is necessary since M² factor would be different for two orthogonal directions to the beam axis for non-circular beams. For example, for diode bars, M² is low for the fast axis and high for the slow axis.

For any laser beam, the product of the beam radius (w_0R) and the far-field divergence (θ) is a constant, and the ratio,

where w_0Real and q_Real are the beam waist and far field divergence of the real beam respectively. M² is an accurate indication of the propagation characteristics of the beam.

There are some other important points related to M²:

• The value of M² is always greater than or equal to 1 and ranges from 1 for a diffraction-limited TEM₀₀ laser beam, to several hundred for a distorted, poor quality beam.
• M² = 1 only occurs for single-mode TEM₀₀ Gaussian beams.
• Helium neon lasers typically have an M² factor that is less than 1.1. For ion lasers, the M² factor is typically between 1.1 and 1.3. Collimated TEM₀₀ diode laser beams usually have an M² ranging from 1.1 to 1.7. For high-energy multimode lasers, the M² factor can be as high as 3 or 4. In all cases, the M² factor, which varies significantly, affects the characteristics of a laser beam and cannot be neglected in optical designs.
• Though carefully/optimally designed lasers can achieve the M² ~ 1, high power lasers have very much higher M² value of 100 or even more. Thermal distortions in the active gain media, use of poor optical quality of components, diffraction effects at apertures etc. are the main reasons for the reduction of beam quality. Beam quality also gets affected adversely when the lasers work at higher cavity modes. A pump source with uniform intensity distribution, an optimized cavity design with least sensitivity to thermal lensing, high optical quality components and a gain medium least prone to thermal disturbance are a pre-requisite to a laser system for the generation of high beam quality output.

References

• John Ambroseo : Coherent Laser - THE PHOTONICS DESIGN AND APPLICATIONS HANDBOOK 2001
• LEOT tutorial on lasers
• Coherence-free definition
• Definition of a laser
• Melles Griot - FAQ's
• Melles Griot Guide Book, The Practical Applications of Light, Tutorial on Laser Beam Measurement
• Molecular expressions Microscopy Primer Physics of light and color - Introduction to lasers
• Quantum properties of light
• Rami Arieli The Laser Adventure
• Sam's Laser FAQ - Items of Interest
• WHY LASERS
• Encyclopedia of Laser Physics and Technology
• E. Siegman, New developments in laser resonators, Proc. SPIE, 1224, 2 (1990)
• E. Siegman, Defining, measuring and optimizing laser beam quality, Proc.SPIE, 1868, 2 (1993)
• ISO standard 11146