# II Laser operation Consider the following figures;

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II Laser operation Consider the following figures;
In this section, we discuss how do the laser elements (pump, medium and resonator) work? Consider the following figures; In the above figure ( 3 )

Step (1) An energy from an appropriate pump is coupled into the laser medium. The energy is sufficiently high to excite a large number of atoms from the ground state Eo to several excited states E3. Then the atoms spontaneously decay and back to the ground state Eo. But some of them back by a very fast (radiationless) decay from E3 to a very special level E2.

Step (2) The level E2 labeled as the “upper laser level”, has a long lifetime. Whereas most excited levels in atom decay with lifetime of order 10-8 sec. Level E2 is metastable, with a typical lifetime of order of 10-3 sec. So that the atoms being to pile up at this metastable level (E2), which functions as a bottleneck. N2 grows to a large value, because level E2 decays slowly to level E1 which labeled by “ lower laser level ” and level E1 decays to the round state rapidly, so that N1 cannot build to a large value. The net effect is the population inversion (N2>N1) between the laser levels E1 and E2.

Step (3) When the population inversion has been established, a photon of resonant energy (h=E2-E1) passes by one of the N2 atoms, stimulated emission can be occurred. Then, laser amplification begins. Note carefully that a photon of resonant energy (E2-E1) can also stimulate absorption from E1 to E2. Then the light amplification occurs and there is a steady increase in the incident resonant photon population and lasing continues.

Step )4) One of the inverted N2 atoms, which dropped to level E1 during the stimulated emission process, now decays rapidly to the ground state Eo. If the pump is still operating the atoms is ready to repeat the cycle, there by insuring a steady population inversion and constant laser beam output. Figure (3) shows the same action of figure (2(.

In (a) the laser medium is situated between the resonator (two mirrors) in which most of the atoms are in the ground state (black dots). In (b) An external energy (pumping) raising most of the atoms to the excited levels (as E3). The excited states are shown by circles. During this pumping process, the population inversion is established. In (c) the light amplification process is initiated, in which many of the photons leave through the sides of the laser cavity and are lost. Since the remainder (seed photons) are directed along the optical axis of the laser.

In (d) and (e) As the seed photons pass by the inverted N2 atoms, stimulated emission adds identical photons in the same direction, providing an ever – increasing population of coherent photons that bounce bake and forth between the mirrors. In (f) A fraction of the photons incident on the mirror (2) pass out through it. These photons constitute the external laser beam.

Characteristic of Laser Light
Monochromaticity. The light emitted by a laser is almost pure in color, almost of a single wavelength or frequency. Although we know that no light can be truly monochromatic, with unlimited sharpness in wavelength definition, laser light comes far closer than any other available source in meeting this ideal limit. The monochromaticity of light is determined by the fundamental emission process where atoms in excited states decay to lower energy states and emit light. In blackbody radiation, the emission process involves billions of atoms and many sets of energy-level pairs within each atom. The resultant radiation is hardly monochromatic, as we know.

If we could select an identical set of atoms from this blackbody and isolate the emission determined by a single pair of energy levels, the resultant radiation, would be decidedly more monochromatic. When such radiation is produced by non-thermal excitation, the radiation is often called fluorescence. Figure 1 shows such

Figure (1) fluorescence and its spectral content for a radiative decay process between two energy levels in an atom. (a) Spontaneous decay process between well- defined energy levels. (b) Spectral content of fluorescence in (a), showing line shape and linewidth. an emission process. The fluorescence comes from the radiative decay of atoms between two well-defined energy levels E2 and E1. The nature of the fluorescence, analyzed by a spectrophotometer, is shown in the lineshape plot, a graph of spectral radiant existence ( ) versus wavelength.

Note carefully that the emitted light has a wavelength spread  about a center wavelength 0 , where 0 = c/0 and 0 = (E2 - E1)/h . While most of the light may be emitted at a wavelength 0 , it is an experimental fact that some light is also emitted at wavelengths above and below 0 , with different relative existence, as shown by the lineshape plot. Thus the emission is not monochromatic, it has a wavelength spread given by 0  2 , where  is often referred to the linewidth. When the linewidth is measured at the half maximum level of the lineshape plot, it is called the FWHM linewidth, that is, “ full width at half maximum “

In the laser process, the linewidth  shown in figure (1) is narrowed considerably, leading to light of a much higher degree of monochromaticity. Basically this occurs because the process of stimulated emission effectively narrows the band of wavelengths emitted during spontaneous emission. This narrowing of the linewidth is shown qualitatively in figure (2). To gain a quantitative appreciation for the monochromaticity of laser light, consider the data in table (1), in which the linewidth of a high quality He-Ne laser is compared to the linewidth of the spectral output of a typical sodium discharge lamp and to the linewidth of the very narrow cadmium red line found in the spectral emission of a low-pressure lamp. The conversion from  to n is made by using the approximate relationship where Vl0=C.

The data of table (1) show that the He-Ne laser is 10 million times more monochromatic than the ordinary discharge lamp and about 100,000 times more so than the cadmium red line. No ordinary light source, without significant filtering, can approach the degree of monochromaticity present in the output beam of typical lasers.

Figure (2) Qualitative comparison of linewidths for laser emission and spontaneous emission involving the same pair of energy levels in an atom. The broad peak is the line shape of spontaneously emitted light between levels E2 and E1 before lasing being. The sharp peak is the line shape of laser light between levels E2 and E1 after lasing beings. Table (1) comparison of linewidths Light source Center Wavelength 0 ( A0 ) FWHM Linewidth 0 ( A0 ) FWHM linewidth (HZ) Ordinary discharge lamp 5896 1 9X1010 Cadmium low-pressure lamp 6438 0.013 9.4X108 Helium-neon laser 6328 10-7 7.5X103

Coherence. The optical property of light that most distinguishes the laser from other light source is coherence. The laser is regarded, quit correctly as the first truly coherent light source. Other light source, such as the sun or a gas discharge lamp, are at best only partially coherent. Coherence, simple stated, is a measure of the degree of phase correlation that exists in the radiation field of a light source at different location and different times. It is often described in terms of temporal coherence, which is a measure of the degree of monochromaticity of the light, and a spatial coherence, which is a measure of the uniformity of phase across the optical wavefront.

To obtain a qualitative understanding of temporal and spatial coherence, consider the simple analogy of water waves created at the center of a quite pond by a regular, periodic disturbance. The source of disturbance might be a cork bobbing up and down in regular fashion, creating a regular progression of outwardly moving crests and troughs, as in figure (3). Such a water wave filed can be side to have perfect temporal and spatial coherence. The temporal coherence is perfect because there is but a single wavelength; the crest–to-crest distance remains constant.

As long as the cork keeps bobbing regularly, the wavelength will remain fixed, and one can predict with great accuracy the location of all crests and troughs on the pond's surface. The spatial coherence of the wave filed is also perfect because the cork is a small source, generating ideal waves, circular crests, and troughs of ideal regularity. Along each wave then, the spatial variation of the relative phase of the water motion is zero that is the surface of the water all along a crest or trough is in step or in one phase.

Perfect temporal coherence
Spatial coherence Perfect spatia coherence uniformity of phase phase difference time independent (temporal coherence Figure (3) portion of a perfectly coherent water wave field created by a regularly bobbing cork at S. the wave field contains perfectly ordered wave fronts, C (crests) and T (troughs), representing water waves of a single wavelength

The water wave field described above can be rendered temporally and spatially incoherent by the simple process of replacing the single cork with a hundred corks and causing each cork to bob up and down with a different and randomly varying periodic motion. There would then be little correlation between the behavior of the water surface at one position and another. The wave fronts would be highly irregular geometrical curves, changing shape haphazardly as the collection of corks continued their jumbled, disconnected motions.

It does not require much imagination to move conceptually from a collection of corks that give rise to water waves to a collection of excited atoms that give rise to light. Disconnected, uncorrelated creation of water waves results in an incoherent water wave field. Disconnected, uncorrelated creation of light waves results, similarly, in an incoherent field.

To emit light of high coherence then, the radiating region of a source must be small in extent (in the limit, of course. a single atoms) and emit light of a narrow bandwidth (in the limited, with equal to zero). For real light sources, neither of these conditions is attainable. Real light sources, with the exception of the laser, emit light via the uncorrelated action of many atoms, involving many different wave lengths. The result is the generation of incoherent light. To achieve some measure of coherence with a non-laser source, two notifications to the emitted light can be made. First, a pinhole can be placed in front of the light source to limit the spatial extent of the source. Second, a narrow-band filter can be used to decrease significantly the linewidth  of the light. Each modification improves the coherence of the light given off by the source-but only at the expense of a drastic loss of light energy.

In contrast, a laser source, by the very nature of its production of amplified light via stimulated emission, ensures both a narrow-band output and high degree of phase correlation. Recall that in the process of stimulated emission, each photon added to the stimulated radiation has a phase, polarization, energy, and direction identical to that of the amplified light wave in the laser cavity. The laser light thus created and emitted is both temporally and spatially coherent. In fact, one can describe or model a real laser device as a very powerful, fictitious “point source” located at a distance, giving off monochromatic light in a narrow cone angle. Figure 4 summarizes the basic ideas of coherence for non-laser and laser source.

For typical laser, both the spatial coherence and temporal coherence of laser light are far superior to that for light from other sources. The transverse spatial coherence of a single mode laser beam extends across the full width of the beam, whatever that might be. The temporal coherence, also called “longitudinal spatial coherence,” is many orders of magnitude above that of any ordinary light source. The coherence time tc of a laser is a measure of the average time interval over which one can continue to predict the correct phase of the laser beam at a given point in space. The coherence length Lc, is related to the coherence time by the equationLc =ctc where c is the speed of light.

Thus the coherence length is the average length of light beam along which the phase of the wave remains unchanged. For the He-Ne laser described in table 1 the coherence time is of the order of milliseconds (compared with about 10-11s for light from a sodium discharge lamp), and the coherence length for the same laser is thousands of kilometers (compared with fractions of a centimeter for the sodium lamp).

Improve spatial coherence
Improve temporal coherence

Figure 4. A tungsten lamp requires a pinhole and filter to produce coherent light. The light from a laser is naturally coherent. (a) Tungsten lamp. The Tungsten lamp is an extended source that emits many wavelength. The emission lacks both temporal and spatial coherence. The wave front are irregular and change shape in a haphazard manner. (b) Tungsten lamp with pinhole. An ideal pinhole limits the extent of the tungsten source and improve the spatial coherence of the light. However the light still lacks temporal coherence since all wavelengths are present. Power in the beam has been decreased.

(c) Tungsten lamp pinhole and filter
(c) Tungsten lamp pinhole and filter. Adding a good narrow –band filter further reduces the power but improves the temporal coherence. Now the light is "coherent" but the available power that initially radiated by the lamp. (d) Laser. Light coming from the laser has a high degree of spatial and temporal coherence. In addition, the output power can be very high.

Directionality. When one sees thin, pencil-like beam of a He-Ne laser for the first time, one is struck immediately by the high degree of beam directionality. No other light source, with or without the help of lenses or mirrors, generates, a beam of such precise definition and minimum angular spread. The astonishing degree of directionality of a laser beam is due to the geometrical design of the laser cavity and to the monochromatic and coherent nature of light generate in the cavity. Figure (5) shows a specific cavity design and an external laser beam with an angular spread signified by the angel

The cavity mirrors shown are shaped with surfaces concave toward the cavity, thereby “focusing” the reflecting light back into the cavity and forming a beam waist at one position in the cavity. Figure 5. external and internal laser beam for a given cavity. Diffraction or beam spread, measured by the beam divergence angle f, appears to be caused by an effective aperture of diameter D, located at the beam waist.

The nature of the beam inside the laser cavity and its characteristics outside the cavity are determined by solving the rather complicated problem of eectromagnetic waves in an open cavity. Although the details of this analysis beyond the scope of this discussion, several results are worth examining. It turns out that the beam- spread angel f is giving by the relationship (1) Where  is the wavelength of the laser beam and D is the diameter of the laser beam at its beam waist. One cannot help but observe that Eq. (1) is quite similar to that obtained when calculating the angular spread in light generated by the diffraction of plane waves passing thought a circular aperture .

The pattern consists of a central, bright circular spot, the Airy disk, surrounded by a series of bright rings. The essence of this phenomenon is shown in figure (6). The diffraction angle , tracking the Airy disk, is given by (2)

Figure (6). Fraunhofer diffraction of plane waves through a circular aperture. Beam divergence angle is set by the edges of the Airy disk.

where  is the wavelength of the collimated
where  is the wavelength of the collimated. Monochromatic light and D is the diameter of the circular aperture. Both Eqs. (1) and (2) depend on the ratio of a wavelength to a diameter. They differ only by a constant coefficient. It is tempting, then, to think of the angular spread  inherent in laser beams and given in Eq. (1) in terms of diffraction. If we treat the beam waist as an effectives circular aperture located inside the laser cavity, then by controlling the size of the beam waist we control the diffraction or beam spread of the laser. The beam waist, in practice, is determined by the design of the laser cavity and depends on the radii of curvature of the two mirrors and the distance between the mirror.

Therefore, one ought to be able to build lasers with a given beam waist and, consequently, a given beam divergence or beam spread in the far field, that is, at sufficiently great distance L from the diffracting aperture that L >> area aperture/. Such is indeed the case. With the help of Eq. (1), one can now develop a feel for the low beam spread, or high degree of directionality, of laser beams. He-Ne lasers (632.8 nm) have an internal beam waist of diameter near 0.5 mm. Equation (1) then yields

This is a typical laser-beam divergence, indicating that the beam width will increase about 1.6 cm every 1000cm. Since we can control the beam waist D by laser cavity design and “select” the wavelength by choosing different laser media, what lower limit might we expect for the beam divergence? How directional can lasers be? If we design a laser with a beam waist of 0.5 cm diameter and a wavelength of 200 nm, the beam divergence angle  becomes about radian,. This beam would spread about 1.6cm every 320m.

Clearly, if beam waist size is at our command and lasers can be built with wavelength below the ultraviolet, there is no limit to how parallel and directional the laser beam can be made. The high degree of directionality of the laser, or any other light source, depends on the monochromaticity and coherence of the light generated. Ordinary sources are neither monochromatic nor coherent. Lasers, on the other hand, are superior on both counts, and as a consequence generate highly directional, quasi- collimated light beams.

Laser Source Intensity
Laser Source Intensity. It has been that a 1-mW He- Ne laser is hundreds of times “brighter” than the sun. As difficult as this may be to imagine, calculations for luminance or visual brightness of a typical laser, compared to the sun, substantiate these claims. To develop an appreciation for the enormous difference between the radiance of lasers and thermal sources we consider a comparison of their photon output rates (photons per second).

Small gas lasers typically have power outputs P of 1mW
Small gas lasers typically have power outputs P of 1mW. Neodymium-glass lasers, such as those under development for the production of laser-induced fusion, boast of power outputs near 1014 W!. Using these two extremes and an average energy of J per visible photon (E=h), the photon output of laser (P/h) varies from 1016 photons/s to 1033 photons/s. For comparison, consider a broadband thermal source with a radiating surface equal to that of the beam waist of a 1-mW He-Ne laser with diameter of 0.5 mm, an area of A=2X103 cm2.

Let the surface emit radiation at a wavelength of 633 nm with a linewidth of =100nm (or=7x1013HZ) and temperature T=1000K. The photon output rat for the broadband source can be calculated from the equation (3) Substituting the values given above into Eq. (3), we find that the thermal photon output rat is only about 109 photons/s! This value is 7 orders of magnitude smaller than the photon output rat of low-power 1-mW He-Ne laser and 24 orders of magnitude smaller than a powerful neodymium-glass laser. The comparison is summarized in figure 7.

Figure (7). Comparison of photon output rates between a low-power He-Ne gas laser and a hot thermal source of the same radiating surface area. (a) 1-mW He-Ne laser (=633nm), A=2x10-3cm2, o=633nm  =100nm. Note that the laser emits all of the photons in a small solid angle (2x10-6 sr) compared with the 2 solid of the thermal source.

We see also from figure 7 that the He-Ne laser emits photons/s into a very small solid angle of about 2X10-6sr. whereas the thermal emitter, acting as a Lambertian source, radiates 109 photons/s into a forward, hemispherical solid angel of 2 sr. If we were to ask how many thermal photons/second are emitted by the thermal source into a solid angel equal to that of the laser, we would find the answer to be 320 photons/s: The comparison between 1016 photons/s for the laser source and 320 photons/s for the thermal source is now even more dramatic.

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