This version reflects the comments of the core participants as reviewed and incorporated in accordance with CORD's FIPSE-supported Curriculum Morphing Project.

MODULE 1-9
SPATIAL CHARACTERISTICS
OF LASERS

(1) The spatial distribution of the irradiance of a laser beam is of prime importance in many applications. For example, laser drilling requires beams of a particular diameter so that holes of the proper size can be drilled. Laser ranging requires well­collimated beams that diverge slowly as they travel away from the laser. Almost all applications require the uniform spatial distribution of irradiance produced by the Gaussian,or TEM₀₀ mode.

(2) This module discusses (1) transverse electromagnetic modes in lasers and how they can be controlled, (2) methods for the determination and control of laser­beam divergence, and (3) methods for the determination of beam diameter at a known distance from the laser or at the focal point of a lens. In the laboratory, the student will scan a laser beam with a small­aperture detector to determine the beam profile and will measure beam diameter at two points and calculate beam divergence.

Upon completion of this module, the student should be able to:

1. Sketch selected transverse electromagnetic modes of a laser.
2. Explain briefly the origin of transverse electromagnetic modes in a laser, and explain how unwanted higher­order modes can be eliminated.
3. Draw and label a diagram of the irradiance of the TEM₀₀ mode as a function of distance across the beam.
4. Explain the meaning of the following terms:
   • a. Spot size.
     b. Beam diameter at 1/e² points.
5. List three reasons for which the TEM₀₀ mode is the most important mode in practical applications of lasers.
6. Given the wavelength of a laser and the diameterof its output aperture, calculate the diffraction­limited beam divergence.
7. Given the initial diameter and divergence angle of a laser beam, calculate its diameter at a given distance.
8. Given the diameter (or spot size) of a laser beam and the diameter of a circular aperture upon which the beam is centered, calculate the fraction of the power transmitted through the aperture.
9. Given the beam diameters at two distances from a laser, determine the beam divergence angle.
10. Explain briefly the difference between the "nearfield" and the "far field" of a laser.
11. Draw and label a diagram that illustrates the shape of the optical surfaces of the output coupler of a gas laser,and explain how the coupler reduces beam divergence.
12. Given the divergence angle of a laser beam and the focal length of a positive lens used to focus the beam, calculate the diameter of the focused spot.
13. Given the appropriate equipment, scan a HeNelaser beam with a small aperture detector; plot the beam profile; and determine the beam diameter.
14. Given the appropriate equipment, measure the transmission of a HeNe laser beam through a circular aperture placed in the beam at two points; and calculate beam diameter at both points and the beam divergence angle.

TRANSVERSE ELECTROMAGNETIC MODES

(3) The longitudinal modes of a laser discussed in Module1­7, "Optical Cavities and Modes of Oscillation,"describe the variations in the electromagnetic field along theoptical axis of the laser cavity. A complete description of theE­M field requires that variations in directions perpendicular to the optical axis also be considered. Electromagnetic field variations perpendicular to the direction of travel of the wave are called "transverse electromagnetic modes," or "TEM modes" as shown in Figure 1.

Fig. 1 Transverse electromagnetic modes

(4) Figure 1 illustrates the irradiance patterns produced by lasers operating in various transverse modes. The general mode is specified as TEM_mn, where m is the number of dark bands (white areas in Figure 1) crossing the horizontal axis and n is the number of dark bands (white areas) crossing the vertical axis. Thus, TEM₂₁ (Figure 1f) has two vertical bands (shown as white)  crossing the x­axis and one horizontal band (shown as white) crossing the y­axis.

(5) The centers of the dark bands (white bands) in the intensity patterns of the TEM modes actually are nodes in the electric field within the laser cavity. The electric fields of two modes within the cavity of a vertically polarized laser are depicted in Figure 2. Figure 2a shows the electric field of the TEM₀₀ mode in a plane perpendicular to the optical axis of the cavity that contains an antinode of the longitudinal mode at one instant of time. The electric field is upward at all points within this plane. The curve drawn on the plane represents the magnitude of the electric field along the x­axis of the plane.The field is maximum at the center of the cavity and decreases uniformly toward the edges of the cavity aperture.

Fig. 2 Electric fields of transverse
modes in a laser cavity

(6) The same curve is the solid line in Figure 3a. After a time equal to one-half the period of the wave, the direction of the electric field in this plane will be pointed downward, as indicated by the dotted line in Figure 3a. The boundaries of the cavity aperture are nodes of this transverse standing wave.

Fig. 3a

Fig. 3b

Fig. 3c

Fig. 3 Electric field and irradiance of transverse modes

(7) Figure 2b displays the electric field distribution of the TEM₁₀ mode. In this case, the field is upward on one side of the cavity and downward on the other. The field is also represented by the solid line in Figure 3b. One­half cycle later, the direction of the field will reverse, as indicated by the dotted line. This mode has a node in a vertical plane through the optical axis.

(8) Figure 3c gives the electric field pattern of TEM₂₀ as a function of distance across the cavity at two instants of time and the irradiance patterns caused by the three modes.

(9) The mode in Figure 1i is called the "TEM₀₁ quadrature" mode or, more commonly, the "doughnut" mode. This pattern results when TEM₀₁ or TEM₁₀ oscillates in a cavity at the same time with a phase difference of 90°, as often occurs.

(10) A laser will produce an output for all TEM modes for which gain exceeds loss within the laser cavity. Some lasers will lase on several transverse modes at the same time, as indicated by Figure 4. Such simultaneous lasing produces a beam that has dark spots and "hot" spots, i.e., regions of low and high irradiance.

Fig. 4 Multimode output irradiance distribution

(11) If sufficient losses are introduced within the cavity for a particular mode, that mode will cease to oscillate; for example, a vertical scratch through the center of one mirror will cause losses for all modes that do not have a vertical node through their center. In this case, modes TEM₁₀, TEM₃₀, etc., would suffer no loss since such modes have a node or zero electric field at the center of the TEM pattern.

(12) Notice that in Figure 4, TEM₀₀ has a smaller diameter than any other mode. All modes except TEM₀₀ can be eliminated by a cavity­aperture diameter that produces little or no loss for the TEM₀₀ mode, but that introduces greater loss for all higher­order modes. Optical cavities that exhibit high diffraction losses tend to oscillate in the TEM₀₀ mode only. Thus, any cavity can be restricted to TEM₀₀ by installation of a suitable aperture. For most gas lasers, the diameter of the laser tube is chosen only for the purpose of limiting oscillation to TEM₀₀.

(13) A further examination of Figure 2b reveals that the two bright spots of the TEM₁₀ mode are 180° out of phase with one another. In any transverse mode, each bright spot is 180° out of phase with all adjacent bright spots, as illustrated for TEM₂₂ in Figure 5.

Fig. 5 Phase differences in TEM₂₂

(14) TEM₀₀ is termed the "uniphase or pure Gaussian mode" because it is the only transverse mode in which all the light is in one phase at any given time. This uniphase mode is the only mode in which all laser light is spatially coherent, resulting in the following three important characteristics of this mode:

• It has a lower beam divergence than other modes. Lower divergence is important in the transmission of beams over large distances, as, for example, in laser ranging.
• It can be focused to a spot smaller than other existing modes. This is important in an application such as drilling.
• Its spatial coherence is ideal for applications that depend upon the interference of light. Other modes cannot be used because they lack adequate spatial coherence.

(15) Most lasers are designed to operate in TEM₀₀ only. The remainder of this module discusses characteristics that apply to such lasers operating in this mode.

BEAM DIAMETER AND SPOT SIZE

(16) Figure 6 indicates the profile of a TEM₀₀ laser beam. Since the irradiance of the beam decreases gradually at the edges, specification of beam diameter out to the points of zero irradiance is impractical. The "beam diameter" is defined as "the distance across the center of the beam for which the irradiance (E) equals 1/e² of the maximum irradiance (1/e² = 0.135)." The "spot size" (w) of the beam is "the radial distance (radius) from the center point of maximum irradiance to the 1/e² point." These definitions provide standard measures of laser beam size.

Fig. 6 Definitions of beam diameter and spot size (w)

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TRANSMISSION OF A BEAM THROUGH AN APERTURE

(17) If a laser beam is centered upon a circular aperture, the edges of the beam may be truncated as illustrated in Figure 7.

Fig. 7 Transmission through a circular aperture

(18) The fraction of beam power transmitted through the aperture is given by Equation 1.

Equation 1

In some situations, it is useful to be able to calculate the ratio of the aperture radius all of those are lower case R to the beam spot size (w) from a knowledge of the beam power transmitted through a given aperture. In that event, one can rearrange Equation 1A as follows:

Equation 1B
	; Equation 1A
	; (rearrange terms)o
	; take l n of each side and recognize that is by definition
	; solve for
Equation 1B
	; solve for desired ration r/w by taking square root of each side.

where r = Radius of aperture
w = Spot size of laser beam passing through aperture
T = Fractional transmission (T passing/T incident)

(19) Example A and Example B illustrate the use of Equations 1A and 1B.

EXAMPLE A: TRANSMISSION THROUGH AN APERTURE
Given:	A laser beam is centered upon an aperture of a diameter equal to the beam diameter.
Find:	Transmission of the aperture.
Solution:

(20) Figure 8 is a transmission curve based upon Equation 1A. The horizontal scale gives the ratio of aperture diameter (2r) to beam diameter (2w). The vertical scale is calibrated in percent of transmission. The transmission curve can be used with a calibrated aperture to determine the diameter of a laser beam, as illustrated in Example C.

Fig. 8 Percentage of transmitted power through an aperture

(21) The method illustrated in Example C is used for the determination of beam diameter in the laboratory exercise of this module.

BEAM DIVERGENCE

(22) The light emitted by a laser is confined to a rather narrow cone; but, as the beam propagates outward, it slowly diverges or fans out. Figure 9, greatly exaggerated, illustrates the way in which a beam of circular cross-section diverges. At the output aperture of the laser, the beam diameter is d. Its beam divergence angle is , usually expressed in milliradians. In traversing a distance . the beam diverges to a circle of diameter . Simple geometrical considerations indicate that for a small divergence angle, the diameter of the beam at a distance from the output aperture is approximated by Equation 2.

Fig. 9 Divergence of a laser beam

Equation 2

where:

l = Distance from laser output aperture to measurement position. T = Full-angle beam divergence in radians. d = Initial beam diameter. = Beam diameter at measurement position.

(23) Example C illustrates the use of Equation 2 in solving a problem.

EXAMPLE D: BEAM DIAMETER AT A DISTANCE
Given:	A HeNe laser has an output beam diameter 1.5 mm and a full-angle divergence of 1 mrad.
Find:	Beam diameter at 100 m.
Solution:

(24) In the past, some manufacturers have specified the "half­angle beam divergence" given as in Figure 9. The most common practice in recent years has been to specify the "full­angle beam divergence." Specifications must be checked with care in order that these different terms will not be confused.

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MEASURING BEAM DIVERGENCE

(25) The divergence of a CW laser beam can be determined by measurement of the beam diameter at two points (Figure 10). The full angle beam divergence is given by Equation 3.

Fig. 10 Measurement of beam divergence

Equation 3

where:

d1 = Beam diameter at point 1. d2 = Beam diameter at point 2. 1 = Diameter from laser to point 1. 2 = Diameter from laser to point 2.

(26) Equation 3 is valid for small , typical of most lasers.

(27) Example E illustrates the use of Equation 3 in the calculation of beam diameter.

EXAMPLE E: CALCULATION OF BEAM DIAMETER
Given:	A HeNe laser beam has a diameter of 3.5 mm at a distance of 2 m from the laser. At 4 m, the beam has expanded to a diameter of 5.9 mm.
Find:	Beam divergence.
Solution:

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REDUCTION OF THE BEAM DIVERGENCE OF GAS LASERS

(28) Figure 11 illustrates the optical cavity and beam profile of a typical gas laser. The laser beam diverges in the cavity as it strikes the output coupler, and would diverge even more outside the cavity if the second surface of the output coupler were flat. To prevent greater divergence, the second surface is curved slightly more than the first surface to form a positive lens that collimates the beam. The output coupler actually is a positive lens that has a focal length equal to the radius of curvature of its reflective surface.

Fig. 11 Collimation of a laser beam by the output coupler

(29) In most cases, this results in a slight convergence of the beam just past the output aperture. The beam passes through a minimum diameter, or "waist," and then diverges. This external beam waist serves as a "controlling aperture" which determines the beam divergence as the beam continues on past the beam waist.

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THE NEAR AND FAR FIELDS

(30) The discussion of beam divergence accompanying Figure 11 approximates the behavior of the beam near the laser only. The use of Equation 2 and 3 are valid only in the "for fills," at a certain distances from the laser. This range of distances is specified by Equation 4.

Equation 4

where:

= Distance from laser to observation psition. d = Diameter of output aperture. l = Wavelength of laser light.

(31) The region closer to the laser is the "near field" and is defined by the condition given by Equation 5:

Equation 5

(32) The beam divergence in the near field may differ considerably from that in the far field, as illustrated in Figure 11. Why the difference in divergence occurs is beyond the scope of this module. In the "gray area" between the ner field, defined by Equation 5, and the far field, defined by Equation 6, one may use Equation 2 and 3 to obtain reasonable approximations of the laser beam divergence. They involve simple calculations and good "ballpark" results.

(33) Example E is a calculation of near­field and far­field distances for a typical HeNe laser. The "gray area" would be that region between 1.6 meters and 160 meters.

EXAMPLE F: NEAR- AND FAR-FIELD DISTANCES
Given:	The effective output aperture diameter (see Figure 11) of a NeHe laser (l = 632.8 nm) is 1.0 mm.
Find:	Near- and far-field distances.
Solution:

(34) The far-field beam divergence (full angle) of a laser is given by Equation 6.

(35) This equation is the "diffraction­limited beam divergence" (see discussion after Example G) because it is the minimum divergence possible when light of wavelength l is diffracted as it passes through an effective aperture* of diameter d. Example G illustrates the use of this equation.

Equation 6   

EXAMPLE F: CALCULATION OF BEAM DIVERGENCE
Given:	A HeNe laser has an effective output aperture diameter of 1.2 mm. (l = 632.8 nm)
Find:	Beam divergence.
Solution:

*It is well to point out here that the effective aperture "d" in Equation 6 is in truth equal to the diameter of the beam waist located somewhere in the cavity of the laser. That is always the case of the output coupler does not further reshape the output laser beam as it exits the laser. If the output coupler does as shown in Figure 11, then the effective aperture "d" is the diamter of that beam waist formed by the output coupler located external to the laser beam, as shown in Figure 11. It is also well to point out that, for an ordinary light beam passing through a circular hole of diameter "d" (a light beam described as plane waves with a uniform intensity) the correct far-field beam divergence is given by = 2.44 l/d. But, for a TEM₀₀ gaussian laser beam, the correct far-field beam divergence from a cavity is precisely as given in Equation 6. Do not confuse the two situations. They are distinctly different.

A NOTE ON DIFFRACTION—LIMITED OPTICS

(35a) When light passes through circular openings such as pin holes or apertures, light spread out and diverges. When light is focused by a lens, the light does not focus to a geometrical point; instead it focuses to a tiny spot of some diameter, surrounded by alternate bright and dark rings, the entire image referred to as an AIRY diffraction pattern. The wave theory of light explains the behavior of the spreading of light passing through aperturer as well as the focusing of light as AIRY patterns rather than geometrical points. This is generally handled under the concepts of diffraction of lightwaves.

(35b) As such, diffraction theory set a lower limit on the amount of beam divergence that occurs when a laser beam passes through an effective aperture. Thus, any real optical system, containing imperfections in optical lenses, variation in the index of refraction along the atmosphere path of propagation and so on, the divergence is greater than that predicted by Equation 6, . In the same way, the spot size of a focused laser beam, as discussed in the next section, is predicted by diffraction theory to be of a value given by Equation 7, =f. However, for real optical systems and real lenses, the focused spot is in fact, larger than that predicted by Equation 7.

(35c) So diffraction-limited optics sets the ideal limit for such results as expected beam divergence of expected focal spot size. When you use relationships such as Equation 6 and 7, to calculate beam divergence or focused spot size , be aware that you are obtaining the "best" values possible in view of light diffraction. In fact, for your "real" optical systems, the beam divergence and spot sized will both be larger than the equations predict. Real optical systems are therefore poorer in performance than those limited only by diffraction. We often refer to such real systems as many-times diffraction limited. For example, if Equation 6 predicts a beam divergence of = 1 milliradian for your systems, but you actually measure =5 milliradians, you can conclude that your system is 5-time diffraction limited.

THE FOCUSING OF LASER BEAMS

(36) Figure 12 illustrates a laser beam focused by a positive lens. The beam is focused to a radius called the "diffraction­limited spot size." The diameter of this spot is given approximately by Equation 7.

Fig. 12 Focusing of a laser beam

Equation 7

d' = f

where:

d' = Diameter of focused spot. f = Focal length of lens. = Full-angle beam divergence.

(37) Example H illustrates the use of Equation 7.

EXAMPLE H: DIAMETER OF FOCUSED LASER BEAM
Given:	An Nd:YAG laser has a beam divergence of 2.0 mrad. The beam is focused by a lens of focal length 2.5 cm.
Find:	Diameter of focused spot.
Solution:	d' = f d' = (2.5 × 10–2 m)(2.0 × 10–3 rad) d' = 5.0 × 10–5 m d' = 50 mm

(38) Example I is a more complicated problem that involves the use of several equations presented in this module.

1. Sketch the following TEM modes:
   • a. TEM₁₀
     b. TEM₀₁
     c. TEM₃₄
     d. TEM₂₁
2. Explain how unwanted, higher­order modes can be eliminated in a laser.
3. Draw and label a diagram of the beam profile of the uniphase  (Gaussian) mode. Indicate spot size and beam diameter on the diagram.
4. List three reasons for which most laser applications require the TEM₀₀ mode.
5. Explain the difference between the near­field and far­field regions for details of beam propagation.
6. Draw and label a diagram that illustrates the shape of the output coupler of a gas laser, and explain how it reduces beam divergence.
7. A Nd:YAG laser (l = 1.06 m) has an effective aperture diameter of 2.5 mm. Determine the beam divergence.
8. A CO₂ laser has an effective output beam diameter of 1.5 cm and a divergence angle of 2.0 mrad. Determine the beam diameter at the following distances:
   • a. 10 m
     b. 100 m
     c. 1000 m
9. A 10­mW laser has a spot size of 2.5 mm. Determine the percent of power transmitted when this beam is centered on apertures of the following diameters:
   • a. 2.5 mm
     b. 4.0 mm
     c. 5.0 mm
     d. 6.0 mm
10. A laser beam has a diameter of 1.2 cm at a distance of 10 m from the laser and a diameter of 7.5 cm at a distance of 35 m. Determine the beam divergence angle.
11. The beam from the laser in Exercise 10 is to be focused to a spot size of 30 m. What focal length lens is required?
12. A HeNe laser has an effective output aperture diameter of 1.1 mm and a power of 2.0 mW. The beam enters a human eye with a lens focal length of 1.7 cm. Determine the irradiance of the focused spot on the retina.
13. An argon ion laser operating at 488 has an aperture diameter of 1.0 mm. Determine the beam diameter at 100 m.

HeNe laser
Optical power meter
Lab jack
Calibrated linear translator
Fiber optic
Optical bench with two carriers
Calibrated apertures*
Meter stick
Two plane first surface mirrors (optional)
Two sheets graph paper

*Calibrated apertures may be constructed of aluminum sheets in which holes have been drilled. Apertures of several known diameters in the range of 2-6 mm should be available.

See Idea Bank

In this laboratory, the student will measure, by two different methods, the diameters of a HeNe laser beam at two distances from the laser and will calculate the beam divergence of the laser.

Read all instructions before beginning the experiment.

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BEAM DIAMETER BY TRANSMISSION THROUGH AN APERTURE

The following procedure may be used to determine the diameter of a CW laser beam by measurement of transmission through a calibrated aperture.

1. Measure the laser output power; record in Data Table 1.
2. Place an aperture in the laser beam, and adjust the aperture position for maximum transmitted power as measured with an optical power meter. Measure and record the transmitted power.
3. Determine percent transmission of aperture. Record. (If transmission is outside the range of 50% to 90%, select another aperture to bring it within this range.)
4. Determine and record relative beam diameter from Figure 8.
5. Determine and record beam diameter as in Example C.

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BEAM DIAMETER BY SCANNING BEAM PROFILE

See Idea Bank

The following procedure can be used to determine the diameter of a CW laser beam by scanning the beam with a small­aperture, fiber­optic probe and by plotting the beam profile. (See Figure 13)

1. Attach one end of the fiber­optic probe securely to a linear translation stage. Attach the other end to the detector of an optical power meter with an aperture mask to ensure that only the light transmitted through the fiber reaches the detector.
2. Position the translation stage in such a manner that the input of the fiber optic is at least two beam diameters away from the edge of the laser beam.
3. Measure the background power, and record in Data Table 2.
4. Position the input of the fiber optic vertically in such a manner that its height is the same as that of the center of the beam.
5. Position the fiber optic horizontally at one edge of the laser beam. Record the position indicated on the translator and the measured power at that position.
6. Adjust the translator to move the fiber optic toward the center of the beam by approximately one­tenth the beam diameter. Record the new position and power. Continue this process until the fiber optic reaches the other edge of the beam.
7. Subtract the background power from each reading to find the corrected beam power. Record.
8. Plot power versus distance across the beam on a sheet of graph paper. The plot should be similar to that in Figure 6.
9. Divide the maximum power achieved by e² to locate the 1/e² points.
10. Measure beam diameter to 1/e² points on the graph paper. Record.

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EXPERIMENTAL PROCEDURES

1. Set up the equipment as depicted in Figure 13. Distance should be at least 2 m. The optional first­surface mirrors may be used to fold the beam if space limitations are a problem.

   

   Fig.. 13 Experimental setup

2. Measure and record (₁) in Data Table 2.
3. Determine beam diameter by method of transmission through an aperture. Record all data.
4. Remove aperture, and replace with linear translator and fiber optic.
5. Determine beam diameter by scanning method. Record all data.
6. Increase distance from laser by at least 2 m, and repeat all measurements and calculations.
7. Calculate divergence angle for each method, and compare the results.

DATA TABLE 1. BEAM DIAMETER AND BEAM DIVERGENCE
BY TRANSMISSION THROUGH APERTURES.

DATA TABLE 2. BEAM DIAMETER BY THE SCANNING OF BEAM PROFILE

References

Goldman, Leon and Rockwell, R. James Jr. Lasers in Medicine. New York: Gordon and Breach, 1971.

McAleese, Frank G. The Laser Experimenter's Handbook. Blue Ridge Summit, PA: TAB Books, 1979.

O'Shea, Donald; Callen, W. Russell; and Rhodes, William T. Introduction to Lasers and Their Applications. Reading, MA: Addison­Wesley Publishing Co., 1978.

Pressley, Robert J. CRC Handbook of Lasers, with Selected Data on Optical Technology. Cleveland, OH: The Chemical Rubber Company, 1971.

Ready, John F. Effects of High­Power Laser Radiation. New York: Academic Press, 1971.

______. Industrial Applications of Lasers. New York: Academic Press, 1978.

______, ed. Lasers in Modern Industry. Dearborn, MI: Society of Manufacturing Engineers, 1979.

Siegman, A.E. An Introduction to Laser and Masers. New York: McGraw­Hill Book Company, 1971.

Silicon Photodetector Design Manual. Santa Monica, CA: United Detector Technology.

Verdeyen, Joseph T. Laser Electronics. Englewood Cliffs, NJ: Prentice Hall, Inc., 1981.

Wilson, Leroy E., et al, eds. Electronic Transition Lasers. Cambridge, MA MIT Press, 1977.

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