SPATIAL CHARACTERISTICS OF LASERS

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 well collimated 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.
TRANSVERSE ELECTROMAGNETIC MODES:
Optical Cavities and Modes of Oscillation, "describe the variations in the electromagnetic field along the optical axis of the laser cavity. A complete description of the E-M 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
Figure 1 illustrates the irradiance patterns produced by lasers operating in various transverse modes. The general mode is specified as TEMmn, 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 x-axis and one horizontal band (shown as white) crossing the y-axis.
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 x-axis 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
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. One half 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.
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.
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.
A laser will produce an output for all TEM modes for which gain exceeds loss within the laser cavity. Some lasers will laser 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
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.
Notice that in Figure 4, TEM00 has a smaller diameter than any other mode. All modes except TEM00 can be eliminated by a cavity aperture diameter that produces little or no loss for the TEM00 mode, but that introduces greater loss for all higher order 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.
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
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:
1-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.
2-It can be focused to a spot smaller than other existing modes. This is important in an application such as drilling.
3-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.
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:
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" (?) 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 (?)
TRANSMISSION OF A BEAM THROUGH AN APERTURE:
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
The fraction of beam power transmitted through the aperture is given by Equation 1.
Equation 1

where: T = Fractional transmission.
r = Radius of aperture.
? = Spot size (radius of beam to 1/e2 points).
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)