محاضرة -6

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., regio