HIGHER ORDER GAUSSIAN LASER BEAMS

HIGHER ORDER GAUSSIAN LASER BEAMS

In the real world, the truly 100-percent, single transverse mode, Gaussian laser beam (also called a pure or fundamental mode beam) described by equations 36.7 and 36.8 is very hard to find. Low-power beams from helium neon lasers can be a close approx- imation, but the higher the power of the laser, and the more com- plex the excitation mechanism (e.g., transverse discharges, flash-lamp pumping), or the higher the order of the mode, the more the beam deviates from the ideal.

To address the issue of higher order Gaussian beams and mixed mode beams, a beam quality factor, M2, has come into general use. A mixed mode is one where several modes are oscillating in the res- onator at the same time. A common example is the mixture of the lowest order single transverse mode with the doughnut mode, before the intracavity mode limiting aperture is critically set to select just the fundamental mode. Because all beams have some wavefront defects, which implies they contain at least a small admixture of some higher order modes, a mixed mode beam is also called a “real” laser beam.

For a theoretical single transverse mode Gaussian beam, the value of the waist radius–divergence product is (from equation 36.11):

 

It is important to note that this product is an invariant for trans- mission of a beam through any normal, high-quality optical system (one that does not add aberrations to the beam wavefront). That is, if a lens focuses the single mode beam to a smaller waist radius, the convergence angle coming into the focus (and the divergence angle emerging from it) will be larger than that of the unfocused beam in the same ratio that the focal spot diameter is smaller: the product is invariant.

For a real laser beam, we have

 

where W0 and V are the 1/e2 intensity waist radius and the far-field half-divergence angle of the real laser beam, respectively. Here we have introduced the convention that upper case symbols are used for the mixed mode and lower case symbols for the fundamental mode beam coming from the same resonator. The mixed-mode beam radius W is M times larger than the fundamental mode radius at all propagation distances. Thus the waist radius is that much larger, contributing the first factor of M in equation 36.16.

The second factor of M comes from the half-angle divergence, which is also M times larger. The waist radius–divergence half-angle product for the mixed mode beam also is an invariant, but is M2 larger. The fundamental mode beam has the smallest divergence allowed by diffraction for a beam of that waist radius. The factor M2 is called the “times-diffraction-limit” number or (inverse) beam quality; a diffraction-limited beam has an M2 of unity.

For a typical helium neon laser operating in TEM00 mode, M2 < 1.05. Ion lasers typically have an M2 factor ranging from 1.1 to 1.7. For high-energy multimode lasers, the M2 factor can be as high as 30 or 40. The M2 factor describes the propagation char- acteristics (spreading rate) of the laser beam. It cannot be neglected in the design of an optical train to be used with the beam. Trunca- tion (aperturing) by an optic, in general, increases the M2 factor of the beam.

The propagation equations (analogous to equations 36.7 and 36.8) for the mixed-mode beam W(z) and R(z) are as follows:

 

and

The Rayleigh range remains the same for a mixed mode laser beam:

 

Now consider the consequences in coupling a high M2 beam into a fiber. Fiber coupling is a task controlled by the product of the focal diameter (2Wf) and the focal convergence angle (vf). In the tight focusing limit, the focal diameter is proportional to the focal length f of the lens, and is inversely proportional to the diameter of the beam at the lens (i.e., 2Wf ∝ f/Dlens).

The lens-to-focus distance is f, and, since f x vf is the beam diameter at distance f in the far field of the focus, Dlens ∝ fvf. Com- bining these proportionalities yields

 

for the fiber-coupling problem as stated above. The diameter- divergence product for the mixed-mode beam is M2 larger than the fundamental mode beam in accordance with equations 36.15 and 36.16.

There is a threefold penalty associated with coupling a beam with a high M2 into a fiber: 1) the focal length of the focusing lens must be a factor of 1/M2 shorter than that used with a fundamen- tal-mode beam to obtain the same focal diameter at the fiber; 2) the numerical aperture (NA) of the focused beam will be higher than that of the fundamental beam (again by a factor of 1/M2) and may exceed the NA of the fiber; and 3) the depth of focus will be smaller by 1/M2 requiring a higher degree of precision and stability in the optical alignment.