CHAPTER 03

Can light be spatially confined and transported in free space without angular spread?
Although the wave nature of light precludes the existence of such an idealization, light
can take the form of beams that come as close as possible to spatially localized and
nondiverging waves.
A plane wave and a spherical wave represent the two opposite extremes of angular
and spatial confinement. The wavefront normals (rays) of a plane wave are parallel to
the direction of the wave so that there is no angular spread, but the energy extends
spatially over the entire space. The spherical wave, on the other hand, originates from
a single point, but its wavefront normals (rays) diverge in all directions.
Waves with wavefront normals making small angles with the z axis are called
paraxial waves. They must satisfy the paraxial Helmholtz equation derived in Sec. 2.2C.
An important solution of this equation that exhibits the characteristics of an optical
beam is a wave called the Gaussian beam. The beam power is principally concentrated
within a small cylinder surrounding the beam axis. The intensity distribution in any
transverse plane is a circularly symmetric Gaussian function centered about the beam
axis. The width of this function is minimum at the beam waist and grows gradually in
both directions. The wavefronts are approximately planar near the beam waist, but they
gradually curve and become approximately spherical far from the waist. The angular
divergence of the wavefront normals is the minimum permitted by the wave equation
for a given beam width. The wavefront normals are therefore much like a thin pencil of
rays. Under ideal conditions, the light from a laser takes the form of a Gaussian beam.
An expression for the complex amplitude of the Gaussian beam is derived in Sec.
3.1 and a detailed discussion of its physical properties (intensity, power, beam radius,
angular divergence, depth of focus, and phase) is provided. The shaping of Gaussian
beams (focusing, relaying, collimating, and expanding) by the use of various optical
components is the subject of Sec. 3.2. A family of optical beams called Hermite-
Gaussian beams, of which the Gaussian beam is a member, is introduced in Sec. 3.3.
Laguerre-Gaussian and Bessel beams are discussed in Sec. 3.4.
3.1 THE GAUSSIAN BEAM
A. Complex Amplitude
The concept of paraxial waves was introduced in Sec. 2.2C. A paraxial wave is a plane
wave e -jkz (with wavenumber k = 2n/A and wavelength A) modulated by a complex
envelope Ah) that is a slowly varying function of position (see Fig. 2.2-5). The complex
amplitude is
U(r) = A(r) exp( -jkz). (3.1-1)
The envelope is assumed to be approximately constant within a neighborhood of size A,
so that the wave is locally like a plane wave with wavefront normals that are paraxial
rays.
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82 BEAM OPTICS
For the complex amplitude U(r) to satisfy the Helmholtz equation, V2U + k2U = 0,
the complex envelope A(r) must satisfy the paraxial Helmholtz equation (2.2-22)
v;A -j2k; =0, (3.1-2)
where V; = J2/Jx2 + a2/~y2 is the transverse part of the Laplacian operator. One
simple solution to the paraxial Helmholtz equation provides the paraboloidal wave for
which
p2=.X2+y2 (3.1-3)
(see Exercise 2.2-2) where A, is a constant. The paraboloidal wave is the par-axial
approximation of the spherical wave U(r) = (A,/r) exp(-jkr) when x and y are much
smaller than z (see Sec. 2.2B).
Another solution of the paraxial Helmholtz equation provides the Gaussian beam. It
is obtained from the paraboloidal wave by use of a simple transformation. Since the
complex envelope of the paraboloidal wave (3.1-3) is a solution of the paraxial
Helmholtz equation (3.1-2), a shifted version of it, with z - ,$ replacing z where 5 is a
constant,
4
A(r) = - P2
4(z) exp [-j k- 1
h?(z) ’
4.4 = z - 6, (3.1-4)
is also a solution. This provides a paraboloidal wave centered about the point z = 5
instead of z = 0. When ,$ is complex, (3.1-4) remains a solution of (3.1-2), but it
acquires dramatically different properties. In particular, when 6 is purely imaginary,
say 5 = -jz, where za is real, (3.1-4) gives rise to the complex envelope of the
Gaussian beam
The parameter z. is known as the Rayleigh range.
To separate the amplitude and phase of this complex envelope, we write the
complex function l/q(z) = l/(z + jz,) in terms of its real and imaginary parts by
defining two new real functions R(z) and W(z), such that
1 1 .A -=--
4-4 R(z) hV2(z)
(3.1-6)
It will be shown subsequently that W(z) and R(z) are measures of the beam width and
wavefront radius of curvature, respectively. Expressions for W(z) and R(z) as functions
of z