Lecture 33
High Frequency Solutions,
Gaussian Beams
33.1 High Frequency Solutions
High frequency solutions are important in many real-world applications. This occurs when the
wavelength is much smaller than the size of the structure. This can occur even in microwave
interacting with reflector antennas for instance. It is also the transition from waves regime
to the optics regime in the solutions of Maxwell’s equations. Often times, the term “quasi-
optical” is used to describe the solutions in this regime.
33.1.1 Tangent Plane Approximations
We have learnt that reflection and transmission of waves at a flat surface can be solved in
closed form. The important point here is the physics of phase matching. Due to phase
matching, we have the law of reflection, transmission and Snell’s law [52].1
When a surface is not flat anymore, there is no closed form solution. But when a surface is
curved, an approximate solution can be found. This is obtained by using a local tangent-plane
approximation when the radius of curvature is much larger than the wavelength. Hence, this
is a good approximation when the frequency is high or the wavelength is short. This is similar
in spirit that we can approximate a spherical wave by a local plane wave at the spherical wave
front when the wavelength is short.
When the wavelength is short, phase matching happens locally, and the law of reflection,
transmission, and Snell’s law are satisfied approximately as shown in Figure 33.1. The tangent
plane approximation is the basis for the geometrical optics (GO) approximation [31, 167]. In
GO, light waves are replaced by light rays. The reflection and transmission of these rays at
an interface is estimated with the tangent plane approximation. This is also the basis for lens
or ray optics from which lens technology is derived. It is also the basis for ray tracing for
high-frequency solutions [168, 169].
1This law is also known in the Islamic world in 984 [166].
331
332 Electromagnetic Field Theory
Most of these problems do not have closed-form solutions, and have to be treated with ap-
proximate methods. In addition to geometrical approximations mentioned above, asymptotic
methods are also used to find approximate solutions. Asymptotic methods implies finding
a solution when there is a large parameter in the problem. In this case, it is usually the
frequency. Such high-frequency approximate methods are discussed in [170–174].
Figure 33.1: In the tangent plane approximation, the surface where reflection and refraction
occur is assumed to be locally flat. Hence, phase-matching is approximately satisfied, and
hence, the law of reflection, transmittion, and Snell’s law.
33.1.2 Fermat’s Principle
Fermat’s principle (1600s) [52,175] says that a light ray follows the path that takes the shortest
time between two points.2 Since time delay is related to the phase delay, and that a light ray
can be locally approximated by a plane wave, this can be stated that a plane wave follows the
path that has a minimal phase delay. This principle can be used to derive law of reflection,
transmission, and refraction for light rays. It can be used as the guiding principle for ray
tracing.
2This eventually give rise to the principle of least action.
High Frequency Solutions, Gaussian Beams 333
Figure 33.2: In Fermat’s principle, a light ray, when propagating from point A to point C,
takes the path of least delay.
Given two points A and C in two different half spaces as shown in Figure 33.2. Then the
phase delay between the two points, per Figure 33.2, can be written as
P = ki · ri + kt · rt (33.1.1)
As this is the shortest path according to Fermat’s principle, another other path will be longer.
In other words, if B were to move to another point, a longer path will ensue, or that B is the
stationary point of the path length or phase delay. Specializing (33.1.1) to a 2D picture, then
the phase delay as a function of xi is stationary. In this Figure 33.2, we have xi + xt = const.
Therefore, taking the derivative of (33.1.1) with respect to xi, one gets
∂P
∂xi
= 0 = ki − kt (33.1.2)
The above yields the law of refraction that ki = kt, which is just Snell’s law. It can also be
334 Electromagnetic Field Theory
obtained by phase matching. Notice that in the above, only xi is varied to find the stationary
point and ki and kt remain constant.
33.1.3 Generalized Snell’s Law
Figure 33.3: A phase screen which is position dependent can be made. In such a case, one
can derive a generalized Snell’s law to describe the diffraction of a wave by such a surface
(courtesy of Capasso’s group [176]).
Metasurfaces are prevalent these days due to our ability for nano-fabrication and numerical
simulation. One of them is shown in Figure 33.3. Such a metasurface can be thought of as a
phase screen, providing additional phase shift for the light as it passes through it. Moreover,
the added phase shift can be controlled to be a function of position due to advent in fabrication
technology and commercial software for numerical simulation.
To model this phase screen, we can add an additional function Φ(x, y) to (33.1.1), namely
that
P = ki · ri + kt · rt − Φ(xi, yi) (33.1.3)
Now applying Fermat’s principle that there should be minimal phase delay, and taking the
derivative of the above with respect to xi, one gets
∂P
∂xi
= ki − kt − ∂Φ(xi, yi)
∂xi
= 0 (33.1.4)
The above yields that the generalized Snell’s law [176] that
ki − kt = ∂Φ(xi, yi)
∂xi
(33.1.5)
It yields the fact that the transmitted light can be directed to other angles due to the
additional phase screen.
High Frequency Solutions, Gaussian Beams 335
33.2 Gaussian Beam
We have seen previously that in a source free space
∇2A + ω2μεA = 0 (33.2.1)
∇2Φ + ω2μεΦ = 0 (33.2.2)
The above are four scalar equations with the Lorenz gauge
∇ · A = −jωμεΦ (33.2.3)
connecting A and Φ. We can examine the solution of A such that
A(r) = A0(r)e−jβz (33.2.4)
where A0(r) is a slowly varying function while e−jβz is rapidly varying in the z direction.
(Here, β = ω√μ.) This is primarily a quasi-plane wave propagating predominantly in the
z-direction. We know this to be the case in the far field of a source, but let us assume that
this form persists less than the far field, namely, in the Fresnel as well.
Taking the x component of (33.2.4), we have3
Ax(r) = Ψ(r)e−jβz (33.2.5)
where Ψ(r) = Ψ(x, y, z) is a slowly varying envelope function of x, y, and z.
33.2.1 Derivation of the Paraxial/Parabolic Wave Equation
Substituting (33.2.5) into (33.2.1), and taking the double z derivative first, we arrive at
∂2
∂z2
[Ψ(x, y, z)e−jβz ] =
[ ∂2
∂z2 Ψ(x, y, z) − 2jβ ∂
∂z Ψ(x, y, z) − β2Ψ(x, y, z)
]
e−jβz (33.2.6)
Consequently, after substituting the above into the x component of (33.2.1), we obtain an
equation for Ψ(r), the slowly varying envelope as
∂2
∂x2 Ψ + ∂2
∂y2 Ψ − 2jβ ∂
∂z Ψ + ∂2
∂z2 Ψ = 0 (33.2.7)
When β → ∞ , or in the high frequency limit,
∣
∣
∣
∣2jβ ∂
∂z Ψ
∣
∣
∣
∣
∣
∣
∣
∣
∂2
∂z2 Ψ
∣
∣
∣
∣ (33.2.8)
In the above, we assume the envelope to be slowly varying and β large, so that |βΨ|
|∂/∂zΨ|. And then (33.2.7) can be approximated by
∂2Ψ
∂x2 + ∂2Ψ
∂y2 − 2jβ ∂Ψ
∂z ≈ 0 (33.2.9)
3Also, the wave becomes a transverse wave in the far field, and keeping the transverse component suffices.
336 Electromagnetic Field Theory
The above is called the paraxial wave equation. It is also called the parabolic wave equation.4
It implies that the β vector of the wave is approximately parallel to the z axis, and hence,
the name.
33.2.2 Finding a Closed Form Solution
A closed form solution to the paraxial wave equation can be obtained by a simple trick5. It
is known that
Ax(r) = e−jβ|r−r′|
4π|r − r′| (33.2.10)
is the solution to
∇2Ax + β2Ax = 0 (33.2.11)
if r 6 = r′. If we make r′ = −ˆzjb, a complex number, then (33.2.10) is always a solution to
(33.2.10) for all r, because |r − r′| 6 = 0 always. Then
|r − r′| = √x2 + y2 + (z + jb)2
≈ (z + jb)
[
1 + x2 + y2
(z + jb)2 + . . .
]1/2
≈ (z + jb) + x2 + y2
2(z + jb) + . . . , |z + jb| → ∞ (33.2.12)
And then
Ax(r) ≈ e−jβ(z+jb)
4π(z + jb) e−jβ x2+y2
2(z+jb) (33.2.13)
By comparing the above with (33.2.5), we can identify
Ψ(x, y, z) = A0
jb
z + jb e−jβ x2+y2
2(z+jb) (33.2.14)
By separating the exponential part into the real part and the imaginary part, and writing
the prefactor in terms of amplitude and phase, we have
Ψ(x, y, z) = A0
√1 + z2/b2 ej tan−1( z
b )e−jβ x2+y2
2(z2+b2) z e−bβ x2+y2
2(z2+b2) (33.2.15)
The above can be rewritten as
Ψ(x, y, z) = A0
√1 + z2/b2 e−jβ x2+y2
2R e− x2+y2
w2 ejψ (33.2.16)
4The paraxial wave equation, the diffusion equation and the Schrodinger equation are all classified as
parabolic equations in mathematical parlance [34, 43, 177, 178].
5Introduced by Georges A. Deschamps of UIUC [179].
High Frequency Solutions, Gaussian Beams 337
where
w2 = 2b
β
(
1 + z2
b2
)
, R = z2 + b2
z , ψ = tan−1 ( z
b
)
(33.2.17)
For a fixed z, the parameters w, R, and ψ are constants. Here, w is the beam waist which
varies with z, and it is smallest when z = 0, or w = w0 =
√ 2b
β . And R is the radius of
curvature of the constant phase front. This can be appreciated by studying a spherical wave
front e−jβR, and make a paraxial wave approximation, namely, x2 + y2 z2 to get
e−jβR = e−jβ(x2+y2+z2)1/2
= e−jβz
(
1+ x2+y2
z2
)1/2
≈ e−jβz−jβ x2+y2
2z ≈ e−jβz−jβ x2+y2
2R (33.2.18)
In the last approximation, we assume that z ≈ R in the paraxial approximation. The phase
ψ changes rapidly with z.
A cross section of the electric field due to a Gaussian beam is shown in Figure 33.4.
Figure 33.4: Electric field of a Gaussian beam in the x − z plane frozen in time. The wave
moves to the right as time increases; b/λ = 10/6 (courtesy of Haus, Electromagnetic Noise
and Quantum Optical Measurements [74]).
33.2.3 Other solutions
In general, the paraxial wave equation has solution of the form6
Ψnm(x, y, z) =
( 2
πn!m!
)1/2
2−N/2
( 1
w
)
e−(x2+y2)/w2
e−j β
2R (x2+y2)ej(m+n+1)Ψ
·Hn
(
x√2/w
)
Hm
(
y√2/w
)
(33.2.19)
6See F. Pampaloni and J. Enderlein [180].
338 Electromagnetic Field Theory
where Hn(ξ) is a Hermite polynomial of order n. The solution can also be express in terms
of Laguere polynomials, namely,
Ψnm(x, y, z) =
( 2
πn!m!
)1/2
min(n, m)! 1
w e−j β
2R ρ2
− e−ρ2/w2
e+j(n+m+1)Ψejlφ
(−1)min(n,m)
( √2ρ
w
)
Ln−m
min(n,m)
( 2ρ2
w2
)
(33.2.20)
where Lk
n(ξ) is the associated Laguerre polynomial.
These gaussian beams have rekindled recent excitement in the community because, in
addition to carrying spin angular momentum as in a plane wave, they can carry orbital
angular momentum due to the complex transverse field distribution of the beams.7 They
harbor potential for optical communications as well as optical tweezers to manipulate trapped
nano-particles. Figure 33.5 shows some examples of the cross section (xy plane) field plots
for some of these beams.
Figure 33.5: Examples of structured light. It can be used in encoding more information in
optical communications (courtesy of L. Allen and M. Padgett’s chapter in J.L. Andrew’s book
on structured light [181].
7See D.L. Andrew, Structured Light and Its Applications and articles therein [181].
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