microoptics chapter 1

7/29/2019 MicroOptics Chapter 1

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UNIVERSITY OF EASTERN FINLAND

DEPARTMENT OF PHYSICS AND MATMEMATICS

MICRO- AND NANOPHOTONICS

Jari Turunen

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A TALBOT CARPET

Lecture notes

Third Edition (2013)

JOENSUU 2013


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2 1. Harmonic fields

Let us represent the real (square-integrable) field f(r)(r, t) in the form of a temporal

Fourier integral

f(r)(r, t) =

f(r)(r, )exp(it) d. (1.2)

Since f(r)(r, t) is real, the Fourier components

f(r)(r, ) =1

2

f(r)(r, t)exp(it) dt (1.3)

satisfy the conditionf(r)(r, ) =

f(r)(r, )

(1.4)

indicating that the negative frequency components contain no information that is notalready contained in the positive ones. Let us now define

f(r, ) = |f(r, )| exp {iarg[f(r, )]} =

2f(r)(r, ) if 00 if < 0

(1.5)

and write

f(r, t) =

0

f(r, )exp(it) d. (1.6)Evidently, Eqs. (1.1) and (1.2) are satisfied and hence the quantity in Eq. (1.6) representsthe complex analytic signal. Furthermore, we have explicitly

f(r)(r, t) =0

|f(r, )| cos {arg[f(r, )] t} d. (1.7)This is a spectral representation of the real field in terms of its positive frequency com-ponents 0.

A strictly monochromatic field oscillates at a single frequency, say, = 0. Thus thecomplex analytic signal takes the form

f(r, t) = f(r, 0)exp(i0t) (1.8)with

f(r, ) = f(r, 0)( 0). (1.9)The real field may be written as

f(r)(r, t) =

1

2[f(r, 0) exp(i0t) + f(r, 0) exp(i0t)]

= |f(r, 0)| cos {arg[f(r, 0)] 0t} (1.10)and its Fourier representation is

f(r)(r, ) =1

2[f(r, 0)( 0) + f(r, 0)( + 0)] . (1.11)

Fields of this form are called harmonic; while strictly harmonic fields can never be encoun-tered in nature, they can be well approximated by radiation originating from continuous-wave single-mode lasers.

When considering strictly monochromatic fields, we will often suppress the explicitdependence of spectral fields quantities f(r, 0) on 0 to shorten the notation.


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1.1 Preliminaries 3

1.1.2 Macroscopic Maxwells equations

Using complex analytic signal representations for all position- and time-dependent quan-tities, we denote by E(r, t) and B(r, t) the fundamental field vectors associated withelectromagnetic waves, namely the electric field and the magnetic induction.1 We denotethe primary sources of the field, i.e., the free charge and current densities, by (r, t) andJ(r, t), respectively. Further, the polarization and magnetization of the (macroscopic)medium are denoted by P(r, t) and M(r, t). Then the fields E(r, t) and B(r, t) satisfy, ifthe medium is at rest,2 Maxwells equations [1]

E(r, t) = tB(r, t), (1.12)

B(r, t) = 0J(r, t) + M(r, t) + 0

tE(r, t) +

tP(r, t)

, (1.13)

E(r, t) = 10 [ (r, t) P(r, t)] , (1.14)and

B(r, t) = 0, (1.15)where 0 is the permittivity and 0 is the permeability of vacuum. According to Eq. (1.13), B is affected by true currents, magnetization currents, and currents due to the ratesof change of the electric field and polarization. Similarly, in view of Eq. (1.14), both true

charges and polarization charges contribute to E.In principle the polarization and magnetization are nonlinear functions of the electricand magnetic fields. At the high frequencies in the optical region, magnetization is typi-cally very small even if the field strength is large (it can often be neglected, or at least alinear dependence on B can be assumed). However, if intense laser illumination is used,polarization must generally we written in the form of a series

P(r, t) = (1)(r)E(r, t) + (2)(r)E2(r, t) + (2)(r)E3(r, t) + . . . , (1.16)

where the (generally position-dependent) constants (j) are susceptibilities of differentorders. If it is sufficient to retain only the first term in the right-hand-side of Eq. (1.16),

we are in the domain of linear optics; the second and third terms give rise to nonlineareffects of second and third order, respectively [2]. In this text we restrict the attention tothe linear domain, i.e., we assume that only the linear susceptibility (1) differs appreciablyfrom zero.

Maxwells equations can be simplified (at least formally if not in physical sense) byintroducing two new field quantities, the electric displacement

D(r, t) = 0E(r, t) + P(r, t) (1.17)

and the magnetic fieldH(r, t) = 10 B(r, t)

M(r, t) (1.18)

1The term induction is of historical origin.2Additional terms must be included if the medium is in motion, but we do not consider such situations

in this text.


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to describe the effect of the medium. With the aid of these new field vectors, Eqs. (1.13)

and (1.14) are transformed into the form

H(r, t) = J(r, t) + tD(r, t), (1.19)

D(r, t) = (r, t), (1.20)while Eqs. (1.12) and (1.15) remain unchanged.

In addition to Maxwells equations, we need so-called constitutive relationsdescribe theeffect of the medium in the field vectors. Such relations generally depend on the frequency in a much simpler way than they depend on time. Hence it is of advantage to tranform

Maxwells equations into the frequency domain using Fourier integral representations ofthe analytic signals in the form of Eq. (1.6). Performing the temporal integrations, wethen have from Eqs. (1.12), (1.19), (1.20), and (1.15) the set

E(r, ) = iB(r, ), (1.21)

H(r, ) = J(r, ) iD(r, ), (1.22) D(r, ) = (r, ), (1.23)

B(r, ) = 0. (1.24)If we let the media be temporally dispersivebut assume them to be spatially non-dispersive,the constitutive equations depend on frequency but not on the spatial neighborhood ofthe observation point r. It is then customary to write them in the form (applicable tolinear media only)

D(r, ) = 0r(r, )E(r, ), (1.25)

H(r, ) = 10 1r (r, )B(r, ), (1.26)

J(r, ) = (r, )E(r, ), (1.27)

where r is the relative permittivity of the medium, r is its relative permeability, and is its electrical conductivity. In anisotropic media, where the optical properties depend

on direction, these scalar constants must be replaced by 3 3 tensors. However, in thistext, we will not consider anisotropic materials.By applying the constitutive relations (1.25)(1.27), we may cast Eqs. (1.22) and (1.23)

into the form 1r (r, )B(r, ) = i /c2 r(r, )E(r, ), (1.28)

[r(r, )E(r, )] = (r, ), (1.29)where

r(r, ) = r(r, ) + i(r, )/0 (1.30)

is known as the complex relative permittivity of the medium and

c = (00)1/2 (1.31)

is the speed of light in vacuum.


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1.1 Preliminaries 5

As already pointed out, we can often (especially at optical frequencies) write r(r, ) =

1 since the materials can be considered non-magnetic. This obviously simplifies Eq. (1.28)and it also allows us to effectively consider H instead ofB as the basic magnetic fieldvector since they differ only by the constant 0. We will follow this convention in much ofthe rest of the text since it is common practise in optics literature to desribe fields withthe aid of vectors E and H.

1.1.3 Measures of energy and flux

It is a simple matter to show (see Problem 8.1.1) that Maxwells equations (1.12) and(1.19) lead to3

S

E(r)(r, t) H(r)(r, t)

n dS+

V

E(r)(r, t) J(r)(r, t) dV

+

V

E

(r)(r, t) tD

(r)(r, t) + H(r)(r, t) tB

(r)(r, t)

dV = 0 (1.32)

where V is an arbitrary volume, S is its surface, and n is the unit normal vector onS. This result is known as the energy law for electromagnetic fields. It is exact since itwas derived directly from Maxwells equations. Here we consider real fields, instead ofcomplex analytic signals associated with them, since nonlinear functions of field quantitiesare involved.

Let us now assume that the field is either harmonic or quasimonochromatic, meaningthat its frequency spectrum is confined over a narrow band in the neighborhood of a centerfrequency 0. We may then regard the permittivity, permeability, and conductivity in theconstitutive relations (1.25)(1.27) as constants across the frequency interval in question4

and write

D(r)(r, t) = 0r(r)E

(r)(r, t), (1.33)

H(r)(r, t) = 10

1r (r)B

(r)(r, t), (1.34)

J(r)(r, t) = (r)E(r)(r, t). (1.35)

Equations (1.33) and (1.34) allow us to write

E(r)(r, t)

tD

(r)(r, t) =1

2

t

E

(r)(r, t) D(r)(r, t)

, (1.36)

H(r)(r, t)

tB

(r)(r, t) =1

2

t

H

(r)(r, t) B(r)(r, t)

. (1.37)

Then the quantity

EV(t) =

V

[we(r, t) + wm(r, t)] dV, (1.38)

3Using Eq. (1.1) one can immediately see that Maxwells equations are valid for real fields in the same

form as for complex analytic signals.4This amounts to the same thing as assuming that the materials are temporally non-dispersive, which

however is not true for real materials over a substantial spectral range: both r and are generallycomplex and depend on .


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1.2 Plane-wave representation of harmonic fields 7

Considering the energy density at some setting frequency 0, the detector measures

under the above-stated conditions the time average

we(r, t) = 02

limT

1

2T

TTE

(r)(r, t) E(r)(r, t) dt (1.42)

where we have used the constitutive relation (1.33) with r(r) = 1 since we are measuringthe field in free space. Using the first form of Eq. (1.10) we then obtain, after some simplealgebra,

we(r, t) = 04E(r, 0) E(r, 0)

+08

[E(r, 0) E(r, 0) + E(r, 0) E(r, 0)] limT

sin(20T)20T

(1.43)

and the second term on the right-hand-side of Eq. (1.43) vanishes when the limit is taken.We are therefore left with the observable quantity

we(r, t) = 04

|E(r, 0)|2 . (1.44)

In a similar fashion, assuming r(r) = 1, we may derive

wm(r, t) =0

4 |H(r, 0)|2

(1.45)

and

S(r, t) = 12{E(r, 0) H(r, 0)} , (1.46)

where again denotes the real part. This expression actually holds also in dielectric andmetallic (non-magnetic) media.

1.2 Plane-wave representation of harmonic fields

Throughout the rest of this Chapter we assume that the field under consideration is strictly

harmonic at frequency 0 (which we will omit for brevity of notation). However, unlessstated otherwise, the medium in which the field propagates is takes to be non-magneticwith complex relative permittivity defined in Eq. (1.30). Here the quantities r and may depend on both position and frequency, and they are generally complex. In dielectricmaterials = 0 and r is nearly real, except close to atomic or molecular resonances of thematerial (where electronic transitions between energy levels can take place). In metals, differs from zero and is predominantly real, at least in the infrared region and at longerwavelengths. However, in all cases we may define the complex refractive index

n(r, ) = r(r, ) = n(r, ) + i(r, ) (1.47)

where n(r, ) and (r, ) are both real functions of position and frequency. Clearly, if = 0 and r is real, n(r, ) =

r(r, ) represents the real refractive index of a dielectric

material (far from resonances).


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1.2.1 Harmonic plane waves

Let us consider harmonic fields in a non-magnetic homogeneous medium with n(r, 0) = nand r(r, 0) = 1. Substituting from Eqs. (1.21) and (1.26) into Eq. (1.28), we obtain

[ E(r)] = k20n2E(r). (1.48)If we use the vector identity

( E) = ( E) 2E (1.49)and assume that the (source) charge density vanishes in the region under consideration,i.e., (r) = 0 in Eq. (1.23), we see that the electric field vector must satisfy the Helmholtz

equation2E(r) + k20n2E(r) = 0. (1.50)

Here we used Eq. (1.47) and denoted k0 = 0/c = 2/0, where 0 is the vacuumwavelength of the monochromatic field in question. The simplest solution of Eq. (1.50) isthe harmonic plane wave

E(r) = A exp (ik r) , (1.51)where A is called the polarization vectorand k the wave vector. Substituting this expres-sion in Eq. (1.50), we see that the wave vector must satisfy the condition

kk = k2

0

n2. (1.52)

If we write, in cartesian wave-vector coordinates, k = (kx, ky, kz) and take = (kx, ky) asa real-valued spatial-frequency vector, Eq. (1.52) fixes

kz =

k20

n2 2 k2x k2y + i2k20n1/2 = kzr + ikzi. (1.53)Here kzr and kzi are the real and imaginary parts of kz, respectively. Hence we may writethe space-dependent part of the electric field vector in the form

exp (ik r) = exp[i(kxx + kyy + kzrz)]exp(kziz) , (1.54)

which is a product of rapidly oscillating and real-exponential parts. When consideringfields in the positive half-space, we always choose the branch of the square root thatgives a positive value for kzr. In lossy media with > 0 we choose the branch such thatalso kzi > 0 so that the field amplitude decreases exponentially in the z-direction. Inamplifying media with < 0 we take kzi < 0, which implies that the field amplitudeincreases exponentially with z.

Lossless dielectrics represent an important special case worth separate consideration.Writing = 0 we obviously have, from Eq. (1.53),

kz = kzr =

k20n2 k2x k2y

1/2if k2x + k

2y k20n2

ikzi = i

k2

x + k2

y k2

0n21/2

if k2

x + k2

y > k2

0n2

(1.55)

when considering plane waves in the positive half-space. Then we have homogeneous waveswith real-valued kz and exponentially decaying evanescent waves with purely imaginary


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0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

kzr

f

(a)

0 1 2 3 4 5

0

1

2

3

4

5

kzi

f

(b)

Figure 1.1: (a) Real and (b) imaginary parts of kz in lossy media. Solid line:

n = 1.5 + i1.0 (high loss). Dashed line: n = 1.5 + i0.1 (smaller loss). Dotted line:

n = 1.5+i0.01 (nearly dielectric). The dotted line resembles the result of Eq. (1.55).

0 1 2 3 4 50.0

0.1

0.2

0.3

0.4

0.5

k

zr

f

(a)

0 1 2 3 4 50

2

4

6

8

kzi

f

(b)

Figure 1.2: (a) Real and (b) imaginary parts of kz in metal-like media. Solid line:

n = 0.5+i1.0 (poor metal). Dashed line: n = 0.5+i2.0 (rather poor metal). Dotted

line: n = 0.5 + i5.0 (decent metal).

It follows from simple vector algebra that the time-averaged Poynting vector, defined inEq. (1.46), is given by

S(r, t) = 12k0

00

|A|2 {k} . (1.66)

Hence, for the special case of homogeneous plane waves with {k} = k, the Poyntingvector is parallel to the wave vector. Since (k A) = 0 in view of Eq. (1.24) andwe already saw that k A = 0, the direction of S(r, t) and k is perpendicular to theoscillation direction of the electric and magnetic fields. Therefore homogeneous plane

electromagnetic waves are often called transverse waves.Returning again to the general case, all waves with kzi = 0 have surfaces of constant

amplitude in planes z = constant but the surfaces of equal phase = constant are given


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with f > n. Thus, when f

n, the decay distance becomes infinite. However, it

reaches subwavelength scale when f >

n2 + 1/(2)2. In vacuum, for example, thissubwavelength decay threshold is f 1.0126, which is only slightly above the evanescent-wave threshold f = 1. Therefore it can generally be said that evanescent waves do notcontribute significantly to the field at distances greater than a few wavelengths.8

In media with complex refractive index the expressions for the wavelength and theeffective decay distance read as

=

20

(n2 2 f2)2 + (2n)2

1/2

+ (n2 2 + f2)1/2 (1.74)

andzd =

0

2

(n2 2 f2)2 + (2n)21/2 (n2 2 f2)1/2 . (1.75)Figures 1.3 and 1.4 illustrate the wavelength and effective decay distance, as a function ofthe normalized radial spatial frequency, of plane waves in several dielectric and metallicmedia. It is noteworthy that the wavelengths of plane waves in some metals can besubstantially larger than 0 for small values of f. This occurs for metals with n < 1 forsmall values of f since, for a homogeneous wave with f = 0, we have = 0/n.

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

/0

f

(a)

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

zd

/0

f

(b)

Figure 1.3: (a) Wavelength and (b) effective penetration depth of plane waves in

dielectric media with n = 1 (vacuum). n = 1.45 (SiO2), and n = 2.2 (TiO2).

Figure 1.5 illustrates the different types of plane waves. Here we consider the real(monochromatic) field at t = 0, i.e.,

E(r)(r, 0) = A cos(kxx + kyy + kzrz)exp(kziz) , (1.76)

assuming that A = y and ky = 0 so that the electric field has only a single scalarcomponent that varies only in the xz plane. In (a) we assume n = 1.45 and kx = 0.5k0

8The small range of evanescent fields is another property of evanescent waves, which is highly sig-nificant in nanophotonics. This property implies that observations of evanescent waves must be made atdistances within around one vacuum wavelength of the region where they are created.


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0 1 2 3 4 50

1

2

3

4

5

/0

f

(a)

0 2 4 6 8 100.00

0.01

0.02

0.03

0.04

0.05

zd

/0

f

(b)

Figure 1.4: (a) Wavelength and (b) effective decay distance of plane waves inmetals at 0 = 633 nm with n = 1.45 + i7.5 (Aluminium, solid), n = 0.20 + i3.09

(Gold, dashed), n = 3.68 + i3.62 (Chromium, dotted).

to illustrate a homogeneous wave propagating in a dielectric. In (b) and (c) we illustrateevanescent waves in the same dielectric by choosing kx = 1.5k0 (just above the thresholdof becoming evanescent) and kx = 2.5k0, respectively; soon after the evanescent-wavethreshold is reached, the effective decay distance in a dielectric becomes less than onewavelength. Plane waves in metals are illustrated in (d)(f): we assume n = 1.45 + i7.5,

which corresponds roughly to the complex refractive index of aluminium. In (d) wehave kx = 0, which generated a decaying homogeneous wave. In (e) and (f) we choosekx = 1.5k0 and kx = 2.5k0 to illustrate inhomogeneous waves in metal. Clearly, waves inmetals decay appreciably in a scale that is a small fraction of the vacuum wavelength.

1.2.2 Polarization of a plane wave

Let us next look more closely at the concept of polarization or propagating plane waves.We saw that only two components of the polarization vector A are independent and thatk A = 0. Hence generality is not sacrificed if we assume k = z and consider Ax andAy as the independent components (in this case Az = 0). If we set Ax = ax exp (ix) andAy = ay exp (iy), the real representations of the transverse components of the electricfield read

E(r)x (z, t) = Erx = ax cos( + x) , (1.77)

E(r)y (z, t) = Ery = ay cos( + y) , (1.78)

where we have abbreviated the notation and written = k0nz . These are in generalparametric equations of an ellipse, defined by the evolution of the tip of the electric fieldvector, which is known as the polarization ellipse.

We may now use standard trigonometric identities and simple manipulations to putthe two equations in a form

E2rxa2x

+E2rya2y

2ErxEryaxay

cos = sin2 , (1.79)


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0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

x/0

z/0

(a)

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

x/0

z/0

(b)

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

x/0

z/0

(c)

0 0.5 1 1.5 2

0

0.02

0.04

0.06

0.08

0.1

x/0

z/0

(d)

0 0.5 1 1.5 2

0

0.02

0.04

0.06

0.08

0.1

x/0

z/0

(e)

0 0.5 1 1.5 2

0

0.02

0.04

0.06

0.08

0.1

x/0

z/0

(f)

Figure 1.5: Amplitude of the real field E

(r)

y (x,z, 0) for (a) a propagating homoge-neous plane wave, (b,c) evanescent plane waves, (d) a homogeneous decaying wave

and (e,f) inhomogeneous plane waves. Note the 20 magnified z-scale in the lowerrow compared to the upper row. White means unit field amplitude, the darkest

shade means 1, and the middle shade represents zero amplitude.

where = y x. This expression describes more explicitly a rotated ellipse. Note that,unlike Eqs. (1.77) and (1.78), the expression (1.79) no longer contains information on thedirection of rotation of the tip along the ellipse. A simpler axpression for tje polarizationellipse is obtained by rotating the coordinate axes such that they are parallel to the majorand minor axes as shown in Fig. 1.6. In this new coordinate system we have

Er = Erx cos + Ery sin , (1.80)

Er = Erx sin + Ery cos , (1.81)and the equation of the ellipse is simply

E2ra2

+E2rb2

= 1. (1.82)

Let us now multiply Eq. (1.79) by a

2

xa

2

y and Eq. (1.82) by a

2

b

2

and compare the resultingexpressions. We then see that

a2x = a2 cos2 + b2 sin2 , (1.83)


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a2y = a2 sin2 + b2 cos2 , (1.84)

and also2axay cos =

a2 b2 sin(2), (1.85)

axay sin = ab. (1.86)

We have therefore related the properties of the ellipse in the two coordinate systems.

x

y

a

b

ax

ay

a0

Figure 1.6: The polarization ellipse with major and minor semi-axes of length a

and b, inclined at an angle with respect to the cartesian coordinate axes.

It is useful to derive explicit expressions for the properties of the elliple in the rotatedsystem to see its main features more clearly. To this end, we define angles and bywriting9

ax = a0 sin , ay = a0 cos (1.87)

and tan2 = b2/a2. (1.88)

Adding Eqs. (1.83) and (1.84), we first have the relations

a2 + b2 = a2x + a2y = a

20. (1.89)

Further, subtracting Eq. (1.84) from Eq. (1.83) and inserting into Eq. (1.85), we see that

tan(2) =2axay

a2x a2ycos = tan(2)cos . (1.90)

9

The angle is widely used in ellipsometry, along with , to characterize the state of polarization ofa plane wave. In fact these two angles are often referred to as ellipsometric angles: see, e.g., Ref. [8],noting that the notation there are different from those adopted here. We will return to the subject ofellipsometry in Sect. 3.2.3.


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To obtain the last form, we have used the identity tan(2x) = 2 tan x/ (1 + tan2 x). This

expression gives the polarization azimuth rotation angle, defined in the range 0 < ,which is the angle between the major axis of the ellipse with respect to the x axis asillustrated in Fig. 1.6. The ellipticity angle [/4, /4) defines the ratio of thelengths of the major and minor axes of the ellipse. Using Eqs. (1.88), (1.86), (1.87), andthe identity sin(2x) = 2 tan x/ (1 + tan2 x) we have

sin(2) =2axay

a2x + a2y

sin = sin(2)sin . (1.91)

The angles and , together with a, clearly define the orientation, shape, and size of thepolarization ellipse uniquely.

Since the tip of the electric field vector defines an ellipse, a monochromatic plane waveis said to be elliptically polarized in general. If sin > 0, also 0 < /2 and the waveis said to be right-handed elliptically polarized. In this case the tip is seen to rotate ina clockwise sense if observed from the direction in which the wave propagates. In theopposite case, i.e., when sin < 0 and /4 < 0, the polarization is left-handed andthe observer sees a counter-clockwise rotation of the tip.

The ellipse is known to have two degenerate states, namely a line and a circle. If = 0or = , Eq. (1.79) is seen at once to reduce to a line, i.e.,

Ery = ayax

Erx, (1.92)

where the positive sign corresponds to = 0 and the negative sign to = . Then thelight is said to be linearly polarized, having a slope ay/ax with respect to the x axis. If,on the other hand, ax = ay = a0 and = /2, Eq. (1.79) reduces to the equation of acircle:

E2rx + E2ry = a

20. (1.93)

Then wave is said to be right circularly polarized (RCP) if = /2 and left circularlypolarized (LCP) if = /2.

We just saw that any monochromatic plane wave is necessarily elliptically polarized.Thus light with no preferential polarization direction, i.e., unpolarized light, cannot be

monochromatic. To describe such light, it is necessary to make use of statistical concepts,and we will return to this later.It is often convenient to use a shorthand notation for polarization properties of a plane

wave by writing the real electric field in the form

E(r)(z, t) = {Jexp(i)} , (1.94)

where the column vector

J=

AxAy

=

ax exp (ix)ay exp (iy)

= exp (ix)

axay exp (i)

(1.95)

(with = y x as before) is known as the Jones vector.10 It contains all relevantinformation about the polarization properties of a plane wave. An arbitrary state of

10Not to be confused with the electric current density, which was denoted by the same symbol.


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an ordinary differential equation

d2

dz2Aj(, z) + k

2zAj(, z) = 0, (1.104)

where kz is defined in Eq. (1.53). The solutions of this equation are linear combinations ofterms of the form exp (ikzz). Considering waves that propagate (or decay) in the positivez-directions, we disregard the solution with the negative sign and write the general solutionin the form

Aj(, z) = Aj(, z0)exp(ikzz) , (1.105)

where z = z z0 and z = z0 is an arbitrary reference plane.In view of the preceding results, the general solution of our original Helmholtz equationcan be written as

Vj(, z) =

Aj(, z0)exp(ikzz)exp(i ) d2. (1.106)

This is a continuous superposition of plane waves. If we again consider A1 = Ex andA2 = Ey as fixed, we have

A3(, z0) = k1z [kxA1(, z0) + kyA2(, z0)] . (1.107)

in analogy with Eq. (1.64). Writing C0 = k10 0/0, we also have

A4(, z0) = C0 k1z

kxkyA1(, z0) +

k2y + k2z

A2(, z0)

, (1.108)

A5(, z0) = C0 k1z

k2x + k

2z

A1(, z0) + kxkyA2(, z0)

, (1.109)

andA6(, z0) = C0 [kyA1(, z0) kxA2(, z0)] (1.110)

in analogy with Eq. (1.65). Using Eq. (1.103) at z = z0, we can evaluate A1 and A2 if V1and V2 are known at that plane. Then Eqs. (1.106) are applied to obtain the remainingcomponents A3 to A6, and finally all scalar components of the electromagnetic field can becalculated from Eq. (1.106) anywhere in the half-space z > z0. In conclusion, we now havea complete solution of the electromagnetic wave propagation problem in a homogeneousmedium, expressed in terms of (homogeneous and inhomogeneous) plane waves.

The functions Aj(, z0) are known as the angular spectraof the scalar field components,and field expansions in Eq. (1.106) are known as angular spectrum representations ofthe associated field components. This nomenclature arises from the interpretation of thespatial frequency vector as a vector defining the propagation direction of a homogeneousplane wave. To see this, we define polar coordinates (, ) and spherical polar angles (, )by equations

kx = cos = k0n sin cos ky = sin = k0n sin sin

kz = k0n cos

(1.111)

and = || = k2x + k2y1/2 = k0n sin . (1.112)


20/41


Now the angles and define uniquely the direction of the wave vector k provided that

the medium is dielectric and sin < 1. Similarly, we will often employ spherical polarcoordinates also for the position vector:

x = cos = r sin cos y = sin = r sin sin z = r cos

(1.113)

with = || = x2 + y21/2 = r sin (1.114)

and

r = |r| = 2 + z21/2 = x2 + y2 + z21/2 . (1.115)The notations just defined are illustrated in Fig. 1.7, which also shows the unit positionvector s = r/r.

(a) (b)

x

y

z

kx

ky

kz

rsk

z = z0

Figure 1.7: Notations used for (a) wave-vector coordinates of a propagating wave

and (b) position coordinates, with definitions of spherical polar angles in dielectric

media.

If we use polar coordinates for spatial frequencies but cartesian components for theposition vector, the angular spectrum representation of the field components takes theform

Vj(x,y,z) =

0

20

Aj(,,z0)exp[i (x cos + y sin ) + ikzz] d d (1.116)

where

Aj(,,z0) =1

(2)2

Vj(x,y,z)exp[

i (x cos + y sin )] dx dy (1.117)

and k2z = k2on

2 2, with sign conventions in taking the square root as described above.If we use polar coordinates also for the transverse position vector, the angular spectrum


21/41


representation reads as

Vj(,,z) =

0

20

Aj(,,z0)exp[i cos( ) + ikzz] d d (1.118)

with

Aj(,,z0) =1

(2)2

0

20

Vj(,,z0)exp[i cos( )] d d. (1.119)

Finally, if we use spherical polar coordinates for both spatial frequencies and position co-ordinates, assume a purely dielectric medium, and include only propagating plane waves,the angular spectrum representation takes the form

Vj(,,z) = k2

/20

20

sin cos Aj(,,z0)

exp {ik [ sin cos( ) + zcos ]} d d (1.120)with

Aj(,,z0) =1

(2)2

0

20

Vj(,,z0)exp[ik sin cos( )] d d. (1.121)

Explicit expressions for Aj and Vj in terms of A1 and A2 are obtained straightforwardlyusing Eqs. (1.107)(1.110). We will find these last results very useful when dealing with

focusing and imaging problems of electromagnetic fields.

1.2.4 Far-zone fields

The angular spectrum representation allows us to propagate all scalar field componentsover any finite distance in a homogeneous medium. The behavior of the field is, however,simplified considerably if we proceed sufficiently far from the source. In fact, we willsee that if r in a homogeneous dielectric with n = n, the radiation field can beinterpreted as a geometrical (electromagnetic) wave with a well-defined local propagationdirection.

Let us write the right-hand-side of Eq. (1.106) explicitly in terms of the cartesian

coordinates of the wave vector and the unit position vector and make use of the upperbranch of Eq. (1.55). We then have

Vj(rs) =

Aj(kx, ky, z0)exp[ir (kxsx + kysy + kzsz)] dkx dky. (1.122)

where the argument of the exponential term is real. This integral that appears here is ofthe general form

F(u) =

f(x, y)exp[iug(x, y)] dx dy. (1.123)

In the asymptotic limit u , the integral in Eq. (1.123) can be evaluated by the methodof stationary phase. If g(x, y) is real, its limiting value is (see Sect. 3.3.3 in Ref. [7])

F(u) =i2f(xc, yc)

| 2|exp [iug(xc, yc)]

u, (1.124)


22/41


where xc and yc are the (transverse) coordinates of a critical point (of the first kind).

These coordinates may be determined by solving the equations

xg(x, y) =

yg(x, y) = 0 . (1.125)

Furthermore,

=

1 if > 2 and 01 if > 2 and < 0i if < 2

(1.126)

and

=

2

x2 g(x, y) |(xc,yc) , (1.127) =

2

y2g(x, y) |(xc,yc) , (1.128)

=2

xyg(x, y) |(xc,yc) , (1.129)

where |(xc,yc) means evaluation at the critical point.The integral on the right-hand-side of Eq. (1.122) is precisely of the form of that in

Eq. (1.123) if we make the substitutions u r, (x, y) (kx, ky), f(x, y) A(kx, ky),and g(x, y)

g(kx, ky) with

g(kx, ky) = sxkx + syky + szkz. (1.130)

Let us calculate the partial derivatives of g(kx, ky). Recalling that kz = (k2 k2x k2y)1/2,

where we have denoted k = k0n for brevity, and that s is a unit vector, we find find thatthe coordinates of the critical point are to be determined from equations

sx sz kxkz

= 0, (1.131)

sy sz kykz

= 0. (1.132)

The (only) solution iskxc = ksx, kyc = ksy, kzc = ksz. (1.133)

Therefore we haveg (kxc, kyc) = k (1.134)

and = k1 1 + (sx/sz)2 , (1.135) = k1 1 + (sy/sz)2 , (1.136)

=

k1sxsy/s

2z . (1.137)

Hence 2 = (ksz)2 > 0, which implies that = 1 in Eq. (1.126), and we haveVj(rs) = i2kszAj(ksx, ksy, z0)U(r) , (1.138)


23/41


where

U(r) = exp (ikr)r

(1.139)

is the expression for a diverging scalar spherical wave. Hence we have established that eachcomponent of the electromagnetic field behaves, in the far zone, as a diverging sphericalwave modulated by the angular spectrum of plane waves Aj(ksx, ksy, z0) associated withthis particular field component.

If we now use representations of the form of Eq. (1.138) for the transverse componentsV1 and V2 of the electric field vector, Eq. (1.107) implies that the longitudinal componentof the electric vector takes the form

V3(rs) = i2k [sxA1(ksx, ksy, z0) + syA2(ksx, ksy, z0)] U(r) , (1.140)

Furthermore, in view of Eqs. (1.108)(1.110), the transverse scalar components of themagnetic field are given by

V4(rs) = i2k2C0

sxsyA1(ksx, ksy, z0) +

s2y + s

2z

A2(ksx, ksy, z0)

U(r) , (1.141)

V5(rs) = i2k2C0

s2x + s2z

A1(ksx, ksy, z0) + sxsyA2(ksx, ksy, z0)

U(r) , (1.142)

and the longitudinal component is

V6(rs) = i2k2C0sz [syA1(ksx, ksy, z0) sxA2(ksx, ksy, z0)] U(r) . (1.143)

The time-averaged Poynting vector in the far zone is obtain after somewhat lengthy but

straightforward calculation from the definition in Eq. (1.46), using the field representationjust derived. The result is (see Problem 8.1.8)

S(rs, t) = J(s) sr2

, (1.144)

where

J(s) = J0|szA1 (ksx, ksy, z0)|2 + |szA2 (ksx, ksy, z0)|2

+ |sxA1 (ksx, ksy, z0) + syA2 (ksx, ksy, z0)|2

(1.145)

and

J0 =n

20

0 (2k)2

. (1.146)

This result has important implications. If we consider the direction of S(rs, t) as thepropagation direction of the field, we see that this direction is always that of the unitposition vector s of the observation point. On the other hand, we may now safely interpretthe magnitude of S(rs, t) as the optical intensity in a local sense since the field atposition rs is fully determined by a single plane-wave component in the angular spectrum,with wave vector k parallel to s.

The quantity J(s) is known as radiant intensity. It expresses the angular distribu-tion of the radiation pattern in the far zone: using the spherical polar angles defined inEq. (1.113) we have

J(s)/J0 = cos2 |A1 (ksx, ksy, z0)|2 + |A2 (ksx, ksy, z0)|2

+sin2 |cos A1 (ksx, ksy, z0) + sin A2 (ksx, ksy, z0)|2 (1.147)


24/41


with sx = sin cos and sy = sin sin . Hence the radiation pattern in the far zone is

seen to depend critically on the relation between A1 and A2, i.e., on the polarization ofthe field at z = z0.

The derivation presented above is not formally valid if n is complex because then thefunction g(kx, ky) defined in Eq. (1.130) is no longer real. However, if we allow the criticalpoint have complex coordinates, the same procedure can be carried through with thecritical point kxc = k0nsx, kyc = k0nsy and the final result is the same as Eqs. (1.138)(1.143), but with k = k0n. We will, however, justify numerically in Sect. 1.3.1 that, inweakly absorbing of amplifying media with || n, k = k0n can be used in Eq. (1.138)everywhere except in the expression (1.139), which must be written in the form

U(r) = exp (ik0nr)r

(1.148)

to take absorption or amplification properly into account.

1.2.5 Polarization of harmonic fields

We saw above that, in the far zone, the electromagnetic field in a dielectric medium isa diverging spherical wave modulated by the angular spectrum of the field at z = z0.The scalar components of the angular spectrum depend only on the spherical polar angles and or, equivalently, on the angles and , Therefore, the field can be consideredlocally as a plane wave propagating in this precise direction. Hence it is meaningful todefine the local state of polarization of the field in the far zone by considering a Jonesvector

J(, ) =

A(, )A(, )

= A(, ) + A(, ), (1.149)

where and are unit vectors in the two orthogonal and directions, and we havedropped the dependence of A(, ) and A(, ) on z0 for brevity.

As indicated by the notation, the complex amplitudes A = a exp (i) and A =a exp (i) may generally depend on propagation direction. At least in principle, thevalues of a, a and can be measured using a goniometric setup to determine thelocal state of polarization experimentally. If it is found that a

= 0 for all and , the

angular spectrum is said to be azimuthally polarized. If, on the other hand, a = 0 for all

and , the angular spectrum is said to be radially polarized. Since the unit vectors and are orthogonal and the polarization vector does not have a component in the directionof s, the angular spectrum may always be expressed as a superposition of its azimuthallyand radially polarized components as indicated in Eq. (1.149). Its representation in thepolar (,,z) coordinate system is (A, A, Az) with

A = A cos , (1.150)

Az =

A sin . (1.151)

In the cartesian coordinate system we get

A1 = A cos A sin = A cos cos A sin , (1.152)


25/41


A2 = A sin + A cos = A cos sin + A cos , (1.153)

A3 = A sin . (1.154)In writing these transformations, we have dropped the explicit dependence of angularspectra on the angles and for brevity. Using the two orthogonal components of theangular spectrum, the radiant intensity assumes the simple form

J(s) = J0 cos2 |A(ksx, ksy, z0)|2 + |A(ksx, ksy, z0)|2 (1.155)

since the interference term vanishes (see Problem 8.1.9).The polarization state of the field in any point of space can be determined by inserting

Eqs. (1.152) and (1.152) into the angular spectrum representations of the components V1,V2, and V3 of the electric field vector. In general the field is elliptically polarized; theellipticity, orientation of the major axis, and the plane defined by the tip of the electricfield can vary with position. All these properties can, of course, be calculated if the scalarcomponents V1, V2, and V3 are known (see Ref. [3] for details). Some specific polarizationconfigurations of the field are of particular interest. For example, we may express the fieldin polar coordinates such that

E(r) = E(r) + E(r) + zEz(r), (1.156)

where

E(r) = V1(r)cos + V2(r)sin , (1.157)E(r) = V1(r)sin + V2(r)cos , (1.158)

and Ez(r) = V3(r). If it is then found that E = 0 across some plane (or throughoutthe entire space), the field in called azimuthally polarized in that particular plane (orglobally). Analogously, if E = 0, the field is said to be radially polarized. It shouldbe stressed that radial or azimuthal polarization state of the angular spectrum does notnecessarily imply that the field itself would be radially or azimuthally polarized, althoughexamples of fields that behave in this manner can be found (see, e.g., Ref. [9]).

1.2.6 Paraxial fields

Fields emitted by a number of practical light sources, such as many lasers, propagate inthe form of narrow beams. Thus they effectively contain only homogeneous plane waves,which have appreciable complex amplitudes only within a narrow cone centered at the zaxis. In this situation we may use the first two non-vanishing terms of the Taylor seriesto approximate kz as

kz k0n ||2

2k0n. (1.159)

This is known as the paraxial approximation of the wave vector.Using the approximation in Eq. (1.159) allows us to obtain, from Eq. (1.102), the

general paraxial propagation formula in media of arbitrary refractive index in the form

Vj(, z) = exp (ik0nz)

Aj(, z0)expa(z) ||2 exp (i ) d2 (1.160)


26/41


with

a(z) = iz2k0n

= n2 + 2

z2k0

+ i nn2 + 2

z2k0

. (1.161)

To put this formula in a more commonly appearing form, we us now replace with inEq. (1.103) and insert the result in Eq. (1.160). The spatial-frequency integrations canbe evaluated with the identity

exp

ax2 + bx dx = a

exp

b2

4a

, (1.162)

which is valid for complex a and b provided that {a} > 0. Then

exp

a(z) ||2 + i ( ) d2 = a(z)

exp

(

)2

4a(z)

. (1.163)

Hence, if{a(z)} > 0, which implies > 0,

Vj(, z) =k0n exp(ik0nz)

i2z

Vj(, z0)exp

ik0n

2z( )2

d2. (1.164)

It must be stressed that this Fresnel propagation formula is in principle not valid in gainmedia since its derivation required the condition > 0. Thus, when dealing with gain

media, it is safer to use Eq. (1.160).By expanding the exponential within the integral of Eq. (1.164) we obtain an alterna-

tive form of the Fresnel propagation formula:

Vj(, z) = V0jk0n exp (ik0nz)

i2zexp

ik0n

2z||2

Vj(

, z0)exp

ik0n

2z||2

exp

ik0n

z

d2. (1.165)

For sufficiently large values of z, the contribution of the quadratic term within theintegral becomes negligible. We may then approximate the Fresnel propagation formulawith the Fraunhofer propagation formula

Vj(, z) = V0jk0n exp (ik0nz)

i2zexp

ik0n

2z||2

Vj(, z0)exp

ik0n

z

d2.

(1.166)This formula is also obtained from Eq. (1.138) with k = kon if we approximate

r z+ ||2

2z(1.167)

in the exponential factor of Eq. (1.139), r

z in its denominator, and sx

x/z,sy y/z in the arguments of Aj .

According to the Fraunhofer formula, the propagated field in a purely dielectric mediumwith real refractive index is just a scaled Fourier transform of the field at z = z0. This is a


27/41


good approximation also in media with low loss, i.e., if

n. Then we may approximate

n by n in the latter (quadratic) term in Eq. (1.159). As a result, we may replace n by nin all but the first factor of the Fresnel and Fraunhofer formulas.

The paraxial approximation, Eq. (1.159), is valid if kx kz and ky kz. Hence, inview of Eqs. (1.107) and (1.110), the longitudinal components of the electromagnetic fieldvanish to first approximation. According to Eqs. (1.108) and (1.109), we further have

A4(, z0) k0nC0A2(, z0) (1.168)and

A5(, z0) k0nC0A1(, z0) (1.169)

Thus Hx Ey and Hy Ex, and the field may be called transverse electromagnetic inthe same sense as a plane wave. However, it can be shown numerically that the paraxialapproximation can be reasonably accurate up to field angles of the order of 30 (seeSect. 1.3.1). In this case the longitudinal components are no longer negligible. If weapproximate kz k0n, the angular spectrum of the longitudinal electric field takes theform

A3(, z0) (k0n)1 [kxA1(, z0) + kyA2(, z0)] . (1.170)To the same degree of approximation, we have

A6(, z0) (k0n)1 C0 [kyA1(, z0) kxA2(, z0)] (1.171)for the angular spectrum of the longitudinal magnetic field.

1.3 Propagation of some model fields

We will next proceed to consider some special types of fields with substantial practicalimportance. This will also allow us to consider some of the approximations made abovein more quantitative terms.

1.3.1 Gaussian fields

Let us assume that, at the plane z = z0, the transverse components of the electric field

are of the Gaussian form

Vj(, z0) = V0j exp

||

2

w20

= V0j exp

k0n

2zR||2

, (1.172)

where j = 1, 2, V0j and w0 are constants, and we have introduced a parameter

zR =1

2k0nw

20 =

n

0w20. (1.173)

If we introduce Eq. (1.172) into Eq. (1.103) and use again the integral formula (1.162), it

follows that

Aj(, z0) = A0j exp

1

4w20 ||2

= A0j exp

zR

2k0n||2

, (1.174)


28/41

1.3 Propagation of some model fields 27

where A0j = (w20/4) V0j = zRV0j/2k0n.

The far-zone behavior of the field can be examined by inserting Eq. (1.174) intoEq. (1.138) with k = k0n. After some manipulation, we then obtain the result

Vj(rs) =1

2k0

n2 2 w20V0j cos

exp(k0r)r

exp1

4k20

n2 2w20 sin2

exp

i

k0nr 2

exp [i()] , (1.175)

where

() = arctan

n 1

2k20nw

20 sin

2 (1.176)

is the wave front aberration due to absorption, i.e., the departure of the wave front froma shperical shape that results when = 0. It is, however, a simple matter to shownumerically that theis wave aberration is only a small fraction of a radian over the regionwhere the field amplitude is significant, even if|| is as large as 0.1. Thus, for all practicalpurposes, we may set () = 0 in weakly absorbing or amplifying media. Using the sameapproximation elsewhere in Eq. (1.175) allows us to write

Vj(rs) =1

2k0nw

20V0j cos

exp(k0r)r

exp 1

4(k0nw0)

2 sin2 exp ik0nr

2 . (1.177)This amounts to approximating k = k0n k0n everywhere in Eqs. (1.138), except in U(r).This justifies the comments before Eq. (1.148) for Gaussian fields in weakly absorbing oramplifying media, and we may expect that the same applies to all well-behaved fields.

The evaluation of the integral in the exact propagation formula (1.102) is possible onlynumerically. However, a simple closed-form solution is available in paraxial approxima-tion. Inserting from Eq. (1.174) into Eq. (1.160) and defining a new parameter

q(z) = z izR (1.178)we obtain

Vj(, z) = exp(ik0nz)

exp

iq(z)

2k0n||2

exp (i ) d2. (1.179)

The integral can again be evaluated using the identity (1.162), which gives

Vj(, z) = V0jzR

iq(z)exp

ik0n

z+

||22q(z)

, (1.180)

provided (formally) that

iq(z)

2k0n > 0 (1.181)

or, more explicitly

z+1

2k20w

20

n2 + 2

> 0. (1.182)


29/41


This condition is, of course, valid for all lossy media. In gain media with negative values

of, it implies an upper bound for propagation distances z, which however is very largefor small values of.

A paraxial expression for the field in the far zone is obtained by inserting fromEq. (1.172) into the Fraunhofer propagation formula. This results, after some simpli-fication, into

Vj(, z) = V0jzR

izexp

ik0n

z+

||22z

exp

zRw0z

2||2

. (1.183)

Precisely the same results can are obtained from Eq. (1.180). To this end, we note that

if z |zR|, we may write1

q(z) 1

z

1 + i

zRz

1

z. (1.184)

If we use the second approximation in the factor in front of the exponential of Eq. (1.180)and the first approximation inside the exponential itself, we arrive at Eq. (1.183). It isagain easy to show numerically that if || n, the complex zR can be replaced by itsreal part in Eq. (1.183) without making an appreciable error.

It is worthwhile to consider the result of the Fresnel propagation formula for Gaussianfields, Eq. (1.180), in more detail. Let us first consider the case of purely dielectric media,which is usually a reasonable approximation. Then, if we write

ik0n

2q(z)= 1

w2(z)+

ik0n

2R(z)= 1

w2(z)+

izRw20R(z)

, (1.185)

a straightforward calculation gives

w(z) = w0

1 +

z

zR

2(1.186)

and

R(z) = z+z2Rz

, (1.187)

where of course zR = k0nw20/2. For large values of z we may approximate

w(z) w0zzR

=2z

k0nw0=

0z

nw0(1.188)

andR(z) z. (1.189)

Writing also q(z) = |q(z)| exp [i(z)] with |q(z)| =

z2 + z2R and

(z) = arctan (z/zR) (1.190)

we obtain a more explicit form of Eq. (1.180), i.e.,

Vj(, z) = V0jw0

w(z)exp[i(z)] exp

||

2

w2(z)

exp

ik0n||2

2R(z)

. (1.191)


30/41


Evidently, w(z) is the characteristic (or 1/e) width of

|V(, z)

|and R(z) is the radius of

curvature of the parabolic wave front. The evolution of the Gaussian beam in a dielectricmedium is illustrated in Fig. 1.8. The dashed line illustrates the asymptotic (far-zone)limit, where the beam width and the radius of curvature both grow linearly with z asindicated in Eqs. (1.188) and (1.189). The angle 0 is given by

0 = limz

w(z)

z=

0nw0

. (1.192)

Furthermore, Eq. (1.188) shows that at z = zR the beam width has increased by a factorof

2, and Eq. (1.189) shows that the radius of curvature at this propagation distance

reaches its minimum value R(zR) = 2zR.

y

z

0

z0

w0

w(z)R(z)

Figure 1.8: Illustration of the evolution of a Gaussian field in a homogeneous

dielectric medium.

The Gaussian fields provide us a possibility to investigate more quantitatively someof the approximations made above. Let us first consider the distances at which theasymptotic approximation r is reasonable. The asymptotic formula (1.138) nowgives

V(a)j (rs) = in (w0/0)cos exp

(nw0/0)2 sin2 exp (i2nr/0)r/0

, (1.193)

where we have introduced the spherical field angle . The exact angular spectrum repre-sentation for a rotationally symmetric case is obtained from Eq. (1.118) with the aid ofthe Bessel-function identity

J0(x) =1

2

20

exp(ix cos ) d. (1.194)

to carry out the angular integration. The general result is

Vj(, z) = 2

0

Aj(, z0)J0()exp(ikzz) d. (1.195)

Inserting from Eq. (1.174), we then have

Vj(rs) =1

2w20

0

exp

1

4w20

2

J0(r sin )exp

ir cos

k20n

2 2

d. (1.196)


31/41


Some representative convergence curves curves ofV(a)j (rs) towards Vj(rs), as r

, are

shown in Fig. 1.9. Here we have chosen sin = 0.1, w0 = 2, n = 1.5, and plotted therelative amplitude error

A = 1 V

(a)j (rs)

Vj(rs)

(1.197)as well as the absolute phase error

= arg

V(a)j (rs)

arg[Vj(rs)] (1.198)

for several values of . These values were chosen small enough to allow a significantfield amplitude at distances up to r = 200. It is seen from the figures that uniform

convergence towards the asymptotic limit is achieved at large distances for all values of in terms of both field amplitude and phase.

60 80 100 120 140 160 180 200

0.00

0.01

0.02

0.03

0.04

0.05

(a)

r/0

A

60 80 100 120 140 160 180 200

0.06

0.05

0.04

0.03

0.02

0.01

0.00

(b)

r/0

Figure 1.9: Comparison of the exact angular spectrum and stationary-phase propa-

gation techniques. (a) Relative amplitude error and (b) phase error of the asymptotic

result as a function of r for field angle = arcsin 0.1 if n = 1.5 and = 0 (solid

lines), = 0.001 (dashed lines) and = 0.01 (dotted lines).

The rate of convergence towards the asymptotic limit depends critically on the valueof w0. For small values, the asymptotic limit can be achieved (at least for most practicalpurposes) at distances of the order of tens of wavelengths or less, as shown in Fig. 1.9.To obtain more quantitative results, valid in the paraxial domain, we may consider theevolution of the propagation parameters w(z) and R(z) in more detail. The relative beamwidth error

w =w(z) w0z/zR

w(z) 1

2

zRz

2(1.199)

is less than 1% if z > 7zR. The absolute phase error at || = w(z), defined as

1/e = k0n2z

k0n

2R(z)w2(z) = zR

z

, (1.200)

is less that 2/10 already at distances z > 1.6zR. This implies that the field at thosedistances is a good approximation of a spherical wave with a Gaussian amplitude profile,


32/41


with its origin at r = (0, 0, z0), in view of a commonly used wave aberration criterion that

the absolute optical path error should be less than, say, /8 over the region where the fieldamplitude differs significantly from zero. The observations se just made on asymptoticconvergence of relative amplitude and absolute phase are in agreement with the non-paraxial results presented in Fig. 1.9, which also shows that the absolute phase reaches agood approximation of its asymptotic value well before the amplitude distribution doesthe same.

In Sect. 1.2.6 we did not specify the range of validity of the paraxial approximation.This is indeed difficult since it depends on the chosen criterion. Let us now have acloser look at this problem by comparing the exact and paraxial expressions for far-zonedistributions of Gaussian fields in dielectric media, i.e., Eqs. (1.177) and (1.183) with

= 0. If we write them in slightly different forms

Vj(rs) = V0jzRir

cos exp (ik0nr)exp

zRw0

2sin2

(1.201)

and

Vj(, z) = V0jzR

izexp

ik0n

z+

||22z

exp

zRw0

2 ||2z2

, (1.202)

we see immediately that Eq. (1.202) follows from Eq. (1.201) in the paraxial approximation

(1.167), which also implies cos = z/r 1 and sin = ||/r ||/z. In particular,we wish to compare these expression at an arbitrary plane z = constant and thereby writer2 = ||2 + z2 in Eq. (1.201) so that the exact expression in the far zone reads as

Vj(, z) =V0jzRz

i||2 + z2 exp

ik0n

||2 + z2

exp

zRw0

2 ||2||2 + z2

.(1.203)

Hence, view of the exact expressions, the far-field amplitudes differ from Gaussian bothon the surface of any sphere r = constant and across any plane z = constant. The sameconclusion also hold before the far zone is reached, although analytic expressions are notavailable there.

Figure 1.10 shows a comparison of far-zone distributions of two Gaussian fields withwavelength-scale waist sizes in free space (n = 1) at a distance z = 250, which islarge enough to ensure that the far-zone condition is valid. It can be concluded that thedifference between the exact and paraxial results is insignificant if w0 > 0, i.e., if thefar-zone field is effectively confined within an angular range < 30. This divergence-based esitmate for the validity of the paraxial approximation is useful also for otherwell-behaves fields. However, for fields of wider divergence, the error made in using theparaxial approximation becomes significant at smaller fields angles, as indicated by theresults of Fig. 1.10 for w0 = 0/2.

The absolute phase error encountered in applying the paraxial approximation at a

plane z = constant in the direction defined by field angle may be expressed as

(z, ) = k0nz

1 +

1

2tan2 cos1

, (1.204)


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0 5 10 15 20 25 30 35

0.00

0.02

0.04

0.06

0.08

0.10

0.12

||/0

|

V1

(|

|

,z)/V01

|

0 5 10 15 20 25 30

0.0

0.5

1.0

1.5

2.0

2.5

(b)

[]

()

Figure 1.10: (a) Comparison of the exact (solid lines) and paraxial (dashed lines)far-zone field distributions for Gaussian waves in free space. The narrower pair

corresponds to waist radius w0 = 0 and the wider pair to w0 = 0/2. (b) Compar-

ison of the absolute phase error as a function of field angle on a plane surface with

z = 100 (solid line) and on a spherical surface of radius r = 100 (dashed line).

and on a spherical surface r = constant as

(r, ) = k0nrcos +1

2

sin tan

1 . (1.205)

These expressions depend linearly on z and r, respectively, which indicates that thephase error can become arbitrarily large in the far zone and therefore the exact non-paraxial propagation laws should be used to determine the phase. Figure 1.10 showsthat the errors at z = 250 as a function of the field angle. As discussed above, thisdistance is aleady large enough for the far-zone expression to hold for highly divergentfields. We see that at least for reasonable values of the phase error is still only asmall fraction of a radian; since a closed-form expression that gives the correct phaseis available, there is no necessary reason to use the paraxial formula at large distancesand field angles. Moreover, we may expect that at smaller distances the paraxial Fresnel

formula for Gaussian beams (and other well-behaved fields) holds for both amplitude andphase, provided that the divergence is less than 30. Finally, it is easy to extablishnumerically that if w0 > 100, propagation to distances of the order of 10

60 can bedescribed by the paraxial formulation without introducing a phase error larger than 1 radat a field angle satisfying sin 1/e = w0/zR, where the field amplitude has dropped to 1/eof its axial value.

So far we have only considered the transverse (electric) components of the Gaussianfield. Evidently, in general, the longitudinal components also play a role, which can beevaluated using the techniques presented in Sections 1.2.3 and 1.2.4. The effect of thesecomponents is included in, e.g., Eq. (1.145) for the radiant intensity. Thus, for example,

if A2 = 0, we have

J(, ) cos2 + sin2 cos2 exp 12

(k0nw0)2 sin2

(1.206)


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and the radiant intensity exhibits rotational non-symmetry, which is due to the longitu-

dinal field components. Numerical investigation of this expression shows, however, thatthe asymmetry is negligible unless w0 0 or smaller. Such a weak polarization depen-dence of the radiation pattern is common to fields that contain plane-wave componentspropagating mainly at small angles. We will, however, see in the following subsection thatfor certain types of fields the effects due to longitudinal field components can be muchmore significant. We finally stress that these effects do not (completely) vanish in theparaxial approximation, which in Eq. (1.206) means cos 1 and sin . Thus thecommonly held belief that paraxial approximation is equivalent with scalar modeling ofwave propagation11 is incorrect.

1.3.2 Conical fields

Field consisting of plane waves with wave vectors confined on a single cone are calledconical fields. Both scalar [10, 11] and electromagnetic [9, 12, 13] conical fields have havebeen studied extensively and shown to have many interesting properties and applications,reviewed in Ref. [14]. We will find such conical field useful in a number of examples tofollow, and now proceed to investigate some of their basic properties.

Considering the angular spectrum representation in term,s of the components of theelectric angular spectrum in the orthogonal basis, we obtain the general class of conicalwaves by demanding that the angular spectrum vanishes for all but one single value ofthe angle , say 0. Mathematically, we then write

A(, ) = A()( 0) (1.207)

and

A(, ) = A()( 0), (1.208)thus allowing the radial and azimuthal components A and A to vary arbitrarily inthe azimuthal direction. According to Problem 8.1.13, the cartesian components of theangular spectrum spectrum are then

A1(, )A2(, )A3(, )A4(, )A5(, )A6(, )

= A()( 0)

cos cos cos sin sin

C0 sin C0 cos

0

+ A()( 0)

sin cos

0C0 cos cos C0 cos sin

C0 sin

(1.209)

with C0 = C0k = n

0/0.The expressions for the cartesian components of the electric and magnetic field can now

be obtained by inserting Eq. (1.209) into the angular spectrum representation. Becauseof the symmetry of the cone, it will prove convenient to use polar position coordinates

11By this one usually means that consideration of a single scalar component is sufficient for adequatedescription of field propagation in terms of intensity and phase; polarization effects can of course not bedescribed using a scalar model.


35/41


(, ) and employ Eq. (1.120). To achieve a more compact notation, we introduce two

new constants = k sin 0, (1.210)

= k cos 0, (1.211)

which are in fact the radial and longitudinal components of any wave vector on the conedefined by the angle 0. In this way we obtain

Vj(,,z) = exp (iz)

20

Aj()exp[i cos( )] d, (1.212)

where the azimuthal angular spectra can be expressed in the form

A1()A2()A3()A4()A5()A6()

= A()

(/k)cos (/k)sin

/kC0 sin C0 cos

0

+ A()

sin cos

0C0(/k)cos C0(/k)sin

C0/k

. (1.213)

The result (1.212) reveals a key property of conical fields. We see that the z-dependenceof all field components of imaginary-exponential form, separable from the transverse de-pendence defined by the integral. Therefore all quantities related to energy and flux (the

time-averaged electric and magnetic energy densities and the magnitude of the Poyntingvector) are independent on z. Therefore all conical fields may be called propagation-invariant fields.

The limit of a conical wave at 0 0 is a plane wave propagating in the directionof the z-axis,12 which of course is the simplest example of a propagation-invariant field.In the highly paraxial region the longitudinal components of the electric and magneticvectors are seen to be negligible with (at least the largest) transverse components, andthus the field electromagnetic field itself is essentially transverse.

Let us examine the nature of conical fields by looking at a certain special case in moredetail. To begin with, since the spherical components of the angular spectrum of any

conical field are periodic with period 2, we may (quite generally) express them in theform of Fourier series

A() =

m=am exp (im) , (1.214)

A() =

m=am exp (im) , (1.215)

with Fourier coefficients

am =1

2 2

0

A()exp(im) d, (1.216)12In this limit 0 so that the field vanishes, in principle. However, this is only a matter of re-

normalization; the energy on the cone vanishes when the cone radius approaches zero, but the functionalform of the field approaches that of a plane wave.


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am =

1

220 A()exp(im) d. (1.217)

Let us first choose the spherical components of the angular spectrum such that A1() =exp (im) and A2() = 0. This amount to assuming that the electric field is polarizedin the xz-plane, the magnetic field is linearly polarized in y-direction, and in the highlyparaxial limit the electric field is linearly x-polarized. Moreover, only a single Fouriercomponent with a fixed integer value ofm is non-zero in Eqs. (1.216) and (1.217). Then,if we use the integral formula

2

0

exp[i(m x cos )] d = 2imJm(x), (1.218)

where Jm(x) is a Bessel function of the first kind and order m, we have

V1(,,z) = V0m exp (iz) Jm()exp(im) , (1.219)

where V0m = 2im. Also V3 and V5 are non-zero, but V3 is negligible in the highly

paraxial limit , and in this limit we also have V5 V1. Hence the properties of thefield can be studied by inspection of V1 alone.

In view of Eq. (1.219), the field V1 has a helical phase profile for all values of m otherthan m = 0. A field with this property is said to have vortex of order m at the optical

axis = 0. This can be interpreted as a rotation of the electric field about the opticalaxis upon propagation, and such rotation has important mechanical effects. The electricenergy density at the highly paraxial limit is readily seen to be proportional to J2m(),i.e., to the square of the Bessel function of order m, and therefore fields of this type arereferred to as Bessel fields. Transverse profiles of some lowest-order paraxial Bessel fieldsare illustrated in Fig. 1.11 and some properties of the fundamental J0 Bessel field aregiven in Table 1.1.

Table 1.1: Some properties of the ring system associated with the fundamental J0Bessel field (adopted from Ref. [14]).

lobe n:o 1 2 3 4 5 q 1outer radius/ 2.405 5.520 8.654 11.791 14.931 (q 1/4)width/ 3.115 3.134 3.137 3.140 energy 4.897 6.187 6.252 6.268 6.274 2

It is seen from Fig. 1.11 that the fundamental Bessel field has a central maximumsurrounded by fainter concentric rings, while the higher-order Bessel fields are hollowin the sense that the axial field amplitude is zero. Both types of fields are useful inmanipulation and trapping of particles. The transverse scale of the field depends on the


37/41


xxx

xxx

y

y

y

y

y

y

(a)

(b)

(c)

(d)

(e)

(f)

0

0

0

0

0

0

0

0

0

0

0

0

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

20

20

20

20

20

20

20

20

20

20

20

20

202020202020

202020202020

Figure 1.11: Transverse field distributions of Bessel fields of different order. Left:

J0 field. Middle: J1 field. Right: J2 field. Upper row: amplitude |V1(x, y)|; whiteindicates the maximum of the J0 field, to which the other scales are normalized.Lower row: phase; white and black indicate extreme values of and , respectively(adopted from Ref. [14]).

parameter , i.e., on the cone angle. As indicated in Table 1.1, the radius of the centralmaximum is

0 2.405

0.38 0n sin 0

. (1.220)

Hence it can be smaller than /2 for large values of0, indicating that the field is localizedin a tube of wavelength-scale diameter. However, while the functional form of V1 isindependent on the scale, the longitudinal components becomes important for large valuesof the cone angle and cannot be neglected (see Problem 8.1.14). This component has azero on axis and it exhibits azimuthal nonsymmetry, which causes the electric energydensity of the total field to be rotationally non-symmetric as well.

Several other special cases of electromagnetic propagation-invariant fields have beendiscussed in, e.g., Ref. [14]. For example, a purely azimuthally polarized fundamentalBessel field has no longitudinal component and a rotationally symmetric transverse energydensity distribution proportional to J21 , but it does not have a vortex.

1.3.3 Electric dipole radiation

An electric dipole is usually an adequate model for radiation from electromagnetic pointsources, i.e., for sources with dimensions much smaller than the wavelength in a medium.


38/41


A radiating monochromatic dipole is described by a vector potential13

a(r) = ikpU(r), (1.221)where k = k0n, the constant (real-valued) vector p is known as the electric dipole moment,and

U(r) =exp (ikr)

r(1.222)

is a scalar spherical wave. The magnetic and electric field vectors can generally be derivedfrom the vector potential using the formulas [15]

H(r) = 10 a(r) (1.223)

andE(r) =

i

kn

00

H(r) = ickn

[ a(r)] . (1.224)

Inserting from Eqs. (1.221) and (1.222) into Eqs. (1.223) and (1.224) we obtain (aftersome vector algebra)

H(r) =k2

0sp

1 1

ikr

U(r) (1.225)

and

E(r) =c

n

k2 (sp) s+ [3s (s p) p]

1

r2 ik

r

U(r), (1.226)

where s is the unit position vector as before.The electric field in Eq. (1.226) is seen to consist of a superposition of three terms

with different (inverse) powers of r, whereas the magnetic field in Eq. (1.225) containsonly two such terms. Clearly, only the term proportional to U(r) survives in the far zone,and we therefore see immediately that the field there is an outgoing spherical wave with

H(r) =k2

0U(r) sp (1.227)

and

E(r) =1

n0

0H(r)

s (1.228)

so that E H s in agreement with the general conclusions of Sect. 1.2.4. In thenear-field, where r 0, the term diverging as U(r)/r2 dominates in Eq. (1.226); sincethe magnetic field does not have a term of this magnitude, the field in the near zoneof the dipole is predominantly electric and one often speaks of the static zone. In theintermediate zone, where r and are of the same order of magnitude, all terms must beretained to obtain an accurate representation of the field.

We may again use the time-averaged Poynting vector to characterize the intensity anddirection of energy flow in the the far zone. A simple calculation shows that

S(rs, t) |p|2

sin2

s

r2 , (1.229)13A capital A is usually used to denote the vector potential in the literature but we use a lower-case

a to avoid confusion with the angular spectrum.


39/41


where is the angle between p and the observation direction s. Thus we have the familiar

1/r2 decay of field amplitude and we see that the dipole does not radiate in the (positiveand negative) directions of the dipole-moment vector p.

It is straightforward to show (see Problem 8.1.16) that the angular spectrum of thediverging scalar wave U(r) is given by

A(kx, ky) =i

2kz(1.230)

and the angular spectrum representation of the scalar diverging spherical wave in thehalf-space z > z0 is therefore of the form

U(r) =i

2

1

kz exp[i(kxx + kyy + kzz)] dkx dky. (1.231)

This result is known as Weyls formula.The angular spectrum representation of electric dipole radiation into the positive half-

space z > z0 can be readily evaluated using Weyls formula. In view of Eqs. (1.221),(1.222), and (1.231), the vector potential clearly has the representation

a(r) =

k

2kzp exp[i(kxx + kyy + kzz)] dkx dky. (1.232)

Applying Eqs. (1.225) and (1.226), we can calculate the cartesian components of the

magnetic and electric angular spectra. The results can be organized in a compact form

E(r) =

A(e)p exp[i(kxx + kyy + kzz)] dkx dky (1.233)

and

H(r) =

A(m)p exp[i(kxx + kyy + kzz)] dkx dky (1.234)

if we consider p as a column vector and define matrices A(e)(kx, ky) and A(m)(kx, ky) as

A(e)

(kx, ky) =

A(e)0

kz k

2 k2x kxky kxkzkxky k

2

k2

y kykzkxkz kykz k2 k2z

(1.235)

and

A(m)(kx, ky) =A(m)0

kz

0 kz kykz 0 kx

ky kx 0

, (1.236)

where A(e)0 = c/2n and A

(m)0 = k/20.

It is sometimes necessary to treat also dipole fields propagating towards the negativehalf-space. This can be accomplished by replacing kz with kz in expressions (1.233)(1.235) and writing

|z

|instead of z in the exponentials of the angular spectrum rep-

resentations. In addition, a dipole positioned at an arbitrary location (z0, y0, z0) can bedescribed by replacing x by x = x x0 and y by y = y y0 in the exponentials of theangular spectrum representations.


40/41

References

[1] W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (AddisonWesley, Reading, MA, 1978).

[2] R. W. Boyd, Nonlinear Optics (Academic Press, Boston, 1992).

[3] M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

[4] C. Iaconis, V. Vong, and I. A. Walmsley, Direct interferometric techniques forcharacterizing ultrashort optical pulses, IEEE J. Sel. Top. Quantum Electron. 4,

285294 (1999).

[5] R. Trebino, ed., Frequency-Resolved Optical Gating: the Measurement of UltrashortOptical Pulses (Kluwer, 2002).

[6] I. A. Walmsley and C. Dorrer, Characterization of ultrashort electromagneticpulses, Adv. Opt. Photon. 1, 308437 (2009).

[7] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Uni-versity Press, Cambridge, 1995).

[8] H. G. Tompkins and E. A. Irene, Eds. Handbook of Ellipsometry (William AndrewPubl., Norwich, NY, 2005).

[9] P. Paakkonen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, General vectorialdecomposition of electromagnetic fields with application to propagation-invariantand rotating fields, Opt. Express 10, 949959 (2002).

[10] J. Durnin, Exact solutions for nondiffracting beams. I. The scalar theory, J. Opt.Soc. Am. A 4, 651654 (1987).

[11] J. Durnin, J. J. Miceli, and J. H. Eberly, Diffraction-free beams, Phys. Rev. Lett.

58, 1499-1501 (1987)[12] J. Turunen and A. T. Friberg, Self-imaging and propagation-invariance in electro-

magnetic fields, J. Eur. Opt. Soc. Part I: Pure Appl. Opt. 2, 5160 (1993).

39


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40 REFERENCES

[13] K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia,

Orbital angular momentum of a high-order Bessel light beam, J. Opt. B: QuantumSemiclass. Opt. 4, S82S89 (2002).

[14] J. Turunen and A. T. Friberg, Propagation-invariant optical fields, in Progress inOptics, Vol. 54, E. Wolf, ed. (Elsevier, Amsterdam, 2009), 188.

[15] J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1961).

microoptics chapter 1

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