Angular Spectrum Representation of Optical Fields

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Chapter 2

Angular Sp ectrum

Representation of Optical Fields

The angular sp ectrum representation is a mathematical technique to describ e optical elds

in homogeneous media. Optical elds are describ ed as a sup erp osition of plane waves and

evanescentwaves which are physically intuitive solutions of Maxwell's equations. The angu-

lar sp ectrum representation is found to b e a very p owerful metho d for the description of laser

b eam propagation and light fo cusing. Furthermore, in the paraxial limit, the angular sp ec-

trum representation b ecomes identical with the framework of Fourier optics which extents

its imp ortance even further. In the subsequent sections we will apply the angular sp ectrum

representation to gain an understanding of the limits of far eld optics.

2.1 Basic Formalism

Under angular sp ectrum representation we understand the series expansion of an arbitrary

eld in terms of plane waves with variable amplitudes and propagation directions. Assume we

know the electric eld Eratany p oint r =x; y ; z in space. For example, Er can b e the

solution of an optical scattering problem as shown in Fig. 2.1 for which E = E + E .In

inc scatt

the angular sp ectrum picture, we draw an arbitrary axis z and consider the eld E in a plane

z = const transverse to the chosen axis. In this plane we can evaluate the two dimensional

Fourier transform of the eld E as

Z Z

i [k x + k y ]

x y

E k ;k ; z = Ex; y ; z e dx dy ; 2.1

x y

1 1

2 2. ANGULAR SPECTRUM REPRESENTATION OF OPTICAL FIELDS

where x; y are the Cartesian transverse co ordinates and k ;k the corresp onding spatial

x y

frequencies or recipro cal co ordinates. Similarly, the inverse Fourier transform reads as

Z Z

i [k x + k y ]

x y

Ek ;k ; z e dk dk : 2.2 Ex; y ; z =

x y x y

Notice, that in the notation of Eqs. 2.1 and 2.2 the eld E =E ;E ;E and its Fourier

x y z

^ ^ ^ ^

transform E =E ; E ; E representvectors. Thus, the Fourier integrals hold separately

x y z

for eachvector comp onent.

So far wehave made no requirements ab out the eld E, but we will assume that in

the transverse plane the medium is homogeneous, isotropic, linear and source-free. Then,

a time-harmonic, optical eld with angular frequency ! has to satisfy the vector Helmholtz

equation

2 2

r + k E r=0; 2.3

where k is determined by k =!=c n and n = " is the index of refraction. In order to

get the time-dep endent eld Er;twe use the convention

i! t

Er;t= RefE re g : 2.4

Inserting the Fourier representation of Er Eq. 2.2 into the Helmholtz equation and de n-

ing

2 2 2

k k k k w ith Imfk g 0 ; 2.5

z z

x y

we nd that the Fourier sp ectrum E evolves along the z-axis as

ik z

^ ^

Ek ;k ; z =E k ;k ;0e : 2.6

x y x y

Einc

Escatt

z = const.

Figure 2.1: In the angular sp ectrum representation the elds are evaluated in planes z =

const p erp endicular to an arbitrarily chosen axis z .

2.1. BASIC FORMALISM 3

The `' sign sp eci es that wehavetwo solutions that need to b e sup erimp osed: the `+' sign

refers to a wave propagating into the half-space z>0 whereas the `' sign denotes a wave

propagating into z<0. Eq. 2.6 tells us that the Fourier sp ectrum of E in an arbitrary image

plane lo cated at z = const can b e calculated bymultiplying the sp ectrum in the object plane

at z =0 by the factor expik z . This factor is called pr opag ator in recipro cal space. In

Eq. 2.5 we de ned that the square ro ot leading to k renders a result with p ositive imaginary

part. This ensures that the solutions remain nite for z !1. Inserting the result of Eq. 2.6

into Eq. 2.2 we nally nd for arbitrary z

Z Z

i [k x + k y k z ]

x y z

Ek ;k ;0e dk dk Ex; y ; z = 2.7

x y x y

which is known as the angular spectrum representation. In a similar way,we can also represent

the magnetic eld H by an angular sp ectrum as

Z Z

i [k x + k y k z ]

x y z

H k ;k ;0e dk dk ; 2.8 Hx; y ; z =

x y x y

By using Maxwell's equation H =i! rEwe nd the following relationship

^ ^

between the Fourier sp ectra E and H

^ ^ ^

H = Z [k =k E k =k E ] ; 2.9

x y z z y

^ ^ ^

[k =k E k =k E ] ; H = Z

z x x z y

^ ^ ^

H = Z [k =k E k =k E ] ;

z x y y x

=" " is the wave imp edance of the medium. Although the angular sp ec- where Z =

o o "

tra of E and H ful ll Helmholtz equation they are not yet rigorous solutions of Maxwell's

equations. We still have to require that the elds are divergence free, i.e. rE = 0 and

rH = 0. These conditions restrict the k vector to directions p erp endicular to the eld

vectors k E = k H = 0.

For the case of a purely dielectric medium with no losses the index of refraction n is a

real and p ositive quantity. The wavenumber k is then either real or imaginary and turns

the factor expik z into an oscillatory or exp onentially decaying function. For a certain

k ;k pair we then nd two di erentcharacteristic solutions

x y

i [k x + k y ] i jk j z 2 2 2

x y z

P lane waves : e e ; k + k k

x y

2.10

i [k x + k y ] jk jjz j 2 2 2

x y z

E v anescent w av es : e e ; k + k >k

x y

Hence, we nd that the angular sp ectrum is indeed a sup erp osition of plane waves and

evanescent waves. Plane waves are oscillating functions in z and are restricted by the condi-

2 2 2 2 2 2

tion k + k k . On the other hand, for k + k >k we encounter evanescentwaves with

x y x y

an exp onential decay along the z axis. Fig. 2.2 shows that the larger the angle b etween the

4 2. ANGULAR SPECTRUM REPRESENTATION OF OPTICAL FIELDS

k-vector and the z axis is, the larger the oscillations in the transverse plane will b e. A plane

2 2

wave propagating in direction of z has no oscillations in the transverse plane k + k = 0,

x y

whereas, in the other limit, a plane wave propagating at a right angle to z shows the highest

2 2 2

spatial oscillations in the transverse plane k + k = k . Even higher spatial frequencies

x y

are achieved byevanescentwaves. In principle, an in nite bandwidth of spatial frequencies

can b e achieved. However, the higher the spatial frequencies of an evanescentwave are, the

stronger the eld decay along the z axis will b e. Therefore, practical limitations make the

bandwidth nite.

So far we know from Eq. 2.6 how the Fourier sp ectrum of E propagates along the z

axis. Let us now determine how the elds themselves evolve. For this purp ose we denote

0 0

the transverse co ordinates in the ob ject plane at z =0asx ;y and in the image plane

at z = const as x; y . The elds in the image plane are describ ed by the angular sp ectrum

Eq. 2.7. We just have to express the Fourier sp ectrum Ek ;k ; 0 in terms of the elds in

x y

the ob ject plane. Similar to Eq. 2.1 this Fourier sp ectrum can b e represented as

Z Z

0 0

0 0 i [k x + k y ] 0 0

x y

Ex ;y ; 0 e dx dy : 2.11 E k ;k ;0 =

x y

After inserting into Eq. 2.6 we nd the following expression for the eld E in the image plane

z = const:

1 1

Z Z Z Z

0 0

i [k xx + k y y k z ] 0 0 0 0

x y z

Ex; y ; z = e dx dy dk dk : 2.12 Ex ;y ;0

x y

1 1

This equation describ es an invariant lter with the following impulse resp onse propagator

a) z b) z c) ky

k k x x plane waves 2 2 2 kx +ky = k

ϕ kz kx

x evanescent waves

Figure 2.2: a Representation of a plane wave propagating at an angle ' to the z axis. b

Illustration of the transverse spatial frequencies of plane waves incident from di erent angles.

2 2 1=2

The transverse wavenumber k + k dep ends on the angle of incidence and is limited to

x y

the interval [0 :: k ]. c The transverse wavenumb ers k , k of plane waves are restricted to a

x y

circular area with radius k .Evanescentwaves ll the space outside the circle.

2.2. PARAXIAL APPROXIMATION OF OPTICAL FIELDS 5

in direct space

Z Z

0 0

0 0 i [k xx +k y y k z ]

x y z

H x x ;y y ; z = e dk dk : 2.13

x y

The eld at z = const: is represented by the convolution of H with the eld at z = 0. The

2 2 2

lter H is an oscillating function for k + k

x y

2 2 2

for k + k >k .Thus, if the image plane is suciently separated from the ob ject plane,

x y

the contribution of the decaying parts evanescentwaves is zero and the integration can b e

2 2 2

reduced to the circular area k + k

x y

2 2 2

ltered representation of the original eld at z = 0. The spatial frequencies k + k >k of

x y

the original eld are ltered out during propagation and the information on the high spatial

variations gets lost. Hence, there is always a loss of information on the way of propagation

from near- to far- eld and only structures with lateral dimensions larger than

= 2.14 x

k 2 n

can b e imaged with sucient accuracy. In general, higher resolution can b e obtained bya

higher index of refraction of the emb o dying system substrate, lenses, etc. or by shorter

wavelengths. Theoretically, resolutions down to a few nanometers can b e achieved by us-

ing far-ultraviolet radiation or X-rays. However, X-rays do cause damage to many samples.

Furthermore, they are limited by the p o or quality of lenses and do not provide the wealth

of information of optical frequencies. The central idea of near- eld optics is to increase the

bandwidth of spatial frequencies by retaining a part of the evanescent comp onents of the

source elds.

2.2 Paraxial approximation of optical elds

In many optical problems the light elds propagate along a certain direction z and spread

out only slowly in the transverse direction. Examples are laser b eam propagation or optical

waveguide applications. In these examples the wavevectors k =k ;k ;k in the angular

x y z

sp ectrum representation are almost parallel to the z axis and the transverse wavenumb ers

k ;k are small compared to k .We can then expand the square ro ot of Eq. 2.5 in a series

x y

q 2 2

k + k

x y

2 2 2

k = k 1 k + k =k k ; 2.15

x y

2 k

This approximation considerably simpli es the analytical integration of the Fourier integrals.

In the following we shall apply the paraxial approximation to nd a description for weakly

fo cused laser b eams.

2.2.1 Weakly fo cused laser b eams

We consider a fundamental laser b eam with a linearly p olarized, Gaussian eld distribution

in the b eam waist

0 2 0 2

x +y

0 0

Ex ;y ; 0 = E e ; 2.16 o

6 2. ANGULAR SPECTRUM REPRESENTATION OF OPTICAL FIELDS

where E is a constant eld vector in the transverse x,y plane. Wehavechosen z =0 at

the b eam waist. The parameter w denotes the b eam waist radius. We can calculate the

Fourier sp ectrum at z =0 as

Z Z

0 2 0 2

x +y

0 0

i [k x + k y ] 0 0

w x y

Ek ;k ;0 = e dx dy E e

x y o

2 2

k +k

x y

e ; 2.17 = E

which is again a Gaussian function. Wenow insert this sp ectrum into the angular sp ectrum

representation Eq. 2.7 and replace k by its paraxial expression in Eq. 2.15

Z Z

2 2

w iz

o + i [k x + k y ] k ikz +k

x y

4 2 k

Ex; y ; z =E e dk dk ; 2.18 e e

o x y

This equation can b e integrated and gives as a result the familiar paraxial representation of

a Gaussian b eam

2 2

ikz x +y

E e

2 2

w 1+2 iz=kw

o o

: 2.19 Ex; y ; z = e

1 + 2 i z=kw

2 2 2

To get a b etter feeling for a paraxial Gaussian b eam we set = x +y , de ne a new parameter

z as

z = ; 2.20

1 2 2

expax + ibx dx = =a expb =4a and

2 2 3=2

x expax + ibx dx = ib expb =4a=2a

|E|

1/e ρ z w(z)

θ Å2k/wo

2 zo

Figure 2.3: Illustration and main characteristics of a paraxial Gaussian b eam. The b eam has

a Gaussian eld distribution in the transverse plane. The surfaces of constant eld strength

form a hyp erb oloid along the z -axis.

2.3. POLARIZED ELECTRIC AND POLARIZED MAGNETIC FIELDS 7

and rewrite Eq. 2.19 as

i [kz z +k =2Rz ]

w z

e e E; z =E 2.21

w z

with the following abbreviations

2 2 1=2

w z = w 1 + z =z beam w aist 2.22

2 2

Rz = z 1 + z =z w av ef r ont r adius

z = arctan z=z phase cor r ection

2 2

x + y for which The transverse size of the b eam is usually de ned by the value of =

the electric eld amplitude is decayed to a value of 1=e of its center value

jEx; y ; z j = jE0; 0;zj =1=e : 2.23

It can b e shown, that the surface de ned by this equation is a hyp erb oloid whose asymptotes

enclose an angle

= 2.24

with the z axis. From this equation we can directly nd the corresp ondence b etween the

numerical ap erture NA = n sin and the b eam angle as NA 2n=k w . Here we used

the fact that in the paraxial approximation, is restricted to small b eam angles. Another

prop erty of the paraxial Gaussian b eam is that close to the fo cus, the b eam stays roughly

collimated over a distance 2z . z is called the Rayleigh range and denotes the distance from

o o

2. It is imp ortant to notice the b eam waist to where the sp ot has increased by a factor of

that along the z axis = 0 the phases of the b eam deviate from those of a plane wave. If

at z !1 the b eam was in phase with a reference plane wave, then at z ! +1 the b eam

will b e exactly out of phase with the reference wave. This phase shift is called Gouy phase

shift and has practical implications in nonlinear confo cal microscopy [1]. The 180 phase

change happ ens gradually as the b eam propagates through its fo cus. The phase variation is

describ ed by the factor z in Eq. 2.22. The tighter the fo cus the faster the phase variation

will b e.

A qualitative picture of a paraxial Gaussian b eam and some of its characteristics are

shown in Fig. 2.3 and more detailed descriptions can b e found in other textb o oks Ref. [2, 3].

It is imp ortant to notice that once the paraxial approximation is intro duced, the eld E

do es not ful ll Maxwell's equations anymore. The error b ecomes larger the smaller the b eam

waist radius w is. When w b ecomes comparable to the reduced wavelength =n wehave

o o

to include higher order terms in the expansion of k in Eq. 2.15. However, the series expan-

sion converges very badly for strongly fo cused b eams and one needs to nd a more accurate

description. We shall return to this topic at a later stage.

2.3 Polarized electric and p olarized magnetic elds

Any propagating optical eld can b e split into a p olarized electric PE and a p olarized

magnetic PM eld

PE PM

E = E + E : 2.25

8 2. ANGULAR SPECTRUM REPRESENTATION OF OPTICAL FIELDS

For a PE eld, the electric eld is linearly p olarized in the transverse plane. Similarly, for a

PM eld the magnetic eld is linearly p olarized in the transverse plane. Let us rst consider

a PE eld for whichwe can cho ose E =E ; 0;E . Requiring that the eld is divergence

x z

free rE =0we nd that

^ ^

E k ;k ;0= E k ;k ;0; 2.26

z x y x x y

PE PE

which allows us to express the elds E ; H in the form

Z Z

1 1

i [k x + k y k z ] PE

x y z

[k n k n ]e dk dk ; 2.27 E k ;k ;0 E x; y ; z =

z x x z x y x x y

2 k

Z Z

1 1

PE 2 2

H x; y ; z = [k k n +k + k n 2.28 E k ;k ;0

x y x y x x y

x z

2 Z kk

" z

i [k x + k y k z ]

x y z

k k n ]e dk dk ;

y z z x y

where n , n , n are unit vectors along the x, y , z axes. To derive H we used the relations

x y z

in Eq. 2.9.

To derive the corresp onding PM elds we require that H =0;H ;H . After fol-

y z

lowing the same pro cedure as b efore one nds that in the PM solution the expressions for

the electric and magnetic elds are simply interchanged

Z Z

Z 1

PM 2 2

E x; y ; z = H k ;k ;0 [k + k n k k n + 2.29

y x y x x y y

y z

2 kk

i [k x + k y k z ]

x y z

k k n ]e dk dk ;

x z z x y

Z Z

1 1

i [k x + k y k z ] PM

x y z

[k n k n ]e dk dk : 2.30 H k ;k ;0 H x; y ; z =

z y y z x y y x y

2 k

It is straight forward to demonstrate that in the paraxial limit the PE and PM solutions

are identical. In this case they b ecome identical with a TEM solution.

The decomp osition of an arbitrary optical eld into a PE and a PM eld has b een

achieved by setting one transverse eld comp onent to zero. The pro cedure is similar to the

commonly encountered decomp osition into transverse electric TE and transverse magnetic

TM elds for which one longitudinal eld comp onent is set to zero see Problem 2.4.

2.3.1 Rigorous description of Gaussian b eams

In the paraxial approximation the Gaussian b eam is a TEM wave, i.e. simultaneously a PE

and a PM wave. Therefore, the magnetic eld has the same form as the electric eld and

its eld vector is p erp endicular to the propagation direction z and the eld E.Ifwe assume

that a Gaussian b eam is characterized byhaving a Gaussian eld distribution at the b eam

waist we can apply the angular sp ectrum framework to generalize the results in Section 2.2.1.

However, dep ending whether we assume a Gaussian distribution for the electric or magnetic

2.3. POLARIZED ELECTRIC AND POLARIZED MAGNETIC FIELDS 9

eld we arriveatPE or PM solutions. Let us rst discuss the PE b eam for whichwe require

a Gaussian eld distribution at the b eam waist c.f. Eq. 2.16. Wecho ose the x axis of our

co ordinate system along the constant eld vector E , i.e. E = E n . The corresp onding

o o o x

Fourier sp ectrum has b een determined in Eq. 2.17 as

2 2

k o +k

x y

E k ;k ;0 = E e ; 2.31

x x y o

PE PE

This sp ectrum can now b e inserted into Eqs. 2.27 and 2.28 to nd the elds E , H for

the p olarized electric Gaussian b eam.

To derive the PM solution we require a Gaussian eld distribution for the magnetic eld

at the b eam waist which yields the following sp ectrum

2 2

+k k

y x

H k ;k ;0 = H ; 2.32 e

y x y o

Here, wechose H to p ointin y direction, i.e. H = H n . Similar to the case b efore, we

o o o y

PM PM

insert the sp ectrum into Eqs. 2.29 and 2.30 to nd the elds E , H for the p olarized

magnetic Gaussian b eam.

Although we found the generalized elds for a Gaussian b eam, we are left with expres-

sions with two in nite integrals that cannot b e solved analytically. It is straight forward see

Section 2.7 to transform the integrals to cylindrical co ordinates where it is p ossible to ana-

lytically integrate the angular dep endence. This waywe are left with a single, semi-in nite

integral which has to b e solved numerically.Weavoid this pro cedure at this p oint since

a rigorous description of a Gaussian b eam is of limited practical imp ortance. A Gaussian

b eam is generated in a confo cal resonator with spherical end mirrors. Most resonators use

slowly converging b eams for which the paraxial approximation is sucient. Furthermore,

the eld distribution near the fo cus of a strongly fo cused laser b eam is determined by the

b oundary conditions of the fo cusing element such as a high f numb er lens. Therefore, the

generalized Gaussian b eams presented in this section are not adequate to describ e the fo cal

elds outside the laser cavity. But even for the case of a strongly converging confo cal cavity

we do not knowhow to sup erimp ose the two indep endent PE and PM solutions. Barton and

Alexander [4] use a an equal sup erp osition but in principle any combination is a legitimate

solution. In Section 2.6 we will present a formalism to accurately describ e the fo cal elds of

a strongly fo cused laser b eam.

2.3.2 Higher order laser b eams

The elds of the fundamental Gaussian mo de can b e used to derive higher order b eam mo des.

The most commonly encountered higher b eam mo des are Hermite-Gaussian and Laguerre-

Gaussian b eams. The former are generated in cavities with rectangular end mirrors whereas

the latter are observed in cavities having circular end mirrors. In the transverse plane, the

elds of these mo des extend over larger distances and have sign variations in the phase. Since

the fundamental Gaussian mo de is a solution of a linear homogeneous partial di erential

equation, namely the Helmholtz equation, any combinations of spatial derivatives of the

fundamental mo de are also solutions to the same di erential equation. Zauderer [5] p ointed

10 2. ANGULAR SPECTRUM REPRESENTATION OF OPTICAL FIELDS

out, that Hermite-Gaussian mo des E can b e generated from the fundamental mo de E

according to

n m

@ @

H n+m

E x; y ; z =w Ex; y ; z ; 2.33

nm o

n m

@x @y

where n; m denote the order and degree of the b eam. Laguerre-Gaussian mo des E are

n;m

derived in a similar wayas

@ @ @

ik z L n 2n+m ik z

+ i Ex; y ; z e E x; y ; z = k w e : 2.34

nm o

@z @x @y

Thus, once an accurate solution for the fundamental Gaussian mo de is obtained, any higher

order mo de can b e generated by simply applying Eqs. 2.33 and 2.34. It can b e shown, that

Laguerre-Gaussian mo des can b e generated as a sup erp osition of a nite numb er of Hermite-

Gaussian mo des and vice versa. The two sets of mo des are therefore not indep endent. Note

that the parameter w only represents the b eam waist for the Gaussian b eam and that for

higher order mo des the amplitude E do es not corresp ond to the eld at the fo cal p oint.

Fig. 2.4 shows the elds in the fo cal plane z = 0 for the rst four Hermite-Gaussian mo des.

As indicated by the arrows, the p olarizations of the individual maxima are either in phase

a) b)

y 500 nm

c) d)

Figure 2.4: IntensityjEj in the fo cal plane z = 0 of the rst four Hermite-Gaussian

mo des. a 00 mo de fundamental Gaussian mo de, b 10 mo de, c 01 mo de, and d

11 mo de. The wavelength and b eam angle are = 800nm and =28:65 , resp ectively. The

arrows indicate the p olarization direction of the individual lob es. A linear scaling is used

between contourlines.

2.3. POLARIZED ELECTRIC AND POLARIZED MAGNETIC FIELDS 11

or 180 out of phase with each other. The elds in Fig. 2.4 have b een calculated b eyond

the paraxial approximation by an equal sup erp osition of PE and PM solutions. As a conse-

quence, the resulting elds of these b eams p ossess longitudinal eld comp onents E , H as

z z

shown in Fig. 2.5 and Fig. 2.6. While the longitudinal electric eld of the fundamental Gaus-

sian b eam is always zero on the optical axis it shows two lob es to the sides of the optical axis.

Displayed on a cross-section through the b eam waist, the two lob es are aligned along the

p olarization direction. The longitudinal electric eld of the Hermite-Gaussian 10 mo de, on

the other hand, has it's maximum at the b eam fo cus with a much larger eld strength. This

longitudinal eld qualitatively follows from the 180 phase di erence and the p olarization

of the two corresp onding eld maxima in Fig. 2.4, since the sup erp osition of two similarly

p olarized plane waves propagating at angles ' to the z axis with 180 phase di erence also

leads to a longitudinal eld comp onent. It has b een prop osed to use the longitudinal elds

of the Hermite-Gaussian 10 mo de to accelerate charged particles along the b eam axis in

linear particle accelerators [6]. The longitudinal 10 eld has also b een applied to image the

spatial orientation of molecular transition dip oles [7]. In general, the 1; 0 mo de is imp ortant

for all exp eriments which require the availability of a longitudinal eld comp onent. We shall

see in Section 2.7 that the longitudinal eld strength of a strongly fo cused higher order laser

b eam can even exceed the transverse eld strength.

x 10

a)b) c)

x 1µm

Figure 2.5: Fields of the fundamental Gaussian b eam depicted in the p olarization plane

x; z . The wavelength and b eam angle are = 800 nm and =28:65 , resp ectively. a Time

dep endentpower density; b Total electric eld intensityjEj ; c Longitudinal electric eld

intensityjE j . A linear scaling is used b etween contourlines. z

12 2. ANGULAR SPECTRUM REPRESENTATION OF OPTICAL FIELDS

The commonly encountered doughnut mo des with a circular intensity pro le can b e de-

scrib ed by a sup erp osition of Hermite-Gaussian or Laguerre-Gaussian mo des. Linearly p o-

L L

larized doughnuts are simply de ned by the elds E or E . An azimuthally p olarized

01 11

doughnut mo de is a sup erp osition of two p erp endicularly p olarized E elds and a radially

p olarized doughnut mo de is a sup erp osition of two p erp endicularly p olarized E elds.

2.4 Angular sp ectrum representation of a dip ole

Strongly lo calized sources such as dip oles are most conveniently describ ed in a spherical

co ordinate system. The corresp onding solutions of the wave equation are called multip oles.

In order to couple these solutions with the angular sp ectrum picture we need to express the

lo calized sources in terms of plane waves and evanescentwaves. Let us start with the vector

p otential A of an oscillating dip ole with its axis aligned along an arbitrary z -axis. The vector

p otential can b e expressed as a one-comp onentvector eld as see section ??

2 2 2

ik x +y +z

ik Z e

n ; 2.35 Ax; y ; z =Ax; y ; z n =

z z

2 2 2

x + y + z

where Z is the wave imp edance of the medium. Besides a constant factor, the expression

on the right hand side corresp onds to the scalar Green's function c.f. Eq. ??. According

x 3

a)b) c)

x 1µm

Figure 2.6: Fields of the Hermite-Gaussian 10 mo de. Same scaling and de nitions as in Fig. 2.5.

2.5. FARFIELDS IN THE ANGULAR SPECTRUM REPRESENTATION 13

to Eq. ?? and Eq. ?? the electric and magnetic elds are obtained from A as

Ex; y ; z = i! 1 + rr Ax; y ; z 2.36

r Ax; y ; z : 2.37 Hx; y ; z =

Thus, the electromagnetic eld of the dip ole can b e constructed from the function expikr =r ,

2 2 2 1=2

where r =x + y + z is the radial distance from the dip ole's origin. To nd an angular

sp ectrum representation of the dip ole's electric and magnetic eld we need rst to nd the

angular sp ectrum of the function exp ikr =r . This is not a trivial task and cannot b e derived

with the framework wehave established in Section 2.1. The reason is that the function

expikr =r is singular at r = 0 and therefore not divergence free at its origin. The homogeneous

Helmholtz equation 2.3 which formed the basis for our angular sp ectrum representation is

not valid in the present case. Nevertheless, using a more careful analysis combined with

complex contour integration it is p ossible to derive an angular sp ectrum representation of

the function exp ik r =r . Since the derivation can b e found in other textb o oks [3,9]we state

here only the result which reads as

Z Z

2 2 2

ik x +y +z ik x+ik y +ik jz j

x y z

e e i

dk dk ; 2.38 =

x y

2 2 2

2 k

x + y + z

where we require that the real and imaginary parts of k stay p ositive for all values of k ;k

z x y

in the integration. This equation is known as the Weyl identity [10].

2.5 Far elds in the Angular Sp ectrum Representation

Consider an optical eld in the ob ject plane z = 0 with the angular sp ectrum representation

Z Z

i [k x + k y k z ]

x y z

E k ;k ;0e dk dk : 2.39 Ex; y ; z =

x y x y

We are interested in the asymptotic far-zone approximation of this eld, i.e. in the evaluation

of the eld in a p oint r = r with in nite distance from the ob ject plane. The dimensionless

unit vector s in direction of r is given by

x y z

s =s ;s ;s = ; ; ; 2.40

x y z

r r r

2 2 2 1=2

where r =x + y + z is the distance of r from the origin. To calculate the far eld

E we require that r !1 and rewrite Eq. 2.39 as

y k k

z x

s + s s ] ikr [

x y z

k k k

dk dk : 2.41 E s ;s ;s = lim Ek ;k ;0e

x y 1 x y z x y

kr!1

2 2 2

k +k k

x y

Because of their exp onential decay,evanescentwaves do not contribute to the elds at in nity.

2 2 2

We therefore rejected their contribution and reduced the integration range to k + k

x y

14 2. ANGULAR SPECTRUM REPRESENTATION OF OPTICAL FIELDS

The asymptotic b ehavior of the double integral as kr !1 can b e evaluated by the metho d

of stationary phase. For a clear outline of this metho d we refer the interested reader to

chapter 3.3 of Ref. [3]. Without going into details, the result of Eq. 2.41 can b e expressed as

ikr

E s ;s ;s =ik s Eks ;ks ;0 : 2.42

1 x y z z x y

This equation tells us that the far elds are entirely de ned by the Fourier sp ectrum of the

elds E k ;k ; 0 in the ob ject plane if we replace k ! ks and k ! ks . This simply

x y x x y y

means that the unit vector s ful lls

k k k

y z x

; ; ; 2.43 s =s ;s ;s =

x y z

k k k

which implies that only one plane wave with the wavevector k =k ;k ;k of the angular

x y z

sp ectrum at z = 0 contributes to the far eld at a p oint lo cated in the direction of the

unit vector s. The e ect of all other plane waves is canceled by destructiveinterference.

Combining Eqs. 2.42 and 2.43 we can express the Fourier sp ectrum E in terms of the far eld

ikr

ir e

E k ;k ;0= 2.44 E k ;k

x y 1 x y

This expression can b e substituted into the angular sp ectrum representation Eq. 2.39 as

ikr

ir e 1

i [k x + k y k z ]

x y z

Ex; y ; z = 2.45 dk dk E k ;k e

x y 1 x y

2 k

2 2 2

k +k k

x y

Thus, as long as evanescent elds are not part of our system then the eld E and its far eld

E form essentially a Fourier transform pair. The only deviation is given by the factor 1=k .

1 z

In the approximation k k , the two elds form a p erfect Fourier transform pair. This is

the limit of Fourier optics.

As an example consider the di raction at a rectangular ap erture with sides 2 L and 2 L

x y

in an in nitely thin conducting screen whichwecho ose to b e our ob ject plane z = 0. A

plane wave illuminates the ap erture at normal incidence from the back. For simplicitywe

assume that the eld in the ob ject plane has a constant eld amplitude E whereas the screen

blo cks all the eld outside of the ap erture. The Fourier sp ectrum at z = 0 is then

Z Z

+L +L

y x

0 0

i [k x + k y ] 0 0

x y

Ek ;k ;0 = e dx dy

x y

L L

y x

2L L sink L sink L

x y x x y y

= E ; 2.46

k L k L

x x y y

With Eq. 2.42 wenow determine the far eld as

ikr

sinks L e sinks L 2L L

y y x x x y

; 2.47 E s ;s ;s =ik s E

1 x y z z o

ks L ks L r

x x y y

2.6. FOCAL FIELDS OF TIGHTLYFOCUSED LASER BEAMS 15

which, in the paraxial limit k k , agrees with Fraunhofer di raction.

Eq. 2.42 is an imp ortant result. It links the near- elds of an optical problem with the

corresp onding far elds. While in the near- eld a rigorous description of elds is necessary,

the far elds are well approximated by the laws of geometrical optics.

2.6 Fo cal elds of tightly fo cused laser b eams

The limit of classical light con nementisachieved with highly fo cused laser b eams. Such

b eams are used in uorescence sp ectroscopytoinvestigate molecular interactions in solutions

[11] and the kinetics of single molecules on interfaces [12, 7]. Highly fo cused laser b eams also

playa key role in confo cal microscopy, where resolutions on the order of =4 are achieved.

In optical tweezers, fo cused laser b eams are used to trap particles and to move and p osition

them with high precision [13]. All these elds require a theoretical understanding of laser

b eams.

The elds of a fo cused laser b eam are determined by the b oundary conditions of the

fo cusing optical element and the incident optical eld. In this section we will study the

fo cusing of a paraxial optical eld by an aplanatic optical lens as shown in Fig. 2.7. In our

theoretical treatmentwe will follow the theory established by Richards and Wolf [14,15].

The elds near the optical lens can b e formulated by the rules of geometrical optics. In this

approximation the niteness of the optical wavelength is neglected k !1 and the energy

is transp orted along lightrays. The average energy density is propagated with the velo city

v = c=n in direction p erp endicular to the geometrical wavefronts. To describ e an aplanatic

lens we need two rules of geometrical optics: 1. the sine condition and 2. the intensity

law. These rules are illustrated in Fig. 2.8. The sine condition states that each optical ray

which emerges from or converges to the fo cus F of an aplanatic optical system intersects

its conjugate ray on a sphere of radius f Gaussian reference sphere, where f is the fo cal

length of the lens. Under conjugate ray one understands the refracted or incidentray which

n1 n2

Einc

Figure 2.7: Fo cusing of a laser b eam by an aplanatic lens.