Holographic optical tweezers

Dept. of MechatronicsYong-Gu Lee

1. Joseph W. Goodman, Introduction to Fourier optics 2nd Edition, McGraw-Hill 19962. Eric R. Dufresne et al, Computer-generated holographic optical tweezer arrays, Review of scientific instruments Vol 72 No 3, pp 1810-

1816

The Helmholtz equation

Let the light disturbance at position and time be represented by the scalar function ( , ).

For a monochromatic wave, the scalar field may be written explicitly as

( , ) cos 2

where and

t u t

u t A t

A

P P

P P P

P P

2

are the amplitude and phase respectively, of the wave at position ,

while is the optical frequency. A more compact form of above is found by using complex

notation, writing

( , ) Re . j tu t U e

P

P P

2 22

2 2

(3-10)

If the real disturbance ( , ) is to represent an optical wave, it must satisfy the scalar wave

equation

u t

n uu

c t

P

0 (3-12)

at each source-free point. If (3-10) is substituted in (3-12), it follows must obey the time

in

U

2 2

dependent equation

0. (3-13)

Here is termed the wave number and is given by

22

The rel

k U

k

k nc

ation (3-13) is known as the Helmholtz equation.

Green’s theorem

Calculation of the complex disturbance U at any observation point in space can be accomplished

with the help of the mathematical relation known as Green's theorem.

Let and G be any two complex-valueU P P

2

d functions of position, and let be a closed surface

surrounding a volume . If , , and their first and second partial derivatives are single-valued

and continuous within and on , then we have

S

V U G

S

U G 2

v

where signifies a partial derivative in the outward normal direction at each point on .

This theorem is in many respects the prime foundation of scalar diffraction

S

G UG U dv U G ds

n n

Sn

theory.

The integral theorem of Helmholtz and Kirchhoff

The Kirchhoff formulation of the diffraction problem is based on a certain integral theorem

which expresses the solution of the homogeneous wave equation at an arbitrary point in terms

of the values of the solution and its first derivative on an arbitrary closed surface surrounding

that point. This theorem had been derived previously in acoustics by H. von Helmholtz.

Let the point of observation be denoted , and let denote an arbitrary closed surface

surrounding . The problem is to express the optical disturbance at in terms of its values

on the surface . To solve this problem, we follow K

S

S

0

0 0

P

P P

irchhoff in applying Green's theorem

and in choosing as an auxiliary function a unit-amplitude spherical wave expanding about the

point . Thus the value of Kirchhoff's G at an arbitrary point is gi0 1P P

01

01

01

ven by

G ,

where we adopt the notation that is the length of the vector pointing from to .

jkre

r

r

1

01 0 1

P

r P P

To be legitimately used in Green's theorem, the function (as well as its first and second

partial derivatives) must be continuous within the enclosed volume . Therefore to exclude

the discontinuity a

G

V

t a small spherical surface , of radius , is inserted about the point

. Green's theorem is then applied, the volume of integration being that volume lying

between and , and the surface o

S

V

S S

0

0

P

P

f integration being the composite surface

.

Note that the "outward" normal to the composite surface points outward in the conventional

sense on , but inward (towards ) on .

Within the volume

S S S

S S

0P

2 2

, the disturbance , being simply an expanding spherical wave,

satisfies the Helmholtz equation

0.

V G

k G

2 2 2 2

v v

(3-18)

Substituting the two Helmholtz equations (3-13) and (3-18) in the left-hand side of Green's

theorem, we find

0.U G G U dv UGk GUk dv

1

Thus the theorem reduces to

0

or

. (3-19)

Note that, for a general point on , we ha

S

S S

G UU G ds

n n

G U G UU G ds U G ds

n n n n

S

P

01

01

01

01 01

ve

G

and

G 1cos , (3-20)

where cos , represents the cosine of the angle between the outward

jkr

jkr

e

r

en jk

n r r

n

1

101

01

P

Pr

r

normal and the

vector joining to . For the particular case of on , cos , and these

equations become.

G 1G and .

jk jk

n

S n

e ejk

n

01 0 1 1 01

11

r P P P r

PP

2

0 0

Letting become arbitrarily small, the continuity of (and its derviatives) at allows

us to write

1lim lim 4 4 .

Substitution of

jk jk

S

U

UG U e eU G ds U jk U

n n n

0

00 0

P

PP P

01 01

01 01

this result in (3-19) (taking account of the negative sign) yields

1.

4

This result is known as the integral theorem of Helmholtz and Kirchhoff; it pla

jkr jkr

S

U e eU U ds

n r n r

0P

ys an important

role in the development of the scalar theory of diffraction, for it allows the field at any point

to be expressed in terms of the "boundary values" of the wave on any closed surface s0P urrounding

that point.

Application of the integral theorem

Consider now the problem of diffraction of light by an aperture in an infinite opaque screen.

A wave distrubance is assumed to impinge on the screen and the aperture from the left, and

the field at the point behind the aperture is to be calculated. The field is assumed to be

monochromatic.

To find the field at the point , we apply the integral theorem of Helmholtz and Kirchhoff,

being careful to c

0

0

P

P

hoose a surface of integration that will allow the calculation to be performed

successfully. Following Kirchoff, the closed surface is chosen to cosnsist of two parts, as shown

in the figure. Let a pl

S

1

2

ane surface, , lying directly behind the diffracting screen, be joined

and closed by a large spherical cap, , of radius and centered at the observation point .

The total closed surface is sim

S

S R

S0P

1 2

01

1 2

01

ply the sum of and . Thus applying (3-21).

1

4

where, as before,

.

S S

jkr

S S

U GU G U ds

n n

eG

r

0P

2As increases, approaches a large hemispherical shell. It is tempting to reason that,

since both and will fall off as 1/ , the integrand will ultimately vanish, yielding

a contribution of zero f

R S

U G R

2

2

rom the surface integral over . However the area of integration

increases as , so this argument is incomplete. It is also tempting to assume that, since

the disturbances are propagating with finite

S

R

2

velocity / , will ultimately be so large

that the waves have not yet reached , and the integrand will be zero on that surface.

But this argument is incompatible with our assumption of monochromatic

c n R

S

2

disturbances,

which must (by definition) have existed for all time. Evidently a more careful investigation

is required before the contribution from can be disposed of.

Examining this problem in more

S

2detail, we see that, on ,

and, from (3-20),

G 1 jkG

where the last approximation is valid for large .

jkR

jkR

S

eG

R

ejk

n R R

R

?

2

2

2

2

The integral in question can thus be reduced to

where is the solid angle subtended by at . Now the quantity is uniformly

bounded on . Therefore t

S

U UG UjkG ds G jkU R d

n n

S RG

S

0P

2he entire integral over will vanish as R becomes arbitrarily

large, provided the disturbance has the property

lim 0

uniformly in angle. This requirement is known as the Sommerfeld ra

R

S

UR jkU

n

2

diation condition and

is satisfied if the disturbance vanishes at least as fast as a diverging spherical wave. It

guarantees that we are dealing only with outgoing waves on , rather than incoming w

U

S

2

2 2

aves,

for which the integral over might not vanish as . Since only outgoing waves will

fall on in our problem, the integral over will yield a contribution of precisely zero.

S R

S S

2

1

Having disposed of the integration over the surface , it is now possible to express the

disturbance at in terms of the disturbance and its normal derivative over the infinite

plane immediately b

S

S0P

1

ehind the screen, that is,

1 (3-23)

4

The screen is opaque, except for the open aperture which will be

S

U GU G U ds

n n 0P

1

denoted as . It therefore

seems intuitively reasonable that the major contribution to the integral (3-23) arises from

the points of located with the aperture , where we would expect the integrand S

to be

largest. Kirchhoff accordingly adopted the following assumptions.

1. Across the surface , the field distribution and its derivative are exactly the same

as they would be in the absence of t

UU

n

1

he screen.

2. Over the portion of that lies in the geometrical shadow of the screen, the field distribution

and its derivative are identically zero.

These conditions are commonly known as the K

S

UU

n

irchhoff boundary conditions. The first allows

us to specify the disturbance incident on the aperture by neglecting the presence of the screen.

The second allows us to neglect all of the surface of integration except that portion lying

directly within the aperture itself. Thus (3-23) reduced to

1

4

While the Kirchhoff boundary conditions simplify the results considerably, it is important to

realize that neither can be exactly true. The presence of the screen will i

U GU G U ds

n n

0P

nevitably perturb the

fields on to some degree, for along the rim of the aperture certain boundary conditions

must be met that would not be required in the absence of the screen. In addition, the sh

adow

behind the screen is never perfect, for fields will inevitably extend behind the screen for a

distance of several wavelengths. However, if the dimensions of the aperture are large

compared with a wavelength, these fringing effects can be safely negelected, and the two

boundary conditions can be used to yield results that agree very well with experiement.

A further simplification of the expressi

01

01

01

on is obtained by noting that the distance

from the aperture to the observation point is usually many optical wavelengths, and therefore,

since k 1/r ,Eq. (3-20) becomes

G 1cos ,

U r

n jkn r

0

101

P

Pr

01 01

01 01

jk cos , (3-25)jkr jkre e

nr r

01r

The Rayleigh-Sommerfield formulation of diffraction

The Kirchhoff theory has been found experimentally to yield remarkably accurate results

and is widely used in practice. However, there are certain internal inconsistencies in the

theory which motivated a search for a more satisfactory methematical development.

The difficulties of the Kirchhoff theory stem from the fact that boundary conditions must

be imposed on both the field strength and its normal derivative. In particular, it is a well

-known theorem of potential theory that if a two-dimensional potential function and its

normal derivative vanish together along any finite curve segment, then that potential

function must vanish over the entire plane. Similarly, if a solution of the three-dimensional

wave equation vanishes on any finite surface element, it must vanish in all space. Thus

the two Kirchhoff boundary conditions together imply that the field is zero everywhere

behind the aperture, a result which contradicts the know physical situation.

The inconsistencies of the Kirchhoff theory were removed by Sommerfeld, who eliminated

the necessity of imposing boundary values on both the disurbance and its normal derivatives

simultaneously.

1

1

4

The conditions for validity of this equation are:

1. The scalar theory holds.

2. Both U and G satisfy the homogeneous scalar wave equation

3. The Sommerfeld radiation c

S

U GU G U ds

n n 0P

ondition is satisfied

Suppose that the Green's function G were modified in such a way that, while the development

leading to the above equation remains valid, in addition either or vanishes over G

Gn

1

the

entire surface .

Sommerfeld pointed out that Green's functions with the required properties do indeed exist.

Suppose G is generated not only by a point source located at , but also simultaneousl

S

0P y by

a second point source at a position which is the mirror image of on the opposite side

of the screen. Let the source at be of the same wavelength as the source at , and

and suppose tha

0 0

0 0

P P

P P

01 01 01 01

01 01 01 01

t the two sources are oscillating with a phase difference. The Green's

function in this case is given by

G .jkr jkr jkr jkre e e e

r r r r

1P

Clearly such a function vanishes on the plane aperture , leaving the following expression

for the observed field:

1.

4

We refer to this solution as the first Rayleigh-Sommerfeld solutio

I

GU U ds

n

0P

01 01

01

01 01 01 01

n.

To specify this solution further let be the distance from to . The corresponding normal

derivative of is

1 1cos , cos ,

Now for

jkr jkr

r

G

G e en jk n jk

n r r r r

0 1

1 01 01

P P

P r r

01

01

01 01

01 01

01

01

on , we have

cos , cos ,

and therefore on that surface

12cos ,

For , the second term above can be dropped, leaving

2 cos ,

jkr

jkr

r r

n n

G en jk

n r r

r

G ejk n

n r

1 1

01 01

1 01

1 01

P S

r r

P r

P r

, (3-35)

which is just twice the normal dervative of the Green's function G used in the Kirchhoff analysis, i.e.

2G

n

1P G

n

1P

With this result, the first Raylegih-Sommerfeld solution can be expressed in terms of the more

simple Green's function used by Kirchoof

1.

2

An alternative and equally valid Green's funct

I

GU U ds

n

0P

01 01 01 01

01 01 01 01

ion is found by allowing the two point source to

oscillate in phase, giving

.

It can be shown that the normal derivative of this function vanishes across t

jkr jkr jkr jkre e e eG

r r r r 1P

01

he screen and aperture,

leading to the second Rayleigh-Sommerfeld solution,

1.

4

It can be shown that, on and under the condition that , is twice the Kirchoff Green's

function .

II

UU G ds

n

r G

G

0P

2 .

This leads to an expression for in terms of the Green's function used by Kirchoff,

1.

2

Let the Green's function be substituted for G in Eq. (3-23). Using (3-25), it follo

II

II

G G

U

UU Gds

n

G

0

0

P

P

ws directly

that

01

1

01

1

01

1

01

01

01

01

12 cos ,

4

2cos ,

2

1cos ,

where it has bee assumed that r . The Kirchhoff boundary conditions may now be a

jkr

I S

jkr

S

jkr

S

eU U jk n ds

r

jk eU n ds

k r

eU n ds

j r

0 01

01

01

P r

r

r

01

01

pplied

to , yielding the general result

1cos , (3-41)

Since no boundary conditions need to be a

jkr

I

U alone

eU U n ds

j r

0 1 01P P r

01

101

pplied to , the inconsistencies of the Kirchoff theory

has been removed.

If the alternative Green's function of (3-37) is used, the result can be shown to be

1.

2

jkr

II S

U

n

U eU ds

n r

10

PP (3-42)

We now specialize Eq. (3-41) and Eq. (3-42) to the case of illumination with a diverging spherical wave,

The illuminati

2

on of the aperture in all cases is a spherical wave diverging from a point source at

position .P

21

01 21

121

01 21

21

Using G we obtain

cos ,

This result is known as the Rayleigh-Sommerfeld diffraction fomula. Using . and assuming

, the corresponding resu

jkr

jk r r

I

eU A

r

A eU n ds

j r r

G

r

0 01

P

P r

01 21

201 21

2

lt is

cos ,

where the angle between and is greater than .2

jk r r

I

A eU n ds

j r r

n

0 1

1

P r

r

The Huygens-Fresnel principle in rectangular coordinates

As shown in Fig. 4.10, the diffracting aperture is asumed to lie in the , plane and at normal

distance z from it. The z axis pierces both planes at their origin.

According to Eq. (3-41), the Huygens-F

01

01

resnel principle can be stated as

1cos (4-8)

ˆwhere is the angle between the outward normal n and

jkreU U ds

j r

0 1P P

01

01

01

201

the vector pointing from to .

The term cos is given exactly by

cos ,

and therefore the Huygens-Fresnel principle can be written

, , d (4jkr

r

z

r

z eU x y U d

j r

0 1P P

01

2 2201

-9)

where the distance is given exactly by

. (4-10)

There have been only two approximations in reaching this expression. One is the approximati

r

r z x y

01

on

inherent in the scalar theory. The second is the assumption that the observation distance is many

wavelengths from the aperture, . We now embark on a series of additional approximations.r

The Fresnel approximation

01 1 0

To reduce the Huygens-Fresnel principle to a more simple and usable expression, we introduce

approximations for the distance between and . The approximations are based on the

binomial expansio

r P P

2

n of the square root in Eq. (4-10). The binomial expansion of the squre root

is given by

1 11+b 1 , (4-11

2 8b b

01

01

)

where the number of terms needed for a given accuracy depends on the magnitude of b.

To apply the binomial expansion to the problem at hand, factor a z outside the expression for

, yielding

1

r

xr z

2 2

. (4-12)

Let the quantity b in Eq. (4-11) consist of the second and third terms under the square r

y

z z

2 2

01

oot in

(4-12). Then, retaining only the first two terms of the expansion (4-11), we have

1 11 .

2 2

x yr z

z z

The question now arises as to whether we need to retain all the terms in the approximation (4-13),

or whether only the first term might suffice. The answer to this question depends on which of the

sever 201 01

01

al occurrences of is being approximated. For the r appearing in the denominator of Eq. (4-9),

error introduced by dropping all terms but z is generally acceptably small. However for the r

appea

r

7

ring in the exponent, errors are much more critical. First they are multiplied by a very large

number k, a typical value for which might be greater than 10 in the visible region of the spectrum

(e.g. 75 10 ). Second, phase change of as little as a fraction of a radian can change

the value of the exponential significantly. For this reason we retain both terms of the binomial

approximation in t

meters

2 2

2

he exponent. The resulting expression for the field at (x,y) therefore becomes

, , d (4-14)

wh

kjkzj x yz

eU x y U e d

j z

ere we have incorporated the finite limits of the aperture in the definition of , , in

accord with the usual assumed boundary conditions.

Equation (4-14) is readily seen to be a convolution, expressib

U

2 2

2

le in the form

, , , d

where the convolution kernel is

h x,ykjkz

j x yz

U x y U h x y d

ee

j z

2 2

2 2 2 2

2

2

2 2

Another form of the result (4-14) is found if the term is factored outside the integral signs,

yielding

, , d

kj x yz

k kjkz j x yj x y jzz z

e

eU x y e U e e d

j z

(4-17)

which we recognize (aside from the multiplicative factors) to be the Fourier transform of the

product of the complex field just to the right of the aperture and a quadratic phase exponential.

We refer to both forms of the result (4-14) and (4-17), as the Fresenl diffraction integral. When

this approximation is valid, the observer is said to be in the region of Fresenl diffration, or equivalently

in the near field of the aperture.

Fourier opticsfA planar array of optical tweezers can be described by the intensity distribution, I (x,y),

of laser light in the focal plane of a microscope's objective lens. This pattern is determined

by the electri

in

c field of light incident at its input plane, as depicted in Fig. 2. Suppose that

the input plane is illuminated by monochromatic light of wavelength . Its wave

front at the input plane, E ( , ), con

,

tains both phase and amplitude information

( , ) , ,

where the amplitude, , , and phase, , , are real valued functions. The electric

field in the focal plane has a similar form,

(

iniin in

in in

f

E A e

A

E

2 2

,

2 2f

, , 2

,

, ) , ,

so that I , ( , ) , . These fields are related by the Fourier transform

pair

( , ) ( , ) d , where 2

( , ) and

fi x yf

f f

k kj x y j x y jkf

i x y i x yf in f f

in

x y A x y e

x y E x y A x y

kE x y e E e d e e

f

F E

, 1

,

(4)

( , ) ( , ) dy ( , ) (6)2

where is the

kj x y

i x yin f ff

x y

kE e E x y e dx F E x y

f

f

f

2 focal length of the lens and k= is the wave number of the incident light.

The additional phase profile, ( , ), due to the lens geometry does not contribute to I ,

and may be ignored without los

x y x y

s of generality.

,

,

, ,

, ,

2

, ,

( , ) ( , ) d 2

= ( , ) d d 2 2

( , ) d2

kj x y

i x yf in f

k kj x y j x y

i x y i x y f f f

x y

k kj x y j x y

i x y i x y f f f

x

kE x y e E e d

f

k ke e E x y e dx y e d

f f

ke e E x y e e dx y

f

, ,

2

, ,

, ,

2

, ,

, ,

d

( , ) d d2

( , ) d d2

2

y

k kj x y j x y

i x y i x y f f f

x y

kj x x y y

i x y i x y f f

x y

d

ke e E x y e e d dx y

f

ke e E x y e d dx y

f

k

f

22 2

, , 2 2

,

2

, ,

,

2

,

( , ) d d

( , ) d2 2 2

2

k kj x x j y y

i x y i x y f f f

x y

i x y i x y f

x y

i x y i

e e E x y e d e dx y

k k ke e E x y x x y y dx y

f f f

ke e

f

2

, 2( , ) ( , )x y f ff

E x y E x yk

, 1

, ( , ) ( , ) d ( , )

2

kj x y

i x yin f ff

x y

kE e E x y e dx y F E x y

f

inverse Fourier transform with variable 2

k

fx x

Phase-only holograms

0i ,0 0

Obtaining a desired wave front in the focal plane requires introducing the appropriate wave

front in the input plane. Most lasers, however, provide only a fixed wave front,

E , =A , e .

Shaping E

in0 , into E , involves modifying both the amplitude and phase at the

input plane. Changing the amplitude with a passive optical element necessarily diverts power

from the beam and diminishes trappin

in0

g efficiency. Fortunately, optical trapping relies on the beam 's

intensity and not on its phase. We can exploit this redundancy by setting A (r) , =A , and

modulating only the phase of the input bea

i ,in0

m to obtain the desired trapping configuration.

After passing through a phase modulating hologram, the electric field in the input plane has

a modified wave front

E , =E , e in

i , i ,

(8)

where e is the imposed phase profile. Calculating the phase hologram, e , needed to

project a desired patte

in in

rn of traps is not particularly straightforward, as a simple example

demonstrates.

0

f0

In a typical application of holographic optical tweezer arrays, the undiffracted beam, E ( , ),

projects a single optical tweezer into the center of the focal plane with output wave

front E (x,y), and

the goal is to create displaced copies of this tweezer in the focal plane.

One possible wave front describing an array of N optical tweezers at positions , in the focal

plane is a superposition of i ix y

fi 0

1

i1

0 0 0

single (nonoverlapping) tweezers

, E , ,

where the normalization 1 conserves energy. , may be written as a convolution

, , , , , of ,

Nf

i ii

Nf

i

f f f f

E x y x x y y

E x y

E x y E x y T x x y y dx dy E x y T x y E x

i1

i ,in 1 1 10 0 0

with a lattice

function

, .

Equations (6) and (8) relate , to the associated input wave front

2E , e , , , ,

by the Fourier convolution theore

in

N

i ii

f

f f

y

T x y x x y y

E x y

fF E x y T x y F E x y F T x y

i , 1

m. The phase modulation needed to achieve the array of

optical tweezers then follows from Eq. (8).

2e , ,

in fF T x y

iindependent of the form of the single tweezer. The phases of the complex weights, , must

be selected so that , is a real-valued function. Unfortunately, the resulting system

of equations has no

in

analytic solution. Still greater difficulties are encountered in designing

more general systems of optical traps, including tweezers which trap out of the focal plane

or mixed arrays of conventional and vortex tweezers. Rather than deriving solutions for

particular tweezer configurations, we have developed more general numerical methods

which we apply in the following sections to creating planar arrays optical tweezers.

Adaptive-Additive algorithmAdaptive-Additive (AA) is an iterative numerical technique which explores the space of degenerate

phase profiles, ( , ), to find a phase modulation of the incident laser beam encoding any

desired int

f x yensity profile in the focal plane. To facilitate calculation and fabrication, both

the input and output planes are discretized into M M square arrays of pixels. Optimal spatial

resolution requires pixe

f

ls in the focal plane to be one half wavelength on a side,

= /2. The number, M, of pixels on a side then depends on the desired dimensions of the trapping

array. Lengths in the input and focal planes

in f in

are related by Eqs. (4) and (6), so that the corresponding

pixel size in the input plane is = / M 2 / . If is inconveniently small, then L1

and L2 can be chosen so that a more amenable pixel si

f f M

1

f

1

i ,1 0

ze in the input plane corresponds to in the

focal plane.

The AA algorithm, depicted in Fig. 3, starts with an arbitrary initial guess for ,

and an initial input wave front E , =E , ein

in

in

1 ,1 1 1

2

1 1

. The Fourier transform of

this wave front is the starting estimate for the output electric

( , ) E , ( , ) ,The corresponding intensity in the output plane,

( , ) ( , ) is unl

fi x yf in f

f f

E x y F A x y e

I x y A x y

ikely to be a good rendition of the desired intensity pattern,

?

2

1

2f

2

1 121

,1 1

( , ) ( , ) . The error

1( , ) ( , )

is reduced by mixing a proportion, , of the desired amplitude into the field in the focal plane

( , ) ( , ) 1 ( , ) .

In

f

f

Mf f

i i i ii

i x yf f f

I x y A x y

I x y I x yM

a

E x y aA x y a A x y e

2

1

,1 1

verse transforming ( , ) yields the corresponding field in the input plane,

( , ) ( , ) . At this point, the amplitude in the input plane no longer matches

the actual laser profile,

in

f

iin in

E x y

E A e

2

1 0

,2 0

so we replace ( , ) with ( , ). The result is an improved

estimate for the input field: E ( , ) ( , ) . This completes one iteration of the

AA algorithm. Subsequent iterations lead to

in

in

iin

A A

E e

n

n n-1 n

monotonically improving estimates, , ,

for the desired phase modulation. The cycle is repeated until the error, , in the

th iteration converges to within an acceptable tolerance:

( )/ < .

inn

n

The phase and amplitude fields are computed as arrays of double-precision numbers, and

their Fourier transforms calculated with fast Fourier transform (FFT) routines. Starting

from random input phases, 1

6

n

, , uniformly distributed in the range 0 to 2 ,

the AA algorithm typically requires eight iterations to converge within 10 of an acceptably

accurate local minimum of using an intermediate

in

value for the mixing parameter, a=0.5.

0 ,E

( , ) ( , ) , , , , , ( , ) ( , )in in f f f f in inn n n n n n n nE E x y A x y x y E x y E

,( , ) ,iniin inE A e ,( , ) ,

fi x yf fE x y A x y e

1

( , ) ( , )

( , ) ( , )

f in

in f

E x y F E

E F E x y

1 1 1 1 1 1 1, ( , ) , , , , , ( , ) ( , )in in f f f f in inE E x y A x y x y E x y E

0 ,E , ,fA x y a 1 ( , ),inA

1 2 2 2 2 2 2 2( , ) ( , ) , , , , , ( , ) ( , )in in f f f f in inE E x y A x y x y E x y E

, ,fA x y a 2 ( , ),inA

F F-1

F-1F

0 ,E , ,fA x y a ( , ),innA

F-1F

Guess

Calc Error

Calc Error

Calc Error

Ref. Hydrodynamic coupling and optical patterning of many-particle colloidal systems, Eric R. Dufresne, PhD Thesis, 2000, The University of Chicago

Ref. Hydrodynamic coupling and optical patterning of many-particle colloidal systems, Eric R. Dufresne, PhD Thesis, 2000, The University of Chicago