Macroscopic Theory of Optical Momentum

Macroscopic Theory of Optical Momentum

Brandon A. Kemp

College of Engineering

Arkansas State University

Jonesboro, AR 72467, USA

[email protected]

Abstract

Light possesses energy and momentum within the propagating electromagnetic

fields. When electromagnetic waves enter a material, the description of en-

ergy and momentum becomes ambiguous. In spite of more than a century of

development, significant confusion still exists regarding the appropriate macro-

scopic theory of electrodynamics required to predict experimental outcomes and

develop new applications. This confusion stems from the myriad of electromag-

netic force equations and expressions for the momentum density and flux. In

this review, the leading formulations of electrodynamics are compared with re-

spect to how media are modeled. This view is applied to illustrate how the

combination of electromagnetic fields and material responses contribute to the

continuity of energy and momentum. A number of basic conclusions are deduced

with the specific aim of modeling experiments where dielectric and magnetic me-

dia are submerged in media with a differing electromagnetic response. These

conclusions are applied to demonstrate applicability to optical manipulation

experiments.

Keywords: Optical momentum, Abraham momentum, Minkowski

momentum, Lorentz force, optical trapping, optical tweezers, stress tensor,

negative index materials, macroscopic electrodynamics

1

Contents

1 Introduction 3

2 Classical Electrodynamics 6

2.1 Electrodynamics of Moving Media . . . . . . . . . . . . . . . . . 7

2.2 The Quasi-Stationary Approximation . . . . . . . . . . . . . . . . 12

2.3 Electrodynamics of Quasi-Stationary Media . . . . . . . . . . . . 16

3 Optical Force, Momentum, and Stress 18

3.1 Electromagnetic Force . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Electromagnetic Momentum . . . . . . . . . . . . . . . . . . . . . 22

3.3 Electromagnetic Stress . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Momentum and Stress in Dispersive Media 35

4.1 Field and Material Contributions . . . . . . . . . . . . . . . . . . 36

4.2 Time-Averaged Energy and Momentum Propagation . . . . . . . 39

5 Discussion 41

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Application to Optical Manipulation . . . . . . . . . . . . . . . . 43

Appendix A Scattering by a Cylinder 48

Appendix B Scattering by a Sphere 51

2

1. Introduction

Light possesses energy and momentum. This statement should be obvious

any time we feel the warmth of the sun or look at the tail of a comet. Light

leaves the sun, travels long distances through space, and does work on ponder-

able media at remote distances. Conservation principles ensure that the energy

and momentum are not lost only to be regained after the elapsed time required

for light to reach its destination. Therefore, energy and momentum must be ac-

counted for within light itself. Maxwell (1891) first applied electromagnetic wave

theory to predict and calculate the radiation pressure of light. This prediction

was soon verified by the observation of light pressure on reflectors in vacuum by

Lebedew (1901) and Nichols and Hull (1903). Poynting (1905) quantitatively

validated the theory with detailed measurements of radiation pressure. Such

experiments were relegated to scientific validation of the electromagnetic wave

theory until the invention of the laser.

Ashkin (1970) first demonstrated the manipulation of small dielectric parti-

cles by laser light when particles suspended in water were drawn into the axis

of a laser beam and accelerated in the direction of propagation. Rough calcula-

tions for the radiation pressure were used for designing the experiments which

involved a laser focused upon spheres a few wavelengths in diameter. In addition

to radiation pressure, the particles experienced a trapping force that pulled the

particles into the laser beam. This force was reversed when the relative index

of refraction was inverted. Air bubbles, with dielectric permittivity less than

that of the surrounding water, were observed to be pushed out of the beam,

while the radiation pressure remained in the beam propagation direction. The

trapping force and radiation pressure are commonly known as the gradient force

and the scattering force, respectively, due to the separation based on the ray

optics approximation for large particles or the Rayleigh approximation for small

particles.

Optical levitation [Ashkin and Dziedzic (1971, 1976)] and radiation pressure

on a liquid surface [Ashkin and Dziedzic (1973)] were among the many experi-

3

ments that followed. Lasers were also proposed as a means to trap atoms [Ashkin

(1978); Ashkin and Gordon (1979)], and single atoms were eventually trapped

and cooled by optical tweezers [Chu, Holberg, Bjorkholm, Cable, and Ashkin

(1985); Chu, Bjorkholm, Ashkin, and Cable (1986a); Chu, Bjorkholm, Ashkin,

Gordon, and Holberg (1986b)]. Colloidal dielectric particles were also trapped

in three dimensions using a single highly focused laser beam [Ashkin, Dziedzic,

Bjorkholm, and Chu (1986)]. Such optical tweezers result when a the gradient

force is larger than the scattering force. Optical manipulation has since found

many application in physics, biology, chemistry, and medicine [Ashkin (1997,

2000)], and optical tweezers have become a necessary tool for many researchers

in microbiology [Ashkin and Dziedzic (1987)].

The description of optical manipulation was further expanded by Burns,

Fournier, and Golovchenko (1989) when optical binding was first observed. Op-

tical binding can be used to create optical matter, a lattice of dielectric particles

held together by multiple particle interactions within an optical field [Burns,

Fournier, and Golovchenko (1990)]. The optical scattering, gradient, and bind-

ing forces provide a generic description for the optical manipulation of colloidal

particles. In the past few years, there has been a renewed interest in the theory

of optical momentum transfer to matter. Much of this can be attributed to the

growth in nanotechnology, where optical forces can play a significant role in the

dynamics of material systems [Grier (2003); Dholakia and Zemnek (2010)].

In spite of the simple language used to describe optical manipulation, de-

tailed modeling of optical forces can be a daunting task. There are two steps

involved when employing a macroscopic model of classical electromagnetics.

First, the total electromagnetic fields must be determined. Second, the electro-

magnetic force is calculated from the total fields. The first task is accomplished

by a variety of analytical and numerical solutions (e.g. FEM, FDTD), which

are generally well known. The second task is shrouded in controversy. A quick

review of the literature reveals a number of electromagnetic force density equa-

tions. The most commonly used are listed in Table 1 along with the momentum

continuity formulation to which they are inherently tied [Kemp (2011)]. Al-

4

though the list in Table 1 is not exhaustive, it illustrates the source of confusion

surrounding optical forces as each listed force density may predict different force

distributions in matter under certain conditions [Mansuripur, Zakharian, and

Wright (2013)]. The myriad of force densities are related to the dispute over

the momentum of light in media. This will become evident in Section 2. For

now, it is sufficient to say that we may calculate the total force on an object

by either integrating a force density over the volume or by considering the total

change in optical momentum.

Formulation Force Density [N/m3]

Minkowski − 1

2E2∇ǫ − 1

2H

2∇µ + ρE + J × ¯B

Chu[

ρ −(

∇ · P)]

E −(

∇ · µ0M)

H +(

J + ∂P

∂t

)

× µ0H − µ0∂M

∂t× ǫ0E

Einstein-Laub[

ρ +(

P · ∇)]

E +(

µ0M · ∇)

H +(

J + ∂P

∂t

)

× µ0H − µ0∂M

∂t× ǫ0E

Abraham − 1

2E2∇ǫ − 1

2H

2∇µ + ρE + J × ¯B + ∂

∂t

{

GM − GA

}

Amperian −(

∇ · P)

E + ∂P

∂t× B +

(

∇ × M)

× B + ρE + J × B

Table 1: Leading electromagnetic force densities applied to linear media.

The confusion over applicability and interpretation of the various electro-

magnetic force equations is related to a dispute regarding the momentum of

electromagnetic waves known as the Abraham-Minkowski debate. The debate

is more than a century old and relates to the correct form of optical momen-

tum in media [Minkowski (1908); Abraham (1909, 1910)]. Since, it has been

suggested that the debate is linked to the wave-particle duality of a photon

[Leonhardt (2006)]: the Minkowski momentum

pM = ~k = n~ω

c(1)

is related to the reduced wavelength in matter and the Abraham momentum

pA = mv =1

n

~ω

c(2)

is related to Einstien’s mass equivalent m = ~ω/c2, where ~ is the reduced

Plank’s constant, k is the wavenumber, n is the refractive index, ω is the an-

gular frequency, and c is the speed of light in vacuum. Both experiment and

5

theory have been applied in attempt to resolve the debate over the past century,

which have been reviewed by Brevik (1979), Pfeifer, Nieminen, Heckenberg, and

Rubinsztein-Dunlop (2007), Barnett and Loudon (2010), Baxter and Loudon

(2010), Milonni and Boyd (2010), and Kemp (2011).

This correspondence presents a macroscopic view of optical momentum and

forces with three fundamental aims. First, a description of how media are mod-

eled within classical electrodynamics illuminates the differences between the

various optical force and momentum densities in Section 2. Second, interpre-

tation of these differences allows the identification of the kinetic momentum of

light and how materials contribute to the electromagnetic stress in Sections 3

and 4. Finally, using these formulations and interpretations, it is shown how

optical manipulation experiments can be modeled based on the developed con-

clusions in Section 5. The overall goal is to present a classical framework for

predicting experiments involving optical momentum transfer to matter, such as

involved with the growing applications in optical manipulation.

2. Classical Electrodynamics

To address energy, momentum, and charge conservation in electrodynamics,

we use the subsystem mathematical framework. This is necessary since there

are a number of formulations of classical electrodynamics (e.g. Minkowski, Chu,

and Amperian) commonly used today which differ in how matter is modeled.

Since each formulation separates the field and material responses differently,

there are energy, momentum, and charge continuity equations presented for the

various formulations used.

The energy and momentum continuity are expressed for any given subsystem

j by the equations [Penfield and Haus (1967)]

ϕj(r, t) = −∇ · Sj(r, t)−∂Wj(r, t)

∂t(3a)

fj(r, t) = −∇ · ¯Tj(r, t)−∂Gj(r, t)

∂t(3b)

where fj is the force density, ϕj is the power density, ¯Tj is the momentum flux

6

or stress tensor, Sj is the power flux, Gj is the momentum density, and Wj is

the energy density. This indicates that each j may represent any subsystem

such as electromagnetic, hydrostatic, thermodynamic, kinetic etc. Closing the

overall system, such that

∑

j

ϕj(r, t) = 0 (4a)

∑

j

fj(r, t) = 0, (4b)

is is a statement of energy and momentum conservation. Additionally, we find

that charge is generally a conserved quantity in any formulation of electrody-

namics and is written as

0 = −∇ · Jj(r, t)−∂ρj(r, t)

∂t, (5)

where Jj(r, t) and ρj(r, t) denote a current density and charge density for a

given formulation, respectively.

2.1. Electrodynamics of Moving Media

Multiple formulations of Maxwell’s classical field theory have been advanced

over the years. Each differ in how ponderable media are modeled. Here, we re-

view the three most commonly used today; the Chu, Amperian, and Minkowski

formulations. The reader is directed to Kong (2005), Kemp (2011), and Penfield

and Haus (1967) for additional information regarding these and other, less used,

formulations.

2.1.1. Minkowski Formulation

The Maxwell-Minkowski equations

∇× H(r, t)− ∂

∂tD(r, t) = J (r, t) (6a)

∇× E(r, t) + ∂

∂tB(r, t) = 0 (6b)

∇ · D(r, t) = ρ(r, t) (6c)

∇ · B(r, t) = 0, (6d)

7

combine the field and material contributions within the time and space depen-

dent field vectors E(r, t), H(r, t), D(r, t), and B(r, t), which contain the response

of the material via macroscopic constitutive relations. The free charge density

ρ(r, t) and free current density J (r, t) are the only field-free quantities present

when considering the Minkowski formulation. In a linear, isotropic medium,

which is effectively dispersionless at frequencies of interest, the constitutive re-

lations D(r, t) = ǫE(r, t) and B(r, t) = µH(r, t) yield the terms for the momen-

tum continuity equation in the Minkowski formulation

fM (r, t) = −1

2E(r, t) · E(r, t)∇ǫ− 1

2H(r, t) · H(r, t)∇µ

+ ρ(r, t)E(r, t) + J (r, t)× B(r, t) (7a)

¯TM (r, t) =1

2

[

D(r, t) · E(r, t) + B(r, t) · H(r, t)] ¯I

− D(r, t)E(r, t)− B(r, t)H(r, t) (7b)

GM (r, t) = D(r, t)× B(r, t), (7c)

where− 12 E(r, t)·E(r, t)∇ǫ is the well-known Helmholtz force. The corresponding

energy continuity equations are defined by

ϕM (r, t) = J (r, t) · E(r, t) (8a)

SM (r, t) = E(r, t)× H(r, t) (8b)

WM (r, t) =1

2

[

D(r, t) · E(r, t) + B(r, t) · H(r, t)]

. (8c)

In Section 4, the consequences of dispersion on the momentum of the wave

are explored. A statement of charge conservation results by taking the diver-

gence of Eq. (6a) and utilizing Eq. (6c). Charge conservation in the Minkowski

formulation is in terms of the free charge and currents

0 = −∇ · J (r, t)− ∂ρ(r, t)

∂t. (9)

Eq. (7) and Eq. (8) are derived using the predefined constitutive relations

with Eq. (6). This indicates that the desired momentum and energy terms,

along with the f and ϕ terms, depend on the constitutive relations used in

8

modeling the media. Thus, when using the Minkowski formulation, the combi-

nation of both field and material contributions are present in the description of

the momentum and energy of the electromagnetic subsystem.

2.1.2. Chu Formulation

The Maxwell-Chu equations

∇× HC(r, t)− ǫ0∂

∂tEC(r, t) = Je(r, t) (10a)

∇× EC(r, t) + µ0∂

∂tHC(r, t) = −Jh(r, t) (10b)

ǫ0∇ · EC(r, t) = ρe(r, t) (10c)

µ0∇ · HC(r, t) = ρh(r, t) (10d)

separate the electric field EC(r, t) and magnetic field HC(r, t) from the material

response and are, therefore, often referred to as the EH representation. The Chu

formulation models the material responses by equivalent electric current density

Je(r, t), magnetic current density Jh(r, t), electric charge density ρe(r, t), and

magnetic charge density ρh(r, t). These quantities for moving media with local

velocity field v(r, t) are defined as

Je(r, t) ≡∂

∂tPC(r, t) +∇× [PC(r, t)× v(r, t)] + JC(r, t) (11a)

Jh(r, t) ≡ µ0∂

∂tMC(r, t) + µ0∇× [MC(r, t)× v(r, t)] (11b)

ρe(r, t) ≡ −∇ · PC(r, t) + ρC(r, t) (11c)

ρh(r, t) ≡ −µ0∇ · MC(r, t), (11d)

where MC(r, t) is the magnetization, PC(r, t) is the polarization, JC(r, t) is

the free current density, and ρC(r, t) is the free charge density of the given

medium. Here, the subscript C indicates that the values involved within the

Chu formulation differ from similar terms in other common formulations. The

momentum continuity equation in the EH representation is defined by the terms

9

feh(r, t) = ρeEC + ρhHC + Je × µoHC − Jh × ǫ0EC (12a)

¯Teh(r, t) =1

2

[

ǫ0EC · EC + µ0HC · HC

] ¯I − ǫ0EC EC − µ0HCHC (12b)

Geh(r, t) = ǫ0µ0EC × HC , (12c)

and the corresponding terms in the energy continuity equation are

ϕeh(r, t) = Je · EC + Jh · HC (13a)

Seh(r, t) = EC × HC (13b)

Weh(r, t) =1

2

[

ǫ0EC · EC + µ0HC · HC

]

, (13c)

where the subscript eh denotes quantities resulting from the EH representation

which are quadratic in the fields. In these equations, the exact response of

the material has been left open. That is, the material could be nonlinear,

anisotropic, or dispersive. The Chu formulation combines the bound and free

electric and magnetic charges to conserve the charge within the system. Charge

conservation can be expressed in terms of the total charge and current densities

0 = −∇ · Je(r, t)−∂ρe(r, t)

∂t(14a)

0 = −∇ · Jh(r, t)−∂ρh(r, t)

∂t. (14b)

2.1.3. The Amperian formulation

The Maxwell-Amperian equations

1

µ0∇× BA(r, t)− ǫ0

∂

∂tEA(r, t) = Jeb(r, t) (15a)

∇× EA(r, t) +∂

∂tBA(r, t) = 0 (15b)

ǫ0∇ · EA(r, t) = ρeb(r, t) (15c)

∇ · BA(r, t) = 0 (15d)

separate the electric EA(r, t) and magnetic BA(r, t) fields from the material

response and is, therefore, often referred to as the EB representation. The

10

Amperian formulation models the material responses by equivalent electric cur-

rent density Jeb(r, t) and electric charge density ρeb(r, t). These quantities for

moving media are defined as

Jeb(r, t) ≡ ∂

∂t

[

PA(r, t)− µ0ǫ0MA(r, t)× v(r, t)]

+ ∇× [MA(r, t) + PA(r, t)× v(r, t)] + JA(r, t) (16a)

ρeb(r, t) ≡ −∇ ·[

PA(r, t)− µ0ǫ0MA(r, t)× v(r, t)]

+ ρA(r, t), (16b)

where MA(r, t) is the magnetization, PA(r, t) is the polarization, JA(r, t) is

the free current density, and ρA(r, t) is the free charge density of the given

medium. Here, the subscript A indicates that the values involved within the

Amperian formulation differ from similar terms in other common formulations.

The momentum continuity equation in the EB representation is defined by the

terms

feb(r, t) = ρebEA + Jeb × BA (17a)

¯Teb(r, t) =1

2

[

ǫ0EA · EA + µ−10 BA · BA

] ¯I − ǫ0EAEA − µ−10 BABA (17b)

Geb(r, t) = ǫ0EA × BA, (17c)

and the corresponding terms in the energy continuity equation are

ϕeb(r, t) = Jeb · EA (18a)

Seb(r, t) = µ−10 EA × BA (18b)

Web(r, t) =1

2

[

ǫ0EA · EA + µ−10 BA · BA

]

, (18c)

where the subscript eb denotes quantities resulting from the EB representation

which are quadratic in the fields. In these equations, the exact response of

the material has been left open. That is, the material could be nonlinear,

anisotropic, or dispersive. Charge conservation is expressed in terms of the

total charge and current density

0 = −∇ · Jeb(r, t)−∂ρeb(r, t)

∂t. (19)

11

2.2. The Quasi-Stationary Approximation

While the Abraham-Minkowski debate originated out of relativistic consider-

ations, the primary differences between the theories can be studied independent

of material motion [Nelson (1991)]. For simplicity, this paper presents a quasi-

stationary analysis of electrodynamics. That is, we apply a limiting process

such that v → 0 and m → ∞ simultaneously such that the kinetic energy 12mv2

of the material approaches zero while the momentum mv remains, in general,

a nonzero vector. This stationary approximation is commonly applied in the

literature and in text book calculations [Kong (2005)] where optical momentum

is imparted to material while the optical energy is conserved. This is illustrated

in the following two examples of optical energy incident from vacuum onto a

perfect reflector. First, an example is given within the framework of classi-

cal electrodynamics [Daly and Gruenberg (1967); Sheppard and Kemp (2014)].

Second, a simple argument of a Doppler shifted photon is given [Kemp and

Grzegorczyk (2011)]. Both illustrate that energy is not conserved in the quasi-

stationary approximation. However, the approximation is useful for modeling

many experimental configuration where the kinetic energy of the material is

much smaller than the energy of the electromagnetic waves.

2.2.1. Plane Wave Example

Consider the electromagnetic fields resulting from a plane wave incident

in the −z direction upon a perfect electrical conductor at z = vt. Complex

notation is used to represent time-harmonic fields such that, for example, the

time-harmonic electric field E is related to the complex field E by

E(r, t) = ℜ{

E(r)e−iωt}

, (20)

where ℜ{} is the real-part operator. Average values of derived quantities, which

are quadratic in the fields, are computed directly from the complex fields. For

example, the time-averaged vacuum Poynting power is given by

〈S〉 = 1

2ℜ{

E × H∗

}

, (21)

12

Figure 1: A plane wave incident upon a perfect electrical conductor. The Fresnel reflection

coefficient of the reflector in the stationary frame is R = −1.

where ∗ denotes complex conjugation. The incident and reflected electric and

magnetic fields are

Ei = xE0

[

e−i(kiz+ωit) +Rei(krz−ωrt)]

(22a)

Hi = −yH0

[

e−i(kiz+ωit) −Rei(krz−ωrt)]

, (22b)

where H0 = E0

√

ǫ0/µ0, ki = ωi/c, and kr = ωr/c. The boundary condition

z× (E + v× B) = 0 at z = vt yields the phase matching condition −kiv−ωi =

krv − ωr and the reflection coefficient

R =β + 1

β − 1, (23)

where β = v/c. Note that the Doppler effect is a consequence of phase matching

at a moving boundary

ωr = ωi

(

1 + β

1− β

)

. (24)

Since the incident region is vacuum, there is no ambiguity as to the applicable

energy and momentum equations. The incident and reflected power flow are

〈Si〉 =1

2ℜ{

Ei × H∗i

}

= −z1

2E0H0 = −z〈Si〉 (25a)

〈Sr〉 =1

2ℜ{

Ei × H∗i

}

= z1

2E0H0|R|2 = z〈Sr〉. (25b)

13

The total electromagnetic power flow into a closed volume at the reflector in-

terface is

〈Pelec〉 = 〈Si〉 − 〈Sr〉 = −4〈Si〉c

β

1− β2. (26)

The change in stored electromagnetic energy is

〈Pstored〉 = −v

2ℜ{

ǫ02E · E∗ +

µ0

2H · H∗

}

= −2〈Si〉β1 + β2

1− β2. (27)

Therefore, the mechanical work that must be done to the system to keep the

reflector at constant velocity is

〈Pmech〉 = 〈Pstored〉 − 〈Pelec〉 = 2〈Si〉β(1 + β)

(1− β). (28)

Likewise, the force on the reflector is

〈Felec〉 = −z [〈Ti〉+ 〈Tr〉] + v [〈Gi〉+ 〈Gr〉] , (29)

where the z components of the momentum densities and the zz components of

the stress tensors are

〈Gi〉 = =1

2ℜ{

ǫ0µ0Ei × H∗i

}

=〈Si〉c

(30a)

〈Gr〉 = =1

2ℜ{

ǫ0µ0Er × H∗r

}

=〈Sr〉c

(30b)

〈Ti〉 = =1

2ℜ{ǫ02Ei · E∗

i +µ0

2Hi · H∗

i

}

=〈Si〉c

(30c)

〈Tr〉 = =1

2ℜ{ǫ02Er · E∗

r +µ0

2Hr · H∗

r

}

=〈Sr〉c

. (30d)

The mechanical force required to keep the reflector from accelerating is

〈Fmech〉 = −〈Felec〉 = z2〈Si〉c

(1 + β)

(1− β). (31)

Obviously, v · 〈Fmech〉 = 〈Pmech〉 so that the mechanical force and work are

required to keep the reflector at constant velocity. The case of v → 0 produces

the well known result 〈Felec〉 = −z2〈Si〉/c, as is obtained from the stationary

reflector case treated in most electromagnetic textbooks [Kong (2005)].

Since 〈Pmech〉 is proportional to v, it will go to zero under a stationary ma-

terial approximation, but limv→0〈Fmech〉 = z2〈Si〉/c. If the mechanical balance

14

force is removed, the reflector will accelerate from v = 0 and the kinetic energy

of the mirror will be increased. However, this problem is typically treated us-

ing a stationary media approximation such that the Doppler shift is ignored.

It is equivalent to taking a quasi-stationary approximation where v → 0 while

m → ∞ such that the kinetic momentum of the material mv remains a nonzero

number and the kinetic energy of the material 12mv · v goes to zero.

2.2.2. Doppler Shifted Photon Example

Next, consider a photon with energy ~ω normally incident upon a mirror

initially at rest as in Fig. 1. The reflected photon is red shifted due to the

Doppler effect described in Eq. (24). For simplicity, assume that |v| ≪ c so that

ω′ = ω(

1 +v

c

)

, (32)

where ω′ is the Doppler-shifted frequency of the reflected photon and m is the

mass of the mirror, and v = zv is the the velocity of the mirror after reflecting

the photon. The non-relativistic Doppler shift is applied since assuming that

the mass of the mirror is much greater than the effective mass of the photon

(i.e. m ≫ ~ω/c2). Energy conservation states that the initial energy of the

photon is equal to the sum of the final kinetic energy of the photon plus the

final kinetic energy of the mirror

~ω =1

2mv2 + ~ω

(

1− v

c

)

. (33)

Solving Eq. (33) yields solutions for the mirror energy and momentum

1

2mv2 = −~ω

v

c(34a)

mv = −2~ω

c. (34b)

If, however, we let v → 0 a priori as in the quasi-stationary approximation, the

calculated kinetic energy of the mirror will be zero while the momentum after

reflection will be unchanged.

15

2.3. Electrodynamics of Quasi-Stationary Media

Under the quasi-stationary approximation, the velocity terms in Maxwell’s

equations go to zero. That is, v → 0 in all of the equations of Section 2.1. Also,

the field vectors in each formulation are the same. That is EC = EA = E , andlikewise for all other fields and sources. Because of this, we may simply state

the polarization and magnetization in terms of the field vectors such that

P(r, t) = D(r, t)− ǫ0E(r, t) (35a)

µ0M(r, t) = B(r, t)− µ0H(r, t), (35b)

which are general enough to include causal (i.e. lossy and dispersive) and

anisotropic media.

Obviously, the Minkowski, Chu, and Amperian force densities given in Ta-

ble 1 correspond to the corresponding formulations for stationary media. There-

fore, the stationary force densities in Table 1 may be used only when the velocity

of the material can be safely ignored, such as when relativistic effects are in-

significant. Two other equations, the Einstein-Laub and the Abraham force

densities, are also given in Table 1. Both have been discussed extensively (see

for example reviews by Pfeifer et al. (2007) and Milonni and Boyd (2010). Here,

Eqs. (35) are used to rearrange terms in the formulations given in Section 2.1

to arrive at the Einstein-Laub and the Abraham force densities. These math-

ematical rearrangements are exact and consistent with Maxwell’s equations.

However, such excercises shouldn’t be taken as basis for proof or interpretation

[Grzegorczyk and Kemp (2008)].

Einstein and Laub (1908) proposed a formulation which models both dielec-

tric and magnetic response as effectively bound dipoles. This differs from the

Chu formulation, which models media as effectively bound electric and magnetic

charges, and the Amperian formulation, which models media as bound electric

charges and infinitesimal current loops. However, at the macroscopic level, these

microscopic viewpoints are effectively lost, and the real difference is in which

field terms give W , S, G, and ¯T defining the electromagnetic subsystem for the

16

formulation. Using vector calculus identities to write the force on bound electric

charges and bound magnetic charges

−(∇ · P)E = (P · ∇)E − ∇ · (PE) (36a)

−µ0(∇ · M)H = µ0(M · ∇)H − µ0∇ · (MH) (36b)

allows for substitution into the Chu force given in Table 1. The tensor terms

are then moved to the other side of the momentum continuity equation such

that

[

ρ+(

P · ∇)]

E +(

µ0M · ∇)

H+

(

J +∂P∂t

)

× µ0H − µ0∂M∂t

× ǫ0E

= − ∂

∂t

[

ǫ0µ0E × H]

−∇ ·[

1

2

(

ǫ0E · E + µ0H · H) ¯I − DE − BH

]

, (37)

where the relations in Eqs. (35) have been used. Equation (37) is the momentum

continuity equation for the Einstein-Laub formulation. The energy continuity

equation is reported as identical to Chu [Mansuripur et al. (2013)].

Likewise, terms can be rearranged from the Minkowski formulation to arrive

at the Abraham formulation. This is accomplished by simply adding the term

known as the Abraham force ∂(GM − GA)/∂t to the Minkowski force density

and subtracting the same term from the momentum density to arrive at the

momentum continuity equation

−1

2E2∇ǫ − 1

2H2∇µ+ ρE + J × ¯B +

∂

∂t

(

GM − GA

)

= − ∂

∂t

[

ǫ0µ0E × H]

−∇ ·[

1

2

(

D · E + B · H) ¯I − DE − BH

]

, (38)

where GM is the Minkowski momentum density and GA is the Abrahammomen-

tum density given in Table 2. It should be pointed out that a symmetric form

of the Abraham tensor has been perpetuated in the literature [Kemp (2011),

Brevik (1979)]. However, the asymmetric stress tensor given in Eq. (38) was

originally presented by Abraham (1909). The energy continuity equation is

reported as identical to Minkowski [Pfeifer et al. (2007)].

The leading electromagnetic momentum continuity equations in stationary

media are defined by the force densities in Table 1 along with the momentum

17

Formulation Momentum Density Stress Tensor

Minkowski D × B 1

2

(

D · E + B · H) ¯I − DE − BH

Chu ǫ0µ0E × H 1

2

(

ǫ0E · E + µ0H · H) ¯I − ǫ0EE − µ0HH

Einstein-Laub ǫ0µ0E × H 1

2

(

ǫ0E · E + µ0H · H) ¯I − DE − BH

Abraham ǫ0µ0E × H 1

2

(

D · E + B · H) ¯I − DE − BH

Amperian ǫ0E × B 1

2

(

ǫ0E · E + µ−1

0B · B

)

¯I − ǫ0EE − µ−1

0BB

Table 2: Leading electromagnetic momentum densities and stress tensors.

densities and stress tensors in Table 2. The form of the energy and momentum

continuity equations are given in Eq. (3). The Minkowski, Chu, and Amperian

energy terms have been given in Section 2.1. The Einstein-Laub energy con-

tinuity terms are identical to the Chu formulations, and the Abraham energy

continuity terms are identical to the Minkowski formulation. In general, torque

density can be calculated as τ = r× f , with the exception of the Einstein-Laub

torque density which is supplemented by the terms P × E +µ0M× H [Einstein

and Laub (1908); Mansuripur et al. (2013)].

3. Optical Force, Momentum, and Stress

The electromagnetic force may be computed directly from the force density

or by considering spatial variations of the stress tensor and temporal variations

of the momentum density. In the following sections, consideration is given to

these contributions to the total force. In certain cases, different force densities

give identical predictions for the total force on a material object, and these

cases are defined. Additionally, there are situations where either the momentum

density G or the stress tensor ¯T may be isolated. These situations are treated

individually to explore the nature of G and ¯T for nondispersive media. This

general nature will be used later to model optical manipulation experiments.

Finally, this section is concluded with a presentation of optical momentum in

causal media, which must include dispersion and loss in the material model

contribution.

18

3.1. Electromagnetic Force

The force distribution inside an object depends upon the equation used to

compute the force density. In general, the equation takes the form of Eq. (3b)

where the force densities f(r, t) are given in Table 1 and the corresponding values

for G(r, t) and ¯T (r, t) are given in Table 2. The total electromagnetic force

F (t) on an object results from integration of an electromagnetic force density

f(r, t) over the volume V . Equivalently, one may choose to apply the divergence

theorem to reduce the contribution of an electromagnetic stress tensor ¯T to an

integral over the surface A with outward pointing area element dA enclosing the

volume V so that the total force is calculated equivalently by

F (t) = −∫

V

dV∂

∂tG(r, t)−

∮

A

dA · ¯T (r, t). (39)

In the following sections, this equation is studied in regards to situations when

the various formulations give equivalent results.

3.1.1. Equivalence of Total Force

The equivalence of total force for different formulations and calculation meth-

ods has been demonstrated or proven by a number of researchers. For exam-

ple, see reports by Kemp, Grzegorczyk, and Kong (2005); Barnett and Loudon

(2006); Loudon and Barnett (2006); and Mansuripur (2008). To show which

total force equations are equivalent at any point in time, refer to the diagram

in Fig. 2 (a) depicting an object surrounded by vacuum. If we choose to inte-

grate the force density f(r, t) over the volume completely enclosing the object

as depicted by the dashed line, the total force is given by Eq. (39), where the

tensor reduces to the vacuum Maxwell stress tensor

¯Tvac =1

2(ǫ0E · E + µ0H · H)− ǫ0E E − µ0HH. (40)

This results because the surface of integration is outside the material. The mo-

mentum density G must still be chosen based on the formulation. Therefore,

any formulations which share a common momentum density will produce iden-

tical results for the total force on an object at all points in time. For example,

19

Figure 2: Diagram for illustration of total force calculation. (a) The total force on an object in

vacuum is computed by volume integration of a force density within a region which completely

encloses the material object. The volume is enclosed by the surface depicted by the dashed

line. (b) The total force on an object embedded is computed in a similar way by considering

a thin vacuum region between the two materials.

the Chu, Einstein-Laub, and Abraham force densities are equivalent in terms

of total force. Since the Amperian formulation differs only in the modeling of

magnetic media, it will also produce identical results for total force for dielectric

materials.

As a second case, consider an object which is submerged or embedded inside

another medium as depicted in Fig. 2 (b). In this case, the total force will still

be the same for two formulations if their momentum densities are identical, but

it is important to clarify how the force is computed. If the force is computed

by integrating the force density f(r, t) over a volume, there will generally be

both volume and surface forces. The surface forces arise due to material discon-

tinuities at the boundaries [Mansuripur (2004), Kemp, Grzegorczyk, and Kong

(2006a)]. Therefore, there may be initial ambiguity as to which material, the

embedded object or the embedding medium, to which the surface forces should

20

be assigned. To alleviate this ambiguity, imagine there is a thin layer of vacuum

in the boundary. A volume integration procedure can be applied, as before,

to the embedded object. Since the surface of the integration is in the vacuum

region, the tensor reduces to the vacuum stress tensor and the force separation

for the two media is unambiguous. Allowing the vacuum region to vanish, or

mathematically taking the limit of the gap to approach zero, yields the origi-

nal field problem with the forces correctly separated [Kemp and Grzegorczyk

(2011)]. In this case, however, there will generally be forces on both the embed-

ded object and the surrounding medium. The conclusion remains the same; any

formulations which share a common momentum density will produce identical

results for the total electromagnetic force on a material object at any point in

time.

3.1.2. Equivalence of Total Time-Average Force

Systems are often modeled under the influence of time-harmonic excitation.

In these situations, it is the time-average, or cycle-average, force that is the

observable quantity. The total average force on an object due to time-harmonic

excitation is

〈F 〉 = −∮

A

dA · 〈 ¯T 〉. (41)

The momentum density does not contribute to the time-average force of time-

harmonic fields. This is because the the ∂G/∂t term is comprised of terms such

as ∂/∂t[cos2(ωt)], ∂/∂t[sin2(ωt)], and ∂/∂t[cos(ωt) sin(ωt)], which all vanish un-

der time-averaging.

The total average force can be computed by integrating an average force

density over a volume that includes the entire object. All such computations

will yield identical results regardless of the force density applied, assuming it is

consistent with Maxwell’s equations. To see why, consider again the diagram

in Fig. 2 (a). Choosing a volume which completely encloses the material object

allows for the force to be computed by a surface integration along a contour

which lies completely in the vacuum region as depicted by the dashed lines. In

21

this case, the momentum density G does not contribute and all formulations

reduce to the divergence of the time-averaged vacuum Maxwell stress tensor,

which is given by

〈 ¯Tvac〉 =1

2ℜ{

1

2(ǫ0E · E∗ + µ0H · H∗)− ǫ0EE∗ − µ0HH∗

}

. (42)

A similar argument holds for a material object which is embedded inside

another medium. It is only necessary to consider separation of the surface

force contributions between the embedded material and the embedding medium

as described in the previous section and depicted in Fig. 2 (b). The total

time-average force on a material object is unambiguous. Any formulation will

produce an identical value for the total time-average electromagnetic force on a

material object. Because of this, it is impossible to determine the time-domain

force equation from a time-average force equation, and such assumptions lead

to serious errors [Kemp (2013)].

3.2. Electromagnetic Momentum

In this section, contributions of the electromagnetic momentum density

G(r, t) are isolated. This is accomplished by considering pulses of finite ex-

tent. When the volume V of integration is extended such that the material

and the pulse are completely contained, the fields on the surface A are zero.

Therefore, the stress tensor is also zero on A and Eq. (39) reduces to

F (t) = −∫

V

dV∂

∂tG(r, t). (43)

3.2.1. Comparison of Momentum Densities

It has been established that the different predictions in total force on a mate-

rial object arises from the variety of momentum density terms for G(r, t), not the

stress tensor terms ¯T (r, t), and the leading momentum densities have been given

in Table 2. Only three unique momentum densities result from the five leading

formulations reviewed. First, the Abraham momentum density, GA = ǫ0µ0E×Hshows up in the Chu, Einstein-Laub, and Abraham formulations. Second, the

22

Livens momentum density GL = ǫ0E × B (see references by Livens (1918), Nel-

son (1991), and Scalora, Aguanno, Mattiucci, Bloemer, Centini, Sibilia, and

Haus (2006)) is included in the Amperian subsystem. Third, the Minkowski

momentum density GM = D × B completes the Minkowski momentum con-

tinuity equation. For each form, the momentum of an electromagnetic pulse

transmitted into a medium can be written in terms of vacuum quantities. In

this way, the various forms can be compared and contrasted [Kemp (2011)].

For comparison, consider a plane wave pulse normally incident upon a lin-

ear, transparent material with index of refraction n ≡ c√ǫµ. For simplicity, the

material is considered to be nondispersive in the frequency range of interest and

the quasi-stationary approximation is applied. The incident pulse is described

by the electric field E = xE0(z, t) cos(k0z − ωt). Here, E0(z, t) is an envelope

function for the harmonic wave cos(k0z − ωt) and k0 is the wavenumber in the

incident vacuum region. The incident energy densityWi(z) = ǫ0E20 (z, t) and mo-

mentum density Gi(z, t) = ǫ0µ0Ei(z, t)×Hi(z, t) = zWi(z, t)/c are unambiguous

since the wave is incident from vacuum. The incident optical energy is defined

as the volume integral over the entire incident pulse. The one-dimensional pulse

varies only in z so that the incident energy is

Ei ≡∫∫∫

V

dVWi(z) =

∫

dzWi(z) (44)

and has units [J/m2]. Similarly, the unambiguous incident momentum is deter-

mined by integrating the incident momentum density at an instant prior to the

field interaction with the material. It is given by

pi ≡∫∫∫

V

dV Gi(z) = z

∫

dzWi(z)

c= z

Ei

c, (45)

and has units [N · s/m2].

We may also arrive at quantities for the reflected energy density Wr(z) =

R2Wi(z) and reflected momentum density Gr(z) = R2Wi(z)/c, where R2 is the

reflectivity [Kong (2005)]. The reflected energy in terms of the incident energy

is

Er = R2Ei, (46)

23

and the reflected momentum in terms of the incident momentum is

pr = −R2pi. (47)

In order to express the energy and momentum in the material in terms of unam-

biguous vacuum quantities, the excitation energy and momentum are defined

as

Eexc ≡ Ei − Er = Ei

(

1−R2)

(48)

pexc ≡ pi − pr = pi(

1 +R2)

, (49)

which represent the difference between the incident and reflected pulses. No

assumptions are made as to the nature of momentum propagation or transfer in

the material. The intent, rather, is to study the difference between Abraham,

Livens, and Minkowski momenta in media by developing precise relations for

the different momenta in terms of the vacuum quantities.

The transmitted Abraham momentum density is

GA(z) = ǫ0µ0Et × Ht = zη0ηT 2Gi(z) = z

Wi(z)

c(1−R2), (50)

where η0 is the wave impedance of free space, η is the wave impedance of the

material, T is the transmission coefficient, and the relation (η0/η)T2 = 1 − R2

resulting from the boundary conditions has been used. The Livens momentum

density is given by

GL(z) = ǫ0Et × Bt = ǫ0µ0Et × Ht + ǫ0µ0Et × Mt

= zWi(z)

c(1−R2)− ǫ0µ0Mt × Et, (51)

where the last term is the “hidden momentum” as coined by Shockley and James

(1967). Similarly, the transmitted Minkowski momentum density is

GM (z) = ǫµEt × Ht = zn2 η0ηT 2Gi(z) = zn2Wi(z)

c

(

1−R2)

. (52)

The spatial length of the transmitted pulse is decreased proportional to the

factor n−1 due to the change in velocity. The Abraham momentum of the

24

transmitted pulse is

pA =

∫

z

dzGA(z, t) = z1

n

Ei

c

(

1−R2)

= z1

ncEexc. (53)

The Livens momentum of the transmitted pulse reduces to

pL =

∫

z

dzGL(z, t) = z1

ncEexc −

∫

z

dzǫ0µ0Mt × Et, (54)

which differs from the Abraham form by the hidden momentum. Finally, the

Minkowski momentum density yields the momentum

pM =

∫

z

dzGM (z, t) = z1

n

n2Ei

c

(

1−R2)

= zn

cEexc (55)

transmitted into the medium. Consequently, both the direction and magnitude

of the momentum transferred to the surface depend upon this choice according

to Loudon (2004). In more exact mathematical terms, this means that the

subsystem required to close the overall system depends upon which formulation

of electromagnetics is applied.

3.2.2. Field Momentum

The various forms for the momentum of light differ in how the material

response is included. The total system may be subdivided in a number of ways

such that portions of the material response are included in the electromagnetic

subsystem. It is desirable to determine the kinetic momentum of light, which

is the momentum which contains only field contributions without contributions

from the mass of the material. The thought experiment by Balazs (1953) allows

for the determination of the kinetic momentum by studying the center-of-mass

displacement of a material slab as an electromagnetic pulse passes through. A

number of authors have presented versions of this thought experiment. For

example, see the works of Loudon (2004), Scullion and Barnett (2008), Barnett

(2010), Kemp (2011), and Griffiths (2012).

Here, the analysis is simplified by considering a non-dispersive magneto-

dielectric material that is impedance-matched to the surrounding vacuum. The

slab is comprised of a material with index of refraction n = c√ǫµ and wave

25

impedance η =√

µ/ǫ =√

µ0/ǫ0. An electromagnetic wave pulse depicted in

Fig. 3 has an initial free space momentum Ei/c. The pulse is delayed with

respect to the free space path by the distance L = (n − 1)d by the slab of

thickness d since the velocity in the material is v = c/n. In order to maintain

uniform motion of the center-of-mass energy, the required kinetic momentum of

the material while the pulse overlaps spatially with the slab is

pm =Ei

c

(

1− 1

n

)

. (56)

Momentum conservation requires that the electromagnetic contribution to the

total momentum be the difference between the total momentum of the incident

pulse and the material momentum given by Eq. (56). Therefore, the electro-

magnetic momentum of the pulse is the Abraham momentum

pA =1

n

Ei

c. (57)

This analysis is actually very similar to the reasoning applied to arrive at

Eq. (2). The effective mass of the pulse remains constant as it traverses through

a medium requiring that the momentum changes in proportion to the velocity.

The conclusion of the Balazs thought experiment is that the Abraham mo-

mentum density gives momentum contained within the electromagnetic fields.

As previously stated, the Abraham momentum density appears in a number

of subsystems (e.g. Chu and Einstein-Laub). This analysis excludes other

forms, such as the Livens momentum and the Minkowski momentum, as be-

ing the kinetic momentum of light. Of course, the Livens momentum appears

in the Amperian formulation, and it differs from the Abraham momentum by

the hidden momentum defined in Section 3.2.1. The topic of hidden momen-

tum continues to receive significant attention [Mansuripur (2012); Griffiths and

Hnizdo (2013); Saldanha (2013)]. The fact that a hidden momentum term must

be added to the Livens momentum in the Balazs thought experiment does not

imply that the Amperian formulation violates momentum conservation. It does

mean, however, that the Amperian formulation is not considered as a possible

choice for the kinetic subsystem of light. It must be concluded that the macro-

26

Figure 3: Diagram of Balazs thought experiment for determining the kinetic momentum

of light. An electromagnetic wave (a) traveling through an impedance-matched magneto-

dielectric slab is delayed by a length L = (n−1)d in comparison to (b) an identical, unimpeded

pulse traveling through vacuum.

27

scopic formulation of electrodynamics presented as the Amperian formulation

(i.e. EB representation) does not provide the field only contribution to energy

and momentum. Although, microscopic reasoning can be applied to justify the

inclusion of the hidden momentum model [Griffiths and Hnizdo (2013)], the hid-

den momentum term cannot be simply added to the momentum density in the

Amperian formulation without being accounted for equally in either the force

density term or stress tensor term of the formulation. This mathematical rigour

is necessary to ensure that the macroscopic momentum continuity equation is

consistent with Maxwell’s equations. As previously demonstrated, such math-

ematical rearrangements defines a new subsystem, but this author is unaware

of such a formally proposed formulation and interpretation. Such a subsys-

tem would effectively include the Abraham momentum density with a modified

version of the Amperian force density and/or stress tensor. It should also be

reiterated that the results of this section make no claims regarding the micro-

scopic nature of matter response. Although the different formulations provide

conceptual views of material response (e.g. effective dipoles, bound charges, or

current loops), at the macroscopic level these details are lost [Kong (2005); Pen-

field and Haus (1967)], and the important features of the macroscopic theories

pertain to which macroscopic fields appear in the continuity equations.

3.3. Electromagnetic Stress

In this section, contributions of the electromagnetic stress tensor are isolated.

This is accomplished by computing the time-average force of time-harmonic

waves. As previously shown, such computations nullify contributions from the

momentum density term.

3.3.1. Momentum Flow Into Media

As a first example, consider an electromagnetic wave normally incident from

vacuum upon a dielectric boundary at z = 0. It is desirable to study the momen-

tum flow from the Chu and Einstein-Laub formulations as leading contenders

for the kinetic subsystem of electrodynamics [Kemp (2011)]. However, in this

28

analysis, no distinction is made between the two. The fields in the incident

region are

E0 = xEi

[

eikz +Re−ikz]

(58a)

H0 = yHi

[

eikz −Re−ikz]

. (58b)

The excitation power flow is the power available to the medium

〈Sexc〉 = −n · 12ℜ{

E0 × H∗0

}

=1

2EiHi

(

1− |R|2)

=(

1− |R|2)

〈Si〉, (59)

where 〈Si〉 = 12ℜ{EiHi} is the power flow of the incident wave. The reflection

coefficient is R = (n− 1)/(n+ 1). Therefore, the excitation power flow can be

written as

〈Sexc〉 =[

(n+ 1)2

(n+ 1)− (n− 1)

2

(n+ 1)

]

〈Si〉 =4n

(n+ 1)2< Si >= n|T |2〈Si〉, (60)

where n|T |2 = 4n/(n+ 1)2 is the transmissivity [Kong (2005)]. The excitation

energy is completely defined in terms of the vacuum fields. Therefore, the

excitation power flow does not include material contributions. Likewise, the

excitation momentum flow is the momentum available to the medium

〈Texc〉 =1

2ℜ{

ǫ02|E0|2 +

µ0

2|H0|2

}

=1

2ℜ{

EiHi

c

(

1 + |R|2)

}

, (61)

and it is unambiguously defined in terms of the Tzz component of the free space

Maxwell stress tensor in the incident region. Substituting for |R|2 gives

〈Texc〉 =[

(n+ 1)2

(n+ 1)2 +

(n− 1)2

(n+ 1)2

]

〈Si〉c

=2n2 + 2

(n+ 1)2〈Si〉c

. (62)

Finally, the excitation momentum flux reduces to

〈Texc〉 =1

2

(

n+1

n

) 〈Sexc〉c

. (63)

Inside the dielectric, both the Chu and Einstein-Laub formulations yield the

electromagnetic stress

〈Teh〉 =1

2ℜ{

ǫ02|Et|2 +

µ0

2|Ht|2

}

=1

2ℜ{

1

2

(

n+1

n

)

EtHt

c

}

=1

2

(

n+1

n

) 〈Sexc〉c

. (64)

29

The electromagnetic stress appears as the average of n/c and 1/nc times the

excitation power flow. In this form, the transmitted momentum flow may

be written as the average of the transmitted Abraham and Minkowski mo-

menta [Mansuripur (2004)]. However, it is noted that the transmitted momen-

tum flow given in Eq. (64) is equal to the excitation momentum in Eq. (63).

Therefore, the EH representation of the stress tensor in media yields a momen-

tum flow which is equal to the excitation momentum flow at normal incidence.

Both the Chu and Einstein-Laub formulations yield zero surface pressure in this

case as the net momentum flow is conserved.

As a second example, consider an impedance-matched material such that

n = c√µǫ and η =

√

µ/ǫ =√

µ0/ǫ0. The reflectivity is zero in this case. The

excitation momentum flow is simply given by the incident momentum flow

〈Texc〉 =1

2ℜ{

ǫ02|E0|2 +

µ0

2|H0|2

}

=1

2

EiHi

c=

〈Si〉c

. (65)

The transmitted momentum flow given by either the Chu or Einstein-Laub

formulations is

〈Teh〉 =1

2ℜ{

ǫ02|Et|2 +

µ0

2|Ht|2

}

=1

2

EtHt

c=

〈Si〉c

. (66)

In this case, the EH representation yields zero force on the surface since the

momentum flow is conserved.

3.3.2. Stresses in Dielectric Fluids

At normal incidence upon a planar boundary of dielectric and magnetic

media, the EH representation provides for conserved momentum flow at the

boundary. However, this is only a special case. In general, oblique incidence

or curved surfaces give rise to surface forces. Furthermore, wave interference

produces volume forces due to field variations [Kemp et al. (2006a)]. These

forces, in turn, produce material stresses in addition to the electromagnetic

stresses. Before considering the stresses due to interference fringes, it is illus-

trative to consider the electromagnetic and material stress contained within a

plane wave. A time-harmonic, unbounded plane electromagnetic wave exerts

30

zero time-average force on a dielectric fluid. However, a time-average material

stress exists within a propagating plane wave.

Consider a wave with a planar front which is ramped up from zero to electric

and magnetic field amplitudes E0 and H0. The velocity of the wave in the

dielectric fluid is v = c/n, where n = c√µ0ǫ. The electric and magnetic fields

are

E(z, t) = E0(z, t) cos(kz − ωt) (67a)

H(z, t) = H0(z, t) cos(kz − ωt), (67b)

where E0(z, t) = xE0g(z − vt) and H0(z, t) = yH0g(z − vt). The function

g(z − vt) is the ramp function which is slowly varying in comparison to ω and

approximately linear over the spatial length L so that its temporal duration is

L/v. For z > vt + L, the electric and magnetic field amplitudes are zero, and

for z < vt, the electric and magnetic field amplitudes are E0 and H0. Before the

planar front arrives at a region of the material, the electromagnetic and material

stresses are zero. The average electromagnetic stress in the region z < vt is given

by the 〈Tzz〉 component of the Chu or Einstein-Laub stress tensors

〈Πe(z)〉 =1

2ℜ{

ǫ02E2

0 +µ0

2H2

0

}

=1

2

(

n+1

n

) 〈S0〉c

, (68)

where 〈S0〉 = 12E0H0.

The material stress is determined by closing the system. Inside the fluid, the

electromagnetic force is balanced by an equal and opposite force of the material

fe + fm = 0. The material stress is written as −fm = fe = ∇Πm. Considering

the one-dimensional problem, the material stress at t = 0 is

Πm(z) =

∫ L

0

dzfe · z =

∫ L

0

dz

(

∂P∂t

× µ0H)

· z. (69)

Because the ramp is approximately linear, the ∂P/∂t term is constant over

the span [0, L] and can be approximated as P0v/L, where P0 = ǫ0(n2 − 1)E0.

The µ0H term is linear, so the integration yields 12µ0HL. The material stress

reduces to

Πm(z) =

(

P0

Lv

)

(µ0

2H0L

)

=1

2

(

n− 1

n

)

S0

c. (70)

31

Taking the time-average in the region z < vt (i.e. once the time-harmonic plane

wave has been well established) yields

〈Πm(z)〉 = 1

2

(

n− 1

n

) 〈S0〉c

. (71)

This gives the material response contribution to the average stress of a plane

wave. Adding the material and electromagnetic components gives the total

stress or momentum flow associated with the plane wave 〈Π(z)〉 = n〈S0〉/c,which is consistent with the Minkowski momentum. For comparison, evalua-

tion of the Minkowski stress tensor in the region z < vt yields the total stress

associated with the fields and material response

〈ΠMin(z)〉 =1

2ℜ{

ǫ

2E2

0 +µ0

2H2

0

}

= n〈S0〉c

. (72)

3.3.3. Radiation Pressure

Jones and Richards (1954) and later Jones and Leslie (1978) measured the

pressure on submerged mirrors due to optical pressure as a function of refrac-

tive index n =√

ǫ/ǫ0. The conclusion of the JRL experiments is that the

observed pressure on the submerged mirror is proportional to n. The generality

of this conclusion has since been questioned by Mansuripur (2007), who cor-

rectly pointed out that the electromagnetic force on the mirror depends upon

the type of mirror used. The force on a submerged perfect reflector indeed de-

pends upon the phase of the reflection coefficient Rmirror = eiφ. For example,

a perfect electrical conductor (PEC) reflects with a phase φ = π and experi-

ences an electromagnetic pressure of twice the Minkowski momentum, while a

perfect magnetic conductor (PMC) reflects with a phase φ = 0 and experiences

an electromagnetic pressure of twice the Abraham momentum. In this section,

the stress of the material response is added to the electromagnetic stress to de-

duce the observable pressure of light on a submerged reflector with respect to a

submerging fluid [Kemp and Grzegorczyk (2011); Kemp (2012)].

Consider a time-harmonic plane wave E = xE0eikz incident normally from a

dielectric liquid onto a perfect reflector at z = 0. A small vacuum gap of thick-

ness d is introduced so that the reflector is at z = d and the spatial frequency

32

0 20 40 60 80 100 120 140 160 1800

0.5

1

1.5

2

2.5

3

φ [deg.]

Fie

lds

|E1|=Re{1 + R}

|H1|=Re{1 − R}

Figure 4: Normalized fields in a dielectric fluid at the surface of a perfect reflector as a function

of the phase φ of the mirror reflection coefficient (Rmirror = eiφ. The electric field (solid line)

and magnetic field (dashed lines) are out of phase and are plotted as |E1| = ℜ{1 + R} and

|H1| = ℜ{1−R}, respectively.

inside the gap is k0 = ω/c. For the case of a submerged mirror, the air gap

vanishes (d → 0). Inside the dielectric (z < d), the fields form a standing wave

pattern

E1(r) = xEi

(

eikz +Re−ikz)

(73a)

H1(r) = yEi1

η

(

eikz −Re−ikz)

. (73b)

Application of the boundary conditions yield a unique solution for the fields.

The expression for R is [Kemp and Grzegorczyk (2011)]

R =n cos (kd+ φ/2) + i sin (kd+ φ/2)

n cos (kd+ φ/2)− i sin (kd+ φ/2), (74)

where d = 0 for the submerged mirror case. The electric and magnetic fields

form out of phase interference patterns. Figure 4 shows that the electric field

is zero at the surface of a PEC (φ = π) while the magnetic field is zero at the

surface of a PMC (φ = 0).

The electromagnetic stress is determined from the Tzz component of the

time-averaged Chu or Einstein-Laub stress tensor. Substitution of the fields in

the dielectric region z < 0 yields an equation for the time-averaged electromag-

33

netic stress in the dielectric fluid in terms of the index of refraction

〈Πe(z)〉 =[(

n+1

n

)

−(

n− 1

n

)

ℜ{

Re−i2kz}

] 〈S0〉c

. (75)

Here, 〈S0〉 ≡ 12 |E0|2/η = 1

2ℜ{E0H0} is the average incident power.

To close the system, the material response provides an equal and opposite

force to the electromagnetic force. The two subsystems can be written as

〈fe(z)〉 = −〈∇ · ¯Te(z)〉 (76a)

〈fm(z)〉 = −〈∇ · ¯Tm(z)〉 (76b)

where ¯Tm represents the local material stress, and the momentum densities

vanish under time averaging. Since the fluid is nonmagnetic and the wave is

normally incident, the force reduces to a single term

〈fe(z)〉 =1

2ℜ{

− iωP1 × µ0H∗1

}

= −1

2ω (ǫ− ǫ0)µ0ℜ

{

iE1 × H∗1

}

. (77)

The electromagnetic force acts only only in the ±z direction, allowing the use

of scalar quantities. The material stress is the integral of the electromagnetic

force

〈Πm(z)〉 =∫

〈fe(z)〉dz + 〈Π0〉. (78)

Here, the first term is the stress due to the interference of the standing wave

pattern and 〈Π0〉 is the stress due to the individual plane waves. The constant

〈Π0〉 is twice the value given in Eq. (71) since there are counter propagating

plane waves present, each contributing equally to the material response stress.

Therefore,

〈Π0〉 =(

n− 1

n

) 〈S0〉c

, (79)

and the time-average material stress reduces to

〈Πm(z)〉 =[(

n− 1

n

)

+

(

n− 1

n

)

ℜ{

Re−i2kz

}] 〈S0〉c

. (80)

The total stress in the fluid is the sum of the electromagnetic stress in

Eq. (75) and the material response stress in Eq. (80). The sum,

〈Πtotal〉 = 〈Πe(z)〉+ 〈Πm(z)〉 = 2n〈S0〉c

, (81)

34

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

φ [deg.]

Str

ess

[N/m

2 ]

EMMaterialTotal

PEC (φ = 180)PMC (φ = 0)

Figure 5: Stress in a dielectric fluid at the surface of a perfect reflector as a function of the

phase φ of the mirror reflection coefficient (Rmirror = eiφ). The electromagnetic pressure

(triangles) and material pressure (circles) sum to give a constant value 2n〈S0〉/c for the total

stress regardless of reflector phase. The incident momentum is normalized 〈S0〉/c = 1 and the

index of refraction is n = 2.

gives twice the Minkowski momentum as the observed pressure on the mirror

with respect to the submerging fluid regardless of the phase of the reflection

coefficient used. The contributions are plotted in Fig. 5 for n = 2. In the case

of the PEC reflector (φ = π) as used in the JRL experiment, the material stress

does not contribute. This is because the fields are zero at the surface as shown

in Fig. 4, yielding zero material energy and momentum contributions at z = 0.

Therefore, any formulation will directly yield identical results for the pressure

on the mirror, which means the JRL experiment does not actually provide a true

test for the momentum of light. The PMC case (φ = 0) is different. Figure 4

shows that the electric field is nonzero at the mirror surface, and the mate-

rial contributes to both the local energy and momentum of the wave. Adding

these contributions restores the Minkowski pressure to the reflector [Kemp and

Grzegorczyk (2011); Kemp (2012)].

4. Momentum and Stress in Dispersive Media

Real materials exhibit frequency dispersion and losses. This is a necessary

condition of causality [Landau, Lifshitz, and Pitaevskii (1984)]. The inclusion

35

of losses requires that a specific model for P and M be applied. To arrive at

a causal medium model for a dielectric, the equation of motion for a bounded

electron under the action action of an electric field is applied such that

qE = m

[

∂2r

∂t2+ γe

∂r

∂t+ ω2

0 r

]

, (82)

where q is the electron charge, m is the mass of the electron, γe is a damping

constant, ω0 is the resonant frequency, and r is the local displacement. By

defining the polarization P = Nqr as the number density N of such oscillators

times the dipole moments qr, constitutive relations can be derived for a Lorentz

dielectric in the usual way [Kong (2005)]. It has been shown in Section 3 that the

Chu formulation is a reasonable choice for the electromagnetic field subsystem.

For the derivation of the wave energy in materials exhibiting both dielectric and

magnetic response (e.g. metamaterials), the stationary approximation of the

Chu formulation is usually chosen as the starting point [Kong (2005); Cui and

Kong (2004); Ramakrishna and Grzegorczyk (2008)]. Here, the same approach

is taken for energy and momentum. In this section, the dielectric and magnetic

material response are derived using a Lorentz medium model [Loudon, Allen,

and Nelson (1997); Kemp, Kong, and Grzegorczyk (2007)].

4.1. Field and Material Contributions

The material response to the electromagnetic fields is described by the dif-

ferential equations for a Lorentz medium

(

∂2

∂t2+ γe

∂

∂t+ ω2

e0

)

P = ǫ0ω2epE (83a)

(

∂2

∂t2+ γm

∂

∂t+ ω2

m0

)

M = Fω2mpH, (83b)

where the parameters of the equations have their usual meanings [Cui and Kong

(2004)]. To derive the energy of the electromagnetic wave, the material Equa-

tions (83a) and (83b) are dot multiplied by Je and Jh, respectively. The

36

resulting equations

Je · E =1

2ǫ0ω2ep

∂

∂t

{

∂P∂t

· ∂P∂t

+ ω2e0P · P

}

+γe

ǫ0ω2ep

∂P∂t

· ∂P∂t

(84a)

Jh · H =µ0

2Fω2mp

∂

∂t

{

∂M∂t

· ∂M∂t

+ ω2m0M · M

}

+γmµ0

Fω2mp

∂M∂t

· ∂M∂t

(84b)

are then added to the energy conservation equation of the electromagnetic sub-

system given by the Chu formulation. The resulting energy conservation equa-

tion for the electromagnetic wave is in the form of Eq. (3b) with the energy flow

S, energy density W , and energy dissipation ϕ given by

S = E × H (85a)

W =ǫ02E · E +

µ0

2H · H+

1

2ǫ0ω2ep

[

∂P∂t

· ∂P∂t

+ ω2e0P · P

]

+µ0

2Fω2mp

[

∂M

∂t· ∂M

∂t+ ω2

m0M · M]

(85b)

ϕ =γe

ǫ0ω2ep

∂P∂t

· ∂P∂t

+γmµ0

Fω2mp

∂M∂t

· ∂M∂t

. (85c)

The energy flow S in Eq. (85) retains its free-space form even in the presence of

a lossy, dispersive material. Also, the energy density W contains contributions

from the potential energy and kinetic energy of the electric and magnetic dipoles.

Furthermore, the form of Eq. (85b) has been regarded as significant since the

energy density remains positive in negative index of refraction media [Cui and

Kong (2004)]. Furthermore, the energy dissipation term ϕ depends upon the

damping factors γe and γm in Eq. (83). Therefore, ϕ = 0 in the limiting case

of a lossless material, which indicates the energy of the electromagnetic wave is

conserved.

The wave momentum can be derived by a similar method. The material

dispersion Eqs. (83) are dot multiplied by the dyads −∇P and −µ0∇M, re-

spectively. The resulting vector equations are then added to the electromagnetic

37

continuity equation given by the Chu formulation to yield

∇ · ¯Teh +∂Geh

∂t+ feh + E · ∇P + µ0H · M

− ∇P ·(

∂2P∂t2

+ ω2e0P

)

1

ǫ0ω2ep

− µ0∇M ·(

∂2M∂t2

+ ω2m0M

)

µ0

Fω2mp

= ∇P · ∂P∂t

γeǫ0ω2

ep

+∇M · ∂M∂t

γmµ0

Fω2mp

. (86)

Vector calculus allows terms to be rewritten as

−∇P ·(

∂2P∂t2

+ ω2e0P

)

1

ǫ0ω2ep

= ∇ ·[

1

2ǫ0ω2ep

(

∂P∂t

· ∂P∂t

− ω2e0P · P

)]

− ∂

∂t

[

1

ǫ0ω2ep

(

∇P · ∂P∂t

)]

(87a)

−∇M ·(

∂2M∂t2

+ ω2m0M

)

µ0

Fω2mp

= ∇ ·[

µ0

2Fω2mp

(

∂M∂t

· ∂M∂t

− ω2m0M · M

)]

− ∂

∂t

[

µ0

Fω2mp

(

∇M · ∂M∂t

)]

(87b)

feh + E · ∇P + µ0H · M = ∇ ·[

(

P · E + µ0M · H) ¯I − PE − µ0MH

]

+∂

∂t

[

D × B − ǫ0µ0E × H]

. (87c)

By combining Eq. (86) and Eq. (87), the momentum continuity equation for the

wave can be expressed in the form of Eq. (3b) with the stress tensor, momentum

density, and force density

¯T =1

2(D · E + B · H) ¯I − DE − BH+

1

2

[

1

ǫ0ω2ep

(

∂P∂t

· ∂P∂t

− ω2e0P · P

)

+µ0

Fω2mp

(

∂M∂t

· ∂M∂t

− ω2m0M · M

)

+ (P · E + µ0M · H)

]

¯I (88a)

G = D × B − 1

ǫ0ω2ep

∇P · ∂P∂t

− µ0

Fω2mp

∇M · ∂M∂t

(88b)

f = − γeǫ0ω2

ep

∇P · ∂P∂t

− γmµ0

Fω2mp

∇M · ∂M∂t

. (88c)

The expressions in Eqs. (88) define the quantities for the momentum continuity

equation, which complete the wave subsystem along with the energy expres-

siongs in Eqs. (85). The momentum density G contains the Minkowski momen-

tum D × B plus material dispersion terms. Likewise, the momentum flow ¯T is

38

the Minkowski stress tensor plus dispersive terms. Note that the momentum

dissipation term f depends upon the damping factors γe and γm in Eqs. (83).

Therefore, f = 0 in the limiting case of a lossless, unbounded material, which

indicates that the momentum of the electromagnetic wave is conserved.

4.2. Time-Averaged Energy and Momentum Propagation

Consider the propagation of time-harmonic electromagnetic waves or elec-

tromagnetic fields which are contained within a narrow frequency band. The

constitutive parameters follow directly from Eqs. (83) given e−iωt dependence

[Kong (2005)]. The complex permittivity and permeability ,

ǫ(ω) = ǫ0

(

1−ω2ep

ω2 − ω2e0 + iωγe

)

(89a)

µ(ω) = µ0

(

1−Fω2

mp

ω2 − ω2m0 + iωγm

)

, (89b)

are functions of the frequency ω and consist of real and imaginary parts denoted

by ǫ = ǫR + iǫI and µ = µR + iµI . Likewise, the time-average of the squared

polarization and magnetization are

|P |2 =ǫ20ω

4ep

(ω2 − ω2e0)

2+ γ2

eω2|E|2 (90a)

|M |2 =F 2ω4

mp

(ω2 − ω2m0)

2+ γ2

mω2|H |2. (90b)

Similarly, |∂P /∂t|2 = ω2|P |2 and |∂M/∂t|2 = ω2|M |2, which can be applied to

determine the time-average values relating to energy and momentum conserva-

tion given in the previous section.

The time average energy density found from Eq. (85b) is

〈W 〉 =ǫ02

[

1 +ω2ep

(

ω2 + ω2e0

)

(ω2 − ω2e0)

2+ γ2

eω2

]

∣

∣E∣

∣

2

+µ0

2

[

1 +Fω2

mp

(

ω2 + ω2m0

)

(ω2 − ω2m0)

2+ γ2

mω2

]

∣

∣H∣

∣

2. (91)

In the lossless case, γe = 0 and γm = 0 implies that both ǫI = 0 and µI = 0, and

the energy density satisfies the well-known relation [Brillouin (1960); Landau

39

et al. (1984); Jackson (1999); Kong (2005)]

〈W 〉 = 1

4

∂(ǫω)

∂ω

∣

∣E∣

∣

2+

1

4

∂(µω)

∂ω

∣

∣H∣

∣

2. (92)

The extension of Eq. (92) to lossy materials has been criticized due to the pos-

sibility of negative values for negative refractive index metamaterials (i.e. ma-

terials with n < 0). In contrast, the average energy density in Eq. (91) remains

positive for all ω. However, it is the rate of change in energy that appears in

the energy continuity equation, which tends to zero upon cycle averaging. That

is, 〈∂W/∂t〉 = 0, and the resulting conservation equation

−〈∇ · S〉 = 1

2

[

ωǫI |E|2 + ωµI |H |2]

(93)

is generally regarded as the complex Poynting’s theorem, where 〈S〉 = 12ℜ{E ×

H∗} is the time average Poynting power.

Similarly, the average momentum density,

〈G〉 =1

2ℜ{

D × B∗ + kǫ0ωω

2ep

(ω2 − ω2e0)

2 + γ2eω

2|E|2

+ kµ0ωFω2

mp

(ω2 − ω2m0)

2 + γ2mω2

|H |2}

, (94)

is obtained from Eq. (88b). It is simple to show using Eq. (89) that the average

momentum given in Eq. (94) satisfies [Veselago (1968)]

〈G〉 = 1

2ℜ{

ǫµE × H∗ +k

2

(

∂ǫ

∂ω|E|2 + ∂µ

∂ω|H |2

)}

(95)

when the medium is lossless. The time average stress tensor is

〈 ¯T 〉 = 1

2ℜ{

1

2(D · E∗ + B · H∗) ¯I − DE∗ − BH∗

}

(96)

since the dispersive terms in Eq. (88a) tend to zero upon cycle averaging. Since

the average rate of change in momentum density is zero (i.e. 〈∂G/∂t〉 = 0), the

momentum conservation theorem for a monochromatic wave reduces to

−〈∇ · ¯T 〉 = 1

2ℜ{

ωǫIE × B∗ − ωµIH × D∗

}

(97)

where the tensor is the complex Minkowski tensor. The right-hand side of

Eq. (97) is defined as the force density on free currents by Loudon, Barnett,

and Baxter (2005) and Kemp, Grzegorczyk, and Kong (2006b).

40

5. Discussion

Several conclusions can be made from the preceding sections. Each conclu-

sion adds to the logical deduction of how optical manipulation experiments can

be modeled within the macroscopic theory of electrodynamics. These conclu-

sions are listed in the following section followed by a brief discussion of their

applicability to optical manipulation.

5.1. Conclusions

The conclusions are ordered by section number so that a logical flow is

presented and reference to the supporting analysis is maintained.

Conclusions 1: A number of electromagnetic force density equations per-

sist in the optics literature. Each is tied to a particular formulation of elec-

trodynamics. The most commonly used force equations are considered herein

with the goal of modeling modern experiments involving the manipulation of

macroscopic media with light.

Conclusions 2.1: Unique formulations of electrodynamics differ in how

matter is modeled with each resulting in various forms for the electromagnetic

energy and momentum continuity equations. Within media, the force density,

momentum density, and stress tensor are determined by which representation

of Maxwell’s theory is applied. When a medium is in motion, the field vectors

are also unique to a particular formulation. In vacuum, the fields as well as the

energy and momentum continuity equations are unambiguous.

Conclusions 2.2: A quasi-stationary approximation can be applied that al-

lows for the study of optical momentum transfer to media while optical energy

is conserved. This approximation is common and greatly reduces the complex-

ity involved with modeling optical manipulation experiments. Essentially, the

Doppler shift of the electromagnetic wave due to material motion is ignored such

that calculations predict momentum exchange due to reflection or scattering,

but optical energy is conserved.

Conclusions 2.3: The most commonly used electromagnetic force equa-

tions assume stationary media, and are each associated with a momentum con-

41

tinuity equation. The sum of the force density, divergence of the stress tensor,

and rate of change of the momentum density must be zero. Under the quasi-

stationary approximation, the field variables are equivalent among the formu-

lations in vacuum and in media. Terms may be rearranged between the force

density, momentum density, and stress tensor. Such exercises define new elec-

tromagnetic subsystems, but the resulting energy and momentum continuity

equations are left for interpretation.

Conclusions 3.1: Any formulations which share a common momentum

density will produce identical results for the total electromagnetic force on a

material object at any point in time, and all formulations produce identical

values for the total time-average electromagnetic force on a material object. This

statement is true as long as the momentum continuity equations are consistent

with Maxwell’s equations. Additionally, it is impossible to infer a time-domain

force from a time-averaged force.

Conclusions 3.2: The total calculated electromagnetic force on an object

due to an optical pulse of finite extent is due to the time rate of change of

the momentum contained within the pulse. The momentum contained within

a pulse depends upon the momentum density that appears in the momentum

continuity equation. The Abraham momentum gives the kinetic momentum

of light, which is responsible for center of mass translations of media. Any

subsystem that is to be considered as consisting of field only quantities must

include the Abraham momentum density.

Conclusions 3.3: The total average electromagnetic force on an object due

to time-harmonic fields is due to the divergence of the average stress contained

within the fields. The Chu and Einstein-Laub formulations include the Abra-

ham momentum density in the momentum continuity equations. Both predict

conservation of momentum flow due to normal incident transmission and re-

flection at a boundary, both reveal that a plane electromagnetic wave includes

a nonzero average material response stress in addition to the electromagnetic

stress, and both predict an additional material contribution to radiation pres-

sure when an object is submerged in a dielectric fluid. In regards to the last

42

point, the observed optical force on a submerged object is due to the electro-

magnetic force on the object and the electromagnetic force on the submerging

fluid. The pressure difference at the boundary yields the Minkwoski momentum

imparted to the submerged object with respect to the submerging fluid, which is

key to reducing the complexity of modeling optical manipulation experiments.

Conclusions 4.1: Causality requirements for media to be lossy and dis-

persive necessitate the inclusion of material dynamics in the wave energy and

momentum continuity equations. Starting with the Chu formulation and mod-

eling the material response as harmonic oscillators, the energy and momentum

response of the material can be added to the electromagnetic energy and mo-

mentum continuity equation. The resulting wave energy density, power flux,

momentum density, and stress tensor take the Minkowski forms plus additional

material dynamics terms.

Conclusions 4.2: Under time-harmonic excitation, the time-averaged stress

tensor for the electromagnetic wave, which includes both electromagnetic and

material response, reduces to the time-average Minkowski stress tensor. There-

fore, Conclusions 3.2 are valid for dispersive media when the optical excitation

has a narrow bandwidth. Otherwise, the conclusions hold, but the Minkwoski

stress tensor must be supplemented with additional terms due to material dis-

persion.

5.2. Application to Optical Manipulation

The conclusions are applied to model the forces exerted on particles by

an optical field. The modeling process requires two steps. First, the total

fields must be found, which can be done by numerical or analytical techniques.

Second, the force on the particle is calculated. Here, cylinders and spheres are

used to represent particles in two-dimensions and three-dimensions, respectively.

The analytical field solutions for cylinders and spheres are provided in Appendix

A and Appendix B, respectively.

A cylindrical particle is used as a computationally efficient way to model

physical phenomena. A laser can be modeled in two dimensions by a Gaussian

43

Figure 6: Force per unit length (represented by the arrows) on a single infinite cylinder due to

an incident Gaussian beam with NA = 1, E0 = 1, w = 0.5λ0, and wavelength λ0 = 1064 nm.

The background medium is water ǫb = 1.69ǫ0 and the radius of the particle is a = 0.5λ0. The

particle is (a) polystyrene and (b) air.

beam [Shen and Kong (1987)]

Einc = zE0w

2√π

∞∫

−∞

dkxe−

w2k2x

4 eikxx+ikyy, (98)

where k2y = k2b − k2x, k2b = ω2µbǫb, and w is the beam waist. The electric field

in Eq. (98) satisfies the wave equation [Kong (2005)] and is implemented as a

discrete sum of plane waves. The spectrum is actually limited, and the sum is

implemented here within the domain |kx| ≤ NAωc , where NA is the numerical

aperture.

Figure 6 illustrates a model of the original experiments by Ashkin (1970).

44

Figure 7: Classical optical tweezers modeled as a 2-D dielectric particle in water. The arrows

represent the force on a silica particle with index of refraction np = 1.46 and radius a = 50 nm

in water (nb = 1.33). The incident laser is modeled as a Gaussian beam with waist w = 0.2 λ0,

λ0 = 514.5 nm, and spectrum defined by NA = 1.25.

The arrows depict the observed force on either a polystyrene particle or air

bubble in water due to a Gaussian beam described in Eq. (98). The divergence

of the Minkowski stress tensor in the background water is applied to calculate

the time-average force on the 2-D particle (i.e. cylinder) at different points.

This approach is consistent with the results of Sections 3.3 and 4.2. The forces

in Fig. 6 agree with the observed forces by Ashkin (1970); radiation pressure on

both particles is accompanied by a gradient force that attracts the higher index

of refraction particles (polystyrene) into the center of the beam and repels the

lower refractive index air bubbles out of the center of the beam. Since the air

bubbles have a polarization of P = 0, it is actually the kinetic force that pulls

the water into the high field intensity region of the beam. However, the pressure

gradient that is responsible for manipulating the air bubble can be equivalently

computed using Minkowski’s tensor, as demonstrated here.

Figure 7 illustrates classical optical tweezers in two dimensions as originally

demonstrated by Ashkin et al. (1986). The Gaussian beam is focused such that

the so-called gradient force overcomes the scattering force. The parameters

match closely the experimental conditions used in the original experiments to

45

trap silica beads. Therefore, the modeling approach again matches the exper-

imental outcome as the optical force is seen to produce a stable trap near the

beam focus.

Optical manipulation of colloidal particles is typically achieved by creation

of an optical intensity gradient as seen with optical tweezers. One such config-

uration consists of three lasers with 2π/3 incident angle separation [Fournier,

Boer, Delacetaz, Jacquot, Rohner, and Salathe (2004)]. Such a configuration

results in a zero net incident momentum since the vector sum of the incident

momentum vectors is zero, which allows for isolation of the gradient force. This

interference is represented by the background of Fig. 8, where white denotes

regions of high field intensity. The force on a 2-D polystyrene particle in water

is calculated via the divergence of the Minkowski stress tensor as prescribed in

Sections 3.3 and 4.2. Each arrow represents the direction and relative magni-

tude of force on a dielectric cylinder placed at the tail of the arrow. The force

on the small particle with radius a = 0.15λ0 is seen to closely follow the high

intensity gradient of the incident field with stable optical traps occurring in the

high intensity regions. However, larger particles with radius a = 0.30λ0 are

repelled from the high intensity regions and find stable trapping positions in

the dark regions of the incident optical interference pattern. Obviously, this

result contrasts the simple description given by the gradient force obtained in

the Rayleigh approximation. The example demonstrates that such an approxi-

mation yields results which are both quantitatively and qualitatively incorrect

for particles that are on the order of a wavelength. It is also interesting to study

the intermediate particle size regime. For example, the force on a particle with

radius a = 0.225λ0 is also plotted over the background of the incident three

plane wave interference pattern in Fig. 8. An interesting feature of this plot is

that the force field appears to be non-conservative, and a prediction of stable

optical trapping in these vortices has been reported by Grzegorczyk and Kong

(2007).

A concluding example demonstrates the dependence of radiation pressure

on the background fluid. Consider a spherical particle surrounded by either

46

Figure 8: Force per unit length (represented by the arrows) on a single infinite cylinder due

to the interference of three plane waves (represented by the background pattern) of equal

amplitude Ei = 1 [V/m] and wavelength λ0 = 532 nm. The incident angles of the plane

waves are {π/2, 7π/6, 11π/6}. The background medium is water ǫb = 1.69ǫ0, and the cylinder

is polystyrene ǫp = 2.56ǫ0 with radius (a) a = 0.15λ0, (b) a = 0.30λ0, and (c) a = 0.225λ0.

47

50 100 150 200 250 300 350 400 450 500−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10−12

a [nm]

For

ce [N

]

(εb,µ

b)=(ε

0,µ

0)

(εb,µ

b)=−(ε

0,µ

0)

Figure 9: Radiation force on a spherical particle with respect to a submerging background

medium as a function of sphere radius a. The particle has is nonmagnetic with ǫ = 4ǫ0. The

solid line gives the results for a background medium of vacuum (ǫ0, µ0), and the dashed line

depicts the force for a hypothetical inverted vacuum (−ǫ0,−µ0). The incident field is a plane

wave with amplitude E0 = 1 V/m and vacuum wavelength λ0 = 1064 nm.

vacuum or a negative refractive index medium with negative vacuum parameters

(−ǫ0,−µ0). The radiation pressure is reversed according to the theory of Kemp

et al. (2007), which is reproduced in Section 4. Figure 9 shows the force on a

sphere as a function of radius a, which is varied from the Rayleigh regime to

the Mie regime. The fields are calculated by the analytical approach detailed in

Appendix B. The results show that the force on the sphere depends upon the

refractive index of the background medium. The particle in vacuum is pushed,

while the radiation pressure is reversed when the background refractive index

is negative as first predicted by Veselago (1968).

Appendix A. Scattering by a Cylinder

The geometry of the problem consists of an electromagnetic wave incident

upon an infinite cylinder of radius a aligned in the z direction. The cylinder

is characterized by (µp, ǫp) and the background by (µb, ǫb). The incident wave

is assumed to be a plane wave with e−iωt dependence. Many other field distri-

butions, such as a Gaussian beam, can be described by a sum of plane-waves.

Therefore, the total solution for such incident fields can be described as a su-

48

perposition of solutions resulting from a plane-wave. The incident, scattered,

and internal fields are expanded in cylindrical waves given by Nn, Mn, RgNn,

and RgMn. The solution is given by Tsang, Kong, and Ding (2000).

The incident electric field is polarized in the z-direction and propagates in

the plane (kz = 0). The magnetic fields are obtained from Faraday’s law

iωµH(ρ) = ∇× E(ρ) (A.1)

using the identities

∇× Mn = kNn (A.2a)

∇× Nn = kMn. (A.2b)

The incident fields are regular at the origin and are given by

Einc(ρ) = zE0eiki·ρ =

N∑

n=−N

anRgNn(kb, ρ) (A.3a)

Hinc(ρ) =kb

iωµb

N∑

n=−N

anRgMn(kb, ρ), (A.3b)

where the wavenumber in the background medium is given by the dispersion

relation k2b = ω2µbǫb. The scattered fields are

Escat(ρ) =

N∑

n=−N

asnNn(kb, ρ) (A.4a)

Hscat(ρ) =kb

iωµb

N∑

n=−N

asnMn(kb, ρ). (A.4b)

The internal fields are also regular at the origin and can be computed from

Eint(ρ) =

N∑

n=−N

cnRgNn(kp, ρ) (A.5a)

Hint(ρ) =kp

iωµp

N∑

n=−N

cnRgMn(kp, ρ), (A.5b)

where the wavenumber in the cylinder (ρ < a) is given by k2p = ω2µpǫp. The

unknown coefficients (asn, cn) are determined by matching the boundary condi-

tions and the coefficients (an) are known from the incident plane wave expan-

sion Tsang et al. (2000). The expressions for incident, scattered, and internal

49

fields become exact for N → ∞. The vector cylindrical wave functions for this

particular incidence are given by Stratton (1941)

Nn(k, ρ) = zkH(1)n (kρ)einφ (A.6a)

Mn(k, ρ) = ρ

[

in

ρH(1)

n (kρ)

]

einφ

+ φ

[

kH(1)n+1(kρ)−

n

ρH(1)

n (kρ)

]

einφ (A.6b)

RgNn(k, ρ) = zkJn(kρ)einφ (A.6c)

RgMn(k, ρ) = ρ

[

in

ρJn(kρ)

]

einφ

+ φ

[

kJn+1(kρ)−n

ρJn(kρ)

]

einφ. (A.6d)

Here, H(1)n (·) is the Hankel function of the first kind and Jn(·) is the Bessel

function. The coordinates (ρ, θ, φ) represent the point for field evaluation (i.e.

the observer position). The angle φi is used to represent the incident direction

of the illuminating wave.

For a general dielectric and magnetic medium, the boundary conditions give

a 2× 2 system of equations that must be solved for the unknown coefficients,

an = ineinφi

kbE0 (A.7a)

asn =b1m22 − b2m12

m11m22 −m21m12(A.7b)

cn =b1m21 − b2m11

m12m21 −m22m11. (A.7c)

The matrix elements and right-hand-side for the linear system are

m11 = −kbH(1)n (kba) (A.8a)

m12 = kpJn(kpa) (A.8b)

m21 = −kbH(1)n+1(kba) +

n

aH(1)

n (kba) (A.8c)

m22 =µbkpµpkb

[

kpJn+1(kpa)−n

aJn(kpa)

]

(A.8d)

b1 = ankbJn(kba) (A.8e)

b2 = an

[

kbJn+1(kba)−n

aJn(kba)

]

. (A.8f)

50

Appendix B. Scattering by a Sphere

The solution for a plane wave scattering from a Sphere is referred to as Mie

theory and the solution is given by Kong (2005). The incident electric field is

polarized in the x-direction and propagates in the z-direction. The background

is characterized by (ǫb, µb) with a wavenumber kb. The particle is composed of

(ǫp, µp) with wavenumber kp. The incident fields are

Einc = xE0eikbz = xE0e

ikbr cos θ (B.1a)

Hinc = yE0

ηbeikbz = y

E0

ηbeikbr cos θ. (B.1b)

The solution is found by expanding the incident, scattered, and internal fields

as a sum of spherical waves. The unknown coefficients for the scattered and

internal fields are found from the boundary conditions.

The incident wave is expanded in spherical modes using the identity

eikr cos θ ≈N∑

n=0

(−i)−n(2n+ 1)jn(kr)Pn(cos θ), (B.2)

where jn(kr) is a spherical Bessel function and Pn(cos θ) is the Legendre func-

tion. The approximation is exact as N → ∞.

The incident, scattered, and internal waves are decomposed into TM to r

and TE to r waves by using scalar potentials πe and πm respectively. The

potentials are defined by

A = rπe (B.3a)

H = ∇× A = θ1

sin θ

∂

∂φπe − φ

∂

∂θπe (B.3b)

for TM waves and

Z = rπm (B.4a)

E = ∇× Z = θ1

sin θ

∂

∂φπm − φ

∂

∂θπm (B.4b)

for TE waves. The potentials satisfy the Helmholtz equation in spherical coor-

dinates(

∇2 + k2)

π = 0. (B.5)

51

The scaler potentials are

πe =N∑

n=1

Anzn(kr)P1n (cos θ) cosφ (B.6a)

πm =

N∑

n=1

Bnzn(kr)P1n(cos θ) sinφ, (B.6b)

where zn(kr) represents solutions to the Bessel equation. The solutions differ

in the three regions by the following.

• Incident (πe, πm)

zn(kr) = jn(kbr)

An = E0kbωµb

· (−i)−n(2n+ 1)

n(n+ 1)

Bn = −Anωµb

kb

• Scattered (πse , π

sm)

zn(kr) = hn(kbr)

An = E0kbωµb

· an

Bn = −E0 · bn

hn(kbr) = h(1)n (kbr) is the spherical Hankel function of the first kind.

• internal (πie, π

im)

zn(kr) = jn(kpr)

An = E0kpωµp

· cn

Bn = −E0 · dn

The unknown coefficients an, bn, cn, dn are found using the boundary condi-

tions for the tangential electric field and tangential magnetic field.

52

The total fields are:

Er = ik

ωǫ

N∑

n=1

AnP1n(cos θ) [krzn(kr) + 2z′n(kr) + krz′′n(kr)] cosφ

(B.10a)

Eθ =

N∑

n=1

{

iAnk

ωǫ

∂

∂θP 1n(cos θ)

[

1

krzn(kr) + z′n(kr)

]

+ BnP 1n(cos θ)

sin θzn(kr)

}

cosφ (B.10b)

Eφ = −N∑

n=1

{

iAnk

ωǫ

P 1n(cos θ)

sin θ

[

1

krzn(kr) + z′n(kr)

]

+ Bn∂

∂θP 1n(cos θ)zn(kr)

}

sinφ (B.10c)

Hr = −ik

ωµ

N∑

n=1

BnP1n(cos θ) [krzn(kr) + 2z′n(kr) + krz′′n(kr)] sinφ

(B.10d)

Hθ = −N∑

n=1

{

iBnk

ωµ

∂

∂θP 1n(cos θ)

[

1

krzn(kr) + z′n(kr)

]

− AnP 1n(cos θ)

sin θzn(kr)

}

sinφ (B.10e)

Hφ = −N∑

n=1

{

iBnk

ωµ

P 1n(cos θ)

sin θ

[

1

krzn(kr) + z′n(kr)

]

+ An∂

∂θP 1n(cos θ)zn(kr)

}

cosφ. (B.10f)

The spherical Bessel and Hankel functions zn(ξ) are given in terms of the Bessel

and Hankel functions Zn(ξ) by

zn(ξ) =

√

π

2ξZn+ 1

2

(ξ). (B.11)

To evaluate the derivatives, it is helpful to define l = n+2, m = n+1, p = n−1,

and q = n− 2 and use the relation Stratton (1941)

z′n(ξ) =n

2n+ 1zp(ξ)−

m

2n+ 1zm(ξ). (B.12)

53

The second derivative with respect to the argument ξ is

z′′n(ξ) =n

(2n+ 1)(2p+ 1)[pzq(ξ)− nzn(ξ)]

− m

(2n+ 1)(2m+ 1)[mzn(ξ) + lzl(ξ)] . (B.13)

The derivative of the Legendre function with respect to the argument is, in

general, known. Therefore, a useful relation is

∂

∂θPmn (cos θ) = − sin θ

∂

∂ cos θPmn (cos θ). (B.14)

The coefficients as determined from the boundary conditions are

an = gn · kbǫpJn(kpa)J′n(kba)− kpǫbJn(kba)J

′n(kpa)

kpǫbHn(kba)J ′n(kpa)− kbǫpJn(kpa)H ′

n(kba)(B.15a)

bn = gn · kbµpJn(kpa)J′n(kba)− kpµbJn(kba)J

′n(kpa)

kpµbHn(kba)J ′n(kpa)− kbµpJn(kpa)H ′

n(kba)(B.15b)

cn = gn · ikbǫpµp

kbǫpµbJn(kpa)H ′n(kba)− kpǫbµpHn(kba)J ′

n(kpa)(B.15c)

dn = gn · ikpµp

kbµpJn(kpa)H ′n(kba)− kpµbHn(kba)J ′

n(kpa)(B.15d)

where

gn =(−i)−n(2n+ 1)

n(n+ 1). (B.16)

The Ricatti-Bessel functions Jn(ξ) and Hn(ξ) = H(1)n (ξ) are

Zn(ξ) = ξzn(ξ) (B.17a)

Z ′n(ξ) = zn(ξ) + ξz′n(ξ). (B.17b)

Acknowledgments

The author is grateful to Masud Mansuripur helpful discussions and, in par-

ticular, for pointing out discrepancies in the literature regarding the Abraham

stress tensor. The author acknowledges Cheyenne J. Sheppard for proofreading

parts of the manuscript.

This work was sponsored in part by the National Science Foundation EECS

Division of Electrical, Communications, and Cyber Systems (award number

54

ECCS-1150514). The authors would like to thank the Arkansas Science and

Technology Authority for funding provided to the Center for Efficient and Sus-

tainable Use of Resources (CESUR).

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