**2.3 Optical trapping in Ray-optics regime**

**2.3.2 Mathematical formulation of force calculation model in the ray optics**

2.3. Optical trapping in Ray-optics regime

𝑓

Optical axis

lens

Particle

### 1aser b eam

### 𝑆

Fig. 2.4 Schematic diagram representing ray-pencil model.

### 2.3.2 Mathematical formulation of force calculation model in the ray op-

Chapter 2: Force Calculation Model of Optical trapping

𝑃

𝑃^{′} 𝑃^{′′}

𝜔_{0}

𝑟_{𝑃}

𝑟_{𝑆}

Optical axis

Spherical wavefront

Particle

### 𝑆

Fig. 2.5 Schematic diagram depicting the spherical wavefront model.

where,z_{r}is the Rayleigh length and is given by
z_{r}= π ω_{0}^{2}

λ (2.6)

Hereω_{0} is the beam waist and λ is wavelength of light. z_{p} is the distance between S and
P^{′} which is the z coordinate ofPand S is the position of beam waist. The model uses the
approximation|SP^{′}|=|SP^{′′}|, which is the case whenr_{P} is very large or when the wavefront
is almost planar. Net force on the particle is obtained by integrating over all the rays those
are incident on the particle within the cone. The limit of the integration are obtained from
the direction of⃗r_{p} . However, the SWF model provides correct optical trapping force only
under paraxial focusing conditions. On the contrary the ray-pencil model is applicable for both
paraxial and strong focus conditions.

Force due to a single ray

Figure2.6(i) shows a single ray with powerdPincident on a dielectric sphere at an angle of
incidenceα. The incident momentum isn_{med}dP/cwheren_{med} is the refractive index of the
medium. Part of the incident ray gets reflected with powerdP Rand the refracted ray undergoes
infinite number of reflections and refractions in succession with decreasing powers,dP T^{2},
dP T^{2}R,...., dP T^{2}R^{n}. HereRand T are the Fresnel reflection and transmission coefficients,

2.3. Optical trapping in Ray-optics regime

𝛼

𝛽 𝑜

𝑑𝑃

𝑑𝑃𝑅

𝑑𝑃𝑇^{2}
𝑑𝑃𝑇^{2}𝑅

𝜙 𝜓

𝜓 + 𝜙

Incident ray

(i)

𝐹_{𝑠𝑐}

𝐹_{𝑔𝑟𝑎𝑑}

𝐹_{𝑡𝑜𝑡}
𝑦^{′}
𝑧^{′}

𝑜

𝑜 Ԧ𝑟

𝑏 Ԧ 𝑔

𝑥 𝑧

𝛼

(ii)

𝜃

𝜒

Fig. 2.6 Geometry for force calculation in the ray optics regime due to a single incident ray of powerdP. (i) shows the multiple reflections and refractions of the single ray incident on the spherical particle and (ii) shows the geometry to calculate the incident angle when the sphere is shifted by⃗rfrom the focus.

respectively. Due to the momentum change experienced by this single ray a force acts through
the center of the sphere. This force can be divided into two componentsdF_{z}^{′} acting along ˆz^{′}
anddF_{y}′ acting along ˆy^{′}given as [38]

dF_{z}′ =n_{med}dP

c −hn_{med}dP R

c cos(π−2α) +

∞

### ∑

n=0

n_{med}dP T^{2}R^{n}

c cos(ψ+nφ)i

(2.7)

and

dF_{y}′=0−hn_{med}dP R

c sin(π−2α)−

∞ n=0

### ∑

n_{med}dP T^{2}R^{n}

c sin(ψ+nφ) i

(2.8)
We combine the two components in complex plane asdF_{tot} =dF_{z}′ +idF_{y}′, so that we can write

dF_{tot} = n_{med}dP

c (1+Rcos 2α)−in_{med}dP R

c sin 2α−n_{med}dP T^{2}
c

∞ n=0

### ∑

R^{n}e^{−i(ψ+nφ}^{)} (2.9)

Chapter 2: Force Calculation Model of Optical trapping Taking real and imaginary parts we get

dF_{z}′= n_{med}dP
c

h

1+Rcos(2α)−T^{2}[cos(2α−2β) +Rcos(2α)]

1+R^{2}+2Rcos(2β)
i

(2.10) and

dF_{y}′ =−n_{med}dP
c

h

Rsin(2α)−T^{2}[sin(2α−2β) +Rsin(2α)]

1+R^{2}+2Rcos(2β)
i

(2.11) More details about the derivation of the above force components can be found in Appendix A.

The forcedF_{z}′ as given by Eq.2.10is called scattering force which acts in the propagation
direction of the incident ray and the forcedF_{y}′as given by Eq.2.11is called the gradient force
which acts perpendicular to the scattering force. Both the force components depend on the
powerdPcarried by the ray, refractive index of the medium, angle of incidence, and angle of
refraction. To be noted that since the Fresnel transmission and reflection coefficientsRand
T depend on polarization of light [1] the two forces are also polarization dependent. If we
consider the light to be randomly polarized or circularly polarized so that each ray has equal
amount of s and p polarizations then we can write

R(α,β) =1 2

hsin(α−β) sin(α+β)

i2

+1 2

htan(α−β) tan(α+β)

i2

(2.12) and

T(α,β) =1−R(α,β) (2.13)

The net force on the sphere due to a single ray therefore is

d⃗F= (n_{med}/c)[Re(Q_{trap})zˆ^{′}+Im(Q_{trap})yˆ^{′}]dP (2.14)
where,

Q_{trap}=1+Rexp(−2iα)−T^{2}exp[−2i(α−β)]

1+Rexp(2iβ) (2.15)

is called as trapping efficiency. Hence, the force experienced by the dielectric sphere for a given ray can be calculated using Eq.2.14provided the respective(α,β)and(R,T)values are known. In this work we assumeRandT values as given by Eqs.2.12and2.13.

2.3. Optical trapping in Ray-optics regime

Net force due to a tightly focused Gaussian beam

Considering all the rays incident on the sphere the net force on it can be obtained by vectorially addingd⃗F due to each ray. Therefore,

⃗F=

### ∑

^{d}

^{⃗}

^{F}

^{(2.16)}

=

### ∑

^{n}

^{med}

_{c}

^{[Re(Q}

^{trap}

^{)}

^{z}

^{ˆ}

^{′}

^{+}

^{Im(Q}

^{trap}

^{)}

^{y}

^{ˆ}

^{′}

^{]dP}

^{(2.17)}

The elementary power elementdPdepends on the intensity distribution at the entrance pupil of
the objective lens. In this thesis we denote the components of the net force alongx,yandzas
F_{x},F_{y}andF_{z}.

In most of the cases the intensity distribution at the entrance pupil is considered as Gaussian distribution in which case the intensity at the locationρ(x,y)in the pupil plane is given as

I(x,y) =I_{o}exp(−ρ^{2}

2σ^{2}) (2.18)

whereσ is the beam waist andρ=p

x^{2}+y^{2}. We are expressing the power using cylindrical co-
ordinates as the Gaussian distribution has a cylindrical symmetry. The power in the elemental
area dAat (x,y) in the pupil plane isdP = I(ρ)dA, which is also taken as the power in the
corresponding ray passing through(x,y). IfP_{tot} is the net power in the Gaussian beam then

dP= P_{tot}

2π σ^{2}exp(− ρ^{2}
2σ^{2})dA

= P_{tot}

2π σ^{2}exp(− ρ^{2}

2σ^{2})ρdρdϕ (2.19)

since in cylindrical polar coordinates, the elementary areadA=ρdρdϕ. Now, considering the objective lens obeys Abbe’s sine condition, that is

ρ=p

x^{2}+y^{2}= fsinθ (2.20)

Chapter 2: Force Calculation Model of Optical trapping

where,θ is the angle the ray makes with the optical axis of objective lens. Thus the angle made by the ray with thezaxis is decided by the radial distance from the axis in the pupil plane.

Hence

dP= P_{tot}

2π σ^{2}exp(−f^{2}sin^{2}θ

2σ^{2} )f^{2}sinθ cosθdθdϕ (2.21)
Substituting Eq.2.21in Eq.2.17we get [8]

⃗F=

θ0

Z

0 2π

Z

0

n_{med}

c [Re(Q_{trap})zˆ^{′}+Im(Q_{trap})yˆ^{′}] P_{tot}

2π σ^{2}exp(−f^{2}sin^{2}θ

2σ^{2} )f^{2}sinθ cosθdθdϕ (2.22)
In the above Eq.2.22the unit vectors are expressed as

zˆ^{′}= (−sinθcosϕ,−sinθsinϕ,−cosθ) (2.23)

and

yˆ^{′}= zˆ^{′}×(⃗r×zˆ^{′})

|zˆ^{′}×(⃗r×zˆ^{′})| (2.24)
The expression of the incident angleα can be derived using Fig.2.6(ii) as

α =cos^{−1}hg^{2}+b^{2}−r^{2}
2bg

i

(2.25)

where,

g= q

b^{2}−r^{2}+r^{2}(sinχsinθcosϕ+cosχcosθ)^{2}
+r(sinχsinθcosϕ+cosχcosθ)

(2.26)

Using Snell’s law

β =sin^{−1}
n_{med}

n_{s} sinα

(2.27)
where,n_{s}is the refractive index of sphere. The value ofθ0is decided by the numerical aperture
of the focusing lens or the critical angle of incidence on the glass coverslip in case the specimen
slide containing the dielectric sphere uses a glass coverslip, which ever is smaller. Thus the

2.4. Augmented ray-pencil model to calculate optical forces