Concept and theory


The computation of the compliance \(J\) for elastic spheres in the OS can be divided into three main tasks: measuring the deformation \(w\), modeling the optical stress \(\sigma_r\), and computing the GGF from the stress. Several approaches to these problems have been presented in the related literature and are discussed in the following.

Experimentally quantifying deformation

Semimajor and -minor axes of an ellipse fit


Boundary function fitted to the contour


Optical stress profile acting on a prolate spheroid

The optical stress \(\sigma(\theta)\) in dependence of the angle \(\theta\) is a result of the optical forces acting on the surface of the spheroid. The angle \(\theta\) is defined in the imaging plane in a typical OS experiment, with \(\theta=0\) pointing to the right hand fiber.

\(\cos^2\theta\) approximation

Ray optics is used to compute the optical stress acting on a prolate spheroid and a \(\sigma_0 cos^2\theta\) model is fitted to the resulting stress profile with the peak stress \(\sigma_0\) [GAM+01]. The \(\sigma_0 cos^2\theta\) approximation simplifies subsequent computations.

Note that a more general model \(\sigma_0 cos^2n\theta\) with larger exponents (e.g. \(n\) = 2,3,4,…) can also be applied, e.g. for different fibroblast cell lines [AGW+06].

Semi-analytical perturbation approach (Boyde et al. 2009)

  • gaussian laser beam
  • \(a > \lambda\): higher order perturbation theory
  • [BCG09]

Generalized Lorentz-Mie theory (Boyde et al. 2012)


Computation of the GGF

The following derivations are based on the theoretical considerations of Lur’e [Lure64] for a rotationally symmetric deformation of a sphere and their application to the OS by Ananthakrishnan et al. [AGW+06]. Note that a corrigendum has been published for this article in 2008 [AGW+08].

General approach

The GGF connects the measured deformation to the shear modulus \(G\) which, in OS literature, is usually written in the form

\[\frac{w}{r_0} = \frac{\text{GGF}}{G}\]

where \(w\) is the change in radius of the stretched sphere along the stretcher axis and \(r_0\) is the radius of the unstretched sphere. Note that the quantity \(w/r_0\) resembles a measure of strain along the stretcher axis.

The GGF can be computed from the radial stress \(\sigma_r(\theta)\) via the radial displacement \(u_r(r, \theta)\). These quantities can be connected via a Legendre decomposition according to ([Lure64], chapter 6)

\[ \begin{align}\begin{aligned}u_r(r, \theta) &= \sum_n \left[ A_n r^{n+1} (n+1)(n-2+4\nu) + B_n r^{n-1} n \right] P_n(\cos \theta)\\ \frac{\sigma_r(r, \theta)}{2G} &= \sum_n \left[ A_n r^n (n+1) (n^2 - n - 2 - 2\nu) + B_n r^{n-2} n (n-1) \right] P_n(\cos \theta)\end{aligned}\end{align} \]

with the Legendre polynomials \(P_n\) and the Poisson’s ratio \(\nu\). The coefficients \(A_n\) and \(B_n\) have to be determined from boundary conditions. For the case of normal loading, which is given by the electromagnetic boundary conditions in the OS (\(\sigma_\theta=\tau_{r,\theta}=0\)), these coefficients compute to:

\[ \begin{align}\begin{aligned}A_0 = - \frac{s_0}{4G(1+\nu)}\\B_0 = A_1 = B_1 = 0\end{aligned}\end{align} \]

and for \(n>=2\):

\[ \begin{align}\begin{aligned} A_n &= - \frac{s_n}{4Gr_0^n \Delta}\\ B_n &= \frac{s_n}{4Gr_0^{n-2} \Delta} \cdot \frac{n^2 + 2n -1 + 2\nu}{n-1}\\\text{with } \Delta &= n(n-1) + (2n+1) (\nu + 1)\end{aligned}\end{align} \]

Where \(s_n\) is the \(n\text{th}\) component of the Legendre decomposition of \(\sigma_r\)

\[\sigma_r(\theta) = \sum_n s_n P_n(\cos \theta).\]

The radial displacement then takes the form

\[u_r(r, \theta) = \frac{r_0}{G} \left[ \frac{(1-2\nu) s_0}{2(1+\nu)} + \sum_{n=2}^\infty \frac{2s_n}{2n+1} \left(L_n \left(\frac{r}{r_0}\right)^n + M_n \left(\frac{r}{r_0}\right)^{n-2} \right) P_n(\cos \theta) \right]\]

with the coefficients \(L_n\) and \(M_n\) given in [Lure64], chapter 6.6. We measure the displacement at the outer perimeter of the stretched object and on the stretcher axis only; Thus, we set \(r=r_0\) and \(\theta=0\) with \(w=u_r(r_0, 0)\).

To obtain the GGF, we finally compute

\[ \begin{align}\begin{aligned}\text{GGF} &= \frac{G}{r_0} u_r(r_0, 0)\\ &= \left[ \frac{(1-2\nu) s_0}{2(1+\nu)} + \sum_{n=2}^\infty \frac{2s_n}{2n+1} \left(L_n + M_n \right) P_n(\cos \theta) \right].\end{aligned}\end{align} \]


  • Due to the fact hat the stress profile in the OS is rotationally symmetric w.r.t. the stretcher axis, all odd coefficients \(s_n\) are zero.
  • The polar displacement \(u_\theta\) has been omitted here, because it does not represent a quantity measurable in an OS experiment.

Special case: \(\cos^2\theta\) approximation

Following the above approach, the stress profile

\[\sigma_r(\theta) = \sigma_0 \cos^2\theta\]

with the peak stress \(\sigma_0\) can be decomposed into two Legendre polynomials

\[ \begin{align}\begin{aligned}\sigma_r(\theta) &= s_0 P_0(\cos\theta) + s_2 P_2(\cos\theta)\\s_0 &= \frac{1}{3} \sigma_0\\s_2 &= \frac{2}{3} \sigma_0\end{aligned}\end{align} \]

Inserting these Legendre coefficients in the above equation for the GGF yields

\[\text{GGF} = \frac{\sigma_0}{2(1+\nu)} \left[ \frac{1}{3} \left( (1-2\nu) + \frac{(-7 + 4\nu)(1+\nu)}{7+5\nu} \right) + \frac{(7-4\nu)(1+\nu)}{7+5\nu} \cos^2\theta \right].\]

Historically, the relation between strain, stress, and shear modulus was written in the form

\[\frac{w}{r_0} = \frac{\sigma_0 F_\text{G}}{G}\]

with the geometrical factor \(F_\text{G}\) that does not include the peak stress \(\sigma_0\). Hence the term “global geometrical factor” \(\text{GGF} = \sigma_0 F_\text{G}\).

Computation of compliance

A typical OS experiment records the deformation \(w(t)\) over time \(t\). The quantity of interest is the (creep) compliance \(J(t)\). With \(J = 1/G\), it computes to

\[J(t) = \frac{w(t)}{r_0} \cdot \frac{1}{\text{GGF}(t)}.\]

Note that the GGF is now time-dependent, because the optical stress profile \(\sigma_r\), from which the GGF is computed, also depends on the deformation.