Page 58 - Nanotechnology an introduction

account the optically denser nucleus, the term n cA(z) in eqn (5.24) can be replaced by , where n is the
D
κ
refractive index of the nucleus, and D(z) is its cross-sectional area, obviously assumed to be less than A(z). Additional terms could in principle be
added for the endoplasmic reticulum, Golgi body, etc., although probably only the nucleus warrants special treatment, its refractive index being
significantly higher because of the relatively high concentration of phosphorus-containing nucleic acids.
The fundamental property that is measured with total internal reflexion is the phase shift Φ; the complex Fresnel reflexion coefficient can be written
out as

(5.27)
The optical waveguide is constructed by sandwiching a slab of the high refractive index material F between lower refractive index materials, the
mechanically strong support S and the medium C containing the cells. Both these materials are much thicker than 1/s, and hence can be
considered semi-infinite. The condition for the guided waves to propagate is for the different contributions to Φ to sum to zero or an integral multiple
of 2π. This condition can be written as a set of mode equations linking the effective refractive index N of a particular mode with n , n , d and n .
S
C
F
F
Only discrete values of N are allowed, corresponding to the mode numbers m = 0, 1, 2, …, each value of m being represented by two orthogonal
polarizations, transverse magnetic (TM) and transverse electric (TE). The sensitivities ∂N/∂n depend on the polarization, mode number and
C
effective waveguide thickness: below a certain minimum cutoff thickness no propagation can take place and as the thickness increases the
sensitivity rapidly reaches a maximum and then slowly diminishes; the sensitivities decrease for higher order modes for conventional waveguides
[162].
For reverse waveguides (n < n —called “reverse symmetry” waveguides in some earlier literature—(see Section 5.5.4) the sensitivity at the cutoff
C
S
thickness is 1 for all of the modes and slowly decreases above the cutoff thickness [162] and [81]. If n , n and d are known, n can be determined
S
F
C
F
from measurement of a single value of N. According to equation (5.26), any change in n can be interpreted as a change in cell concentration
C
(number per unit area of substratum) c, or in cell size (characterized by the radius R), or in cell shape (if the shape is assumed to be a segment,
then the contact area a will suffice as a characteristic parameter, if it is further assumed that the volume does not change upon spreading), or in cell
refractive index (distribution).
In summary, then, the fundamental measurable quantity output by the optical measurement setup is one (or more) effective refractive indices N.
Further interpretation of N depends on the model chosen, with which the distribution of polarizability within the evanescent field can be linked to
changes in the cover medium refractive index n .
C
5.5.3. Optical Measurement Schemes
The principal approaches to measure the effective refractive indices N with high precision are grating coupling and interferometry. In the former, a
diffraction grating created in the waveguide is used to couple an external light beam incident on the grating with an angle α to the grating normal;
measurement of α yields N according to

(5.28)
where is the diffraction order and Λ is the grating constant. The diffraction grating can be created either by modulating the
topography or the refractive index of the waveguide material. α can be determined by mounting the waveguide on a high precision goniometer with
the grating positioned at its axis of rotation. Photodiodes placed in contact with the ends of the waveguide enable the incoupling peaks to be
determined while α is varied. It is customary to measure propagation in both possible directions and take the average, thereby correcting for any
irregularity in the grating position and yielding the absolute incoupling angle. Alternatively the grating can be used to couple light already
propagating in the waveguide out of it. However, this requires a more elaborate arrangement for introducing the light into the waveguide, and it is
more difficult to measure the absolute incoupling angle. Grating coupling is also called optical waveguide lightmode spectroscopy (OWLS),
because the spectrum of modes can be obtained by scanning α. As well as the high precision and ease with which absolute effective refractive
indices of as many modes as the waveguide will support can be determined, OWLS also enables the incoupling peak shape to be determined; any
broadening beyond the theoretical lower limit of width ( in radians, where L is the illuminated grating length [162]) is highly informative
x
regarding the distribution of objects on the waveguide surface [37].
Variants of the classical incoupling scheme described above include the use of chirped gratings (i.e., with a spacially-varying Λ) and varying λ to
determine N. Such schemes have been devised to eliminate moving parts, especially the expensive and relatively slow high precision goniometer.
However, they are generally less precise and accurate. Nevertheless, it might be worth sacrificing some precision in return for another benefit, such
as high throughput. Hence, incoupling of broadband light and spectral monitoring of the outcoupled light by the same grating provides a very
compact way of measuring relative effective refractive index changes (this scheme is also called the resonant waveguide grating (RWG) [51]).
The three main interferometric techniques that might be used (no measurements with living cells using interferometry have apparently been
reported—often the waveguide geometry is unfavorable; e.g., too narrow to underlie the entire cell, and excessive scattering engendered by the
long waveguides used to increase sensitivity vitiates useful information being obtained) are:
1. Allowing the TE and TM modes to interfere with one another; this is the simplest approach because a planar waveguide without any structuring
can be used (although the optical measuring arrangement is considerably more elaborate than with grating coupling).
2. The Mach–Zehnder interferometer, the integrated optical analog of the Rayleigh interferometer—typically a ridge or channel waveguide is split
into two channels, one of which is the cell substratum, and then recombined.
3. The “dual polarization interferometer” (DPI) in which a sandwich structure consisting of support, reference waveguide, intermediate layer (may
be the same material as the support) and finally the waveguide acting as cell substratum is illuminated at one end, and interference between the
two beams emerging at the other end is recorded in the far field.