Robustness of Lorenz-Mie microscopy against defects in illumination
Abstract.
Lorenz-Mie analysis of colloidal spheres' holograms has been reported to achieve remarkable resolution not only for the spheres' three-dimensional positions, but also for their sizes and refractive indexes. Here we apply numerical modeling to establish limits on the instrumental resolution for tracking and characterizing individual colloidal spheres with Lorenz-Mie microscopy.
§ I. Introduction
In-line holographic video microscopy (HVM) images (1); (2) can be interpreted with predictions of the Lorenz-Mie theory of light scattering (3) to simultaneously track and characterize individual colloidal particles (4). Unlike complementary techniques that focus on measuring the phase of the scattered wavefront (5); (6); (7) numerically reconstructing the scattered light field (1); (2); (8); (9), or interpreting the measured interference pattern with phenomenological models (10); (11); (12), Lorenz-Mie microscopy yields both the three-dimensional positions of individual scatterers and also their sizes and complex refractive indexes. This wealth of time-resolved information has enabled such applications as holographic microrheology (13), nanometer-resolution particle-image velocimetry (14); (15); (16) particle-resolved porosimetry (17), microrefractometry (18) and label-free molecular binding assays (14).
The benefits of Lorenz-Mie microscopy derive not only from the quantity of particle-resolved information it yields, but also from the high precision claimed for each of the extracted parameters (4). Tracking resolution has been verified independently by analyzing the measured trajectories of freely diffusing colloidal spheres (4); (14); (15). These measurements confirm that the resolution for locating a sphere's center can exceed 1 nanometer in the plane and 10 nanometers axially over ranges extending to hundreds of micrometers. Numerical uncertainties in the fit values for colloidal spheres' radii and refractive indexes similarly suggest part-per-thousand resolution in these quantities as well (4); (14); (18). These latter estimates have not been verified independently, however, because no other methods exist to measure the size and refractive index of individual colloidal spheres in situ and with such high resolution.
Here, we establish limits on the tracking and characterization resolution of Lorenz-Mie microscopy by numerically modeling the influence of non-ideal illumination conditions on measured outcomes. The standard analysis (4) treats the incident illumination as a plane wave propagating along the optical axis. Departures from this model in a practical implementation might reasonably be expected to degrade performance. We therefore assess through simulations how uncompensated curvature and tilt of the illuminating beam's wavefronts affect extracted values for particle position, size and refractive index obtained in real-world implementations.


An in-line hologram is created when light scattered by an object interferes with the remainder of the illuminating beam, as shown in Fig. 1. In our implementation, the interference pattern is magnified by a conventional microscope before being recorded with a video camera. Each snapshot in the resulting video stream records the intensity
![]() |
(1) |
at position in the microscope's focal plane due to the
superposition of the time-averaged incident electric field,
, and
the field
scattered by an object centered at position
.
Assuming the object to be small compared with the typical length scale
for variations in the illumination, the scattered field may
be approximated as
![]() |
(2) |
where the scattering function describes
the outgoing wave that is created by illuminating the particle with
the incident wave.
The scattering function
may be expressed in generalized Lorenz-Mie theory
as a series expansion (19); (20)
![]() |
(3) |
in the vector spherical harmonics, ,
,
and
(3).
The expansion coefficients
and
depend on
the structure of the illuminating beam and the properties of the
scatterer.
For the special case of scattering by a sphere,
they reduce to (20)
![]() |
![]() |
(4) | ||
![]() |
![]() |
(5) |
where and
are the familiar Lorenz-Mie coefficients (3) for
scattering of a plane wave by a sphere.
These, in turn, depend on the sphere's radius
, and its complex refractive index,
, relative to that of the
medium
.
The transverse electric (TE) and transverse magnetic (TM)
beam shape coefficients,
and
,
account for the structure of the incident illumination.
Equations (4) and (5)
are the complex conjugate of the expressions in Ref. (20)
because here we assume a time dependence
.
Previous implementations of Lorenz-Mie microscopy
(4); (14); (15)
have treated the incident field as a linearly polarized plane wave,
,
propagating precisely along the optical axis,
for which only the beam shape coefficients with
differ from zero and
are given by
.
This plane-wave approximation also justifies normalizing the measured hologram
by the background intensity
to suppress artifacts due to extraneous scattering centers
in the optical train (4).
The normalized hologram then may be fit to the simplified expression
![]() |
(6) |
to obtain estimates
for the particle's position , its radius
, and
its refractive index,
.
The phenomenological factor
in Eq. (6)
is intended to account
for diffuse scattering due to surface roughness and
for variations in illumination.
It typically has values in the range
.
In the present study, we assess discrepancies in the estimates for
,
and
that arise when the plane-wave approximation,
Eq. (6), is used to interpret holograms
that are formed with more realistic models for the incident illumination.
Specifically, we identify errors arising from divergence and tilt in the
illuminating beam.
These simulations establish limits on systematic errors for these parameters
that can be expected in practical implementations.
Under typical experimental conditions, with
divergence angles smaller than
1
and tilt angles less
than 10
, the resulting errors are found to be an order of magnitude smaller than
numerical uncertainties in the fit parameters (4); (14).
From this we conclude that other factors have a more substantial
influence on measurement errors in Lorenz-Mie microscopy and that
the analysis technique
is robust against this class of illumination defects.
§ II. Influence of beam divergence
To study the effects of diverging illumination, we model the
incident light as a Gaussian beam of vacuum wavelength
and wave number
propagating along
through a medium of refractive index
.
Rather than being collimated as in Fig. 1(a), the
beam diverges from a focus of half-width
at a height
above the focal plane of the microscope, as shown in
Fig. 1(b).
Its electric field is then given in the paraxial approximation by
![]() |
(7) | ||
![]() |
(8) |
Equations (7) and (8) accurately describe
a weakly divergent beam with .
Assuming that the particle is centered on the optical axis at
height
above the focal plane, the
beam shape coefficients are (19)
![]() |
(9) |






We use
Eqs. (7) through (9)
as inputs to
Eqs. (1) through (3)
to compute simulated holograms of spherical particles with specified
radii and refractive indexes at specified heights above
the center of the focal plane.
We then use Eq. (6) to extract estimates
,
and
for each particle's characteristics and position under the
plane-wave approximation.
The differences
,
, and
represent the errors incurred by
fitting the data with a simplified model.
Typical results for the relative errors are plotted in Fig. 2
as a function of the beam divergence angle
![]() |
(10) |
The parameters chosen for these simulations are intended
to mimic the conditions in recently published experiments
(21); (22)
on micrometer-diameter colloidal silica spheres in water with
,
, and
in a medium
with
at
.
Taking the position of the illuminating laser's output coupler to be the center
of curvature, these experiments have
and
, using the manufacturer's specification
for beam divergence.
The corresponding simulations yield relative discrepancies
,
and
.
This is at least one order of magnitude better than the
1
\meter resolution reported for
,
the 10
\meter resolution reported for
and
the part-per-thousand resolution claimed for
(21); (22).
The data in Fig. 2 suggest that
divergence-induced errors scale as when the source is positioned beyond
the Rayleigh range
,
and are maximized at
.
Even in this worst-case configuration, the estimated error is
smaller than the claimed resolution for
,
which is a reasonable value for a well-collimated beam.
§ III. Influence of beam inclination




We now study the influence
of tilted illumination.
In this case, the incident light is a plane wave with wave vector
that makes an angle
with respect
to the
axis as shown in Fig. 1(c).
The axis of inclination is chosen to maximize rotation of the
polarization vector
, thereby maximizing the
influence of the beam's inclination.
Treating the incident beam as a plane wave
is justified by the results of the previous section.
The incident field then has the form
![]() |
(11) |
The scattered field then is given by the plane-wave scattering
function Eq. (3) rotated by an angle
about the particle position, around the direction
.
We therefore rewrite the scattered field as
,
in terms of the scattered field
![]() |
(12) |
that is rotated by about the center of the particle by
the rotation matrix .
As in the previous section, we use Eqs. (11) and
(12) as inputs to Eq. (1)
as a basis for computing simulated
holograms of spherical particles.
These holograms are then fit to Eq. (6)
to obtain the estimated parameters ,
and
.
Figure 3 shows the relative discrepancies for
the radius, index of refraction and axial position
for a particle with and
in a medium with
as a function of the tilt angle of the laser for different
values of the axial position
.
Whereas the errors due to divergence plotted in
Fig. 2 vary smoothly with wavefront curvature,
those due to tilt feature discrete jumps.
This is because inclination distorts a hologram's
circular interference fringes into ellipses, thereby breaking
the radial symmetry implicit in the fitting procedure.
Jumps in Fig. 3 reflect changes in the best circularly
symmetric fit with increasing distortion.
Even after accounting for ellipticity in the inclined images,
the relative discrepancy in all fit parameters is less than a
part per thousand for tilt angles smaller than 1 (17
)
over the experimentally accessible range of the axial position.
This falls within the previously estimated range of errors,
and confirms that inclination of the illumination is not the
dominant source of error in Lorenz-Mie microscopy.
Local inclination due to divergent illumination
similarly causes negligible offsets for particles located off
the optical axis, at least for experimentally relevant values
of the beam divergence.
Although divergence and tilt
do not appreciably influence the resolution
of Lorenz-Mie tracking and characterization, they do introduce
systematic correlations between the estimates for the in-plane
and axial coordinates so that
and
.
Such correlations may be ignored in conventional microscopy
because of the comparatively small depth of focus.
They become apparent, however, over the
large axial range covered by holographic microscopy.
In measurements of positional fluctuations, for example, the
apparent in-plane fluctuations will be augmented by a proportion
of axial fluctuations.
Neither divergence nor tilt causes a measurable dependence
of the apparent axial position,
on the in-plane
coordinates,
and
in the experimentally relevant range
of parameters.
Consequently, the correlation may be removed by subtracting off
linear trends (21); (22).
The observed robustness of Lorenz-Mie microscopy against
errors in optical alignment leaves open the question
of the technique's ultimate resolution.
The principal underlying approximation,
invoked in Eq. (6),
involves normalizing the recorded hologram with
an estimate
for the background illumination.
Correlated artifacts introduced by departures from ideal
normalization can influence fits to Eq. (6).
Equation (6), moreover, does not account
for phase disorder in the illumination, which may introduce additional
correlated artifacts.
Finally, issues such as the signal-to-noise ratio of the recording,
the linewidth of the source and the coherence length of the
illumination have yet to be considered.
The present study thus adds support to the claimed resolution
of Lorenz-Mie microscopy while hinting that substantial additional
gains may yet be achieved.
§ Acknowledgments
This work was supported by the National Science Foundation through Grant Number DMR-0922680.
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