Field-based dynamic light scattering microscopy: theory and numerical analysis Chulmin Joo1,* and Johannes F. de Boer2,3 1

School of Mechanical Engineering, Yonsei University, Seoul, South Korea

2

Department of Physics and Astronomy, VU University, De Boelelaan 1801, HV Amsterdam, The Netherlands 3

e-mail: [email protected]

*Corresponding author: [email protected] Received 12 July 2013; revised 16 September 2013; accepted 9 October 2013; posted 10 October 2013 (Doc. ID 193528); published 31 October 2013

We present a theoretical framework for field-based dynamic light scattering microscopy based on a spectral-domain optical coherence phase microscopy (SD-OCPM) platform. SD-OCPM is an interferometric microscope capable of quantitative measurement of amplitude and phase of scattered light with high phase stability. Field-based dynamic light scattering (F-DLS) analysis allows for direct evaluation of complex-valued field autocorrelation function and measurement of localized diffusive and directional dynamic properties of biological and material samples with high spatial resolution. In order to gain insight into the information provided by F-DLS microscopy, theoretical and numerical analyses are performed to evaluate the effect of numerical aperture of the imaging optics. We demonstrate that sharp focusing of fields affects the measured diffusive and transport velocity, which leads to smaller values for the dynamic properties in the sample. An approach for accurately determining the dynamic properties of the samples is discussed. © 2013 Optical Society of America OCIS codes: (290.0290) Scattering; (290.5820) Scattering measurements; (180.6900) Threedimensional microscopy; (300.6320) Spectroscopy, high-resolution. http://dx.doi.org/10.1364/AO.52.007618

1. Introduction

Dynamic light scattering (DLS), also known as photon correlation spectroscopy (PCS), is a welldeveloped and extensively used method for measuring translational, rotational, and internal motions of small particles over the range of a few nanometers to a few microns in suspension [1–3]. In DLS, a coherent light beam is directed at randomly moving particles. The particles move with different velocities, and the phases of light scattered from the particles vary in time. The light scattered by the particles at a particular detection angle to the incident beam is then collected and measured at a detector. Particles undergoing Brownian motion 1559-128X/13/317618-11$15.00/0 © 2013 Optical Society of America 7618

APPLIED OPTICS / Vol. 52, No. 31 / 1 November 2013

modulate the amplitude and phase of the scattered light, resulting in fluctuations in the scattered light intensity. This fluctuation has a time scale related to the speed of the movement of the particles, and information about the viscoelastic and diffusive properties of the scattering samples can be extracted from the power spectrum or temporal correlation function of the detected intensity fluctuation. DLS information is typically obtained by analyzing the intensity of the scattered light. The theoretical basis and diverse experimental configurations have been developed and applied to studying protein binding and unbinding dynamics [4,5], biopolymer aggregation [6], and sensing smoke aerosols [7]. A combination of light-scattering spectroscopy with high-resolution microscopy has also been investigated to provide a spatial map of diffusive properties of samples in two- or three-dimensional space [8–10].

While the imaging techniques based on intensitybased light spectroscopy are well-established, fieldbased imaging methods that utilize intensity and phase information of scattered light from the microscopic probe volume have not been extensively explored. The feasibility of field-based static light scattering spectroscopy has been demonstrated by use of an interferometer, showing a great sensitivity to the phase front of scattered light [11,12]. Yet there is a significantly smaller amount of work reported on the theory and experimental demonstration of field-based DLS analysis. We recently reported on a combination of a 3D interferometric microscope termed “spectral-domain optical coherence phase microscopy” (SD-OCPM) [13] with field-based DLS, which measured localized diffusive and transport dynamic properties of emulsion particles and components inside the living cell [14]. SD-OCPM is based on a common-path fiber-based low-coherence interferometer that exhibits nanometer-level optical-path length sensitivity (Fig. 1). In a fiber-based SD-OCPM implementation [Fig. 1(a)], the core in the single-mode fiber serves as a detection pinhole. Thus the probe volume is confined to within the confocal and coherence gates. Since the confocal and coherence gates are independent physical confinements for the focal volume, their point-spread functions are multiplicative, leading to

an improvement in the axial resolution [15]. Moreover, SD-OCPM employs the light reflected from a surface along the probe path as a phase reference. Such a common-path interferometer has superior rejection of common-mode phase noise, such as vibrations and thermal drifts. In our implementation, the light reflected from the bottom surface of a coverslip serves as a reference, and the measurement light is backscattered from the probe volume inside a specimen. This feature allows obtaining phase-stable interferometric signals, which is required for accurate F-DLS analysis. The temporal autocorrelation analysis of the measured complex-valued interferometric signals provides statistical measures of diffusive and transport dynamics in scattering samples. In this article, we present a theoretical basis and numerical analysis for field-based dynamic light scattering microscopy. We evaluate the temporal correlation functions for the interferometric signal taking into account the numerical aperture (NA) of the focusing and collection optics. We then develop a simple method to approximate the correlation functions and demonstrate that the diffusive and transport dynamics can be accurately measured with the use of correction factors, which are functions of the NA of the imaging optics. The theory presented here is a combination of Gaussian beam theory in the spatial frequency domain and a generalization of the conventional DLS theory as presented by Cummins et al. [16]. We begin with a derivation of the functional form of the scattered field from a single scatterer and a collection of scatterers. Then we derive the complex-valued temporal autocorrelation for the resultant electric field, from which the mean-squared displacement (MSD) and time-averaged displacement (TAD) of the scatterers are obtained using an approximated correlation function. 2. Theory A. Scattering Field from a Single Particle

Fig. 1. (a) SD-OCPM setup. A broadband light (or low-coherence) source illuminates a fiber-based common-path interferometer. The light coupled to the sample arm is delivered to a specimen via an integrated inverted microscope. The backscattered light is recoupled to the fiber for the subsequent interference spectrum measurement at the detection arm. (b) SD-OCPM detection scheme. SD-OCPM uses the light back-reflected from the bottom surface of a coverslip as a reference to ensure the phase stability for measuring amplitude and phase of sample light scattered from the focal volume [14].

Consider a Gaussian beam incident on a single scatterer inside a probe or scattering volume (Fig. 2). For simplicity, we assume the scatterer is in free space, i.e., n
1. The assumed experimental setup is based on a fiber-based optical coherence microscope, where the probe volume is determined by confocal and coherence gates. Previously, we have experimentally demonstrated a diffraction-limited probe volume of ∼1 femtoliter [14]. We assume that the scattered light from a particle is in the singlescattering regime, since multiple scattered light will have a long optical path length and will be outside the coherence length of the light source. We also assume that the position of the scatterer is close to the origin of the coordinate system (i.e., the center of the beam waist). The normalized Gaussian mode function at the beam waist is given by 1 November 2013 / Vol. 52, No. 31 / APPLIED OPTICS

7619

ZZ Es ⃗rj 


~ s  ⃗qs ; r⃗ j F~ i  ⃗qs d2 qs : E

(5)

Inserting Eq. (3) into Eq. (5), we obtain ZZ ZZ Es ⃗rj 


F~ i  ⃗qi  expi⃗rj · k⃗ i − k⃗ s F~ i  ⃗qs d2 qi d2 qs : (6)

⃗ si ≡ k⃗ s − k⃗ i, we rewrite Eq. (6) as Defining Q Fig. 2. Single scatter is positioned at r⃗ j relative to the center of beam waist. f i x; y and f c x; y represent the incident and collection Gaussian mode functions, respectively. The scattering volume is indicated by the red dashed line.




f i x; y


2

2

1 x y exp − 2 w20 πw0

;

(1)

(2)

Here ⃗qi is the transverse spatial frequency of the wave vector. Denoting r⃗ j as the position vector of the scatterer j, and the scattered field at a transverse spatial frequency ⃗qs is the sum of all the scattered fields arising from all incident plane waves: ZZ ~ F~ i  ⃗qi  expi⃗rj · k⃗ i − k⃗ s d2 qi ; (3) Es  ⃗qs ; r⃗ j 
where k⃗ i and k⃗ s are the incident and scattering wave vectors, respectively, and k⃗
⃗q  kz zˆ . For a given ⃗q, kz can be obtained by the dispersion equation k2
q2  k2z . In Eq. (3), it is assumed that the scattering of the point object is isotropic, and the scattering amplitude is independent of k⃗ i and k⃗ s . The scattering strength, which depends on scattering properties of the particle, was also assumed to be constant and omitted. The scattered field coupled back into the lens is the measurement field Es ⃗rj  ~ s over ⃗qs space, and can be calculated by integrating E multiplied with a weight function determined by the collection efficiency of the lens F~ c  ⃗qs : ZZ ~ s  ⃗qs ; r⃗ j F~ c  ⃗qs d2 qs : E (4) Es ⃗rj 
Since the same lens is used for focusing and collection optics, F~ c  ⃗qs 
F~ i  ⃗qs . Therefore Eq. (4) can be written as 7620

Es ⃗rj 


⃗ si F~ i  ⃗qs d2 qi d2 qs : (7) F~ i  ⃗qi  exp−i⃗rj · Q

B. Scattering Fields from Multiple Particles



where w0 is the Gaussian beam waist radius. The beam waist radius can be expressed by 4f ∕k0 D
2∕k0 NA, where k0 is the free-space wave number, and NA is the numerical aperture of the objective. The Gaussian beam in Eq. (1) can be represented as the sum of its Fourier components (or plane waves) with the amplitudes: F~ i  ⃗qi 
exp−w20 q2i :

ZZ ZZ

APPLIED OPTICS / Vol. 52, No. 31 / 1 November 2013

Each scattering particle j located at position r⃗ j produces an electric field Es ⃗rj  according to Eq. (7). The total electric field Es generated by a set of the particles at r⃗ j positions (j
1; 2; …; N) is then given by Es f⃗rj g; t


N X

nj tEs ⃗rj ;

(8)

j
1

where nj t denotes the occupation number of the scattering particle, j, as specified in [10]. The occupation number nj t equals unity if scattering particle j is in the probe volume (V p ) at time t and zero otherwise. We assume that the positions r⃗ j are statistically independent of each other. The random (diffusive) and directional (convective) motions cause Es to fluctuate and shift in both amplitude and phase. Temporal behavior of the fluctuation can be investigated through the autocorrelation function. C.

Temporal Correlation Function of the Field

We measure the scattered electric field from a probe volume using an interferometer, which allows for detecting the measurement field through the crosscorrelation with a reference electric field. With a constant (time-invariant) reference electric field, ER , the complex-valued interference signal, Ft, can be described as Ft
ER Es f⃗rj g; t
ER

N X

nj tEs ⃗rj :

(9)

j
1

The temporal autocorrelation function is defined as Γτ
hFtF  t  τi;

(10)

where τ denotes the time delay. The Ft at the two times are measured, and the ensemble average denoted by the brackets is taken over all possible f⃗rj g configurations. Here, we assumed that the system is in equilibrium and ergodic, as in typical DLS and fluorescence correlation spectroscopy

analysis [1,17]. Many physical and biological processes in thermodynamic equilibrium have been modeled and analyzed with this assumption [1,17,18]. We substitute Eqs. (7) and (9) into Eq. (10) to obtain N X N X Γτ
jER j2 hnj tnk t  τi j
1 k
1

ZZ ZZ ZZ ZZ ×

hexp−i ⃗rj t · Q⃗ ba  expi ⃗rk t  τ · Q⃗ dc i

× F~ i  ⃗qa F~ i  ⃗qb F~ i  ⃗qc F~ i  ⃗qd d2 qa d2 qb d2 qc d2 qd : (11) The single ensemble average in Eq. (10) can be separated into two ensemble averages (number and phase) in Eq. (11), as the phase fluctuations in the complex exponential factors are uncorrelated with number fluctuations. The summation in Eq. (11) can be separated according to the relationships among the summation indices j, k. Using the notation used by Dzakpasu and Axelrod [10], we have ph num ph Γτ
jER j2 Γnum j≠k Γj≠k  Γj
k Γj
k 

(12)

⃗ ba ihexpi⃗rk t  τ · Q ⃗ dc i
hexp−i⃗rj t · Q

The scattering particles are assumed to be uniformly distributed over the probe volume V p , so ⃗χ⃗r
1∕V p . Thus Eq. (18) becomes Z 1 ⃗ 3 r
1 δQ: ⃗ ⃗ exp−i⃗r · Qd (19) hexp−i⃗r · Qi
Vp Vp Eq. (17) can then be expressed as ⃗ ba ihexpi⃗rk t  τ · Q ⃗ dc i hexp−i⃗rj t · Q


1 ⃗ ba δQ ⃗ dc : δQ V 2p

N X N X

hnj tnk t  τi;

(13)

j
1 k
1

ZZ ZZ ZZ ZZ


hexp−i ⃗rj t · Q⃗ ba  expi ⃗rk t  τ · Q⃗ dc i

× F~ i  ⃗qa F~ i  ⃗qb F~ i  ⃗qc F~ i  ⃗qd d2 qa d2 qb d2 qc d2 qd ; (14) and Γnum j
k


N X hnj tnj t  τi;

(15)

j
1

ZZ ZZ ZZ ZZ Γph j
k


hexp−i ⃗rj t · Q⃗ ba  expi ⃗rj t  τ · Q⃗ dc i

× F~ i  ⃗qa F~ i  ⃗qb F~ i  ⃗qc F~ i  ⃗qd d2 qa d2 qb d2 qc d2 qd:

(20)

Inserting Eq. (20) into the integral of the phase fluctuation term, we get Γph j≠k

Γnum j≠k


(17)

because the particles are statistically independent. ⃗ The term of the form hexp−i⃗r · Qi represents a characteristic function of a random variable r⃗ . We define the probability density function for r⃗ as ⃗χ⃗r, so that Z ⃗ ⃗ 3 r: (18) hexp−i⃗r · Qi
⃗χ⃗r exp−i⃗r · Qd

ZZ ZZ ZZ ZZ

with

Γph j≠k

⃗ ba  expi⃗rk t  τ · Q ⃗ dc i hexp−i⃗rj t · Q




hexp−i ⃗rj t · Q⃗ ba ihexpi ⃗rk t  τ · Q⃗ dc i

× F~ i  ⃗qa F~ i  ⃗qb F~ i  ⃗qc F~ i  ⃗qd d2 qa d2 qb d2 qc d2 qd ZZ ZZ ZZ ZZ 1
2 δQ⃗ ba δQ⃗ dc F~ i  ⃗qa F~ i  ⃗qb  Vp × F~ i  ⃗qc F~ i  ⃗qd d2 qa d2 qb d2 qc d2 qd :

(21)

⃗ ba
The integral has a nonzero value only when Q ⃗ ba
k⃗ b − k⃗ a, we note ⃗ dc
0. From the definition of Q Q ⃗ ⃗ ⃗ ⃗ that kb
ka and kd
kc should be met for nonzero autocorrelation value. This particular configuration is the case in which the incident and scattering vectors are collinear, i.e., the scattering angle is 0 degrees. Since our SD-OCPM operates in the epidetection, this condition cannot be satisfied. Therefore the integral over ⃗q space in Eq. (21) is zero. For the j
k case, the complex exponentials in the phase fluctuation term can be written as ⃗ ba  expi⃗rj t  τ · Q ⃗ dc i hexp−i⃗rj t · Q ⃗ ba  expi⃗rj t  τ · ΔQ ⃗ dc−ba i
hexpiΔ r⃗ j · Q

(16)

⃗ ba ihexpi⃗rj t  τ · ΔQ ⃗ dc−ba i; (22)
hexpiΔ r⃗ j · Q

We first consider the case where j ≠ k. In this case, the complex exponential factors in the phase fluctuation term can be written as

⃗ dc−ba
Q ⃗ dc − Q ⃗ ba . where Δ⃗rj
r⃗ j t  τ − r⃗ j t and ΔQ ⃗ dc−ba i is the same We note that hexpi⃗rj t  τ · ΔQ ⃗ ⃗ form as Eq. (18) except that Q is replaced by ΔQ. Therefore Eq. (22) can be simplified to

D.

Phase Fluctuation Factors

1 November 2013 / Vol. 52, No. 31 / APPLIED OPTICS

7621

⃗ ba  expi⃗rj t  τ · Q ⃗ dc i hexp−i⃗rj t · Q


1 ⃗ ba iδΔQ ⃗ dc−ba : hexpiΔ⃗rj · Q Vp

(23)

Inserting Eq. (23) into the integral in the phase fluctuation factors, we obtain Γph j
k


1 Vp

ZZ ZZ ZZ ZZ

⃗ ba iδΔQ ⃗ dc−ba  hexp−iΔ⃗rj · Q

× F~ i  ⃗qa F~ i  ⃗qb F~ i  ⃗qc F~ i  ⃗qd d2 qa d2 qb d2 qc d2 qd : (24) ⃗ dc
The integral has a nonzero value only when Q ⃗Qba , i.e., Q ⃗ d−Q ⃗ c
Q ⃗ b−Q ⃗ a or ⃗qd − ⃗qc
⃗qb − ⃗qa. Noting this, we insert Eq. (2) into Eq. (24) and arrange the terms to obtain ZZ ZZ Γph j
k


⃗ si iΛ ⃗qs ; ⃗qi d2 qs d2 qi ; hexpiΔ⃗rj · Q

(25)

with

 3 3 Λ ⃗qs ; ⃗qi 
exp −w20 q2i  q2s − ⃗qi · ⃗qs 2 2
 
ZZ  ⃗qi − ⃗qs  2 2 2 ⃗q d qc : × exp −2w0 c  2 The subscripts a and b were replaced with i and s, and the constant factors were left out for simplicity. We assume that each component in Δ⃗rt is a Gaussian random variable with mean and variance of μl τ and σ 2l τ, l
x, y, z, respectively. Equation (25) then becomes ZZ ZZ Γph j
k


exp−Q2si σ 2 τ

⃗ si Λ ⃗qs ; ⃗qi d2 qs d2 qi : × expi ⃗μτ · Q

(26)

Inserting the phase fluctuation factor, Eq. (26), into the temporal autocorrelation function, Eq. (12), we get Γτ
jER j2

ZZZZ N X hnj tnj t  τi exp−Q2si σ 2 τ j
1

⃗ si Λ ⃗qs ; ⃗qi d2 qs d2 qi ; × expi ⃗μτ · Q

(27)

and thus the corresponding normalized autocorrelation function is given by ZZ ZZ Γτ ˆ
Γˆ num τ exp−Q2si σ 2 τ Γτ
Γ0 ⃗ si Λ ⃗qs ; ⃗qi d2 qs d2 qi × expi ⃗μτ · Q 7622

(28)

APPLIED OPTICS / Vol. 52, No. 31 / 1 November 2013

with the number correlation function, Γˆ num τ
PN P N 2 j
1 hnj tnj t  τi∕ j
1 hnj t i. The probe volume (or optical focal volume) in F-DLS is significantly smaller than the total sample volume. We assume that the number fluctuation is negligible compared with the phase fluctuation, so that Γˆ num τ ∼ 1. It may be seen from Eq. (28) that the autocorrelation function for the case of high NA focusing and collection lens is essentially a superposition of the temporal autocorrelation functions ⃗ si , weighted by the factor Λ ⃗qs ; ⃗qi . for each Q We propose that Eq. (28) can be approximated by a representative autocorrelation function Rτ, which has the form ˆ ∼ Rτ
exp−γ D Q2m σ 2 τ expiγ T ⃗μτ · Q⃗ m ; (29) Γτ ⃗ m denotes representative scattering vector, where Q and γ D and γ T are correction factors for diffusive and directional transport dynamics, respectively, that account for the integral in Eq. (28) over the angular dispersion of the incident and detected beams. Using Eq. (29), one can compute the mean-squared displacement defined by MSDl τ
hlt  τ − lt2 i and the TAD of the scatterers in the direction of ⃗ m as Q μτ


tan−1 ImRτ∕ReRτ ; γ T Qm

(30)

MSDτ
σ 2 τ  μ2 τ lnRτR τ 2γ D Q2m   tan−1 ImRτ∕ReRτ 2  : γ T Qm


−

(31)

For instance, in the case of epi-detection configura⃗ m
0; 0; 2k0  as the representation, one may use Q ⃗ tive Qm , but there may be an error associated with this approximation, since the measurement takes the average over the range of scattering vectors determined by NA of the optical system. In Section 3, we demonstrate that this approximation provides the accurate measurement by use of the proper correction factors, γ D and γ T , through numerical simulations. 3. Simulation

Numerical analysis was conducted to obtain insight into F-DLS microscopy and to assess the validity of the approximation given in Eqs. (29)–(31). We first assumed scattering particles with diffusive and direction dynamics and developed the associated fields [Eq. (9)]. We then evaluated the field-based temporal autocorrelation function of the fields directly from Eq. (10). The computed autocorrelation functions

were compared with the model [Eq. (28)] to assess the validity of our derivation. After the validity of Eq. (28) was confirmed, we evaluated the temporal autocorrelation functions as a function of NA according to Eq. (28). The result is the coherent sum of correlation functions for each incident and scattering vector and provides the actual F-DLS information of the dynamic properties of the samples. MSDs and TADs were obtained using the approximation in Eqs. (29) to (31) and were compared with the actual values from the numerical integration of Eq. (28) to investigate the errors associated with the approximation. By quantifying the errors in the diffusive parameters and transport velocity, the correction factors γ D and γ T were determined as a function of NA. For the numerical analysis, a quasi-monochromatic light source at 800 nm was used. Objective lenses with various NAs were used to illuminate and collect the light scattered from the structures. Based on the each NA value, the beam waist was determined accordingly, and the 2D transverse q vectors were digitized in 11 × 11-frequency grids in the −2∕w0 ; 2∕w0  space. A.

Validity of Field Temporal Autocorrelation Model

In order to assess the validity of our derived temporal autocorrelation function [Eq. (28)], scattering particles with Brownian motion only (diffusion coefficient D
5 × 10−3 μm2 ∕s) and both Brownian and direction motion (D
5 × 10−3 μm2 ∕s and z directional velocity
−5 nm∕s) were considered. We obtained the total scattered electric field from the

particles by integrating all the fields with the incident and scattering vectors determined by NAs and used the resultant field to directly compute the field-based temporal autocorrelation function given in Eq. (10). For calculation, we assumed 50 scattering particles in the probe volume and examined the particle motions with a temporal resolution of 100 μs and for 20 s. The obtained correlation functions were then normalized and compared with the results directly calculated from Eq. (28) with the assumed scatterer dynamics. A representative result with NA
0.5 is shown in Fig. 3. Figures 3(a) and 3(b) show the comparison of two correlation functions in terms of amplitude and phase for the particles with Brownian motion only, whereas Figs. 3(c) and 3(d) illustrate the case with both Brownian and directional dynamics. It can be seen that the amplitude ˆ (jΓτj) and phase (ϕ) of the two correlations functions show a great correspondence. The measured rootmean-squared difference for the amplitude and phase were