Scattering information obtained by optical microscopy: Differential dynamic microscopy and beyond

Scattering information obtained by optical microscopy:

Differential dynamic microscopy and beyond

Fabio Giavazzi,^1,

*

Doriano Brogioli,²Veronique Trappe,³Tommaso Bellini,¹ and Roberto Cerbino^1,†

1Dipartimento di Chimica, Biochimica a Biotecnologie per la Medicina, Università degli Studi di Milano, via Fratelli Cervi 93, Segrate, Milano I-20090, Italy

2Dipartimento di Medicina Sperimentale, Università degli Studi di Milano–Bicocca, via Cadore 48, Monza, Milano I-20052, Italy

3Department of Physics, University of Fribourg, Chemin du Musée 3, CH-1700 Fribourg, Switzerland

We describe the use of a bright-field microscope for dynamic light scattering experiments on weakly scat- tering samples. The method is based on collecting a time sequence of microscope images and analyzing them in the Fourier space to extract the characteristic time constants as a function of the scattering wave vector. We derive a theoretical model for microscope imaging that accounts for共a兲the three-dimensional nature of the sample,共b兲 the arbitrary coherence properties of the light source, and共c兲 the effect of the finite numerical aperture of the microscope objective. The model is tested successfully against experiments performed on a colloidal dispersion of small spheres in water, by means of the recently introduced differential dynamic microscopy technique关R. Cerbino and V. Trappe, Phys. Rev. Lett. 100, 188102共2008兲兴. Finally, we extend our model to the class of microscopy techniques that can be described by a linear space-invariant imaging of the density of the scattering centers, which includes, for example, dynamic fluorescence microscopy.

I. INTRODUCTION

Scattering and microscopy are traditionally considered as two complementary techniques for the investigation of soft and biological materials 关1兴. Scattering provides in a single shot a powerful average information from the entire sample, while microscopy allows for a detailed study of the behavior of a small portion of it. As far as dynamic measurements are concerned, both approaches offer interesting possibilities. In the real space it is possible to characterize the motion of moving entities by tracking their position in time. This ap- proach is known as video particle tracking 共VPT兲 关2,3兴. In the reciprocal space one can obtain equivalent information by monitoring the temporal intensity fluctuations associated with the sample dynamics 关4兴. This approach is known as dynamic light scattering共DLS兲.

The possibility of combining scattering and microscopy in a single instrument has been already exploited in the past 共see, for example, Ref. 关5兴 and references therein兲. In most cases a commercial microscope was customized and equipped with a laser source, a suitable optics, and a hard- ware for both the imaging of the sample共VPT兲and the cal- culation of statistical properties of the scattered light共DLS兲. In particular the Fourier transform共FT兲operation, needed to operate in the reciprocal space, was realized by means of an auxiliary lens and the intensity distribution in the back focal plane of the lens was analyzed共Fourier microscopy兲.

It has been recently proposed that equivalent results can be obtained by collecting and processing images close to the sample 关6–9兴. Sequences of real-space images are Fourier analyzed and wave vector resolved information about the system structure and dynamics can be extracted. With refer-

ence to dynamics studies, this approach has been success- fully demonstrated with 共spatially coherent兲 monochromatic 关10,11兴 and quasimonochromatic 关12–14兴 light. The close proximity of the detector to the sample is necessary for the success of the above-mentioned approach. When a sample is illuminated with a large coherent beam, it exists a region—

the deep Fresnel region—where the transverse correlation properties of the scattered light are independent on the dis- tance from the sample关7,8,15,16兴. It is only within the deep Fresnel region that it is possible to recover a scattering pat- tern of the sample by numerical Fourier analysis of the im- ages. In a general way this defines a whole family of tech- niques that we refer to as near-field, or deep Fresnel, scattering关17兴.

In near-field scattering experiments with coherent illumi- nation the deep Fresnel region can be very extended. In the attempt of maximizing the scattering signal at the lowest wave vectors, it is a common practice to operate at large 共from a few millimeters up to 1 m兲 defocusing distances from the sample. At such distances the defocused images are either shadowgrams 关6,10,12–14兴 or speckle images 关7–9,11兴. Very recently it has been attempted to extend the near-field scattering family to progressively incoherent illu- mination. Experiments demonstrated that the requirements in terms of both spatial and temporal coherences can be largely relaxed either for partially coherent x rays关18兴or for bright- field microscopy关19兴. Working with low-coherence sources reduces the extent of the deep Fresnel region and forces col- lecting images closer to the sample. This has been done in Ref. 关19兴 where the microscope object plane was chosen at the sample midplane. The use of low-coherence sources of- fers unexpected advantages, especially when the samples are confined in thin cells. A small value of the longitudinal co- herence length minimizes the tedious Fabry-Perot interfer- ence fringes observed with laser illumination. In addition a close distance to the sample guarantees a good correspon-

*fabio.giavazzi@unimi.it

†roberto.cerbino@unimi.it

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Published in "Physical Review E 80(3): 031403, 2009"

which should be cited to refer to this work.

dence between the image intensity and the local density of scatterers within the sample. A better quantification of this correspondence is of paramount importance in view of future extensions of near-field scattering techniques to systems whose dynamical properties are not uniform in space 共het- erogeneous dynamics兲 关20兴. Moreover, the question about the possible limitations due to the loss of coherence remains to be addressed.

In this paper we present a description of the scattering- based image formation process with partially coherent illu- mination, including the effect of out-of-focus planes in the microscope images. The main focus here is on the applica- tion of such a description to the microscope-based investiga- tion of dynamical systems, with a special reference to the results presented in Ref. 关19兴. Our results suggest that dy- namic microscopy techniques, and in particular differential dynamic microscopy 共DDM兲 关19兴, can be quite flexible di- agnostic tools for the study of dynamical phenomena, in par- ticular for those laboratories that are already equipped with a microscope. We generalize our results to all those cases where the imaging is a linear space-invariant process, includ- ing fluorescence microscopy. The paper structure is as fol- lows. In Sec.IIwe present experimental results on colloidal samples obtained with DDM and provide a two-dimensional 共2D兲model of dynamic microscopy experiments in the Fou- rier space, along the line followed in Ref.关19兴. In Sec.IIIwe give a fully quantitative theory for the three-dimensional 共3D兲bright-field imaging of weak scattering objects, i.e., that do not dephase or absorb light too strongly. In Sec. IVwe treat the more general case of linear space-invariant imaging and provide theoretical results for the case of fluorescence imaging as an example. In Sec. Vwe describe various ap- proaches to the analysis of time series of microscopy images.

II. ROUTE TO THE DESCRIPTION OF DYNAMIC MICROSCOPY EXPERIMENTS IN THE FOURIER SPACE

To introduce dynamic microscopy in the Fourier space, we first present in this section the data obtained by applying the DDM technique to a series of microscopy images, ob- tained by using a Brownian colloidal dispersion as a sample.

However, we stress that DDM is just one of the methods that can be used to perform dynamic microscopy experiments in the Fourier space. Alternative algorithms will be presented in Sec.V. The DDM data presented here will be used through- out the paper to test the theoretical descriptions of scattering- based image formation, which we introduce stepwise.

As a first step we discuss the DDM data in the frame of a simplified two-dimensional model, introduced in Ref. 关19兴.

Our experimental system is a dispersion of polystyrene par- ticles共Duke Scientific, Part No. 3070A, nominal diameter of 73.0⫾2.6 nm兲 which is diluted to 1% w/w such that the interactions between the particles are negligible. A capillary tube with a rectangular section共Vitrocom, Inc.兲is filled with the colloidal dispersion and imaged by means of a commer- cial microscope共LEICA DM IRB兲. The images are acquired with a complementary metal-oxide-semiconductor 共CMOS兲 camera共IDT X-StreamXS-3, 1280⫻1024 pixels, pixel size of 12 ␮m兲. The thickness of the capillary tube along the

optical axis is 100 ␮m and the midplane of the capillary tube is imaged onto the camera sensor. Typical data consist of a sequence of approximately 1000–2000 images, acquired with a sampling rate of 100–400 images/s and with an expo- sure time of 2.5–10 ms.

In Fig. 1 we show two images that are obtained at ⌬t

= 1 s distance with the following experimental parameters:

exposure time of 2.5 ms, numerical aperture of the objective No= 0.85, magnificationM= 40, 2⫻2 binning, and effective pixel sizedpix= 0.6 ␮m. Both images exhibit a strong back- ground due to the transmitted beam and artifacts due to dust particles on the camera sensor and imperfections on the op- tical surfaces. The scattering contributions of the particles are barely visible and can be brought to light by selecting the image in Fig.1 共left兲as a reference and subtracting it from the one taken at later time. This procedure is expected to remove all the time-independent signals and to isolate the signal associated with the particles’ motion. The subtraction procedure can be repeated for different values of ⌬t, as shown in Fig. 2. The inspection of Fig. 2 reveals that the time-independent stray signal present in the original images has been effectively removed from the original images. A small-scale signal is now evident which is purely associated with the particles’ motion. The amplitude of the signal in- creases with ⌬t and a more quantitative assessment of this increase can be obtained by defining the stochastic two- FIG. 1. Microscopy images of a colloidal dispersion acquired 1 s apart in time 共details in the text兲. The size of each panel corre- sponds to about 150 ␮m. Because of the small scattering signal of the particles the two images appear to be identical. The visible signal is due mainly to stray light共for example, dust particles on the optical surfaces of the microscope, on the detector, and on the sur- faces of the sample cell兲.

FIG. 2. Images obtained by subtracting two microscopy images that are taken at time intervals of⌬t= 0.01, 0.1, and 1 s共from left to right兲. The subtraction removes efficiently the stray signal and iso- lates the contribution due to the particles. The average size of the speckles gives an estimate of the microscope resolution. The con- trast in the images increases with⌬t.

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dimensional variable ⌬I共x,t;⌬t兲=I共x,t+⌬t兲−I共x,t兲, where I共x,t兲is the intensity detected at the pixel with positionxon the detector.

In the following we will make the simplifying but not necessary assumption that the dynamics is stationary, i.e., the dynamical properties of the system depend only on the time delay ⌬t but not on the actual time t that is chosen as a reference. We will drop accordingly the dependence on t.

The expectation value 具⌬I共x;⌬t兲典 is then equal to zero and the energy content of the intensity fluctuation ⌬I共x;⌬t兲 can be quantified by its first nonzero moment, i.e., the expecta- tion value␴²共⌬t兲of the variance defined as

␴²共⌬t兲=

冕冕

^{具兩⌬I共x;⌬t兲兩2典dx. 共1兲}

For dynamically active systems,␴²共⌬t兲is expected to in- crease with⌬tas a consequence of the particles’ motion关19兴.

For our Brownian colloidal dispersion, this increase is shown in Fig. 3, where we display data obtained with the highest frame rate 共400 Hz兲 to capture the initial increase in the variance. The signal increases from a small background value, associated with the detector noise, and saturates to a plateau, which is indicative of the complete loss of positional correlation between the particles.

The DDM technique is based on a Fourier analysis of

⌬I共x;⌬t兲and the basic concept behind it is easily explained for a 2D object, i.e., for a sample having negligible thickness along the optical axis. The object refractive index distribu- tion can be Fourier decomposed and each Fourier component can be labeled with a 2D Fourier wave vector qobj. Each Fourier component of the refractive index distribution acts as a periodic diffraction grating, scattering light at an angle

␪= sin⁻¹共qobj/ko兲with respect to the incident wave vectorko. For an imaging system with unitary magnification, the dif- fracted plane wave causes in the image a sinusoidal fringe

pattern with 2D Fourier wave vector q such that q= 2␲^sin␪

␭o

=qobj, 共2兲

where we have used ko= 2␲/␭o 共␭o is the wavelength of light兲. Equation共2兲is the key relation for performing DDM experiments and, more generally, near-field scattering experi- ments. Indeed, it relates to a one-to-one fashion a Fourier component of the measured intensity distribution with the corresponding one in terms of the sample refractive index.

The simplest way of taking advantage of this result would be to study the behavior in time of the Fourier transform of the intensity I共x,t兲, defined as Iˆ共q,t兲

=共1/2␲兲兰兰dxI共x,t兲e^−jq·x, where q=共qx,qy兲. This could be easily done by means of theimage correlation function

G共q,⌬t兲=具Iˆ^ⴱ共q,0兲Iˆ共q,⌬t兲典 共3兲 or thenormalized image correlation function

g共q,⌬t兲=G共q,⌬t兲

G共q,0兲 . 共4兲

Here, the expectation value具¯典 is taken over many sta- tistically independent realizations of the signal. For the case of Brownian motion considered here, we can calculate the 2D correlation function关4兴

g_2D共q,⌬t兲= exp关−⌬t/␶d共q兲兴, 共5兲 where ␶d共q兲=共Dmq²兲⁻¹ is the characteristic diffusion time constant and D_m is the particle diffusion coefficient. How- ever, because of the large and nonhomogeneous background intensity, it was seminally suggested in Ref.关10兴that a more robust statistical estimator is represented by the expectation value of the Fourier power spectrum of⌬I共x;⌬t兲, i.e.,

D共q,⌬t兲⬟具兩⌬Iˆ共q;⌬t兲兩²典. 共6兲 Incidentally D共q,⌬t兲 is the two-dimensional generaliza- tion of the photon structure function used in DLS experi- ments 关21–24兴. To stress the fact that the use of images al- lows us to have a multitude 共typically more than 1.0⫻10⁶兲 of photon structure functions in parallel, we will name it image structure function. In analogy with DLS, the correla- tion and the structure functions are related by

D共q,⌬t兲= 2关G共q,0兲−G共q,⌬t兲兴, 共7兲 which is valid for statistically stationary processes. It is worth noting that G共q,⌬t兲 is a decreasing function of ⌬t from G共q, 0兲 to G共q,⌬t→⬁兲= 0. Equation 共7兲 implies that D共q,⌬t兲 is an increasing function of ⌬t from zero to 2G共q, 0兲. Even if from a theoretical point of view the struc- ture and the correlation functions are equivalent, we will use for the data analysis the structure function approach 共a deeper discussion about this point can be found in Sec.V兲.

Without prior knowledge about the relationship occurring between intensity and refractive index fluctuations, the struc- ture function is generally given by

FIG. 3. The particle rearrangement due to their Brownian mo- tion causes an increase in the variance of the intensity fluctuation

⌬I共x;⌬t兲. With increasing time delay⌬t, the two subtracted images become progressively uncorrelated, such that ␴²共⌬t兲 saturates at large⌬t. Each experimental point is the result of an average ob- tained by using 100 statistically independent ⌬I images with the same⌬t. The line is drawn with no adjustable parameters from the theory presented in Sec.III.

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D共q,⌬t兲=A共q兲关1 −g共q,⌬t兲兴+B共q兲, 共8兲 whereA共q兲is related to the scattering properties of the par- ticles, to the coherence properties of the light source, and to the properties of the microscope objective.B共q兲accounts for the noise of the detection system and is equal to 2 times the power spectrum of the camera noise. As already pointed out in Ref. 关19兴, previous knowledge of A共q兲 and B共q兲 is not necessary for performing DDM experiments. For example, in the case of Brownian motion, we can use Eq. 共5兲 and treat A共q兲,B共q兲, and␶d共q兲as fitting parameters. The information about the dynamics can then be extracted from the analysis of the parameter␶d共q兲.

In Fig.4we showD共q,⌬t兲for different values of⌬t. As expected the energy content increases with ⌬t. Because of the azimuthal symmetry exhibited byD共q,⌬t兲, we consider the azimuthal averageD共q,⌬t兲, whereq=

冑

q_x²+q_y². The result of this averaging operation is shown in Fig. 5, where the azimuthal averageD共q,⌬t兲as a function of the wave vector q is presented for different values of the time delay⌬t. The area under the curve, i.e., the variance␴²共⌬t兲, increases with

⌬t.

The characterization of the dynamics passes through the analysis of the behavior ofD共q,⌬t兲as a function of the time delay⌬t. Typical curves are presented in Fig.6for different wave vectorsq. From a fitting procedure based on Eq.共8兲, it is possible to extractA共q兲,B共q兲, and␶d共q兲. To facilitate the comparison with DLS experiments, we plot in Fig.6共bottom panel兲 the normalized correlation function g共q,⌬t兲= 1

−关D共q,⌬t兲−B共q兲兴/A共q兲. A fair exponential decay is ob- served for the three data sets, as expected for a Brownian system.

The results of the fitting procedure for␶d共q兲are plotted in Fig. 7 共open symbols兲, together with data points obtained with a traditional rotating-arm DLS setup共closed symbols兲.

The DLS data have been acquired on a more dilute sample 共0.001% w/w兲 in the angular range of 20° – 150°. The two data sets agree very satisfactorily with the predictions of the Stokes-Einstein-Sutherland relation D_m=k_BT/共6␲␩^R兲. Here, kB is the Boltzmann constant,T is the absolute tem- perature, ␩ is the solvent viscosity, and 2R is the particle diameter. The result of the theoretical estimate, accounting for the polydispersity of the sample and including a correc- tion for the finite concentration of the colloids is Dm

= 6.0⫾0.2 ␮m²/s, in good agreement with the experimental data. Figure7shows that DDM operates at extremely lowq, typically not accessible with DLS and that the simultaneous use of the two techniques guarantees an access to dynamical information over more than 2 decades in q.

As far as the other two fitting parameters are concerned, we show in Fig.8the results for the amplitudeA共q兲and the FIG. 4. Experimentally determined image structure function

D共q,⌬t兲for the images⌬I共x;⌬t兲in Fig.2:⌬t= 0.01 s,⌬t= 0.1 s, and ⌬t= 1 s. Each image is the result of an average over 2000 statistically independent ⌬I images with the same ⌬t. The black cross at the center of each image is the result of a postprocessing operation and is used to suppress artificially large contributions due to image processing artifacts. The larger energy content of the right image can be appreciated.

FIG. 5. The image structure function D共q,⌬t兲 is plotted as a function of qfor different values of⌬t:⌬t= 0.01 s共open circles, black兲,⌬t= 0.1 s共open squares, dark gray兲, and⌬t= 1 s共close up triangles, light gray兲.

FIG. 6.共Color online兲 共Top兲Growth ofD共q,⌬t兲as a function of

⌬tfor three different values ofq: 0.70共blue down triangles兲, 1.06 共red up triangles兲, and 2.13 ␮m⁻¹共black circles兲.共Bottom兲 Expo- nential decay of the normalized correlation functionG共q,⌬t兲corre- sponding to the same wave vectors.

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background B共q兲 in the wave-vector range 关0.1, 5兴 ␮m⁻¹. The signal A共q兲 can be appreciably discriminated from the background B共q兲 in the wave-vector range 关0.3, 5兴 ␮^m−1. The functionA共q兲has a deep minimum atq= 0, increases to a peak forq⯝1 ␮m⁻¹, and decreases again for larger values ofq. Unfortunately, the 2D model presented in Ref.关19兴and outlined above cannot quantitatively account for the behavior of A共q兲. In this paper we derive a 3D theory that, among other things, predicts the behavior of A共q兲 as a function of the parameters of the microscope. We anticipate here that the theory captures the basic ingredients of the experiments as can be appreciated from Fig.8, where the result of the fitting of theA共q兲data with Eq.共62兲is shown as a dashed line. The agreement of the data with the theory is good and will be discussed in more detail in Sec.III.

A. Toward a 3D model: The Brownian particle Let us consider a Rayleigh scatterer, i.e., a particle whose size is much smaller than the wavelength ␭0 and a mono-

chromatic plane wave e^jk0z impinging on the particle along the optical axis z. The unitary amplitude impinging plane wave is characterized by a wave vector ki and the light quasielastically scattered at an angle ␪ with respect to the incident direction has a wave vector k_s, wherek₀=兩k_s兩=兩k_i兩 共see Fig.10for the wave-vector diagram兲. The interaction of the particle with the plane wave produces a spherical wave propagating away from the particle and centered on it 关25兴.

The resulting scalar field, measured at distance z from the particle center, is given by

U共x,z兲=e^jk0z+Se^jk0r

r , 共9兲

wherer=

冑

_兩x兩²+z² andS is related to the scattering contrast of the particle 关25兴. With a simple change in the reference frame, we can write the intensity pattern in the plane z= 0 due to a particle located in共x0,z₀兲as

I共x兩x0,z₀兲 ⯝1 +2S

z₀cos

冉

^k0兩x2z⁻₀^x0兩2

冊

^{, 共10兲}

where we made the paraxial approximation r=

冑

兩x兩²+z²⯝z +兩x兩²/2z. This pattern has the typical appearance shown in Fig. 9, where it can be appreciated that, while a particle displacement in the共x,y兲plane causes an overall translation of the fringes, a motion alongzis associated with a change in FIG. 7. 共Color online兲Experimentally determined characteristic

decay time ␶d plotted against the wave vector q for the 73 nm particles. Open共black兲 circles are data obtained with DDM. Close 共red兲circles are DLS data. The continuous line in the graph is the theoretical estimate corresponding toD_m= 6.0 ␮m²/s共see text for details兲.

FIG. 8. 共Color online兲 Experimentally determined A共q兲,B共q兲 plotted against the wave vectorqfor a colloidal dispersion of par- ticles with a diameter of 73 nm. The dashed line is a fit of the data forA共q兲as discussed in the text.

FIG. 9. Simulated interference pattern generated at some dis- tance from a Rayleigh scatterer 共particle兲 illuminated by a plane wave, which propagates along thezdirection.共Upper left兲the par- ticle is in 共x₀,y₀,z₀兲; 共upper right兲 the particle has moved in共x₀ +⌬x,y₀+⌬y,z₀兲. The fringes are unchanged and translated in the 共x,y兲 plane by an amount 共⌬x,⌬y兲;共lower left兲 the particle is in 共x₀,y₀,z₀+⌬z兲i.e., further away from the image plane. The fringes are not translated in the 共x,y兲 plane but their shape has changed;

共lower right兲the particle is in共x₀+⌬x,y₀+⌬y,z₀+⌬z兲. The fringes are translated in the 共x,y兲 plane by an amount 共⌬x,⌬y兲 and also modified by virtue of the particle axial displacement⌬z.

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the fringe spacing, which depends on the value of z₀. These fringes do contain precious information about the position of the particles along the optical axis z. In principle this infor- mation allows for the recovery of the three-dimensional po- sition of the particle as routinely done in digital holographic microscopy关26兴.

Here, we use this information in a different way. The two- dimensional Fourier transform of the intensity pattern is given by

Iˆ共q兩x₀,z₀兲 ⯝␦共q兲+2S

k₀e^jq·x0sin共q_zz₀兲 共11兲 where qz⯝q²/2k₀ is the paraxial approximation of the z component of the scattering wave vector Q=k_s−k_i=共q,q_z兲. Indeed, Q=兩Q兩= 2k₀sin共␪/2兲⯝k₀␪,

Q²⯝q²+

冉

2k^q2₀

冊

^{2, 共12兲}

which implies thatq_z⯝q²/2k₀.

It has to be noted that results similar to Eq.共11兲have been already presented before关27–30兴for visible, x-ray, and elec- tron optics. The single-particle model used here allows us to easily interpret in the Fourier space any particle motions. The motion of the particle along thezaxis causes an appreciable phase shift of Iˆ共q兲when an axial displacement on the order of 1/qztakes place, whereas a displacement in the transverse plane has to be on the order of 1/q to cause an appreciable effect. By assuming a Brownian motion of the particle with a diffusion coefficient Dmand inserting the above expression in Eq.共6兲, we obtain

D共q,⌬t兲= 2G共q,0兲关1 −e^{−Dm共q2+qz2兲⌬t}兴

= 2G共q,0兲共1 −e^−DmQ2⌬t兲, 共13兲 whereG共q, 0兲= 4S²/k₀². This model can be easily extended to a collection of noninteracting identical particles and one ob- tains a result identical to Eq.共13兲. This result shows that the dynamics of the Brownian motion of a system of particles can be characterized by investigating the image structure function. The outcome of this investigation performed with coherent light is fully equivalent to a DLS experiment, pro- vided that the dynamics measured at the 2D wave vectorqis correctly associated with the correct 3D wave vectorQ. It is worth noting that for small scattering angles ␪ one has Q

⯝q and the fractional error in the determination of the char- acteristic time of the exponential by assuming that Q⯝q scales as 共Q/2k₀兲²= sin²共␪/2兲. For example for ␪⯝20° the fractional error amounts to about 3%. The advantage of the single-particle model is that it can be easily used to get a semiquantitative understanding of the effects of the limited coherence of the illuminating light. To this purpose we draw in Figs. 11 and 12 two wave-vector diagrams similar to the one reported in Fig. 10 for plane-wave illumination.

These diagrams illustrate the main effect of partially coher- ent illumination on a scattering experiment: different three- dimensional wave vectorsQmay give rise to the same two- dimensional projectionq.

In the case of spatially incoherent illumination the particle is illuminated with a set of plane waves whose mutual phase relationship is a random stochastic variable. In Fig. 11 we can appreciate that the wave vector k₁ gives rise to a scat- tering wave vectorQ₁=共q₁,q_z

1兲 andk₂to a scattering wave vector Q₂=共q2,qz₂兲. The two wave vectors q₁ and q₂ can have the same amplitude provided that the z components have the appropriate value. This means that the two- dimensional wave vector q might correspond to different three-dimensional wave vectorsQ.

A similar effect can be associated with the polychroma- ticity of the source. In Fig. 12the two wave vectorsk₁ and k₂do not differ in direction, as for the previous case, but in amplitude, due to their different wavelengths. Again, the one-to-one correspondence betweenq andQis lost. In gen- eral for everyqit exists a whole set ofqzthat correspond to the same three-dimensional wave vectorQ.

q_z

q k_i

k_s Q

FIG. 10. Diagram showing the relationship between the wave vectors involved in the scattering and imaging processes. The ref- erence system is chosen as follows. Thez axis coincides with the direction of propagation of the incident light, which is characterized by the wave vectork_i. The共x,y兲plane is perpendicular to the inci- dent wave vector.

q_z2

q_z1 q₂

q₁ k₂

k₁ k_s2 k_s1Q₂

Q₁

FIG. 11. Two plane waves forming different angles with respect to the optical axis are characterized by two wave vectorsk₁andk₂ with the same amplitudekbut different directions. They give rise to the same two-dimensional projection q, provided that the compo- nentsq_z1andq_z2are different. Therefore, we expect⌬q⫽0 when- everN_s⫽0.

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B. Lack of spatial coherence

We model the lack of spatial coherence by assuming that the illuminating beam is the 共incoherent兲 superposition of many monochromatic plane waves of the form U_␦k

=e^jk0z+␦k·x. Such a wave forms an angle␦^k/k₀with respect to the zaxis. The summation over the intensity pattern associ- ated with each independent plane wave leads to

I共x兩x0,z₀兲 ⯝1 +2S

z₀

冓

^cos

^冉

^{k兩x2z−0x0兩2−␦k·共x−x0兲}

^冊 冔

^␦k

= 1 +2S

z₀fˆ关⌬ks共x−x₀兲兴cos

冉

^k兩x2z−0^x0兩2

冊

^{, 共14兲}

where we have used as a weight function P共␦k兲

=共1/⌬k_s²兲f共␦^k/⌬k_s兲, and wheref is assumed to be a normal- ized symmetric bell-shaped function with width of ⬃1. In this model ⌬ks/k₀⯝2Ns, where Ns indicates the numerical aperture of the source. This expression leads to a Fourier intensity pattern

Iˆ共q兩x0,z₀兲 ⯝␦共q兲+2S

k₀e^jq·x0f共q/⌬ks兲丢qsin共q¯_zz₀兲, 共15兲 where the symbol 丢qindicates a convolution operation with respect to the variable q. Interestingly, the loss of spatial coherence does not affect the transverse part of Iˆ共q兩x₀,z₀兲 but only the z-dependent term. The function sin共q¯_zz₀兲

= sin共^q_2k^2z₀⁰兲is a chirped function ofq, whose local spatial fre- quency scales as qz₀/k0. This implies that the convolution operation leaves the function sin共q¯_zz₀兲 more or less un- changed at small wave vectors q, where the width of f共q/⌬ks兲 remains small with respect to the local spatial pe- riod of sin共q¯_zz₀兲. For largeq the oscillations of sin共q¯_zz₀兲are smeared out and average to zero. The crossover value of z₀ that marks this transition is

Ls= k₀ q⌬ks

= 1 2qNs

. 共16兲

This value can be interpreted as a q-dependent depth of focus and should be compared with the typical distance trav- eled by a particle during a time interval ⌬t. If

⌬tⰆ␶s⬃ L_s²

D_m⯝ 1

共N_s兲²D_mq², 共17兲 the dynamics is not affected by the limited spatial coherence of the light. For longer times the path becomes longer thanL_s and this imposes a cutoff on the structure function. By taking into account that the Brownian motion acts on time scale on the order of 1/Dmq², the condition for neglecting this spuri- ous dynamics becomes

N_sⰆ1, 共18兲

i.e., as expected the numerical aperture of the light source should be kept small. However, it is worth stressing that the lack of spatial coherence introduces a q-independent effect.

Minimizing the effect of the illumination numerical aperture at one wave vector guarantees that the correct dynamics is measured at every wave vector. Equation 共18兲can be made more quantitative by providing a detailed expression for f共q/⌬ks兲. At this stage this is not worth in view of the com- plete model that will be presented in Sec.III.

C. Lack of temporal coherence

A similar reasoning can be made for describing the effect of the limited temporal coherence that can be modeled by assuming that the illuminating beam is the 共incoherent兲su- perposition of many monochromatic plane waves with differ- ent wavelengths共wave vectors兲U_␦k=e^{j共k0+␦k兲z}. We can again assume a spread P共␦^k兲=共1/⌬k_t兲g共␦^k/⌬k_t兲, whereg is again assumed to be a normalized symmetric bell-shaped function of unitary width. Under this assumption,

Iˆ共q兩x0,z₀兲 ⯝␦共q兲+ 2S

k₀e^jq·x0

冓

^sin

^冋

^¯qz

^冉

^{1 −␦kk0}

^冊

^z0

^册 冔

^␦k

=␦共q兲+2S

k₀e^jq·x0gˆ

冉

^⌬kk₀^t¯q_zz₀

冊

^{sin共q¯zz0兲, 共19兲}

wheregˆ is the Fourier transform of the spectral distribution g. By following a line of reasoning similar to the one out- lined above for the case of limited spatial coherence, it is possible to identify aq-dependent depth of focus

Lt= k₀

¯q_z⌬kt

. 共20兲

In this case if

⌬tⰆ␶t⬃ L_t²

D_m=

冉

⌬^kk⁰_t

冊

²Dm¹¯q_z², 共21兲 the effect of the limited temporal coherence of the source can be neglected. Again, we can compare␶mwith the Brownian time scale and we get the condition

q_z2 q_z1

q₂ q₁ k₂ k₁

k_s2 k_s1 Q₂ Q₁

FIG. 12. Two plane waves having different wavelengths are characterized by two wave vectorsk₁andk₂with different ampli- tudes but with the same direction. If they impinge on the sample along the same direction, they could give rise to the same two- dimensional projectionq, provided that the componentsq_z1andq_z2 are different. Therefore, we expect⌬q⫽0 whenever⌬␭⫽0.

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qⰆ 1

⌬␭ 共22兲

for neglecting the spurious dynamics across Lt. By contrast with the previous case we find a q-dependent condition.

An order of magnitude estimate can be obtained by consid- ering that for typical microscope lamps one has ⌬␭

⯝0.1 ␮m and Eq. 共22兲 becomes qⰆ10 ␮m⁻¹. Again, a more precise estimate would require to model the function P共␦k兲=共1/⌬kt兲g共␦^k/⌬kt兲, which is not done in this section.

The single-particle model just presented succeeds in pointing out the basic ingredients of the effect of partially coherent illumination in dynamic microscopy experiments.

In the next section we will use a more general approach based on Fourier optics arguments. This approach is valid for an arbitrary refractive index distribution of the共weak兲object and accounts also for the properties of the collection optics.

III. MICROSCOPY OF WEAK OBJECTS A. Nemoto-Streibl model of a microscope

An elegant but rather involved treatment of quasimono- chromatic共i.e., the wavelength spread⌬␭is negligible with respect to the peak wavelength␭0兲microscopy with partially coherent light has been proposed by Streibl for 3D weakly scattering objects关31兴. The description is based on the Helm- holtz equation for the scalar field U共x兲 and the object is assumed to be weak in such a way to allow for a first-order Born approximation. A subsequent theory due to Nemoto 关32兴succeeded in showing that Streibl description is consis- tent with the well-known theory of 2D image formation due to Hopkins关33,34兴. If each layer within the object is treated independently with the Hopkins theory, the final image is shown by Nemoto to coincide with the sum of the images of the individual layers. We find convenient to rewrite the Nemoto-Streibl results in the following way:

I共x,t兲=I₀+

冕冕冕

^{dx⬘dz⬘K共x−x⬘,−z⬘兲c共x⬘,z⬘,t兲.}

共23兲 This means that the intensity in the image plane is a linear superposition of all the contributions coming from the differ- ent layers of the object. Each of these contributions is a x convolution of the densityc共x,z,t兲with the transfer function K共x,z兲 that also acts as a weighting function along z. The model of the kernel K共x,z兲 is a challenging task and we report here the results of Nemoto, by using a notation more suitable to our purposes.

The layout of the microscope is presented in Fig.13that describes a 6-f imaging system characterized by unitary magnification. The light emitted by a light source with inten- sity S共q兲 is sent onto the object by using a condenser lens.

The object is described by a complex transmission function f共x,z兲=兩f共x,z兲兩exp关j␾共x,z兲兴. If the object is weak 关i.e., if 兩f共x,z兲兩⯝1 and ␾共x,z兲⯝0兴, one has f共x,z兲= 1 +fA共x,z兲 +jfP共x,z兲, where兩fA共x,z兲兩Ⰶ1 andfP共x,z兲⯝␾共x,z兲. Our ob- ject can be generally described in terms of the density c共x,z,t兲 of the moving particles, where t indicates time, z

tags the particles’ position along the optical axis, and x

=共x,y兲is perpendicular to it. The coefficientsaAandaPare strictly related to the complex refractive index n=nR+jnI

because f共x,z兲= 1 +fA共x,z兲+jfP共x,z兲= 1 +jk₀^⳵_⳵ⁿ_cc共x,z兲. This implies the relationsfA=aAc共x,z兲andfP=aPc共x,z兲. It is evi- dent from Fig. 13that the object is extended along the opti- cal axiszand that the planez= 0 is located somewhere in its proximity. Also, we will temporarily assume that the sample thickness alongzis infinite. The light scattered by the object recombines with the transmitted beam onto a suitable sensor 共typically a pixel detector兲 by means of an optical system 共objective兲 described by a pupil function p共q兲. It is worth noting that the functionS共q兲is directly related to themutual coherence function in the object plane jS共r兲=具U0共x +r兲U₀^ⴱ共x兲典, where U₀共x兲⬅U共x,z= 0兲. The link between the two functions is given by the Van Cittert and Zernike theo- rem 关34兴that is easily expressed in the following form:

JS共q兲=S共q兲, 共24兲 whereJS共q兲is the 2D FT of jS共r兲.

Under the reasonable assumption that bothJ_S共q兲andp共q兲 are real-valued functions we obtain

K˜共q,qz兲=aATAF共q,qz兲+jaPTPF共q,qz兲, 共25兲 where

冉

^TTAFPF^共q,q共q,q^z_z^兲兲

冊

^{=T+共q,qz兲⫾T−共q,qz兲 共26兲}

and

T_⫾共q,qz兲 ⬅

冕冕

^pⴱ

冉

^q⬘+q2

冊

^JS

冉

^q⬘⫾q2

冊

^p

冉

^q⬘−q2

冊

^␦

冉

^qz

+q·q⬘

k₀

冊

^{dq⬘. 共27兲}

In Eq.共25兲we have introduced the 3D Fourier transform K˜共Q兲=共2␲兲^−3/2兰兰兰dXK共X兲e^−jQ·X, where X=共x,z兲 and Q

=共q,qz兲. The two functionsTAF共q,qz兲andTPF共q,qz兲are 3D optical transfer functions共OTFs兲, respectively, for the ampli- tude and the phase of the object. They account for the partial coherence of the light source, the properties of the objective, and the 3D nature of the object.

If we assume in addition that bothJS共q兲andp共q兲are even functions we can write

FIG. 13. 共Color online兲 Simplified layout of an optical microscope.

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冉

^TTAFPF^共共q,q^q,qzz^兲兲

冊

^{=T共q,qz兲⫾T共q,−qz兲, 共28兲}

where we have reduced the knowledge of the two functions TAF and TPF to the knowledge of the single function T共q,q_z兲⬅T₊共q,q_z兲, which is also an even function with re- spect toq. The evenness ofJS共q兲andp共q兲is very plausible in most cases. Indeed, the typical apertures in microscopes are described by circularly symmetric functions and the an- gular distribution of light from the source does also exhibit a symmetric nature.

B. Calculation of the transfer functionT(q ,q_z) An analytical calculation of the transfer functionT共q,q_z兲 is a challenging task for realistic pupils and sources. A nu- merical analysis of the imaging problem can be performed by using the formulas derived here. However, such an analy- sis is outside the aim of the present work, in which we prefer to describe the basic physics of the imaging problem by us- ing convenient Gaussian functions in both cases. This allows us to obtain exact results to guide the experimentalists in choosing optimum working conditions for dynamic micros- copy experiments.

We model the source coherence with JS共q兲= 1

2␲␴c

2k₀²exp

冉

⁻¹²␴^qc²

2k₀²

冊

^共29兲

and the pupil function with

p共q兲= exp

冉

⁻¹²␴^qo²

2k₀²

冊

^{, 共30兲}

where␴cgives an estimate of the numerical aperture of the condenser and␴oof the objective. We define theincoherence parameter

M=␴c

␴o

共31兲 that allows a continuous transition from spatially coherent 共M= 0兲to incoherent共M=⬁兲illumination. This is consistent with the Hopkins 2D theory of image formation 关33兴, where a similar parameter is used.

By inserting Eqs.共29兲and共30兲into Eq.共27兲, we obtain T共q,qz兲= C共q兲

冑

2␲⌬q共q兲exp

冋

⁻¹²

^冉

^{qz⌬q共q兲−¯qz共q兲}

^冊

²

册

^{. 共32兲}

This function is a Gaussian function of the variableqz, cen- tered at

q

¯_z共q兲= q²

2k₀

冉

^{1 + 2M1 2}

冊

^共33兲

and characterized by a width⌬qgiven by

⌬q²共q兲=q²␴^2, 共34兲 where

␴^{2= 1} 2

␴o 2+ 1

␴c 2

= ␴c 2

1 + 2M². 共35兲

Note that ␴⯝␴c if MⰆ1, while ␴⯝0 when MⰇ1. The normalization of Eq. 共32兲 is such that 兰T共q,qz兲dqz=C共q兲, where

C共q兲=

exp

冋

⁻¹²

冉

^qqro

冊

²

册

1 + 2M² 共36兲

is a decreasing function of the wave vectorq that describes the overall frequency modulation introduced by the imaging process. The decrease inC共q兲is characterized by the roll-off wave vector

qro=k₀␴o

冑

^{1 + 2M}1 +M²² 共37兲 that is a measure of the resolution of the microscope. Indeed, the minimum detail that can be resolved by such a micro- scope is given by

x_min=2␲ qro

=

冑

1 + 2M^{1 +M22}

␭0

␴o

. 共38兲

It is quite interesting to note that the resolution increases with the incoherence parameter M in agreement with Hop- kins theory for 2D microscopy 关33兴. More in detail x_min

→␭0/␴o for M→0 and x_min→␭0/共

冑

2␴o兲 for M→⬁. The appearance of the factor

冑

2 is a consequence of the choice of Gaussian functions to model the microscope properties.

Other choices would lead to different numerical factors but always larger than 1.

As far as the contrast of the microscope images is con- cerned, we investigate also the behavior of the amplitude and phase transfer functions in Eq.共25兲. WhileT_AF共q,q_z兲is pro- portional to the sum of two Gaussian functions centered at

⫾q¯_z, TPF共q,qz兲 turns out to be proportional to their differ- ence. The superposition ratio

r_s共q兲=

冉

^⌬¯qqz

冊

²⁼

冋

^{2␴kq0共1 + 2M2兲}

册

^{2 共39兲}

is a quantitative measure of the superposition of the two Gaussian functions. If the ratiorsis small, this superposition is negligible. This happens if q/k₀Ⰷ2␴共1 + 2M²兲. Only for small M it is possible to satisfy the latter condition together with the requirement that the ray is transmitted through the objective共q/k0Ⰶ␴o兲. In this casersis small ifq/k₀Ⰷ␴c.

Equation 共32兲 is an analytical model for describing 3D microscopy of weak objects. In the context of our paper it will serve as a starting point for a more refined description, incorporating also the effect of polychromatic illumination.

C. Effect of polychromatic illumination

The Streibl-Nemoto theory specified to Gaussian pupil and Gaussian mutual coherence function was used in the previous paragraph for the calculation of the transfer func-

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tionT共q,qz兲. In this paragraph we shall include the effect of polychromatic illumination. The latter can be accounted for quantitatively by modeling the source with a Gaussian spec- trum

P共␭兲= 1

冑

2␲⌬␭exp

冋

^{−12共␭⌬␭−␭20兲2}

册

^共40兲

that can be used as a weight function for the individual in- tensity patterns associated with each wavelength. Again this choice is made in order to obtain results in a closed form.

The result of this calculation for the functionT共q,q_z兲can be still written as in Eq. 共32兲. The most general form for the parameters¯q_z共q兲,⌬q²共q兲, andC共q兲is not particularly useful in this context. Here, we report the simpler approximate ex- pressions obtained with a lowest-order expansion in bothM and⌬␭/␭0,

¯q_z共q兲= q²

2ko

冋

^{1 − 2M2−␴1o2}

冉

^kq0

冊

²

冉

^⌬␭␭0

冊

²

册

^{, 共41兲}

⌬q²共q兲=q²

冋

^{共␴oM兲2+14}

冉

^kq0

冊

²

冉

^⌬␭␭0

冊

²

册

^{, 共42兲}

C共q兲=

exp

冋

^{− 121 +共q/q共q/qro兲ro2共⌬␭/␭兲2 0兲2}

册

冑

^{1 +}

冉

^qqro

冊

²

冉

^⌬␭␭0

冊

^{2 , 共43兲}

and the result for the superposition ratio reads

r_s共q兲= 4

冉

^kq0

冊

^{2共␴oM兲2+}

冉

^⌬␭␭0

冊

^{2. 共44兲}

As a by-product of the lowest-order approximation just made, most of the parameters characterizingT共q,qz兲are ex- pressed as a sum of the contributions due to effects of tem- poral and spatial coherence limitations, considered sepa- rately. It is worth pointing out that the functionP共␭兲shall in principle account also for the effect of the spectral sensitivity of the detectors used, whose effect is to weight the spectral intensity components of the image.

D. Application to Brownian motion

One of the advantages of the 3D model is that the inten- sity in the microscope images can be quantitatively linked to the sample properties. As far as dynamics is concerned, one has

G共q,⌬t兲=

冕

^{dqz兩K˜共q,qz兲兩2F˜共q,qz,⌬t兲, 共45兲}

where

F˜共q,qz,⌬t兲=具c˜^ⴱ共q,qz,0兲c˜共q,qz,⌬t兲典 共46兲 is theintermediate scattering function关35兴, which is the spa- tial 3D Fourier transform of the space-time density-density correlation function

F共⌬x,⌬z,⌬t兲⬟具c共0,0,0兲c共⌬x,⌬z,⌬t兲典. 共47兲 The functionF˜共Q,⌬t兲has been the object of intense the- oretical work in the past and it has been calculated for many cases of practical interest such as, for example, that of col- loidal particles undergoing Brownian motion or drifting with uniform velocity 关4兴. For the Brownian motion of small noninteracting particles in a host molecular solvent, F˜共q,q_z,⌬t兲=F₀e^{−Dm共q2+qz2兲⌬t}关4兴and one obtains

G共q,⌬t兲=F₀e^−Dmq2⌬t

冕

^{dqz兩K˜共q,qz兲兩2e−Dmqz2⌬t 共48兲}

for the correlation function and

g共q,⌬t兲=e^−Dmq2⌬t兰dq_z兩K˜共q,q_z兲兩²e^−Dmqz2⌬t

兰dq_z兩K˜共q,q_z兲兩² 共49兲 for the normalized correlation function. By inserting Eqs.

共25兲,共28兲, and共32兲in Eq.共49兲, we obtain gP共q,⌬t兲= e^−Dmq2⌬t

冑

1 +Dm⌬t⌬q²

e^{−关Dm⌬t/共1+Dm⌬t⌬q2兲兴¯qz2}−e^{−共¯qz/⌬q兲2} 1 −e^{−共¯qz/⌬q兲2} ,

共50兲 which is valid for phase objects 共n_I= 0兲, and

g_A共q,⌬t兲= e^−Dmq2⌬t

冑

1 +Dm⌬t⌬q²

e^{−关Dm⌬t/共1+Dm⌬t⌬q2兲兴q¯z2}+e^{−共¯qz/⌬q兲2} 1 +e^{−共¯qz/⌬q兲2}

共51兲 for amplitude 共absorbing兲 objects 共nR= 0兲. Equations 共50兲 and共51兲simplify if the superposition parameterrsⰆ1, i.e., if

⌬qⰆ¯q_z, and in both cases we obtain the 3D generalization of Eq. 共5兲,

g共q,⌬t兲 ⯝g_2D共q,⌬t兲gz共q,⌬t兲, 共52兲 where the factor

gz共q,⌬t兲=e^{−关Dm⌬t/共1+⌬t/␶⌬q兲兴¯qz2共q兲}

冑

1 +⌬t/␶_⌬q

共53兲

quantifies the contribution of the particles’ motion along the microscope optical axis共axial motion兲. Here we have defined a time scale␶_⌬q⬟1/D⌬q².

It is instructive to study two limit cases for the behavior of gz共q,⌬t兲. For coherent illumination we have

g_z,coh共q,⌬t兲= exp关−Dm共q²/2k0兲²⌬t兴 共54兲 and, if we take into account Eq. 共12兲, we obtain g共q,⌬t兲

=e^−DmQ2⌬t. This proves that for coherent illumination the motion along one of the three axes contributes to the decay of the correlation function with a term of the form exp共−Dmq_i²⌬t兲, withi=x,y,z. This symmetry is broken with incoherent illumination because

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