Differential dynamic microscopy

Differential dynamic microscopy (DDM) is an imaging-based optical technique that enables performing light scattering-like experiments by means of a conventional optical microscope. It uses time sequences of microscope images to extract dynamic information about microscopic fluctuations over a range of length scales and time delays, analogous to what is obtained in dynamic light scattering (DLS) experiments.^[1][2] DDM is suitable for typical soft materials such as liquids, gels, suspensions of colloids and polymers, liquid crystals and it has been applied to various biological systems including bacteria and cells.

History and developments

The DDM technique was introduced in 2008 by Cerbino & Trappe as a way to probe wavevector-resolved dynamics of colloidal particles using a microscope equipped with a high-speed camera, demonstrating that dynamic information similar to that obtained in DLS can be extracted from time-lapse images taken in real space.^[1] This foundational paper has accumulated several citations, establishing the unique capability of DDM to combine imaging capabilities with scattering-based wavevector analysis. Shortly thereafter, Giavazzi et al. discussed extensions to other imaging modalities, also based on fluorescence, and clarified the connections of DDM to related imaging-based scattering methods (see below the section Relationship with other imaging-based scattering methods).^[2]

Subsequent studies applied DDM to characterize bacterial motility and dynamics of microorganisms, showing how population-averaged swim speed and other motility parameters can be obtained without single-particle tracking.^[3][4]

DDM was also extended to confocal fluorescence microscopy to characterize concentrated, multiply scattering and actively driven fluorescent systems, broadening the range of accessible samples and imaging modalities.^[5]

In 2014, Giavazzi & Cerbino provided a broader perspective on digital Fourier microscopy for soft-matter dynamics, placing DDM within a wider family of Fourier-space approaches that extract scattering and dynamical information from microscopy image sequences.^[6]

As the technique matured, comprehensive reviews consolidated the field. In 2017, Cerbino & Cicuta published a perspective article highlighting DDM's ability to extract multi-scale activity in complex fluids and biological systems, establishing it as a bias-free tool for quantifying dynamics across different length scales.^[7] Later, in 2022, Cerbino, Giavazzi & Helgeson reviewed the application of DDM specifically for polymer systems, detailing its use in characterizing polymer solutions, gels, and composites.^[8]

Pedagogical introductions and reproducible workflows have been developed to support adoption of DDM across disciplines. A recent Perspective article reviewed DDM as an emerging approach for measuring diffusion coefficients of macromolecules and particles using standard microscopy videos, discussing practical constraints and opportunities for spatially resolved diffusion measurements.^[9] A tutorial-style presentation of DDM, including two example datasets and analysis scripts, is provided by Germain et al. in the American Journal of Physics.^[10]

In 2025, a comprehensive tutorial article titled The Hitchhiker's guide to differential dynamic microscopy was published, providing step-by-step guidance on experiment design, data acquisition and analysis, and emphasizing reproducible workflows supported by open software and example datasets.^[11]

DDM in a nutshell

DDM starts from a microscope movie, i.e. a time sequence of images ${\displaystyle I(\mathbf {r} ,t)}$ acquired at a fixed plane in the sample. The key assumption is that the recorded intensity fluctuations are related to fluctuations of a physical field of interest (often the number density of particles, or fluorescence-labeled structures), possibly blurred by the microscope point spread function (PSF).^[1][2]

To separate spatial scales, each image is analysed in Fourier space: fluctuations are decomposed into modes labelled by a wavevector ${\displaystyle \mathbf {q} }$, corresponding to a characteristic length scale ${\displaystyle \sim 2\pi /q}$. Dynamics is then quantified by comparing images separated by a time delay ${\displaystyle \Delta t}$. In the most common implementation, one computes the Fourier power spectrum of the difference between two images separated by ${\displaystyle \Delta t}$, and averages this quantity over many pairs to improve statistics. This yields the image structure function ${\displaystyle D(q;\Delta t)}$, which increases from a noise/background level at short delays to a plateau at long delays as the fluctuations decorrelate.^[1][2]

Under standard conditions, ${\displaystyle D(q;\Delta t)}$ can be related to the normalized intermediate scattering function ${\displaystyle f(q;\Delta t)}$, i.e. the same dynamical correlation function that is central to DLS. Consequently, by fitting the ${\displaystyle \Delta t}$-dependence at each ${\displaystyle q}$ with appropriate models (diffusion, advection/ballistic motion, mixtures of processes), DDM yields quantitative parameters such as diffusion coefficients, characteristic speeds, or relaxation times.^[1][2][11]

Advantages and limitations of DDM

DDM complements traditional dynamic light scattering (DLS) and multiple particle tracking (MPT) by offering distinct trade-offs suited to specific experimental scenarios.^[6][9][11]

ADVANTAGES OVER DLS

Wavevector range: DDM accesses lower q-values than conventional DLS (typically 0.1–5 μm⁻¹), offering advantages when characterizing larger particles.

Spatial selection: DDM enables region-of-interest (ROI) selection, allowing measurement of spatially heterogeneous samples, and localized dynamics that are challenging or impossible in ensemble DLS measurements.

Optical flexibility: DDM is compatible with various microscope contrast mechanisms (bright-field, phase-contrast, fluorescence, confocal, polarized light, dark-field), whereas DLS typically requires laser illumination.

Dust robustness: DDM is tolerant of stationary dust particles (e.g. on the cell surfaces) and handles multiply-scattering samples well; conventional DLS is highly sensitive to dust and requires stray-light suppression.

Sample volume: DDM requires only microliters of sample; DLS typically requires milliliters.

ADVANTAGES OVER MULTIPLE PARTICLE TRACKING

Statistical power: DDM provides superior ensemble statistics by analyzing all the particles in the field of view simultaneously; MPT requires user-intensive trajectory reconstruction.

Sub-resolution sensitivity: DDM does not require individual particles to be optically resolved, enabling measurements in concentrated and optically dense systems where MPT struggles.

LIMITATIONS AND PRACTICAL CONSTRAINTS

Measurement speed: DDM requires extended acquisition times (minutes to hours) compared to DLS (seconds), making DLS preferred for rapid, routine measurements.

Analysis time: Data analysis is computationally intensive. While traditional CPU-based processing required minutes to hours per dataset, recent GPU-accelerated implementations (e.g. fastDDM) reduce this to seconds, enabling high-throughput workflows.

Commercial maturity: DLS instruments and software are mature and widely available; DDM requires custom setups or adapted commercial microscopes with specialized analysis software.

Optical contrast requirement: DDM performance depends critically on signal contrast. Weakly scattering samples may require careful illumination optimization or fluorescent labeling.

Dynamic range trade-offs: Lowest accessible q-values (longest length scales) correspond to very slow dynamics requiring extended acquisition; highest q-values require high magnification and frame rates, creating practical trade-offs in microscope configuration.

Applicability and working principle

The concentration–intensity proportionality, which ensures the validity of the DDM framework, holds in two primary classes of DDM implementations:

Scattering-based DDM: where image intensity arises from the interference between a strong transmitted beam and weakly scattered light, as in bright field, phase contrast, or polarized imaging. Fluorescence-based DDM: where images result from the incoherent summation of fluorescence emissions from labeled particles or structures, as in fluorescence or confocal imaging. In both classes, convolution with the PSF in real space becomes a simple product in reciprocal (Fourier) space, enabling analysis of given Fourier modes of image intensity to extract dynamics of the corresponding concentration modes. Unlike particle tracking, DDM does not require resolving individual particles, allowing characterization of the dynamics of sub-resolution features. Acquiring images in real space also offers practical advantages over traditional far-field scattering methods, such as flexible choice of imaging modality and robustness to static imaging artefacts.

Data analysis

DDM analysis is based on the differential dynamic algorithm (DDA), developed in the context of optical fluctuation studies.^[12][13] In this approach, pairs of images separated by a delay ${\displaystyle \Delta t}$ are subtracted to suppress static backgrounds and highlight changes due to motion. A two-dimensional fast Fourier transform (FFT) of the difference images yields a Fourier-space representation of the dynamics across spatial frequencies. Averaging the resulting power spectra over many image pairs with the same ${\displaystyle \Delta t}$ (and, for isotropic samples, over wavevector directions) gives the image structure function ${\displaystyle D(q;\Delta t)}$. Theoretical analysis shows that for common DDM modalities^[2]

${\displaystyle D(q;\Delta t)=B(q)+T(q)I(q)[1-f(q;\Delta t)],}$

1

where ${\displaystyle f(q;\Delta t)}$ is the normalized intermediate scattering function, ${\displaystyle I(q)}$ is the static scattering intensity, ${\displaystyle B(q)}$ is a background term associated with detection noise, and ${\displaystyle T(q)}$ is a transfer function that depends on microscope details and imaging conditions.

For simple Brownian motion, one has ${\displaystyle f(q;\Delta t)=\exp(-Dq^{2}\Delta t)}$, where ${\displaystyle D}$ is the diffusion coefficient. Models including directed (ballistic) motion, mixtures of populations, or distributions of speeds are used for active systems such as swimming bacteria.^[3][4][5]

A variant called multi-DDM analyses sub-regions of the image frame, analogous to varying the scattering volume in traditional light scattering. This variant can reveal spatial variations in dynamics and is particularly valuable for studying heterogeneous systems such as biological networks and phase-separating materials where local dynamics vary significantly with position.^[14]

The Hitchhiker's guide to DDM

The 2025 tutorial article The Hitchhiker's guide to differential dynamic microscopy provides a step-by-step practical guide to DDM experiments and analysis, and is associated with open-source software and openly accessible datasets intended for teaching and reproducibility.^[11]

An implementation of DDM analysis used in this tutorial is provided as the open-source Python package fastDDM, with documentation and source code hosted on GitHub.^[15] The fastDDM package incorporates GPU-accelerated algorithms that reduce analysis time from minutes to seconds per dataset, enabling high-throughput analysis and practical applications previously limited by computational constraints.^[11] Worked examples and tutorial notebooks are provided in a companion repository.^[16] Example open datasets used in DDM tutorials are also available via an archived dataset with DOI.^[17]

Applications

DDM was initially used to characterize the dynamics of colloidal particles undergoing Brownian motion in dilute and concentrated suspensions, extracting diffusion coefficients and other hydrodynamic parameters from the wavevector dependence of decorrelation rates.^[1][2]

DDM has been extended to measure dynamics in complex fluids, including aggregation processes of nanoparticles and dynamics of anisotropic colloids.^[18][19]

DDM has been widely applied to quantify bacterial motion and motility, providing ensemble-averaged parameters such as population swim speed, diffusivity, and run–tumble statistics without single-particle tracking.^[3][4] Confocal and fluorescence implementations further enable measurements in fluorescent and optically challenging samples, including concentrated and multiply scattering systems.^[5]

Selected application areas include:

Particle sizing and size distributions: DDM can be used for particle sizing by extracting diffusion coefficients and converting them to hydrodynamic sizes via the Stokes–Einstein equation. This can be extended to multimodal size distributions and mixtures, where DDM provides size information even when individual particles are not resolved.^[1][2][20]

Protein sizing and macromolecular characterization: DDM has been extended to characterize proteins at dilute concentrations of a few milligrams per milliliter, achieving hydrodynamic radius measurements in excellent agreement with commercial DLS instruments while requiring only microliters of sample.^[21]

Turbid and concentrated suspensions: DDM has been used to probe dynamics in turbid suspensions over wide ranges of concentration and optical thickness, enabling analysis of samples challenging for conventional light scattering measurements.^[22]

Microrheology and viscometry: Tracking-free microrheology and microliter viscometry variants have been developed using DDM analysis of tracer fluctuations, enabling determination of viscosity or viscoelastic moduli from small sample volumes.^[23][24][25]

Biological and cytoskeletal network dynamics: DDM has been applied to quantify dynamics in biopolymer and cytoskeletal networks, where it provides a Fourier-space view of relaxation and rearrangement dynamics in microscopy image sequences.^[26]

Liquid crystal phase transitions and orientational ordering: DDM has been applied to probe orientational dynamics and pretransitional fluctuations in liquid crystalline systems. In confined nematic cells, DDM captures changes in director fluctuations and dynamic correlations near phase boundaries, providing insights into collective orientational relaxation and anisotropic dynamics in the nematic regime.^[27][28]

Active and non-equilibrium materials: DDM is increasingly used to quantify multiscale dynamics in active and out-of-equilibrium soft materials, including gels ^[29] and dynamically heterogeneous systems with multiple relaxation regimes. Related reciprocal-space approaches have been applied to glassy colloids to characterize multiscale heterogeneous dynamics in two-dimensional systems,^[30] and to investigate Brownian yet non-Gaussian diffusion of tracer particles in hard-sphere glasses.^[31] In biological active matter, DDM-derived methods have enabled tracking-free quantification of collective cell dynamics, including single-cell displacements and division rates in confluent monolayers,^[32] and have been used to dissect active diffusion and advection driven by cytoskeletal activity in the Drosophila ooplasm.^[33]

Relationship with other imaging-based scattering methods

Scattering-based DDM belongs to the so-called near-field (or deep Fresnel) scattering family,^[34] a family of imaging-based scattering methods.^[35][36] Near field is used here in a similar way to what is used for near field speckles i.e. as a particular case of Fresnel region as opposed to the far field or Fraunhofer region. The near field scattering family includes also quantitative shadowgraphy^[37] and Schlieren.^[12]

References

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2. Giavazzi, F.; Brogioli, D.; Trappe, V.; Bellini, T.; Cerbino, R. (2009). "Scattering information obtained by optical microscopy: Differential Dynamic Microscopy and beyond" (PDF). Phys. Rev. E. 80 (3) 031403. Bibcode:2009PhRvE..80c1403G. doi:10.1103/PhysRevE.80.031403. hdl:10281/36546. PMID 19905112.
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