# Principles and techniques of digital holographic microscopy

**Myung K. Kim**

University of South Florida, Department of Physics, 4202 E. Fowler Avenue, Tampa, Florida33620 mkkim@cas.usf.edu

*SPIE Reviews*. 1, 018005 (May 14, 2010). doi:10.1117/6.0000006

#### Open Access

## Abstract

Digital holography is an emerging field of new paradigm in general imaging applications. We present a review of a subset of the research and development activities in digital holography, with emphasis on microscopy techniques and applications. First, the basic results from the general theory of holography, based on the scalar diffraction theory, are summarized, and a general description of the digital holographic microscopy process is given, including quantitative phase microscopy. Several numerical diffraction methods are described and compared, and a number of representative configurations used in digital holography are described, including off-axis Fresnel, Fourier, image plane, in-line, Gabor, and phase-shifting digital holographies. Then we survey numerical techniques that give rise to unique capabilities of digital holography, including suppression of dc and twin image terms, pixel resolution control, optical phase unwrapping, aberration compensation, and others. A survey is also given of representative application areas, including biomedical microscopy, particle field holography, micrometrology, and holographic tomography, as well as some of the special techniques, such as holography of total internal reflection, optical scanning holography, digital interference holography, and heterodyne holography. The review is intended for students and new researchers interested in developing new techniques and exploring new applications of digital holography.

## Introduction

Digital holography (DH) is an emerging technology of new paradigm in general imaging applications. By replacing the photochemical procedures of conventional holography with electronic imaging, a door opens to a wide range of new capabilities. Although many of the remarkable properties of holography have been known for decades, their practical applications have been constrained because of the cumbersome procedures and stringent requirements on equipment. A real-time process is not feasible, except for photorefractives and other special materials and effects. In digital holography, the holographic interference pattern is optically generated by superposition of object and reference beams, which is digitally sampled by a charge-coupled device (CCD) camera and transferred to a computer as an array of numbers. The propagation of optical fields is completely and accurately described by diffraction theory, which allows numerical reconstruction of the image as an array of complex numbers representing the amplitude and phase of the optical field. Digital holography offers a number of significant advantages, such as the ability to acquire holograms rapidly, availability of complete amplitude and phase information of the optical field, and versatility of the interferometric and image processing techniques. Indeed, digital holography by numerical diffraction of optical fields allows imaging and image processing techniques that are difficult or not feasible in real-space holography.

Holography was invented in 1948 by Dennis Gabor in an effort to improve the resolution of the electron microscope, where the correction of electron lens aberrations posed increasing technical difficulty. Instead of attempting to perfect the electron imaging lens, Gabor dispensed it altogether and realized that the diffraction pattern of the electron beam contained complete information regarding the amplitude and phase of the electron wave. The record of the electron wave diffraction is then used to optically synthesize the object field. This allowed the use of the optics of visible light for image formation, and was much easier and more developed compared to electron optics. He named the new imaging principle holography, for its ability to record the whole optical field ^{1- 3}.

The holography principle was immediately applied to recording and imaging by visible light ^{4}. But it had to wait for two critical inventions before its full potential was to be realized. One was the powerful coherent source of light in lasers to provide high quality interference contrast. The other, due to Leith and Upatnieks ^{5- 7}, was off-axis illumination with a separate reference wave, thus eliminating the problem of the zero-order and twin images of the Gabor's on-axis configuration. Soon many new techniques and applications of holography began to develop. Holography is now a mature field, and an excellent survey is given, for example, in Ref. ^{8}. It was also realized early on that the use of nonplanar structures of the reference beam can lead to various manipulations of the holographic output, ranging from image magnification to more complex image processing, such as encryption, pattern recognition, associative memory, and neural networks ^{9}. Instead of photographic plates or films, real-time holography is possible with photorefractives and other nonlinear optics materials. In fact, we now understand much of nonlinear optics as generalizations of the holography principle, including phase conjugation, four-wave mixing, spectral hole burning, and photon echo ^{10}. The holography process is being developed for other regions of the electromagnetic spectrum, especially x-ray holography, with its prospect of atomic resolution ^{11}, as well as for microholography of living organisms ^{12}.

For many application areas, real-time operation is of critical importance but is difficult with conventional holography. Even photorefractives and other nonlinear optics systems require substantial equipment and technical care to implement them, and have seen only limited practical applications. Digital holography replaces physical and chemical recording processes with electronic ones, and the optical reconstruction process with numerical computation. The propagation of optical fields is completely and accurately described by diffraction theory, and in 1967, Goodman and Lawrence demonstrated the feasibility of numerical reconstruction of an image from a Fourier hologram detected by a vidicon camera ^{13}. Schnars and Jueptner, in 1994, were the first to use a CCD camera directly connected to a computer as the input, and compute the image in a Fresnel holography setup ^{14}. In what is now called digital holography (DH), holographic interference is produced by optical processes in real space, while reconstruction is by numerical computation. Conversely, in computer-generated holography (CGH), the hologram can be produced by numerical computation inside a computer, followed by printing or other outputs to real space ^{15}. Reconstruction is then carried out by optical means. CGH has many interesting properties and applications, such as the ability to arbitrarily prescribe desired amplitude and phase properties of the output optical field starting from fictitious objects. CGH is not a subject of this review ^{16}.

By direct electronic recording of holographic interference, and because of the increasing speed of holographic computation, real-time holographic imaging is now possible, and more importantly, the complete and accurate representation of the optical field as an array of complex numbers allows many imaging and processing capabilities that are difficult or infeasible in real-space holography ^{17}. Various useful and special techniques have been developed to enhance the capabilities and extend the range of applications. In digital holographic microscopy, a single hologram is used to numerically focus on the holographic image at any distance ^{18- 19}. Direct access to the phase information leads to quantitative phase microscopy with nanometer sensitivity of transparent or reflective phase objects, ^{20- 21} and allows further manipulations such as aberration correction ^{22}. Multiwavelength optical phase unwrapping is a fast and robust method for removing $2\pi $-discontinuities compared to software algorithm-based methods ^{23}. A significant constraint of digital holography is the pixel count and resolution of the imaging devices. Suppression of the zero-order and twin images by phase-shifting digital holography allows efficient use of the pixel array ^{24}. Digital Gabor holography, without separate reference beams, is useful for particle imaging applications by providing 4-D space-time records of particle fields ^{25}. Digital holography naturally evolved from the effort to utilize electronic imaging in interferometry, such as in electronic speckle pattern interferometry (ESPI) ^{26}. Metrology of deformations and vibrations is a major application area of digital holography ^{27}. Optical processing, such as pattern recognition and encryption, by digital holography also offers new capabilities ^{28}.

Basic principles of diffraction and general holography are outlined in Sec. 2, and a general description of digital holographic microscopy (DHM) is given in Sec. 3, with an emphasis on the quantitative phase microscopy by DHM. Methods of numerical calculation of diffraction are described and compared in Sec. 4, and a number of main types of interferometer configurations used in digital holography experiments are given in Sec. 5. There are many numerical techniques that lead to the unique and powerful capabilities of digital holography, described in Sec. 6. Then in Sec. 7, a survey is given of the application areas of DHM, as well as special techniques that expand the capabilities and applications of digital holography. This review has an emphasis on microscopy applications of digital holography, and therefore omits some major areas of digital holography development, such as metrology of macroscopic systems and image processing of holographic data ^{17}.

## Basic Theory of Holography

First, some of the main results of scalar diffraction theory are recalled and applied to the description of basic holographic image formation. We start by writing down the Fresnel-Kirchoff diffraction formula for the general problem of diffraction from a 2-D aperture depicted in Fig. 1^{29}.

Given the optical field $E0(x0,y0)$ over the input plane $\Sigma 0$ at $z=0$, the field $E(x,y)$ over the output plane $\Sigma $ at $z$ is, with the wavelength $\lambda =2\pi /k$,

For paraxial approximation, valid for $z3\u2aa2k8[(x-x0)2+(y-y0)2]max2$, the Fresnel PSF is

An alternative approach to describe diffraction is by analysis of the angular spectrum. Given the field $E0(x0,y0)$ at the input plane $\Sigma 0(z=0)$, its angular spectrum is defined as the Fourier transform

It is useful to consider holographic imaging by point sources ^{30- 31}. Referring to Fig. 2, suppose two point sources $E1\delta (x-x1,y-y1,z-z1)$ and $E2\delta (x-x2,y-y2,z-z2)$ emit spherical waves toward the hologram plane $\Sigma 0(x0,y0)$ at $z=0$. Again, $(x,y)$ pairs of expressions are mostly abbreviated with $(x)$ only. Using Fresnel approximation, the fields at $z=0$ are

The results show that the fields $E\xb1(x,y;z)$ are spherical waves centered at $(X\xb1,Y\xb1,Z\xb1)$. Various magnifications can be calculated. The lateral, longitudinal, and angular magnifications are, respectively,

These results are based on the quadratic (Fresnel) approximation in Eq. (4). If higher order terms are included, then one obtains the third-order aberration terms: spherical aberration, coma, astigmatism, field curvature, and distortion ^{30- 31}. With the higher order terms, magnification or wavelength mismatch can introduce aberrations. For the most part, the following theoretical descriptions will be within the Fresnel approximation. If the finite size of the hologram is taken into account, the image point has a finite spread determined by the numerical aperture of the hologram ^{32}.

## Digital Holographic Microscopy

A basic digital holographic microscopy (DHM) setup consists of an illumination source, an interferometer, a digitizing camera, and a computer with necessary programs. Most often a laser is used for illumination with the necessary coherence to produce interference. All different types of lasers have been used, from ubiquitous HeNe lasers and diode lasers, to diode-pumped and doubled YAG lasers (often referred to simply as a solid-state laser), argon lasers, as well as tunable dye lasers and Ti:sapphire femtosecond lasers. For multiwavelength techniques, two or more different lasers can be coupled into the interferometer, or a tunable laser can be employed. There are also low-coherence techniques for the purpose of reducing speckle and spurious interference noise, or generating contour or tomographic images. A short-pulse (picosecond or femtosecond) laser can be used, or a tunable laser can be turned into a broadband source by removing the tuning element. Even an LED typically has 10 $\mu m$ or so coherence length, which can be sufficient for holographic microscopy. DHM using a 10.6-$\mu m$$CO2$ infrared laser ^{33}, deep UV (193 nm), ^{34} and 32-nm soft x-ray ^{35} has been demonstrated.

Two main types of interferometers, the Michelson interferometer for reflective objects and the Mach-Zehnder interferometer for transmissive objects, are depicted in Fig. 3. In each diagram, the light-green beams are the input from the laser, the light blue is the reference beam path, and the light red depicts image formation of an object point. In both designs, the object is illuminated with a plane wave, and the reference arrives at the CCD plane with the same wavefront curvature as the object wave, except for an offset in the angle of incidence for off-axis holography. The Mach-Zehnder types require more components but offer more flexibility in alignment, especially when microscopic imaging optics are used. Interferometers can also include various apertures, attenuators, and polarization optics to control the reference and object intensity ratio. Polarization optics can also be used for the explicit purpose of birefringence imaging. There can also be various types of modulators such as piezo-mounted optics, liquid crystal phase modulators, acousto-optic, or electro-optic modulators to establish modulated signals. Techniques such as the lensless Fourier holography ^{36} configuration can be used for magnification, but in practice achievable magnification is limited and explicit magnification by microscope objective lenses is preferred and necessary. Another lens can be used in the reference arm to match the curvatures of the object and reference wavefronts. There are many versatile techniques in digital holography that compensate for various types of aberrations and imperfections of the optical system (see Sec. 6d), and therefore, in comparison with conventional holography, the optical and mechanical requirements can be significantly less stringent.

Typically a CCD, or more recently CMOS, cameras are used to capture and digitize a holographic interference pattern. The pixel size of these devices is several microns with pixel counts of 1000^{2} or so. These parameters are the main limiting factors in DHM resolution and prescribes the range of applications, but one would expect them to continue to improve in the coming years. The captured hologram pattern is digitized by the camera, or a frame grabber, and input to the computer as a 2-D array of integers with 8-bit or higher grayscale resolution. The main task of the computer is to carry out the numerical diffraction to compute the holographic image as an array of 2-D complex numbers. In addition, the computer program handles a number of other tasks, such as pre- and postprocessing of the images, rendering and storage of images, as well as timing and other necessary controls of the apparatus.

An example of the DHM process is shown in Fig. 4 using a resolution target with field of view (FOV) = $200\xd7150\mu m2$. Figure 4 is the hologram, with detail shown in the inset, where the interference fringes are visible. Figure 4 is the angular spectrum (Fourier transform), showing the zero-order and twin image peaks. One of the twin terms is selected with a numerical bandbass filter (yellow circle). The filtered hologram is then used for numerical diffraction over an appropriate distance, which results in the reconstructed holographic image as a 2-D array of complex numbers. The amplitude and phase images in 4, respectively, are obtained by taking the absolute magnitude and phase of the complex array. The phase image represents a phase profile of the optical field reflected from the object surface or transmitted through a thickness of a transparent object. The phase profile has the precision of a fraction of optical wavelength, and therefore reveals nanometric variations of the surface or the optical thickness of the specimen. In Fig. 4, minute smudges of some kind are visible, apparently some fraction of wavelength thickness, which the amplitude image completely misses. The phase image color scale ranges $2\pi $ from blue to red. The object surface is slightly tilted with respect to the reference wavefront, and such tilt and other aberrations can readily be compensated by numerical techniques described later.

A well-known distinctive feature of holography is the 3-D content of the image information. In DHM, a single hologram is used to reconstruct the optical field at any distance from the hologram, within the limitation of the approximation method used. For example, Fig. 5 shows a hologram of a paramecium. From the single hologram, the image is calculated at various distances, which are then assembled into a video clip in 5. It shows the paramecium image going through a best focus, preceded and followed by more defocused images, emulating the turning of a focusing knob on a conventional microscope. In Fig. 6, a series of holograms of a live paramecium and several euglenas are captured. In the scene, the paramecium and the euglenas swim not only in lateral directions but also in varying depths. In conventional video microscopy,