1. Introduction

Full-field optical metrology of surface structures and thin three-dimensional (3D) structures can be achieved by interferometry or holography. In interferometry, the phase of the reflected signal, proportional to the surface height, is measured by interference with a reference beam. The phase is extracted using either phase-shifting or heterodyne techniques. Phase-shifting can measure height variations smaller than $1$nm in height, and heterodyning is even more sensitive, $<1$pm having been reported. The precise interpretation of the term holography does not seem to be universally agreed, so the distinction between interferometry is not clear. Holography is also an inteferometric technique: one distinction from interferometry is that holography is often performed in the Fresnel regime, or even Fraunhofer regime, whereas interferometry usually refers to interference in the image plane. Another distinction is that holography sometimes refers to the holographic reconstruction process using a physical reconstruction beam, or a digital synthetic simulation of this process.

When using light of a single wavelength, phase-wrapping limits the range of heights that can be measured without ambiguity by interferometry. This problem can be overcome by limiting the temporal coherence, resulting in the visibility of the interference fringes being modulated by a coherence envelope. It is interesting to note that Michelson described measurement of the spectral distribution of a source using interferometry with a moving mirror in the reference arm, effectively by what is now known as Fourier transform spectroscopy [1]:

‘ … there is a close relation between the distribution of light in the source as a function of the wavelength and the corresponding visibility curve.’

Although Michelson recognized the Fourier relationship between the spectral distribution and the optical path difference, he did not, however, seem to mention using the opposite approach, that of measuring the position of the mirror by location of the peak of the visibility. An early paper on white-light interferometry was that by Tomonaga and Ogawa [2], who reported an accuracy of $0.05\mu$m, and reported:

‘Measuring range is now about $6\mu$, and we are planning to widen it to about $15\mu$.’

2. Low coherence interferometry

A review of low-coherence interference microscopy has been given by Sheppard and Roy [3]. Using a low temporal-coherence source, and scanning in depth through the sample, is equivalent to using coherent pulses and measuring the propagation time. In the 1970s time-domain reflectometry was introduced to locate faults in optical fibers [4,5]. Time resolved measurements were also applied to depth measurement in biological and scattering media [6–9]. The alternative frequency-domain (swept source) techniques for optical ranging were introduced in the early 1980s [10,11]. Coherence-domain reflectometry uses a low-coherence source and the visibility curve is extracted. The surfaces of a microscope slide were located using this method [12]. This approach, using phase-shifting to extract the visibility, was also applied to surface metrology [13].

In 1987, Davidson et al. applied low-coherence interferometry to microscopy of surface structures, especially integrated circuits [14]. They used a Linnik interferometer, a Michelson interferometer with microscope objectives in each arm, and called their system the coherence probe microscope. They extracted the visibility by calculating the variance of a series of interference images with different path differences. They recognized that the system could be used to measure the thickness of transparent films. They also explained that the fringe phase information can be used, together with the visibility, to give high sensitivity depth location. Many papers followed, using this basic technique of full-field low coherence interferometry [15–20]. Often the interferometer is of the Mirau variety, which improves stability, but limits the numerical aperture (NA) [15]. For high magnification a Linnik interference microscope therefore offers some advantages [14]. Two matched high-NA microscope objectives are used, so that spherical and chromatic aberration cancel out. The main disadvantage of this type of microscope is that it is very sensitive to vibration and air currents, because it involves a long beam path. However, these vibrations are avoided in optical coherence tomography by employment of heterodyning techniques that reject low frequency variations in signal.

Mirau [15] and Michelson interference microscopes require only one microscope objective. The difference between these two types is that in the Mirau system, the beam splitter and reference mirror are positioned between the objective and test surface, whereas in the Michelson type, a long working distance microscope objective is used so that the beam splitter can be placed between the objective and the object. In these two methods, particularly in the Michelson geometry, the NA is therefore lower than can be achieved in Linnik and confocal systems, so that they are used at a lower magnification.

The most common method for recovering fringe visibility from the sampled data has been by digital filtering in the spatial frequency domain [15]. Here, for each pixel in the sample, the signal in the $z$ direction is Fourier transformed, and then inverse Fourier transformed after shifting back to base-band. In order to accurately record the interference fringes, the steps in optical path difference must be less than or equal to a quarter of the shortest wavelength, which corresponds to a step size $z$ of around $50$nm, so that a large amount of memory and processing time is required.

These requirements can be reduced considerably by modifying the sampling and processing techniques. It should be recognized that according to the sampling theorem, the signal must be sampled at a rate of twice the bandwidth, rather than the sampling rate being determined by the highest frequency in the fringe pattern. This has led to a procedure called (confusingly) sub-Nyquist sampling [17]. Several other approaches have been proposed, including using a Hilbert transform [16], or methods based on communications theory [21].

Alternatively, we can use phase-shifting, which is computationally much less intensive. This involves shifting the phase of the reference wave by three or more known amounts, for each position along the $z$ axis. The corresponding values of the intensity are recorded, and can then be used to evaluate the fringe contrast directly at that axial position [13].

A variety of different phase-shifting algorithms can be used to extract the envelope, based on measurements for three or more values of phase step. A five-step algorithm appears to be good for contrast or visibility measurement, because it has a partial error-correcting property [22]. An exact error compensating five-step algorithm was given by Larkin [23]. This leads to a very efficient five-sample-adaptive (FSA) nonlinear algorithm, in which the visibility of the interference fringes at any given point in the field can be expressed as [23]

Davidson et al. described how using an illumination with low spatial coherence results in a correlation sectioning, in addition to the coherence sectioning resulting from temporal incoherence, the strength of which increases as the NA of the condenser lens is increased [14]. This correlation effect of interferometry was described earlier by Corcoran and by Siegman [24,25]. Basically, integrating over a detector measures the correlation between the signal and reference beam wavefronts. The spatial resolution is thus also increased as the overall point spread function is given by the product of the illumination and imaging point spread functions, similar to the behavior in a confocal system. A Linnik microscope was developed by Gale et al. [26]. In a scanning microscope, confocal sectioning is achieved by using a confocal pinhole, and can be used to give an autofocus image and a surface profile, from the height and position, respectively, of the peak of the defocus signal [27]. Sawatari used the correlation effect to construct a confocal system, in which a reference beam generates a synthetic confocal pinhole [28]. Fujii used the correlation effect to construct a lensless scanning microscope [29].

The sectioning effect of low temporal coherence also results in the rejection of multiply-scattered light, allowing penetration into scattering media such as biological cells and tissues. This mechanism is called coherence gating. In an analogous way, correlation gating can also be used to image effectively through scattering media [30].

3. Confocal gating and angular gating

Confocal gating, in a scanning system with a physical confocal pinhole, is an alternative to correlation gating [31,32]. A review of different gating methods has been presented [33]. Another interesting method is angular gating, where the pupil of the objective is complementary to the condenser pupil [34]. Angular gating can be combined with confocal gating to further improve optical sectioning [35,36]. In confocal profiling, this technique rejects a spurious reflection from the center of curvature, rather than from a locally spherical surface [37].

Confocal interference microscopes have also been constructed, by using a reference beam together with a confocal pinhole [38–41]. Confocal interference microscopes exhibit some differences from ordinary interference microscopes of Michelson, Linnik or Mirau forms, especially from the point of view of aberrations, as wavefront distortion of the reference beam over the aperture of the confocal pinhole can be ignored [42]. Confocal microscopes and confocal interference microscopes can alternatively be based on an optical fiber implementation, where the fiber acts as a coherent detector to act as a synthetic pinhole [43–50]. In this case, only the component of the object and reference beams that match the fiber mode profile are detected. Confocal interference can be combined with a broad-band light source, giving low-coherence interference, or alternatively using a pulsed beam, in both cases resulting in improved axial spatial frequency coverage [42,51].

Some other imaging approaches can also give optical sectioning, such as structured illumination microscopy [52,53], but in this case there is no physical pinhole as there is in confocal microscopy, and so this approach is not so efficient at imaging through a scattering medium.

4. Holography

Turning now to holography, the main advantage of digital holographic microscopy (DHM) is its imaging speed [54]. We note that DHM, although often regarded as giving a three-dimensional 3D image, does not really give one, but rather what is sometimes called a $2 \tfrac {1}{2}$D image [55]. There is no optical sectioning. For example, the two surfaces of a thin film cannot be imaged in depth. This problem can be overcome by holographic tomography, where the illumination or sample are rotated, or a spread of wavelengths, is used [56]. In many cases, the phase is recorded in the image plane, as in tomographic diffractive microscopy [57–60].

5. Tomography

Wolf explained how 3D structure can be determined from interference data [61]. Imaging can be described in terms of scattering and detection of incident plane waves by the object [42,55,57]. Wolf described using a scattering theory based on the first Born approximation. However, in the reflection geometry, light is scattered by changes in refractive index, the gradient being assumed constant over a resolution element according to the Kirchhoff approximation [62–64]. These concepts can be used for modeling and reconstruction of 3D images, of surface structures or 3D objects, by inverse scattering [65–69].

6. Optical coherence tomography

Optical coherence tomography (OCT) is a non-invasive, label-free, interferometric imaging technique that can be used to produce high-resolution cross-sectional images of up to a few millimeters deep into biological/biomedical samples [70]. It has been widely used in inspecting internal microstructure in biological and medical tissues [70–73], especially in ophthalmology [74–76]. The distinction of OCT as compared with earlier low coherence imaging systems is that it is based on a point-scanning technique, usually using a fiber Michelson interferometer illuminated by a broad spectrum source. So, effectively, OCT is a low coherence version of a (fiber-optical) confocal interference microscope. In so-called time-domain OCT, a focused probe beam is scanned over the sample using scanning mirrors, and depth scanning is achieved by the longitudinal translation of a reference mirror. The NA is low enough that the object remains in focus over its depth, with the result that the correlation sectioning is negligible compared with the coherence sectiong. The axial resolution of an OCT image thus depends on the coherence length of the illumination source, which is inversely proportional to the light source’s bandwidth. The transverse spatial resolution of OCT is also limited by the low NA. Unlike full-field low-coherence microscopes, OCT uses a broad-band source with high spatial coherence, such as a superluminescent diode, which can give an axial resolution of $\sim 10$-$15\mu$m. The axial resolution can be further improved, up to $\sim 1$-$3 \mu$m, using a broadband femtosecond laser or a supercontinium light source [77,78]. However, the noise performance of time-domain OCT is limited, and to record full 3D images, image acquisition slow and cumbersome. To improve the noise performance and speed, Fourier-domain OCT (FD-OCT) has been developed, based on either a spectrometer [79,80] or swept-source tunable laser [81], in order to provide depth-resolved information without mechanical axial scanning. Nevertheless, these approaches still suffer from the limited acquisition speed associated with transverse scanning.

7. Optical coherence microscopy

A variant of OCT, in which a high-NA lens is used, is called optical coherence microscopy (OCM) [82]. The axial sectioning is now given by a combination of coherence and correlation effects. The standard technique for extracting the visibility in OCT neglects the inclination of off-axis rays, which results in errors if the NA is increased. To overcome this problem, the technique called interferometric synthetic aperture microscopy (ISAM) was introduced [83]. For high NA, as the depth of focus is now limited, this technique eventually becomes less efficient, and thick samples must be refocused [84].

Full-field low-coherence microscopes avoid the disadvantages of transverse scanning. This approach has been called full-field optical coherence tomography (FF-OCT) [85], and is based on a white-light interference microscope in which the full-field of the sample is illuminated and a transverse (en-face) image can be detected using a 2D sensor array (e.g. CCD or CMOS camera), thereby eliminating the need for complex electromechanical lateral scanning, while significantly improving speed of image acquisition. Depth imaging is performed by scanning the reference mirror or moving the sample axially. In our opinion, the name FF-OCT is somewhat misleading, as white light interference microscopes predate the invention of OCT, and OCT specifically refers to the scanning geometry. Nevertheless, the name is perhaps now well established, so that early low-coherence interference microscopes should be referred to as FF-OCT, as the hardware is identical. The FF-OCT technique is usually actually FF-OCM, and is based on a Linnik-type interference microscope, making use of a spatially incoherent broadband light source (e.g. a halogen lamp or xenon arc lamp) to achieve OCT images with both high transverse and axial resolution [86–90]. The use of a spatially incoherent source in FF-OCM is also cheaper and avoids speckle [87–91]. Vabre et al. [86] and Dubois et al. [87] first demonstrated the generation of ultrahigh spatial resolution images of plant, animal and human tissue using an FF-OCM system using a thermal light source and high-NA objectives, but the imaging speed was relatively slow (1s/image) due to the practical requirements of image averaging to improve the detection sensitivity. Grieve et al. [89] used a high-speed CMOS camera to develop a rapid FF-OCM system, enabling in vivo anterior segment imaging of small animals. More recently, Akiba et al. [90] reported using a halogen white light source and a pair of CCD cameras to result in a video-rate FF-OCM system for in vivo cellular-level blood flow imaging of a Xenopus laevis tadpole.

Most FF-OCM systems use a conventional phase-shifting technique, in which the phase of the reference beam is shifted with a piezoelectric translator (PZT), to extract the visibility and reconstruct en-face tomographic images. However, when using a broadband source, as in FF-OCM, the phase shifts of different spectral components are not exactly the same, resulting in ambiguities in 3D image reconstruction [92,93]. A solution to this problem is to use geometric phase-shifting, which relies on the use of polarization components to render the phase shift achromatic [94–98]. Conventional geometric phase-shifting techniques are based on a rotating wave plate or polarizer using a stepper motor, and are therefore not practical for many applications, particularly in real-time biomedical imaging, because of their relatively slow speed. However, switchable retarding plates based on ferroelectric liquid crystals are suitable for high-speed geometric phase switching. Hariharan et al. have proposed replacing the rotating half-wave plate or the rotating polarizer by a fast switchable ferroelectric liquid-crystal (FLC) phase modulator [99]. With a switching time of less than $80 \mu$s, Roy et al. have demonstrated such an achromatic phase shifter for surface profiling of integrated circuits by use of white-light interferometry [100–102]. In this paper, we demonstrate the use of a switchable geometric phase shifter (GPS) based on FLC technology in a FF-OCM system for imaging biological samples (e.g., onion tissue and HeLa cells).

8. FF-OCM: experimental setup

A schematic diagram of the FF-OCM system for 3D biomedical imaging, using the Linnik geometry and a GPS based on a FLC, (GPS-FLC-FF-OCM), is presented in Fig. 1. The low coherence broadband light source is a collimated tungsten-halogen lamp ($150$W) (current-adjustable). The peak wavelength of the white-light source is $790$nm and the bandwidth is $185$nm. A 3mW He-Ne laser is also provided for finding the interference fringes. A Köhler illumination system, comprising of lenses L1-L3 together with a microscope objective, is used to illuminate the object uniformly. The light is linearly polarized by a polarizer, and is divided into two orthogonally polarized beams using a polarizing beam splitter. The two beams are focused on to the sample and a reference mirror, respectively, through the two identical infinity tube-length $10\times$ $0.25$NA microscope objectives. The two beams from the sample and the reference mirror are recombined together through to a second beam splitter, an analyzer and a tube lens (L4). An interference image is then detected by an NIR enhanced CCD camera (Sony XC-EI50CE, $768 \times 576$, $8$ bits)

 

Fig. 1. Schematic diagram of FLC-FF-OCM system: L1-L4,lens; BS, beam splitter; PBS, polarizing beam splitter; QWP, quarter-wave plate; PZT, piezoelectric translator.

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The phase difference between the two beams is varied by two identical rapid switchable geometric phase-shifters located between the beam splitter and the corresponding microscope objective (Fig. 1). Each phase shifter consists of an achromatic quarter-wave plate (QWP) with its principal axis at $45^\circ$ to the plane of polarization of the beam, followed by a FLC with a retardation of a quarter-wave and a switching angle of $45^\circ$ with its principal axis set at an angle $\theta$ relative to that of the QWP. Using this configuration, the linearly polarized incident light in each beam path is circularly polarized by the QWP and then linearly polarized by the FLC. Each beam is then reflected back to retrace its path back through the FLC and the QWP, so that the polarization state of the beam traverses a closed circuit on the Poincaré sphere [101]. By setting the FLC to an appropriate orientation, the phase difference can be made nearly independent of wavelength. In our FF-OCM system, a $z$-stack of en-face tomographic images is accurately registered in space for generation of a 3D image. To determine the en-face tomographic image, the visibility function of the interference fringes for each sample point is extracted from three sets of phase-shifted image intensity measurements using the formula:

In our previous study, we estimated that the maximum deviations of the phase shift from its value at the design wavelength, are only about $\pm 7^\circ$, so that the corresponding deviations in output amplitude are less than about 1.5% [101].

For a 2D surface structure, the surface height information at each sample point can be extracted by Gaussian fitting to the fringe visibility function of each image pixel for localizing the spatial position of the peak of the visibility curve along the height ($z$) axis.

A customized LabView-based program has been written to automatically control the switching of the FLC devices, the synchronization of image acquisition and recording of the CCD camera, and the translation of the PZT stage, as well as the reconstruction of 2D/3D images of the samples examined.

9. Results

Fig. 2 shows the axial ($z$) response of our geometric FLC-FF-OCM system [103]. The fringe patterns are well resolved. The interferogram envelope (the chained curve) yields a FWHM (full-width at half maximum) of $\sim 1.6 \mu$m in axial resolution, which is close to the theoretically calculated value of $1.5 \mu$m for the tungsten light source used. Note that this value is close to that achievable in conventional OCT with a much more expensive femtosecond pulsed laser. The transverse resolution of the system was evaluated by imaging a USAF resolution target (Fig. 3). The smallest element 6 of group 7 with a resolution of $228$ pairs/mm, can be clearly resolved, demonstrating that a transverse resolution of $\sim 2.2 \mu$m can be achieved by our FLC-controlled FF-OCM system.

 

Fig. 2. Measured axial coherence function for the FLC-FF-OCM system.

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Fig. 3. Image of a USAF resolution target obtained using the FLC-FF-OCM system. The smallest element 6 of group 7, with a resolution of $228$ pairs/mm, can be clearly resolved, demonstrating a transverse resolution of $\sim 2.2 \mu$m.

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We have investigated the utility of the FLC-controlled achromatic phase-shifted FF-OCM technique for high resolution 3D biological imaging. Using the rapid switchable FLC devices to control geometric phase-shifted FF-OCM system, video rate imaging of $40$ ms per frame can be obtained and the three-step phase-shifted images of $30$ sample translations can be completed within $60$s. All image processing can be completed rapidly with high SNR, without the need for repeated image acquisitions or image averaging.

Fig. 4(a) shows an example of a single raw interference microscope image of an onion epithelium acquired by our FF-OCM system [103,104]. The interference fringes on the onion tissue can be clearly seen. The field of view (FOV) is $585\mu$m $\times 439 \mu$m. Figs. 4(b) to (d) show the en-face OCT images of the onion sample at different tissue depths ($5$, $8$ and $11 \mu$m). Subtle differences in cell morphologies and structures of the onion at different tissue depths can be clearly identified, illustrating the ability of our FF-OCM system for rapid optical sectioning of biological samples at high spatial resolutions.

 

Fig. 4. (a): A single raw microscopic image acquired by FLC-FF-OCM system, showing interference fringes on an onion epithelium. (b)-(d): Examples of reconstructed en-face OCM images at different depths ($5 \mu$m, $8 \mu$m and $11 \mu$m, respectively). Field of view: $585\mu$m $\times 439 \mu$m.

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With the FLC-based GPS-FF-OCM system, a series of en-face OCT images of HeLa cells were acquired at successive depths in $0.3 \mu$m step from the surface down to $5.1\mu$m in depth [104]. The 3D volume image and cross-sectional image of HeLa cells are shown in Fig. 5. Cell morphologies such as the cell membrane, nucleus and cytoplasm can be discerned much more clearly as compared with the corresponding conventional image.

 

Fig. 5. 3D image (left) and cross-sectional image (right) of HeLa cells using the FLC-based FF-OCM system. The vertical height of the image is 5.1$\mu$m

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Another main advantage of our FF-OCM technique is the configuration of very fast switchable FLC-controlled geometric phase-shifter (less than $80\mu$s) for rapid FF-OCM imaging, which is crucial for realizing real-time in vivo 3D imaging of living cells and tissues. The image acquisition rate of the system can be further increased with the use of a high-speed CCD or CMOS camera.

10. Conclusions

An historical review of different approaches for imaging and metrology of 3D surface structures is presented. Optical sectioning, and gating to select non-multiply-scattered light, can be achieved by coherence, correlation, and confocal mechanisms.

We have demonstrated the successful use of a geometric phase shifter, based on ferroelectric liquid crystal technology, for full-field OCM on a microscopic scale. The main advantages of our FF-OCM system lie in the achromatic optical sectioning, high speed, high spatial resolution, and the relatively low cost of using a white light source. Other advantages of using a ferro-electric geometric phase shifter are its very short response time ($<100\mu$s) and its freedom from vibration, because it has no mechanical moving parts. By using fast switchable FLC-controlled geometric phase shifter in FF-OCM system, we obtained en-face OCT 3D imaging of biological samples at video rates with the spatial resolution of $1.6 \mu$m (axial) $\times 2.2 \mu$m (transverse).

FF-OCT and FF-OCM are promising techniques for a range of biomedical applications. These include histology, optical biopsy, ophthalmology of both cornea and retina, dermatology, oncology, developmental biology, and study of cell cultures and scaffolds. Although the penetration depth is not as great as in scanning-type OCT, it can give subcellular 3D resolution, and the display of en-face images is often desirable, allowing direct comparison with conventional pathology. Further, the FF-OCT approach has been shown to be compatible with methods for functional imaging, including polarization sensitive FF-OCT [105], spectral imaging [106], elastography [107], photothermal imaging [108] and dynamic imaging [109].