Hilbert Phase Microscope

Hilbert phase microscope (HPM) [23] is an optical interference based technique for quantitative phase imaging by retrieving a full-field phase image from a single spatial interferogram recording.

Typical set-up of Hilbert phase microscope is shown in Fig. 2.15. Details of this set-up are described in reference [24]. A laser beam is divided into two parts. One part in sample arm serves as the illumination field for the inverted microscope. A tube lens images the sample on the CCD via beam splitter cube. The other part of laser beam in reference arm is collimated and expanded by a telescopic system consisting of another microscope objective and the tube lens. This planar reference field interferes with image field with designed tilt angle to produce uniform fringes of an angle of 45º with respect to x and y axes. The intensity of recorded interferogram in one direction is in the form of Eq. (2.6) [25]:

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( ) _{R s}( ) 2 _{R s}( ) cos[ ( )]

I x =I +I x + I I x qx+φ x (2.6)

where I and _R I are, respectively, the reference and sample intensity distribution, _s q is the spatial frequency of the fringes, and φ is the spatial varying phase associated with objects which is aimed to measure.

Figure 2.15 Typical Hilbert phase microscope set-up [24].

First the interferogram is Fourier transformed and high-pass filtered to obtain sinusoidal term ( ) 2u x = I I x_{R s}( ) cos[qx+φ( )]x . Then complex analytical signal is constructed as

The imaginary part of the right-hand side is the Hilbert transform of u x( ) . The relationship below exists according to the properties of Hilbert transform:

( ) tan {Im[ ( )] / Re[ ( )]}x 1 z x z x qx

φ = ⁻ − (2.8)

30 Thus phase associated with object is acquired.

The process of computation of Eq. (2.7) to obtain complex analytic signal is equivalent to performing Fourier transform of two-dimensional sinusoidal signal and suppressing the negative spatial frequencies. From an inverse Fourier transform operation, a two-dimensional complex analytic signal could be obtained. And phase information can be deduced from it.

Figure 2.16 (a) and (c) are Hilbert phase microscope images for quantitative assessment of shape transformation of a red blood cell in a 10s period. (b) and (d) are measured along the profiles indicated by the arrows in (a) and (c) [26].

Hilbert phase microscope has been applied to retrieve phase profile of an optical fibre [25], quantify cell volume and monitor cell dynamic morphology at the millisecond scales and sub-nanometre path-length sensitivity [24], quantify the refractive properties

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of pathology tissue slices [25]. An example of applications is shown in Fig. 2.16. (a) and (c) are images of Hilbert phase microscope for quantitative assessment of shape transformation of a red blood cell in a 10s period. (b) and (d) are measured along the profiles indicated by the arrows in (a) and (c).

The Hilbert phase microscope is a quantitative phase measurement approach based on interference technique and transmission geometry. What is recorded by such technique is the interferogram of image field with the planar reference field. The reconstruction method is isolation of the phase associating the object from the phase of a complex analytical signal constructed with help of Hilbert transform. The final result is quantitative phase or quantitative OPD which is optical thickness related to n and d where n is refractive index and d is the physical thickness of specimen. No specimen preparation is needed.

The Hilbert phase microscope can be used to accurately quantify nanometre-level path-length shifts at the millisecond time scales or less. This is due to its single shot nature so that its acquisition time is limited only by the recording device.

However, the optical system of Hilbert phase microscope is complex. Thus it is difficult to adjust and the phase aberration due to optical elements in the system will be complicate. Furthermore this technique can only reconstruct the phase information at image plane instead of volume as the interferogram is recorded at the image field.

32 2.2.5 Diffraction Phase Microscope

Diffraction phase microscope (DPM) [25] combines the single shot benefit of Hilbert phase microscope with the common path geometry of Fourier phase microscope. Thus, DPM allows for fast imaging rates without compromising phase stability.

The basic setup of the DPM is shown in Fig. 2.17. A grating is placed at image plane of specimen, which generates multiple diffraction orders containing full spatial information of the sample image. L3 and L4 and a pinhole form a 4-f spatial filtering system. This system isolates the 0^th order and 1^st order to generate a common path Mach-Zender interferometer. 0^th order is the reference beam and 1^st order is the object beam and they interfere at CCD plane. From the recorded interferogram, Hilbert transform is used to extract quantitative phase image as in Hilbert phase microscope [27]. Details of this setup are introduced in reference [24].

Figure 2.17 Diffraction phase microscope setup [28].

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DPM technique has been combined with fluorescence microscopy [29] and confocal microscopy [30] techniques to enhance its capability.

DPM has been applied to quantitatively assess the single red blood cell shape and dynamics[31], monitoring cell attack phenomenon [27], red blood cell membrane fluctuation [31], particle tracing [29]. One of the applications is shown in Fig. 2.18. It is the profile of red blood cell. Since red blood cell has uniform refractive index, the phase image Δφ( , )x y is directly proportion to the height profile h x y( , ) where k is a constant.

The scale bar on the right shows cell thickness in microns.

Figure 2.18 Quantitative phase image of a red blood cell of DPM. The scale bar on the right shows cell thickness in microns [29].

Diffraction phase microscope is a quantitative phase measurement approach based on interference technique and transmission geometry. What is recorded by such technique

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is the interferogram of the image field with the planar reference field. The reconstruction method is isolation of the phase associating the object from the phase of a complex analytical signal constructed with help of Hilbert transform. The final result is quantitative phase or quantitative OPD which is optical thickness related to n and d where n is refractive index and d is the physical thickness of specimen. No specimen preparation is needed.

Diffraction phase microscope can be used to accurately quantify nanometer-level path-length shifts in millisecond time with sub-nanometer path-path-length stability. This is due to its single shot nature and common-path interferometer geometry.

However, many optics components such as several lenses, grating and pinholes are used in diffraction phase microscope. This makes the setup of DPM complicate and not easy to adjust and use. Phase aberration due to different optics components will be complicate and severe. Only the phase information at image plane can be reconstructed as the interferogram is recorded at the image field.

2.2.6 Quantitative Differentiation Interference Contrast Microscope DIC microscope is one of the widely used phase imaging technique which is able to capture minute structures of phase objects as we discussed above. However, it is designed for qualitative phase imaging only. The image of DIC microscope is a mixture of amplitude information and phase gradient information wrapped in sinusoidal signal.

But several approaches to achieve quantitative DIC have been proposed and reported.

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Quantitative phase information can be extracted from DIC image— an entanglement of amplitude with phase gradient.

The first step for quantitative DIC is to extract phase gradient from DIC image. Phase-shifting method [29] and approximation methods [32, 33] have been used. In phase shifting method, field of two sheared and orthogonally polarized beams can be presented as caused by the specimen phase gradient. The intensity of final image is

2 2 2

1 2 1 2 2 1 2cos( 2 )

I = +t t =a +a + a a Δ +θ φ (2.11)

In DIC, phase bias can be changed by phase shifting. By incrementing 2φ by / 2π step by step, four images can be obtained. Then the specimen phase difference (phase gradient) can be obtained by

1 /2 3 /2

The next step is to recover phase from its gradient. Many methods have been adopted such as integration, filtering in Fourier domain, 6-frame, 4-frame, 2-frame algorithm, non-iterative and iterative computation.

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Applications include refractive index analysis of optical fiber [33], cell imaging [34].

Figure 2.19 shows an example of cell imaging of quantitative DIC microscope. (a) is a DIC image of a cheek cell. (b) is phase reconstruction at a single plane from (a). (c) is 3D topological view of (b).

Quantitative DIC microscope is based on DIC microscope which is basically a shearing interferometer for phase imaging. Its geometry can be reflection and transmission. What it records is interference intensity image of DIC microscope. Reconstruction method from this image is first to extract phase gradient and then calculate phase from its gradient. Therefore it is an indirect phase measurement technique.

The advantages of this method: It utilizes conventional DIC microscope image as input for calculation. Thus the setup is simple.

The disadvantages of this method: complex calculation from phase gradient to phase, the specimen of DIC should be weak phase object which means the phase shift induced by specimen should be smaller than / 2π . It is not easy to perform real time measurement, as more than one image needs to be recorded.

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Figure 2.19 (a) DIC image of a cheek cell; (b) Phase reconstruction at a single plane from (a); (c) 3D topological view [35].

2.2.7 Spectral-Domain Phase Microscope

Spectral-domain phase microscope [32, 33, 35] is a phase-sensitive functional derivative of spectral-domain optical coherence tomography (OCT) to produce depth-resolved intensity and phase profiles with significantly improved phase stability compared to systems based on time domain OCT [36]. This technique is based on interference. It is also a quantitative phase measurement method. It can generate 3-D

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quantitative phase-contrast image of a specimen simply by scanning the beam laterally as it measures phase profile in depth [37, 38].

Figure 2.20 shows a setup of spectral domain phase microscope which includes common path spectral-domain OCT. A broadband 840nm superluminescent diode (50nm FWHM) is used as light source. Swept laser source can also be used as light source in such technique [37].

Figure 2.20 (a) Schematics of spectral-domain phase microscopy. (b) Sample placed between a coverslip and a microscope slide [36].

As shown in Fig. 2.20 (b), the reflection from the top surface of a coverslip serves as the reference optical field, and the backscattered waves from the sample are the measurement fields. When the thickness of the coverslip is larger than that of the specimen—cell in this case, the interference signal referenced to the top surface of the coverslip can be distinguished and separated easily from interference signals referenced to other surfaces. Details of this setup are introduced in reference [37].

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The information value directly recorded in this technique is interference intensity at point (x, y) where the beam locates on the specimen. It is in the form as Eq. (2.13) [37].

1/2

( , ) ( , )

( ) |_{x y} 2[ _{r s}( )] ( )cos(2 ) |_{x y}

I k = R R z S k k pΔ (2.13)

where k is the free-space wave number; z is the geometrical distance;R and _r R are the _s reference reflectivity and specimen reflectivity at depth z, respectively; S k( )is the source power spectral domain and Δpis the OPD between the reference and the sample beams. The spectral domain phase microscope is of reflection geometry.

To reconstruct phase from the recorded interference intensity, depth profile F z( ) of complex-valued is acquired by performing discrete Fourier transform with respect to 2k [37]. As interference component referenced to the top surface of the coverslip can be separated from interference components referenced to other surfaces, phase information which is a function of z can be obtained from the argument ofF z( ):

whereλ₀ is the center wavelength of the source. The depth-resolved phase measurement is performed as the beam scans laterally across the specimen point by point. We see the OPD at certain layer of depth z with respect to a reference lay—a coverslip this time can be obtained. Therefore the final information got from this technique could be quantitative phase or quantitative OPD which is optical thickness related to n and d where n is refractive index and d is the physical thickness of specimen.

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Figure 2.21 Images of human epithelial cheek cells. (a) Image recorded by a Nomarski microscope (10X;

N.A., 0.3); the bar represents 20μm. (b) Spectral –domain optical coherence phase microscope image, along with the gray scale denoting the OPD in nanometres. (c) Surface plot of (b), showing optically thick structures such as nuclei and subcellular structures in the cell [38, 39].

Such technique has been applied in human epithelial cheek cells imaging [37], in vivo human retinal imaging [37], and cellular dynamics measurement [39], etc.

One example of applications is imaging of human epithelial cheek cells shown in Fig 2.21. (a) is image recorded by a Nomarski microscope, the bar represents 20μm. (b) is spectral –domain optical coherence phase microscope image, along with the gray scale

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denoting the OPD in nanometres. (c) is surface plot of (b), showing optically thick structures such as nuclei and sub-cellular structures in the cell.

No specimen preparation is required in such technique. Furthermore, phase information in certain depth inside the specimen can be isolated out from the whole specimen which means phase sectioning can be performed. However, this technique relies on single point measurement, which for imaging purposes required raster scanning. Thus it is time consuming.

2.2.8 Digital Holography

Digital holography is an interference based technique for quantitative phase imaging. It can simultaneously provide quantitative amplitude and phase images. It also has the capability to numerically reconstruct different object planes without using any optomechanical movement [40].

Digital holography was developed from conventional holography. In conventional holography, the interference between the coherent object and reference waves produces an interference pattern--hologram, which contains the information about not only the intensity of light but also its phase. Conventional holography uses a photographic plate to record the hologram and hologram is developed by photochemical processes. The hologram is then illuminated by the original reference wave. The original object optical field is reproduced by the propagation of diffracted light from the hologram. As the reproduced holographic image retains the information of not only the amplitude but also the phase of object wave, this image is the exact 3D replica of the original object.

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As the conventional processes of holographic recording and hologram photochemical development are complicated and time-consuming, more interests and efforts have been shifted towards digital holography [36]. Main advantages of digital holography over conventional holography are listed below:

• Rapid image acquisition

• Accessibility to quantitative amplitude and phase information

• Various image processing techniques can be applied to the complex field.

Comparing with conventional holography, in digital holography, hologram is sampled by a high resolution CCD array and transferred into a computer as an array of numbers [41]. The recorded digital hologram is multiplied by the digital reference wavefield in the hologram plane and the digital diffracted field in the image plane, which is another numerical array of complex numbers, is determined by the Fresnel-Kirchoff integral to numerically calculate the intensity and the phase distribution of the reconstructed real image array [42-44]. Numerical reconstruction could be performed by Fresnel transform, Huygens convolution, and angular spectrum methods [45].

The principle of digital holography for 3D imaging includes two parts: digital recording and numerical reconstruction which is explained in Fig. 2.22. During the recording process, Fig. 2.22 (a), the illumination wave is scattered and reflected by the object and becomes the object wave at the object plane: , , , where , is the amplitude and , is the phase. In this context, the 3D object profile information is encoded in the phase of object wave , . This is because

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that the object surface topography changes the optical path length of the illumination wave. The object wave propagates from the object plane to the recording device and becomes , , , at the recording plane where , is the amplitude and , is the phase. Since the recording devices can only record intensity information, a reference wave , , , is used to encode the phase information , into the interference pattern—hologram. The hologram , is expressed as

, , , , ,

, , , , , ,

, , , , (2.15)

where * denote the complex conjugate. It can be seen that the phase , is included in the third term of Eq. (2.15). The phase , is thus encoded in the recorded intensity of the hologram.

The numerical reconstruction process is shown in Fig. 2.22 (b). We illustrate it with a reconstruction example in Fig. 2.23. The third term of Eq. (2.15) is extracted from the hologram in the spectrum domain as shown in Fig. 2.23 (a-c) and is combined with the conjugate of the reference wave in space to obtain the term

, , , . Usually the amplitude of the reference wave , is a constant value which can be ignored. Therefore the object wave at the hologram plane , , , can be obtained as in Fig. 2.23 (d). The object wave at the object plane , , , of Fig. 2.23 (e) is achieved by back wave propagation from hologram plane to the object plane which is performed by a digital Fresnel transform. The amplitude image , and phase image ,

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shown in Fig. 2.23 (f) and (g) are obtained. From the phase image , , the OPD value is calculated. According to the relationship between OPD and the object profile in different modes, the reconstruction of 3D profile of Fig. 2.23 (h) is accomplished.

Fig. 2.22 The principle of digital holography for 3D imaging: (a) digital recording and (b) numerical reconstruction.

Illumination Wave

Object

Reference Wave

Recording Device

Object Wave Object Wave

Reconstructed Object Image

Digital Hologram

Object Wave Object Wave

(a)

(b)

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Figure 2.23 An example of the implementation of numerical reconstruction process.

Holographic configurations can be divided into off-axis geometry and in-line geometry [46, 47]. Our group has done leading research in the field of in-line digital holography [48-64]. The problem associated with in-line geometry is the overlapping of zero-order and twin images. Though methods to solve the problem exit, in-line digital holography is still not suitable for real time phase imaging due to the overlapping of different terms

(a) Digital Hologram (b) Hologram Spectrum (c) Spectrum of the third term of Eq. (2.15)

(d) Spectrum of , (e) Spectrum of ,

(f) Amplitude , (g) Phase , (h) 3D Profile

Fourier Transform Extract the 3^rd term of Eq. (2.15)

Multiply with in space Fresnel Transform: Wave propagation from hologram plane to object plane

Inverse Fourier Transform

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and the sensitivity of phase. Therefore in the study of phase measurement, only off-axis configuration will be focused on.

Fig. 2.24 (a) is the schematic geometry of transmission digital holography. One coherent laser beam is split into two parts – the reference beam passes through the beam splitter and illuminates the CCD directly; the object beam illuminates the sample on the stage and passes through it. These two beams interfere at the CCD plane to generate the hologram. Lenses 1 and 2 are used for collimated light. The insets of Fig. 2.24 (a) and (b) show the off-axis geometry in detail.

The schematic geometry of reflection digital holography is shown in Fig. 2.24 (b). Laser beam passes through beam splitter and is divided into two parts. One is object beam which illuminates the specimen. The light reflected from the specimen then travels towards the CCD. The other is reference wave which is reflected by a mirror and then arrives at the CCD to interfere with object wave.

In digital recording, the sampling theory should be satisfied to fully resolve the interference pattern to further acquire reconstructed image of good quality. The hologram is then recorded and digitalized by CCD and transported into computer and saved as a digital hologram.

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Figure 2.24 (a) Schematic geometry of transmission digital holography; (b) Schematic geometry of reflection digital holography.

In numerical reconstruction, two algorithms of wave propagation approaches, Fresnel and convolution, can be used in the propagation calculation based on diffraction theory.

The way to numerically calculates this propagation by Fresnel transformation is shown below [48, 51-54, 56, 57, 59]:

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Δ , Δy Δ Δ , Δ Δ

,

(2.16) where A ; k,l,m,n are integers (-N/2≤ k,l,m,n ≤ N/2); is the reconstruction distance from the hologram plane to the image plane. Δ and ^Δ are sampling intervals in hologram plane and Δ and Δ are sampling intervals in image plane which are equal to

Δ Δ (2.17)

where L is the size of CCD. If digital hologram is not padded in the numerical reconstruction, L is also the size of digital hologram. Δ and Δ defines the transverse resolution in the image plane. Δ , Δy is the reconstructed object wavefront at the image plane. It is an array of complex numbers whose modulus gives the amplitude and the arctangent of the imaginary over real part provides the phase. Phase unwrapping

where L is the size of CCD. If digital hologram is not padded in the numerical reconstruction, L is also the size of digital hologram. Δ and Δ defines the transverse resolution in the image plane. Δ , Δy is the reconstructed object wavefront at the image plane. It is an array of complex numbers whose modulus gives the amplitude and the arctangent of the imaginary over real part provides the phase. Phase unwrapping