2.1.

Digital Holography

Digital holography in off-axis geometry is based on the classic holography principle, with the difference being that the hologram recording is performed by a charge-coupled device (CCD) camera and transmitted to a computer, and the subsequent reconstruction of the holographic image is carried out numerically by the computer.

Suppose that the coordinates of the hologram plane are $(x_H,y_H)$ . The object and reference waves in the hologram plane are written as:

where $O_0(x_H,y_H)$ and $R_0$ are the amplitudes of the object and the reference waves; $j$ is the imaginary unit; $\varphi (x_H,y_H)$ is the phase of the object wave; ${\xi }_r=\sin {\theta }_x∕\lambda$ and ${\eta }_r=\sin {\theta }_y∕\lambda$ are the spatial frequencies of the reference wave in the $x$ and $y$ directions, respectively; $\lambda$ is the wavelength of light; and ${\theta }_x$ and ${\theta }_y$ are the angles between the propagation direction of the object and the reference waves in the $x$ and $y$ directions, respectively.

The recorded intensity $I(x_H,y_H)$ at the hologram plane is the square module of the amplitude superposition of the object and reference waves. It is given by:

The first two terms of Eq. 3 represent the intensities of the reference and object waves, respectively. They do not provide spatial information about the object optical field and form the zero-order term. The last two terms provide spatial frequency of the recorded hologram and are responsible for the virtual and real images, respectively.

The hologram is reconstructed by numerically propagating the optical field along the $z$ direction in accordance with the Fresnel diffraction law. Cuche, Marquet, and Depeursinge¹³ described the method used for hologram reconstruction in detail. The angular spectrum method has a significant advantage in that it has no minimum reconstruction distance requirement.¹⁴ Therefore, it is fit for the analysis of the hologram of biological specimens by employing a microscope objective.

If $E(x,y;0)$ is the wavefront at plane $z=0$ , the angular spectrum $A(\xi ,\eta ;0)=\mathfrak{I}\left\{E(x,y;0)\right\}$ at this plane is obtained by taking the Fourier transform, where $\mathfrak{I}\left\{\right\}$ denotes the Fourier transform; $\xi$ and $\eta$ are corresponding spatial frequencies of $x,y$ directions, respectively; and $z$ is the propagation direction of the object wave. The new angular spectrum $A(\xi ,\eta ;d)$ at plane $z=d$ is calculated from $A(\xi ,\eta ;0)$ as:

The reconstructed complex wavefront at plane $z=d$ is found by taking the inverse Fourier transform aswhere ${\mathfrak{I}}^{-1}\left\{\right\}$ denotes the inverse Fourier transform. The intensity image $I(x,y;d)$ and phase image $\varphi (x,y;d)$ are simultaneously obtained from a single digital hologram by calculating the square module of the amplitude and the argument of the reconstructed complex wavefront:

2.2.

Surface Plasmon Resonance

Surface plasmons are electromagnetic waves associated with longitudinal oscillation of the free electrons on the interface between two media with dielectric constants of different signs, for example, between a noble metal (gold or sliver) and air. The technique of surface plasmon resonance (SPR) uses the excited surface plasmons to probe biological and chemical information of the sample surface. The most widely used method by which to excite surface plasmon waves (SPWs) is prism-based attenuated total reflection. SPW is polarization dependent and is excited only by $p$ -polarized waves. When $p$ -polarized light is incident on the base of a prism with the angle of incidence greater than the critical angle of TIR, an evanescent field will extend from the prism into the noble metal. At a specific angle of incidence, the wave vector of the evanescent wave can match the wave vector of the SPW for a given frequency, and the SPR occurs. The specific angle of incidence is called the SPR angle. At this angle, the intensity of reflected light goes through a minimum, and the phase variation of reflected light is steepest due to surface plasmon resonance. The field associated with the surface plasmon, decaying exponentially, is sensitive to the change in the refractive index of the medium close to the metal-film surface.

The prism-based SPR configuration is shown as Fig. 1 . It is composed of a prism, gold film, and a sample. The gold film is placed onto the prism’s bottom surface, and the sample is attached to the surface of the gold film. The sequence of the optical media from prism to sample is denoted as 1, 2, 3. The symbols $\zeta$ and $\tau$ are the directions of the coordinate system. The reflectivity $r$ of the light amplitude at the interface between two adjacent media is given by:^{15, 16}

where $\theta$ is the angle of incidence; $i$ denotes the $i$ ’th optical medium; $k_{\tau i}$ represents the wave number of the transmitted light in the $i$ ’th optical medium along the $\tau$ direction; ${\epsilon }_i$ is the dielectric constant of the $i$ ’th optical medium; and $n_i$ is refractive index of the $i$ ’th optical medium.

The reflectivity of light amplitude in the SPR system is given as:^{15, 16}

where $d_2$ is the thickness of the gold film. Therefore, the reflectivity $R$ of light intensity and reflection phase $\varphi$ are expressed as:

Calculations are performed with $n_1=1.516$ ( $n_1$ is the refractive index of the prism), $n_2=-13.4+1.4j$ ( $n_2$ is the complex refractive index of the gold film), $n_3=1.335$ or 1.000 ( $n_3$ is the refractive index of the sample), and $\lambda =632.8\mathrm{nm}$ to discuss the relations between reflectivity $R$ , reflection phase $\varphi$ , and the angle of incidence $\theta$ , according to Eqs. 13, 14. The thickness of the gold film $d_2$ is $50\mathrm{nm}$ in the SPR configuration, which is the optimal value discussed by many researchers.^{15, 17}

Figure 2 shows the relations between $R$ , $\varphi$ , and $\theta$ in SPR. It demonstrates that the SPR angles for $n_3=1.000$ and $n_3=1.335$ (the refractive index of a living cell is close to this value) are 43.6 and $71.4\mathrm{deg}$ , respectively. When the incident angle on the gold film/sample interface is close to 43.6 or $71.4\mathrm{deg}$ , there are evident changes both in the reflected light intensity and in its phase.

Physically, the changes of the reflected light in intensity and phase are due to the interaction between the lightwave and the delocalized electrons in gold film in the SPR technique. The states of the delocalized electrons are related to the dielectric constant ${\epsilon }_3$ of the sample near the gold film. Therefore, the change of ${\epsilon }_3$ can indirectly affect the intensity and phase of the reflected light. According to Eq. 11, the intensity and phase images obtained in the SPR technique can indirectly represent the refractive index distribution of sample near the gold film. When the incident angle is close to the SPR angle, the images are most sensitive to the variation of the refractive index of sample near the gold film.

Suppose that the sample can be divided into two layers, whose refractive indices are $n_3=1.380$ and $n_4=1.335$ , and $d_3$ is the thickness of the medium with the refractive index $n_3$ , as shown in Fig. 3 . The relations between $R$ , $\varphi$ , and $d_3$ are analyzed with $n_1=1.67$ , $n_2=-13.4+1.4j$ , and $\theta =60\mathrm{deg}$ (close to the SPR angle) according to Eq. 8, with $i=1,2,3$ , and Eqs. 9, 10 with $i=1,2,3,4$ , and the following equations:¹⁵

Figure 4 is a relation diagram between $R$ , $\varphi$ , and $d_3$ . It indicates that the intensity and phase of the reflected light have a significant change when $d_3$ is less than $100\mathrm{nm}$ , while there are almost no changes when $d_3$ is larger than $100\mathrm{nm}$ . In other words, the intensity and phase of the reflected light are affected by the medium with $n_4$ when $d_3$ is less than $100\mathrm{nm}$ . The intensity and phase images represent the refractive index distribution of a sample within about $100\mathrm{nm}$ near the gold film. Therefore, the spatial resolution in the vertical direction of the sample surface is about $100\mathrm{nm}$ , which is significantly less than a half wavelength for $\lambda =632.8\mathrm{nm}$ .

If the gold film is absent $(d_2=0)$ , this corresponds to the TIR configuration. Figure 5 shows the relations between $R$ , $\varphi$ , and $\theta$ in TIR, according to Eqs. 13, 14. It demonstrates that the phase image can represent the refractive index distribution of a sample when the angle of incidence is greater than the critical angle of TIR while the intensity image cannot represent the refractive index distribution. The phase sensitivity in the TIR technique is lower than that in the SPR technique at about the SPR angle. The relation diagram between $R$ , $\varphi$ , and $d_3$ is shown in Fig. 6 . It indicates that the phase of the reflected light has a significant change when $d_3$ is less than $250\mathrm{nm}$ . Therefore, only the phase image can represent the refractive index distribution of a sample within about $250\mathrm{nm}$ near the prim surface. The spatial resolution in the vertical direction of the sample surface is about $250\mathrm{nm}$ , which is approximately a half wavelength. It is obvious that the SPR technique is better than the TIR technique in spatial resolution. The reason is that the evanescent wave is directly acting on the sample in the TIR technique, while it is indirectly acting on the sample through the gold film in the SPR technique.

Biological or chemical information can be acquired by detecting the changes of the intensity or the phase of reflected light in SPRM. In the traditional SPRM detection method, the light intensity detection method has been studied extensively. However, the refractive index distribution of a sample near the gold film cannot be uniquely determined by the intensity of reflected light in SPRM, for the reason that the intensity of the reflected light is not a single-valued function of the refractive index at a fixed angle of incidence. Figure 7 shows the relations between reflectivity $R$ , reflection phase $\varphi$ , and the refractive index $n_3$ at an angle of incidence $\theta =58\mathrm{deg}$ with $n_1=1.67$ . As shown in Fig. 7, the intensities at A and B are the same, but the refractive indices are different. Some preparatory knowledge is necessary to judge which refractive index more reasonably corresponds to the same intensity.

Digital holography is applied to the SPR technique for reconstructing the wavefront of the reflected light to simultaneously obtain the intensity and phase images of the sample. From the reconstructed complex wavefront, the intensity and phase images can be simultaneously obtained according to Eqs. 6, 7.