2. Phase-shifting interferometry selectively extracting wavelength information: phase-division multiplexing (PDM) of wavelengths

Figure 2 illustrates the schematic of the proposed multiwavelength 3D imaging technique in the case where the number of wavelengths N is two, which was initially presented in 2013 [34–36]. Optical setup is based on phase-shifting digital holography with multiple lasers. Multiple object and reference waves with multiple wavelengths illuminate a monochromatic image sensor simultaneously. The sensor records wavelength-multiplexed phase-shifted holograms I(x,y:α1,α2) by changing the phases of the reference waves. Phase shifts for respective wavelengths α<sup>1</sup> and α<sup>2</sup> are introduced. An object wave at the desired wavelength is selectively extracted from the holograms by the signal processing based on phase-shifting interferometry. As a result, a color 3D image is reconstructed from the selectively extracted object waves. Thus, color 3D imaging can be achieved with grayscale wavelength-multiplexed images. When the number of wavelengths is N, 2N + 1 variables are contained in a wavelengthmultiplexed hologram: the number N of object waves, N of conjugate images, and the sum of the 0th-order diffraction waves. Therefore, five holograms are required to solve the system of equations when N = 2. It is noted that no Fourier transform is essentially required.

Figure 2. Schematic representation of the proposed multiwavelength 3D imaging technique.

Figure 3 describes the principle that wavelength information is selectively extracted by the signal processing in the space domain. As seen in Figure 3, different phase shifts for respective wavelengths are given to object waves with multiple wavelengths, and then wavelength information is separated in the polar coordinate plane. Although this separation is used to extract an object wave from a hologram in general phase-shifting interferometry, in the proposed technique, the separation is utilized to remove not only the conjugate images and 0th-order diffraction wave, but also undesired wavelength information. This means phase-division multiplexing (PDM) of wavelengths. Figure 3 shows the case where specific phase shifts are used [34–36], but this concept is also applicable to the case where arbitrary phase shifts are introduced [39].

2. Phase-shifting interferometry selectively extracting wavelength information: phase-division multiplexing (PDM) of wavelengths

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equations when N = 2. It is noted that no Fourier transform is essentially required.

Figure 2. Schematic representation of the proposed multiwavelength 3D imaging technique.

Figure 2 illustrates the schematic of the proposed multiwavelength 3D imaging technique in the case where the number of wavelengths N is two, which was initially presented in 2013 [34–36]. Optical setup is based on phase-shifting digital holography with multiple lasers. Multiple object and reference waves with multiple wavelengths illuminate a monochromatic image sensor simultaneously. The sensor records wavelength-multiplexed phase-shifted holograms I(x,y:α1,α2) by changing the phases of the reference waves. Phase shifts for respective wavelengths α<sup>1</sup> and α<sup>2</sup> are introduced. An object wave at the desired wavelength is selectively extracted from the holograms by the signal processing based on phase-shifting interferometry. As a result, a color 3D image is reconstructed from the selectively extracted object waves. Thus, color 3D imaging can be achieved with grayscale wavelength-multiplexed images. When the number of wavelengths is N, 2N + 1 variables are contained in a wavelengthmultiplexed hologram: the number N of object waves, N of conjugate images, and the sum of the 0th-order diffraction waves. Therefore, five holograms are required to solve the system of

Figure 3. Principle of phase-division multiplexing (PDM) of wavelengths: separation of multiple wavelengths in the polar coordinate plane.

Figure 4 illustrates optical implementations of the proposed digital holography. Multiple lasers irradiate laser beams with multiple wavelengths simultaneously. A device for shifting the phase of light, such as a mirror with a piezo actuator, a spatial light modulator, or wave plates, is placed in the path of the reference arm. A monochromatic image sensor records the required wavelength-multiplexed phase-shifted holograms sequentially. An optical system based on PDM has the following features: the spectroscopic sensitivity of the optical system can be extended in comparison to the system with a color image sensor; full space-bandwidth product of an image sensor can be used to record object waves with multiple wavelengths; a bright color image can be obtained due to no spectroscopic absorption, while wavelengths filters required in conventional systems absorb light to obtain a color image; and measurement time is shortened by the wavelength-multiplexed recording in comparison with temporal division technique.

Figure 4. Optical implementations of PDM. Optical setups with (a) a mirror with a piezo actuator and (b) a spatial light modulator that has wavelength dependency in phase modulation.

Figure 5 illustrates the image reconstruction algorithm [34–36]. A wavelength-multiplexed phase-shifted hologram I(x,y:α1,α2) is expressed as follows,

$$I(\mathbf{x}, y:\alpha\_1, \alpha\_2) = I\_{\lambda 1}(\mathbf{x}, y:\alpha\_1) + I\_{\lambda 2}(\mathbf{x}, y:\alpha\_2),\tag{1}$$

here Iλ1(x,y:α1) and Iλ2(x,y:α2) are holograms at the wavelengths of λ<sup>1</sup> and λ2, respectively. Eq. (1) means that a recorded monochromatic image is the sum of Iλ1(x,y:α1) and Iλ2(x,y:α2). When the complex amplitude distributions of object waves with different wavelengths are Uλ1(x,y) and Uλ2(x,y), 0th(x,y) is the 0th-diffraction wave, Ar(x,y) is the amplitude distribution of the reference wave, j is imaginary unit, \* means complex conjugate, and L and M are integers, then I(x,y:α1,α2) can be rewritten as follows,

Multiwavelength Digital Holography and Phase-Shifting Interferometry Selectively Extracting Wavelength… http://dx.doi.org/10.5772/67295 211

$$\begin{array}{l} I(\mathbf{x}, y: \alpha\_1, \alpha\_2) = 0th\_{\lambda 1}(\mathbf{x}, y) + Ar\_{\lambda 1}(\mathbf{x}, y) \left\{ \mathcal{U}\_{\lambda 1}(\mathbf{x}, y) \exp(-j\alpha\_1) + \mathcal{U}\_{\lambda 1}\*(\mathbf{x}, y) \exp(j\alpha\_1) \right\} \\ \quad + 0th\_{\lambda 2}(\mathbf{x}, y) + Ar\_{\lambda 2}(\mathbf{x}, y) \left\{ \mathcal{U}\_{\lambda 2}(\mathbf{x}, y) \exp(-j\alpha\_2) + \mathcal{U}\_{\lambda 2}\*(\mathbf{x}, y) \exp(j\alpha\_2) \right\}. \end{array} \tag{2}$$

Only the complex amplitude distributions of object waves with dual wavelengths Uλ1(x,y) and Uλ2(x,y) are derived from five wavelength-multiplexed phase-shifted holograms I(x,y:0,0), I(x, y:α1,α2), I(x,y:-α1,-α2), I(x,y:α3,α4), and I(x,y:-α3,-α4) because five variables are contained in Eq. (2). If the system shown in Figure 4(a) is used to implement the proposed technique by moving the mirror in the reference arm with a piezo actuator at a distance Z in the depth direction, the phase shifts are

$$a\_1 = \frac{4\pi Z}{\lambda\_1},\tag{3}$$

$$
\alpha\_2 = \frac{4\pi Z}{\lambda\_2}.\tag{4}
$$

Here, when Z is equal to Lλ1/2, α<sup>1</sup> is 2πL and α<sup>2</sup> is 2πLλ1/λ2. As a result, the intensity distribution Iλ1(x,y: α1) is not changed and Iλ2(x,y: α2) is changed, unless Lλ1/λ<sup>2</sup> is an integer. In the case where an integral multiple of 2π is utilized for phase shifts, meaning α<sup>2</sup> = 2πM and α<sup>3</sup> = 2πL, Uλ1(x,y) and Uλ2(x,y) are separately derived by the following expressions.

$$\begin{array}{l} \mathcal{U}\_{\Lambda}(\mathbf{x},\mathbf{y}) = \left[ \mathcal{U}(\mathbf{x},\mathbf{y}:\mathbf{0},\mathbf{0}) - \{ I(\mathbf{x},\mathbf{y}:\mathbf{a}\_{1},2\pi\mathbf{M}) + I(\mathbf{x},\mathbf{y}:-\mathbf{a}\_{1},-2\pi\mathbf{M}) \} \right] / \{ 4Ar\_{\Lambda}(\mathbf{x},\mathbf{y}) (1 - \cos\alpha\_{1}) \} \\ \quad + j \{ I(\mathbf{x},\mathbf{y}:-\mathbf{a}\_{1},-2\pi\mathbf{M}) - I(\mathbf{x},\mathbf{y}:\mathbf{a}\_{1},2\pi\mathbf{M}) \} / (4Ar\_{\Lambda}(\mathbf{x},\mathbf{y})\sin\alpha\_{1}), \end{array} \tag{5}$$

$$\begin{array}{l} \mathcal{U}\_{\lambda 2}(\mathbf{x}, \mathbf{y}) = [2I(\mathbf{x}, \mathbf{y} : \mathbf{0}, \mathbf{0}) - \{I(\mathbf{x}, \mathbf{y} : 2\pi \mathbf{L}, \alpha\_4) + I(\mathbf{x}, \mathbf{y} : -2\pi \mathbf{L}, -\alpha\_4)\}] / \{4Ar\_{\lambda 2}(\mathbf{x}, \mathbf{y}) (1 - \cos \alpha\_4)\} \\ \quad + j \{I(\mathbf{x}, \mathbf{y} : -2\pi \mathbf{L}, -\alpha\_4) - I(\mathbf{x}, \mathbf{y} : 2\pi \mathbf{L}, \alpha\_4)\} / (4Ar\_{\lambda 2}(\mathbf{x}, \mathbf{y}) \sin \alpha\_4). \end{array} \tag{6}$$

As shown in Eqs. (5) and (6), subtraction between holograms, which is based on phase-shifting interferometry, is calculated and the unwanted wavelength component Iλ1(x,y) or Iλ2(x,y) is removed. Thus, dual-wavelength information is extracted selectively from five phase-shifted holograms. In this way, multiwavelength information can be separately extracted from 2N + 1 holograms when the number of wavelengths is N. From the extracted complex amplitude distributions on the image sensor plane, a multiwavelength 3D object image is reconstructed by the calculations of diffraction integrals and color synthesis.

Figure 5. Image-reconstruction procedure.

Figure 5 illustrates the image reconstruction algorithm [34–36]. A wavelength-multiplexed

Figure 4. Optical implementations of PDM. Optical setups with (a) a mirror with a piezo actuator and (b) a spatial light

here Iλ1(x,y:α1) and Iλ2(x,y:α2) are holograms at the wavelengths of λ<sup>1</sup> and λ2, respectively. Eq. (1) means that a recorded monochromatic image is the sum of Iλ1(x,y:α1) and Iλ2(x,y:α2). When the complex amplitude distributions of object waves with different wavelengths are Uλ1(x,y) and Uλ2(x,y), 0th(x,y) is the 0th-diffraction wave, Ar(x,y) is the amplitude distribution of the reference wave, j is imaginary unit, \* means complex conjugate, and L and M are

I xð Þ¼ , y : α1, α<sup>2</sup> I<sup>λ</sup>1ð Þþ x, y : α<sup>1</sup> I<sup>λ</sup>2ð Þ x, y : α<sup>2</sup> ; (1)

phase-shifted hologram I(x,y:α1,α2) is expressed as follows,

modulator that has wavelength dependency in phase modulation.

210 Holographic Materials and Optical Systems

integers, then I(x,y:α1,α2) can be rewritten as follows,
