4. Two-step phase-shifting interferometry-merged phase-division multiplexing (2π-PDM)

In a wavelength-multiplexed hologram, 2N + 1 variables are contained. Therefore, 2N + 1 images are needed to extract object waves separately in a general PDM technique. However, 2N wavelength-multiplexed holograms are sufficient to selectively extract object waves with N wavelengths, with the two-step phase-shifting interferometry-merged phase-division multiplexing (2π-PDM) technique [38]. Figure 8 illustrates the basic concept of 2π-PDM. Two main points of 2π-PDM are the utilization of 2π ambiguity of the phase [34, 35] and merger of two-step phaseshifting interferometry [52–56]. As described in section 2, an intensity distribution at a wavelength is not changed when a phase shift is an integral multiple of 2π. We make the best use of this nature to decrease the required number of wavelength-multiplexed images. Also, merging PDM and two-step phase-shifting interferometry is important to satisfy high-quality multiwavelength 3D imaging and acceleration of a recording simultaneously. When recording three wavelengths, six holograms are sufficient with 2π-PDM, as described with an optical implementation in Ref. [38].

Figure 8. Basic concept of 2π-PDM.

3. Numerical simulation

212 Holographic Materials and Optical Systems

resolution were reported in Ref. [36].

amplitude images at the wavelengths of (c) 640 nm and (d) 532 nm.

image.

Numerical simulations were conducted to verify the effectiveness of the proposed technique. Figure 6 shows the amplitude and phase distributions of the object wave at each wavelength. As shown in Figure 6(b), a color object with rough surface was assumed. 640 and 532 nm were assumed as the wavelengths of the light sources. Red and green color components of a standard image "pepper" were used as amplitude images at 640 and 532 nm, respectively. In these simulations, the distance between the object and image sensor was assumed as 200 mm, pixel pitch was 5 μm, resolution was 10 bits, and number of pixels was 512 × 512. Figure 7 shows the images reconstructed by the proposed technique. Faithful images were reconstructed at each wavelength, and crosstalk between object waves with different wavelengths was not seen. The color synthesized image in Figure 7(c) indicates color 3D imaging ability. Thus, the validity of the proposed technique was numerically confirmed. Detailed numerical analyses and an experimental demonstration using an image sensor with 12-bit

Figure 6. Object wave for a numerical simulation. (a) Amplitude and (b) phase distributions of the object wave. Assumed

Figure 7. Numerical results. Reconstructed images at the wavelengths of (a) 640 nm and (b) 532 nm. (c) Color synthesized

The optical setup required for 2π-PDM is the same as that for other PDM techniques. Therefore, the systems in Figure 4 are applicable to 2π-PDM. In 2π-PDM, various types of two-step phaseshifting methods [52–56] can be employed. When merging Meng's two-step method [53] into 2π-PDM, intensity distributions of reference waves Irλ1(x,y) = Arλ<sup>1</sup> 2 (x,y) and Irλ2(x,y) = Arλ<sup>2</sup> 2 (x,y) are sequentially recorded before the measurement by inserting a shutter in the path of the object arm. Figure 9 describes an algorithm for selectively extracting wavelength information in 2π-PDM adopting Meng's technique. In the case of N = 2, a monochromatic image sensor records four wavelength-multiplexed phase-shifted holograms I(x,y:0,0), I(x,y:α1,arb.), I(x,y:2πM,α2), and I(x,y:−2πM,−α2), and intensity distributions of reference waves Irλ1(x,y) and Irλ2(x,y). By making use of 2π ambiguity, both a 0th-order diffraction wave 0thλ2(x,y) and an intensity distribution of a hologram at an undesired wavelength Iλ1(x,y) are removed simultaneously by the subtraction procedure. Therefore, an object wave Uλ2(x,y) is extracted from three holograms, although five variables are contained in each hologram. In the case where α<sup>1</sup> and α<sup>2</sup> > 0, Uλ2(x,y) is derived by

$$\mathcal{U}\_{\lambda2}(\mathbf{x}, \mathbf{y}) = [2I(\mathbf{x}, \mathbf{y}; \mathbf{0}, \mathbf{0}) - \{I(\mathbf{x}, \mathbf{y}; 2\pi\mathbf{M}, \alpha\_2) + I(\mathbf{x}, \mathbf{y}; -2\pi\mathbf{M}, -\alpha\_2)\}] / \{4Ar\_{\lambda2}(\mathbf{x}, \mathbf{y}) (1 - \cos \alpha\_2)\}$$

$$+ j \{I(\mathbf{x}, \mathbf{y}; 2\pi\mathbf{M}, \alpha\_2) - I(\mathbf{x}, \mathbf{y}; -2\pi\mathbf{M}, -\alpha\_2)\} / (4Ar\_{\lambda2}(\mathbf{x}, \mathbf{y}) \sin \alpha\_2). \tag{7}$$

From the extracted object wave Uλ2(x,y) and the amplitude distribution of the reference wave at λ2, the intensity distribution at only λ<sup>2</sup> component Iλ2(x,y:α2) is numerically generated by a computer,

$$I\_{\lambda 2\text{cal}}(\mathbf{x}, y: a\_2) = |\mathcal{U}\_{\lambda 2}(\mathbf{x}, y)|^2 + Ar\_{\lambda 2}(\mathbf{x}, y)^2 + Ar\_{\lambda 2}(\mathbf{x}, y) \{\mathcal{U}\_{\lambda 2}(\mathbf{x}, y) \text{exp}(-ja\_2) \}$$

$$+ \mathcal{U}\_{\lambda 2} \, ^\*(\mathbf{x}, y) \, \text{exp}(ja\_2) \}. \tag{8}$$

If the sum of the intensities of the 0th-order diffraction waves is equal to |Uλ1(x,y)|2 + Irλ1(x,y) + |Uλ2(x,y)|2 + Irλ2(x,y), noiseless multiwavelength 3D imaging can be achieved with 2π-PDM adopting Meng's two-step phase-shifting interferometry, according to the procedures described from here. By using the numerically generated images Iλ2cal(x,y:0) and Iλ2cal(x,y:arb.), intensity distributions at only λ<sup>1</sup> component Iλ1(x,y:0) and Iλ1(x,y:α1) are obtained from I(x,y:0,0) and I(x, y:α1,arb.) as the following expressions:

$$\begin{aligned} I\_{\lambda1}(\mathbf{x}, \mathbf{y}:\mathbf{0}) &= I(\mathbf{x}, \mathbf{y}:\mathbf{0}, \mathbf{0}) \mathbf{-} I\_{\lambda2 \text{cal}}(\mathbf{x}, \mathbf{y}:\mathbf{0}) \\ &= \left| \mathcal{U}\_{\lambda1}(\mathbf{x}, \mathbf{y}) \right|^2 + Ar\_{\lambda1}(\mathbf{x}, \mathbf{y})^2 + Ar\_{\lambda1}(\mathbf{x}, \mathbf{y}) \{ \mathcal{U}\_{\lambda1}(\mathbf{x}, \mathbf{y}) + \mathcal{U}\_{\lambda1}\*(\mathbf{x}, \mathbf{y}) \}, \end{aligned} \tag{9}$$

$$\begin{aligned} I\_{\lambda1}(\mathbf{x}, y: \alpha\_1) &= I(\mathbf{x}, y: \alpha\_1, \mathbf{arb.}) \cdot I\_{\lambda2 \text{cal}}(\mathbf{x}, y: \mathbf{arb.}) \\ &= \left| U\_{\lambda1}(\mathbf{x}, y) \right|^2 + A r\_{\lambda1}(\mathbf{x}, y)^2 + A r\_{\lambda1}(\mathbf{x}, y) \left\{ U\_{\lambda1}(\mathbf{x}, y) \exp(-j\alpha\_1) + U\_{\lambda1} \, ^\*(\mathbf{x}, y) \exp(j\alpha\_1) \right\}. \end{aligned} \tag{10}$$

From the obtained Iλ1(x,y:0) and Iλ1(x,y:α1) and amplitude distribution of the reference wave at λ1, the object wave at λ<sup>1</sup> Uλ1(x,y) can be analytically extracted by using two-step phase-shifting interferometry.

$$\begin{split} \mathcal{U}\_{\Lambda}(\mathbf{x}, \mathbf{y}) &= [\{I\_{\lambda 1}(\mathbf{x}, \mathbf{y} : \mathbf{0}) \mathbf{-} (\mathbf{x}, \mathbf{y})\}] \\ &+ j \{I\_{\lambda 1}(\mathbf{x}, \mathbf{y} : \mathbf{a}\_1) \mathbf{-} I\_{\lambda 1}(\mathbf{x}, \mathbf{y} : \mathbf{0}) \cos a\_1 \mathbf{-} (1 - \cos a\_1) \mathbf{s}(\mathbf{x}, \mathbf{y})\} ] / 2Ar\_{\lambda 1}(\mathbf{x}, \mathbf{y}), \end{split} \tag{11}$$

where,

$$s(\mathbf{x}, y) = \left| \mathcal{U}\_{\lambda 1}(\mathbf{x}, y) \right|^2 + A r\_{\lambda 1}(\mathbf{x}, y)^2 = \left( \frac{\upsilon - \sqrt{\upsilon^2 - 4\mu w}}{2u} \right), \tag{12}$$

$$
\mu = \mathcal{Z}(1 - \cos \alpha\_1),
\tag{13}
$$

$$w = 2\left[ (1 - \cos a\_1) \{ I\_{\lambda 1}(\mathbf{x}, y: \mathbf{0}) + I\_{\lambda 1}(\mathbf{x}, y: a\_1) \} + 2I r\_{\lambda 1}(\mathbf{x}, y) \sin^2 a\_1 \right],\tag{14}$$

Multiwavelength Digital Holography and Phase-Shifting Interferometry Selectively Extracting Wavelength… http://dx.doi.org/10.5772/67295 215

$$\Delta w = I\_{\lambda1}(\mathbf{x}, y:\mathbf{0})^2 + I\_{\lambda1}(\mathbf{x}, y:\mathbf{a}\_1)^2 - 2I\_{\lambda1}(\mathbf{x}, y:\mathbf{0})I\_{\lambda1}(\mathbf{x}, y:\mathbf{a}\_1)\cos a\_1 + 2I r\_{\lambda1}(\mathbf{x}, y)^2 \sin^2 a\_1. \tag{15}$$

U<sup>λ</sup>2ð Þ¼ x; y ½ � 2I xð Þ , y : 0, 0 − f g I xð , y : 2πM, α2Þ þ I xð Þ , y : −2πM, −α<sup>2</sup> =f g 4Ar<sup>λ</sup>2ð Þ x; y ð Þ 1− cos α<sup>2</sup>

From the extracted object wave Uλ2(x,y) and the amplitude distribution of the reference wave at λ2, the intensity distribution at only λ<sup>2</sup> component Iλ2(x,y:α2) is numerically generated by a

If the sum of the intensities of the 0th-order diffraction waves is equal to |Uλ1(x,y)|2 + Irλ1(x,y) + |Uλ2(x,y)|2 + Irλ2(x,y), noiseless multiwavelength 3D imaging can be achieved with 2π-PDM adopting Meng's two-step phase-shifting interferometry, according to the procedures described from here. By using the numerically generated images Iλ2cal(x,y:0) and Iλ2cal(x,y:arb.), intensity distributions at only λ<sup>1</sup> component Iλ1(x,y:0) and Iλ1(x,y:α1) are obtained from I(x,y:0,0) and I(x,

From the obtained Iλ1(x,y:0) and Iλ1(x,y:α1) and amplitude distribution of the reference wave at λ1, the object wave at λ<sup>1</sup> Uλ1(x,y) can be analytically extracted by using two-step phase-shifting

<sup>2</sup> <sup>þ</sup> Ar<sup>λ</sup>1ð Þ <sup>x</sup>; <sup>y</sup>

<sup>v</sup> <sup>¼</sup> 2 1ð Þ <sup>−</sup> cos <sup>α</sup><sup>1</sup> f g <sup>I</sup><sup>λ</sup>1ð Þþ <sup>x</sup>, <sup>y</sup> : <sup>0</sup> <sup>I</sup><sup>λ</sup>1ð Þ <sup>x</sup>, <sup>y</sup> : <sup>α</sup><sup>1</sup> <sup>þ</sup> <sup>2</sup>Ir<sup>λ</sup>1ð Þ <sup>x</sup>; <sup>y</sup> sin <sup>2</sup>

þ j I f <sup>λ</sup>1ð Þ x, y : α<sup>1</sup> −I<sup>λ</sup>1ð Þ x, y : 0 cos α1–ð Þ 1− cos α<sup>1</sup> s xð Þg ; y �=2Ar<sup>λ</sup>1ð Þ x; y ;

<sup>2</sup> <sup>þ</sup> Ar<sup>λ</sup>2ð Þ <sup>x</sup>; <sup>y</sup> <sup>U</sup><sup>λ</sup>2ð Þ <sup>x</sup>; <sup>y</sup> expð Þ <sup>−</sup>jα<sup>2</sup> �

ð Þ x; y expð Þg jα<sup>2</sup> : (8)

<sup>2</sup> <sup>þ</sup> Ar<sup>λ</sup>1ð Þ <sup>x</sup>; <sup>y</sup> <sup>U</sup><sup>λ</sup>1ð Þþ <sup>x</sup>; <sup>y</sup> <sup>U</sup>λ<sup>1</sup>

<sup>2</sup> <sup>¼</sup> <sup>v</sup><sup>−</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>v</sup><sup>2</sup>−4uw <sup>p</sup> 2u !

� �; (14)

u ¼ 2 1ð Þ − cos α<sup>1</sup> ; (13)

<sup>2</sup> <sup>þ</sup> Ar<sup>λ</sup>1ð Þ <sup>x</sup>; <sup>y</sup> <sup>U</sup><sup>λ</sup>1ð Þ <sup>x</sup>; <sup>y</sup> expð Þþ <sup>−</sup>jα<sup>1</sup> <sup>U</sup>λ<sup>1</sup>

� f g ð Þ x; y ;

� �:

(7)

(9)

(10)

(11)

�ð Þ x; y exp jð Þ α<sup>1</sup>

; (12)

α1

þ jIx f g ð Þ , y : 2πM, α<sup>2</sup> −I xð Þ , y : −2πM, −α<sup>2</sup> =ð Þ 4Ar<sup>λ</sup>2ð Þ x; y sin α<sup>2</sup> :

<sup>I</sup><sup>λ</sup>2calð Þ¼ j <sup>x</sup>, <sup>y</sup> : <sup>α</sup><sup>2</sup> <sup>U</sup><sup>λ</sup>2ð Þj <sup>x</sup>; <sup>y</sup> <sup>2</sup> <sup>þ</sup> Ar<sup>λ</sup>2ð Þ <sup>x</sup>; <sup>y</sup>

I<sup>λ</sup>1ð Þ¼ x, y : 0 I xð Þ , y : 0, 0 −I<sup>λ</sup>2calð Þ x, y : 0

�U<sup>λ</sup>1ð Þ <sup>x</sup>; <sup>y</sup> �

�

<sup>2</sup> <sup>þ</sup> Ar<sup>λ</sup>1ð Þ <sup>x</sup>; <sup>y</sup>

<sup>2</sup> <sup>þ</sup> Ar<sup>λ</sup>1ð Þ <sup>x</sup>; <sup>y</sup>

y:α1,arb.) as the following expressions:

¼ �

interferometry.

where,

¼ �

I<sup>λ</sup>1ð Þ¼ x, y : α<sup>1</sup> I xð Þ , y : α1, arb: −I<sup>λ</sup>2calð Þ x, y : arb:

U<sup>λ</sup>1ð Þ¼ x; y ½fI<sup>λ</sup>1ð Þ x, y : 0 –s xð Þ ; y Þg

s xð Þ¼ ; <sup>y</sup> �

�U<sup>λ</sup>1ð Þ <sup>x</sup>; <sup>y</sup> �

�

�

�U<sup>λ</sup>1ð Þ <sup>x</sup>; <sup>y</sup> �

þUλ<sup>2</sup> �

computer,

214 Holographic Materials and Optical Systems

Thus, the object waves at the desired wavelengths are extracted selectively from four wavelength-multiplexed phase-shifted holograms and intensity distributions of the reference waves. In this way, in the case where the number of wavelengths is N, multiwavelength information can be separately extracted from 2N holograms. By applying diffraction integrals to the object waves, amplitude and phase distributions of the object on the desired depth are reconstructed at multiple wavelengths. Therefore, a 3D image and wavelength dependency of the object can be obtained simultaneously.

Figure 9. Algorithm for selectively extracting wavelength information in 2π-PDM.

Note that an arbitrary phase shift at λ<sup>2</sup> is allowable in one of the wavelength-multiplexed, phase-shifted, and monochromatic holograms I(x,y:α1,arb.) in a 2π-PDM algorithm described above. Therefore, 2π-PDM conducts asymmetric phase-shifting and belongs to partially generalized phase-shifting interferometry.
