2. The principle of holography

The word holography is derived from the Greek words "holos," which means whole or entire, and graphein, which means to write. Holography is a method that records and reconstructs both irradiance of each point in an image and the direction in which the wave is propagating at that point. Holography consists of two procedures: recording as shown in Figure 1(a) and reconstruction as shown in Figure 1(b). Because of the development of digital recording process for recording and computer technology for numerical reconstruction, the optical

Figure 1. Principle of digital holography: (a) recording and (b) reconstruction with reference wave.

holography has been replaced by digital holography. The idea of numerical reconstruction was proposed by Goodman and Lawrence [1]. In 1993, Schnars and Juptner [2] used a CCD camera to record a hologram and performed numerical reconstruction in order to reconstruct this digital hologram.

#### 2.1. Numerical reconstruction in digital holography

components for reconstruction has given advantages of digital holography over conventional optical holography. DH enables the extraction of both amplitude and phase information of objects in real time with high resolution. DH technique is less sensitive to external perturbations and has long-term stability in object measurement. DH has merits of wide applications covering particles, living cells, and 3D profiling and tracking of micro-objects or nano-objects. In this chapter, we present recent developments in DH techniques carried out by the author. In Section 2, the principle of holography with highlights on three numerical reconstruction methods, namely, Fresnel approximation, convolution approach, and angular spectrum, is explained. In Section 3, the impact of slightly imperfect collimation of the reference wave in off-axis DH is presented. In Section 4, low-coherence, off-axis digital holographic microscope (DHM) by the use of femtosecond pulse light for measuring fine structure of in vitro sliced sandwiched biological sarcomere sample is described. In Section 5, off-axis terahertz DH using continuous-wave radiation generated from cascade laser source for testing a letter T from paper is

The word holography is derived from the Greek words "holos," which means whole or entire, and graphein, which means to write. Holography is a method that records and reconstructs both irradiance of each point in an image and the direction in which the wave is propagating at that point. Holography consists of two procedures: recording as shown in Figure 1(a) and reconstruction as shown in Figure 1(b). Because of the development of digital recording process for recording and computer technology for numerical reconstruction, the optical

Figure 1. Principle of digital holography: (a) recording and (b) reconstruction with reference wave.

described. Section 6 gives concluding discussions and remarks.

2. The principle of holography

272 Holographic Materials and Optical Systems

In digital holography, the CCD or Complementary Metal-Oxide Semiconductor (CMOS) camera captures the image and transfers it to the computer. This image is saved digitally as a digital hologram. This hologram is digitally accessed and numerically reconstructed by a virtual reference wave, which effectively simulates the reference wave used in the process of recording. The speed of reconstruction procedure depends on the implementation of the numerical reconstruction algorithm and the speed of the computer processing. Because the reference wave has to be generated virtually in the computer, a plane wave or a spherical wave is usually used in the recording process. Figure 1 shows the typical setup of digital holography and the reconstruction with virtual reference wave. Let us consider the coordinate system as shown in Figure 2; then the diffraction by the aperture or hologram in the distance of d along the propagation direction of the wave can be quantitatively described by Fresnel-Kirchhoff integral [3].

If the reference wave is set up to be nominally normal incident to the hologram, then the diffracted light is approximated by the Fresnel-Kirchhoff integral as

$$\Gamma(\eta',\xi') = \frac{i}{\lambda} \int \int \mathcal{U}\_h(\mathbf{x},y)\mathcal{U}\_r(\mathbf{x},y) \frac{\exp\left(-i\frac{2\pi}{\lambda}\rho'\right)}{\rho'}d\mathbf{x}dy\tag{1}$$

where Γ(η′, ξ′) is the diffraction pattern, Uh(x, y) is the digital hologram captured by the CCD camera, λ is the wavelength of the light in the virtual reference beam used in the reconstruction, ρ′ is the distance between a point in the hologram plane and a point in the reconstruction plane, which has the form <sup>ρ</sup>′ <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>x</sup>−ξ′ � �<sup>2</sup> <sup>þ</sup> <sup>y</sup>−η′ ð Þ<sup>2</sup> <sup>þ</sup> <sup>d</sup><sup>2</sup> q , and Ur(x, y) is the function describing the reference wave. In Eq. (1), Γ(η´, ξ´) is the diffraction pattern calculated at a distance d behind the CCD plane (see Figure 2), which means it reconstructs the complex amplitude in the plane of the real image. Therefore, both the intensity and the

Figure 2. Coordinate system for numerical hologram reconstruction.

phase information can be obtained after numerical reconstruction. The reconstructed intensity is written as

$$I(\boldsymbol{\eta'}, \boldsymbol{\xi'}) = \left| \boldsymbol{\Gamma}(\boldsymbol{\eta'}, \boldsymbol{\xi'}) \right|^2 \tag{2}$$

And the reconstructed phase is

$$\varphi(\eta',\xi') = \arctan \frac{\text{Im}\left[\Gamma\left(\eta',\xi'\right)\right]}{\text{Re}\left[\Gamma\left(\eta',\xi'\right)\right]} \tag{3}$$

where Re denotes the real part and Im denotes the imaginary part. The calculated diffraction pattern is the complex amplitude at a distance d behind the CCD plane where the real image is reconstructed. However, the real image could be distorted by the reference wave. To avoid this effect and ensure that an undistorted real image is left, a conjugate reference wave has to be introduced in the reconstruction. Then the calculated diffraction pattern is rewritten as

$$I(\eta,\xi) = \frac{i}{\lambda} \int \int \mathcal{U}\_{\hbar}(\mathbf{x}, y) \mathcal{U}\_{r}^{\*}(\mathbf{x}, y) \frac{\exp\left(-i\frac{2\pi}{\lambda}\rho\right)}{\rho} d\mathbf{x} dy \tag{4}$$

with

$$
\rho = \sqrt{\left(\mathbf{x} - \boldsymbol{\xi}\right)^2 + \left(\mathbf{y} - \boldsymbol{\eta}\right)^2 + d^2} \tag{5}
$$

where U<sup>∗</sup> <sup>r</sup> ð Þ xy is conjugate to the original reference wave Ur(x, y). Both results from Eq. (1) and Eq. (4) are equivalent because Urð Þ¼ xy <sup>U</sup><sup>∗</sup> <sup>r</sup> ð Þ xy . As illustrated above, Eq. (4) is the key formula of digital holography, and it is essential to calculate it numerically to perform numerical reconstruction of a digital hologram. The direct approach of Eq. (4) is not feasible in terms of the calculation complexity and computer run time. Some approximations have to be applied in order to calculate the double integral to make the numerical reconstruction effective and efficient. According to the approximation used in the algorithm, the numerical reconstruction can be classified into three types: Fresnel approximation, convolution approaches, and angular spectrum.

#### 2.1.1. Reconstruction by the Fresnel approximation

In digital holography, the values of the coordinates x and y as well as ξ and η are very small compared to the distance d between the reconstruction plane and the CCD or CMOS device. If now the right hand side of Eq. (5) is expanded to a Taylor series and the fourth term is smaller than the wavelength,

$$\frac{\left[\left(\mathbf{x} - \boldsymbol{\xi}\right)^{2} + \left(\mathbf{y} - \boldsymbol{\eta}\right)^{2}\right]^{2}}{8d^{3}} << \lambda\tag{6}$$

the effect of it and the terms after it are negligible, and they can be removed. Thus the distance ρ can be approximated as

$$\rho = d + \frac{\left(\xi - \mathbf{x}\right)^2}{2d} + \frac{\left(\eta - y\right)^2}{2d} \tag{7}$$

Replacing the denominator in Eq. (4) with d and inserting Eq. (7) into it, the following expression results in the reconstruction of the real image:

$$\begin{split} I(\eta,\xi) &= \frac{i}{\lambda d} \exp\left(-i\frac{2\pi}{\lambda}d\right) \exp\left(-i\frac{\pi}{\lambda d}\left(\xi^2 + \eta^2\right)\right) \\ &\times \int\limits\_{-\kappa \to \infty} \int \mathcal{U}\_r^\*(\mathbf{x}, y) \mathcal{U}\_\hbar(\mathbf{x}, y) \exp\left(-i\frac{\pi}{\lambda d}\left(\mathbf{x}^2 + y^2\right)\right) \exp\left(-i\frac{2\pi}{\lambda d}(\mathbf{x}\xi + y\eta)\right) d\mathbf{x} dy \end{split} \tag{8}$$

This equation is known as the Fresnel approximation or Fresnel transformation due to its mathematical similarity with the Fourier transform.

The intensity is calculated by squaring

$$I(\eta,\xi) = |\Gamma(\eta,\xi)|^2\tag{9}$$

And the phase is calculated by

phase information can be obtained after numerical reconstruction. The reconstructed inten-

where Re denotes the real part and Im denotes the imaginary part. The calculated diffraction pattern is the complex amplitude at a distance d behind the CCD plane where the real image is reconstructed. However, the real image could be distorted by the reference wave. To avoid this effect and ensure that an undistorted real image is left, a conjugate reference wave has to be

; <sup>ξ</sup>′ � � � � � �

> Im Γ η′ ; ξ′ � � � �

Re Γ η′

<sup>r</sup> ð Þ <sup>x</sup>; <sup>y</sup> exp <sup>−</sup><sup>i</sup> <sup>2</sup><sup>π</sup>

<sup>2</sup> <sup>þ</sup> <sup>d</sup><sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>x</sup>−<sup>ξ</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>y</sup>−<sup>η</sup>

<sup>r</sup> ð Þ xy is conjugate to the original reference wave Ur(x, y). Both results from Eq. (1) and

of digital holography, and it is essential to calculate it numerically to perform numerical reconstruction of a digital hologram. The direct approach of Eq. (4) is not feasible in terms of the calculation complexity and computer run time. Some approximations have to be applied in order to calculate the double integral to make the numerical reconstruction effective and efficient. According to the approximation used in the algorithm, the numerical reconstruction can be classified into three types: Fresnel approximation, convolution approaches, and angular

In digital holography, the values of the coordinates x and y as well as ξ and η are very small compared to the distance d between the reconstruction plane and the CCD or CMOS device. If now the right hand side of Eq. (5) is expanded to a Taylor series and the fourth term is smaller

> ð Þ <sup>x</sup>−<sup>ξ</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>y</sup>−<sup>η</sup> <sup>2</sup> h i<sup>2</sup>

<sup>λ</sup> <sup>ρ</sup> � �

<sup>r</sup> ð Þ xy . As illustrated above, Eq. (4) is the key formula

<sup>8</sup>d<sup>3</sup> << <sup>λ</sup> (6)

<sup>2</sup> (2)

; <sup>ξ</sup>′ � � � � (3)

<sup>ρ</sup> dxdy (4)

(5)

; <sup>ξ</sup>′ � � <sup>¼</sup> Γ η′

I η′

; <sup>ξ</sup>′ � � <sup>¼</sup> arctan

introduced in the reconstruction. Then the calculated diffraction pattern is rewritten as

Uhð Þ <sup>x</sup>; <sup>y</sup> <sup>U</sup><sup>∗</sup>

ϕ η′

Γ ηð Þ¼ ; <sup>ξ</sup> <sup>i</sup>

Eq. (4) are equivalent because Urð Þ¼ xy <sup>U</sup><sup>∗</sup>

2.1.1. Reconstruction by the Fresnel approximation

λ −∞ð

−∞

ρ ¼

−∞ð

−∞

q

sity is written as

with

where U<sup>∗</sup>

spectrum.

than the wavelength,

And the reconstructed phase is

274 Holographic Materials and Optical Systems

$$\varphi(\eta,\xi) = \arctan \frac{\text{Im}[\varGamma(\eta,\xi)]}{\text{Re}[\varGamma(\eta,\xi)]} \tag{10}$$

To convert the discrete Fresnel transformation in Eq. (8) to a digital implementation, two substitutions are applied [4]:

$$
\nu = \frac{\xi}{\lambda d}; \quad \mu = \frac{\eta}{\lambda d}; \tag{11}
$$

Therefore Eq. (8) turns into

$$\begin{split} \Gamma(\nu,\mu) &= \frac{i}{\lambda d} \exp\left( -i \frac{2\pi}{\lambda} d \right) \exp\left( -i\pi \lambda d \left( \nu^2 + \mu^2 \right) \right) \\ &\times \mathfrak{F}^{-1} \left\{ \mathcal{U}\_r^\*(\mathbf{x}, y) \mathcal{U}\_h(\mathbf{x}, y) \exp\left( -i \frac{\pi}{\lambda d} \left( \mathbf{x}^2 + y^2 \right) \right) \right\} \end{split} \tag{12}$$

Because the maximum frequency is determined by the sampling interval in the spatial domain according to the theory of the Fourier transform, the relationships among Δx, Δy, Δν, and Δμ are

$$
\Delta \nu = \frac{1}{N \Delta x}; \quad \Delta \mu = \frac{1}{N \Delta y} \tag{13}
$$

With Eq. (13), Eq. (12) can be rewritten as

$$\begin{split} \Gamma(m,n) &= \frac{i}{\lambda d} \exp\left(-i\frac{2\pi}{\lambda}d\right) \exp\left(-i\pi\lambda d \left(\frac{m^2}{N^2\Delta x^2} + \frac{n^2}{N^2\Delta y^2}\right)\right) \\ &\times \mathfrak{F}^{-1}\left\{\mathcal{U}\_r^\*(k,l)\mathcal{U}\_l(k,l)\exp\left(-i\frac{\pi}{\lambda d}\left(k^2\Delta x^2 + l^2\Delta y^2\right)\right)\right\} \end{split} \tag{14}$$

Eq. (14) is known as the discrete Fresnel transform. The matrix Γ is calculated by applying an inverse discrete Fourier transform to the product of U<sup>∗</sup> <sup>r</sup> ð Þ k; l with Uh(k, l) and exp(−iπ(k 2 Δx<sup>2</sup> + l 2 Δy<sup>2</sup> )/(λd)). The calculation can be done very effectively using the fast Fourier transform (FFT) algorithm.

#### 2.1.2. Reconstruction by the convolution approach

This method makes use of the convolution theorem. This approach was introduced to digital holography by Kreis and Juptner [5]. Eq. (4) can be interpreted as a superposition integral:

$$I(\eta,\xi) = \int\_{-\infty}^{-\infty} \int \mathcal{U}\_h(\mathbf{x}, y) \mathcal{U}\_r^\*(\mathbf{x}, y) \mathbf{g}(\eta, \xi, \mathbf{x}, y) d\mathbf{x} dy \tag{15}$$

where the impulse response g(η, ξ, x, y) is given by

$$g(\eta, \xi, x, y) = \frac{i}{\lambda} \frac{\exp\left[-i\frac{2\pi}{\lambda}\rho\right]}{\rho} \tag{16}$$

Eq. (15) can be regarded as a convolution and the convolution theorem can be applied. The convolution theorem states that the Fourier transform of the convolution of two functions is equal to the product of the Fourier transforms of the individual functions. In other words, the convolution of two functions in the spatial domain can be easily obtained through the multiplication of them in another domain, namely, spatial frequency domain.

Applying the convolution theorem to Eq. (4), it is converted to

$$\Gamma(\eta,\xi) = \mathfrak{S}^{-1}\left\{\mathfrak{S}\left(\mathcal{U}\_{\hbar}(\mathbf{x},y).\mathcal{U}\_{r}^{\*}(\mathbf{x},y)\right).\mathfrak{S}(\mathbf{g}(\mathbf{x},y))\right\}\tag{17}$$

Eq. (17) includes two forward Fourier transformations and one inverse Fourier transformation, all of which can be practically implemented via the FFT algorithm. Both Fresnel and convolution methods are practically implemented via the FFT algorithm. In the Fresnel approximation, only a forward FFT is performed. However, two or three FFTs are performed in the convolution approach. In the convolution approach, the pixel sizes in the reconstructed image are equal to that of the hologram. It would seem that a higher resolution could be achieved if a CCD or CMOS device with a smaller pixel size was used in the recording process. For applications that detect very small objects, the convolution approach has more advantages and is more accurate than the Fresnel approximation algorithm.

#### 2.1.3. Reconstruction by the angular spectrum method

With Eq. (13), Eq. (12) can be rewritten as

i λd

2.1.2. Reconstruction by the convolution approach

Γ ηð Þ¼ ; ξ

where the impulse response g(η, ξ, x, y) is given by

exp −i

2π λ d � �

· ℑ<sup>−</sup><sup>1</sup> U<sup>∗</sup>

an inverse discrete Fourier transform to the product of U<sup>∗</sup>

−∞ð

−∞ð

−∞

gð Þ¼ η; ξ; x; y

−∞

plication of them in another domain, namely, spatial frequency domain.

Γ ηð Þ¼ ; <sup>ξ</sup> <sup>ℑ</sup><sup>−</sup><sup>1</sup> <sup>ℑ</sup> Uhð Þ <sup>x</sup>; <sup>y</sup> :U<sup>∗</sup>

Applying the convolution theorem to Eq. (4), it is converted to

and is more accurate than the Fresnel approximation algorithm.

exp <sup>−</sup>iπλ<sup>d</sup> <sup>m</sup><sup>2</sup>

<sup>r</sup> ð Þ k; l Uhð Þ k; l exp −i

Eq. (14) is known as the discrete Fresnel transform. The matrix Γ is calculated by applying

This method makes use of the convolution theorem. This approach was introduced to digital holography by Kreis and Juptner [5]. Eq. (4) can be interpreted as a superposition integral:

> i λ

Eq. (15) can be regarded as a convolution and the convolution theorem can be applied. The convolution theorem states that the Fourier transform of the convolution of two functions is equal to the product of the Fourier transforms of the individual functions. In other words, the convolution of two functions in the spatial domain can be easily obtained through the multi-

Eq. (17) includes two forward Fourier transformations and one inverse Fourier transformation, all of which can be practically implemented via the FFT algorithm. Both Fresnel and convolution methods are practically implemented via the FFT algorithm. In the Fresnel approximation, only a forward FFT is performed. However, two or three FFTs are performed in the convolution approach. In the convolution approach, the pixel sizes in the reconstructed image are equal to that of the hologram. It would seem that a higher resolution could be achieved if a CCD or CMOS device with a smaller pixel size was used in the recording process. For applications that detect very small objects, the convolution approach has more advantages

exp −i <sup>2</sup><sup>π</sup> <sup>λ</sup> <sup>ρ</sup> � �

Uhð Þ <sup>x</sup>; <sup>y</sup> <sup>U</sup><sup>∗</sup>

N2 <sup>Δ</sup>x<sup>2</sup> <sup>þ</sup>

! !

π <sup>λ</sup><sup>d</sup> <sup>k</sup><sup>2</sup>

)/(λd)). The calculation can be done very effectively using the fast Fourier

<sup>Δ</sup>y<sup>2</sup> � � n o � �

n2 N2 Δy<sup>2</sup>

> <sup>Δ</sup>x<sup>2</sup> <sup>þ</sup> <sup>l</sup> 2

<sup>r</sup> ð Þ x; y gð Þ η; ξ; x; y dxdy (15)

<sup>ρ</sup> (16)

<sup>r</sup> ð Þ <sup>x</sup>; <sup>y</sup> � �:ℑð Þ g xð Þ ; <sup>y</sup> � � (17)

(14)

<sup>r</sup> ð Þ k; l with Uh(k, l) and

Γð Þ¼ m; n

276 Holographic Materials and Optical Systems

exp(−iπ(k

2 Δx<sup>2</sup> + l 2 Δy<sup>2</sup>

transform (FFT) algorithm.

Both the Fresnel approximation and the convolution approach suffer the same limitation, i.e., that the object under observation must be placed farther away than some minimum distance. If it is placed inside this distance, the spatial frequency of the detector is too low and aliasing occurs. This minimum distance is given [6]:

$$d\_{\rm min} = \frac{\left(\text{N}\Delta x\right)^2}{N\lambda} \tag{18}$$

where N and Δx are the number and the size of the pixels. However, the angular spectrum method [7] is able to overcome this disadvantage. It is comparable with the other methods in terms of computational efficiency but has the potential of higher accuracy. If the wave field at the plane d = 0 is U0(x,y;0), the angular spectrum A(kx,ky;0) at this plane is obtained by taking the Fourier transform:

$$A\left(k\_x, k\_y; 0\right) = \int \int U\_0(\mathbf{x}, y; 0) \exp\{-i\left(k\_x \mathbf{x} + k\_y y\right)\} dx dy\tag{19}$$

where kx and ky are the corresponding spatial frequencies of x and y. The angular spectrum at the distance d, i.e., A(kx,ky;d), is calculated from A(kx,ky;0) as given by

$$A\left(k\_{\mathbf{x}}, k\_{\mathbf{y}}; d\right) = A\left(k\_{\mathbf{x}}, k\_{\mathbf{y}}; 0\right) \exp(i k\_{z} d) \tag{20}$$

where kz ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 −k<sup>2</sup> <sup>x</sup>−k 2 y q . The reconstructed complex wave field at any plane perpendicular to the propagating z axis is found by

$$\mathcal{U}I(\boldsymbol{\xi},\boldsymbol{\eta};d) = \int \int A(k\_x, k\_y; d) \exp\left[i(k\_x \boldsymbol{\xi} + k\_y \boldsymbol{\eta})\right] dk\_x dk\_y = \mathfrak{S}^{-1}\left[\mathfrak{S}(Ll\_0) \exp(ik\_z d)\right] \tag{21}$$

The resolution of the reconstructed images from the angular spectrum method is the same as that in the hologram plane, which means that the pixel size does not vary with changes of wavelength or reconstruction distance.

#### 3. Impact of imperfect collimation of the reference wave in off-axis DH

In digital holography, the simulation of the plane wave is essential for performing a numerical reconstruction. Conventionally, a shear interferometer is used for producing perfect collimation and hence producing a perfect plane wave. In Section 3, we show that using a slightly imperfect plane wave in digital holography experiments is acceptable [8]. We experimentally proved that by using the Mickelson interferometer, no influence of imperfect collimation of the reference wave in an off-axis digital holography exists, as has been previously claimed. We applied perfect and imperfect collimations to three different surface (flat, spherical, and step height) shapes for height inspection, and the results were almost in good agreement. The samples being tested were mounted in the Michelson interferometer one by one, as shown in

Figure 3. Schematic diagram of the Michelson interferometer.

Figure 3. A laser diode beam passed through a collimating lens of focal lens 100 mm and expanded. The beam splitter splits the collimated beam into two equal beams: one for reference (optical flat of λ/20 flatness) arm and the other for the object (flat, spherical, and step height surfaces) arm. The beam reflected from the reference and the object is recombined at the beam splitter to produce an interference pattern. The perfect collimation that produced a perfect plane wave impinging on the reference was adjusted by a shear interferometer placed between the collimating lens and the beam splitter.

The slightly imperfect collimations were seven equidistance displacements of the collimating lens with 1.5 mm from the perfect collimation and between two successive displacements. The off-axis holograms for the samples under test at perfect and slightly imperfect plane waves were captured and then reconstructed. Details of the reconstruction process are explained in reference [8]. The reconstructed phase for perfect Figure 4(a) and slightly imperfect collimations Figure 4(2–8) and the height line profile for both along the central x-axis are shown in Figure 4(1(b)–1(c)), respectively.

The wrapping phase for the spherical surface for perfect collimations and slightly imperfect collimations is shown in Figure 5(a) and (2–8), respectively. The unwrapping phases of Figure 5b(1–8) are shown in Figure 5c(1–8), and the height line profiles along the central xaxis of Figure 5b(1–8) are shown in Figure 5d(1–8), respectively.

For the third object (step height surface), the off-axis holograms at the perfect and slightly imperfect collimations were captured and reconstructed. The height line profiles along the central x-axis of the reconstructed heights at perfect and slightly imperfect collimations at imaging and Fresnel transform are shown in Figure 6a and b, respectively.

As seen from Figures 4–6, for flat, spherical, and step surfaces, the measured height values of the three tested surfaces were almost consistent at perfect and imperfect collimations. Very small variations may be observed due to noise, which is commonly observed in interferometry measurements. We claim that the variations may be due to the mechanical imperfection of the collimating lens as shown in Figure 7. We claim that when the lens mounting was ideally

Figure 4. (a) Off-axis interferograms. (b) Reconstructed phases of (a). (c) Line height profiles along the central x-axis of the tested flat surface produced at (1) perfect collimation and (2–8) slightly imperfect collimations.

Figure 3. A laser diode beam passed through a collimating lens of focal lens 100 mm and expanded. The beam splitter splits the collimated beam into two equal beams: one for reference (optical flat of λ/20 flatness) arm and the other for the object (flat, spherical, and step height surfaces) arm. The beam reflected from the reference and the object is recombined at the beam splitter to produce an interference pattern. The perfect collimation that produced a perfect plane wave impinging on the reference was adjusted by a shear interferometer placed

The slightly imperfect collimations were seven equidistance displacements of the collimating lens with 1.5 mm from the perfect collimation and between two successive displacements. The off-axis holograms for the samples under test at perfect and slightly imperfect plane waves were captured and then reconstructed. Details of the reconstruction process are explained in reference [8]. The reconstructed phase for perfect Figure 4(a) and slightly imperfect collimations Figure 4(2–8) and the height line profile for both along the central x-axis are shown in

The wrapping phase for the spherical surface for perfect collimations and slightly imperfect collimations is shown in Figure 5(a) and (2–8), respectively. The unwrapping phases of Figure 5b(1–8) are shown in Figure 5c(1–8), and the height line profiles along the central x-

For the third object (step height surface), the off-axis holograms at the perfect and slightly imperfect collimations were captured and reconstructed. The height line profiles along the central x-axis of the reconstructed heights at perfect and slightly imperfect collimations at

As seen from Figures 4–6, for flat, spherical, and step surfaces, the measured height values of the three tested surfaces were almost consistent at perfect and imperfect collimations. Very small variations may be observed due to noise, which is commonly observed in interferometry measurements. We claim that the variations may be due to the mechanical imperfection of the collimating lens as shown in Figure 7. We claim that when the lens mounting was ideally

between the collimating lens and the beam splitter.

Figure 3. Schematic diagram of the Michelson interferometer.

278 Holographic Materials and Optical Systems

axis of Figure 5b(1–8) are shown in Figure 5d(1–8), respectively.

imaging and Fresnel transform are shown in Figure 6a and b, respectively.

Figure 4(1(b)–1(c)), respectively.

Figure 5. (a) Off-axis interferograms. (b) Reconstructed phases of (a). (c) Unwrapping phases of (b). (d) Line height profiles along the central x-axis of the tested spherical surface produced at (1) perfect collimation and (2–8) imperfect collimations.

Figure 6. Line height profiles of the tested step surface at perfect (1) and imperfect (2–8) collimations produced at (a) d = 0.0 mm (imaging scheme) and (b) d = − 500.0 mm (Fresnel transform).

Figure 7. The effect of design of mounting and the adjustment of the collimation lens on the very small height variations: (a) ideal mounting, (b) nonideal mounting.

displaced as shown in Figure 7(a) at position 2 (slightly imperfect collimation), the beams converge toward the reference and the object. The convergent beams would then be canceled out and subsequently have no impact on the height variations. However, it is hard to achieve ideal mounting mechanically. Thus we expect that the effect of nonideal mounting as shown in Figure 7(b) may be the reason of small height variations as shown in Figure 7(b).
