Development of spatial and temporal phase evaluation techniques in digital holography 4

CHAPTER FOUR SIMULATIONS AND EXPERIMENTAL WORK 4.1 Experimental setup for spatial-domain analysis 4.1.1 Experimental setup Figure 4.1 shows a schematic experimental arrangement for re

CHAPTER FOUR SIMULATIONS AND EXPERIMENTAL WORK

4.1 Experimental setup for spatial-domain analysis

4.1.1 Experimental setup

Figure 4.1 shows a schematic experimental arrangement for recording digital holograms in the spatial-domain analysis As can be seen in Fig 4.1, a variable beam splitter separates a He-Ne laser beam (wavelength of 632.8 nm and the power of 30 mw) into two wave paths, i.e., reference and object waves A polarizer is placed in the reference wave path to regulate the intensity A beam splitter cube is placed in front of

a CCD camera to recombine object and reference waves, and a digital hologram is recorded by the CCD camera This CCD has an array of 1392 1040× pixels, and the size of each pixel is 4.65 µm A plate with a central point load at the rear is used as a test sample Detailed description on the equipment and specimen is presented in Section 4.1.2 As this type of the experimental setup is used, the reference wave is considered a spherical wave Hence, the pixels of the CCD can be fully used, and much sharper images can be obtained (Wagner et al., 1999) In addition, since a phase difference map (or wrapped phase map) is required, two digital holograms are recorded at the initial and deformed states of the test specimen, respectively To more clearly demonstrate the experimental layout, an experimental arrangement is shown in Fig 4.2 To avoid or reduce the influence of vibration, a vibration-isolation table is used for this experiment

Figure 4.1 A schematic experimental setup for recording digital holograms: VBS, variable beam splitter; SF, spatial filter

Figure 4.2 An experimental arrangement for recording digital holograms

z

x

y

Polarizer

Mirror

Mirror

Lens VBS

Beam splitter cube

Mirror

Mirror SF

SF

CCD

Computer Plate

CCD Specimen

Vibration-isolation

table Computer

4.1.2 Equipment and specimen

The recording equipment (i.e., CCD camera) in the spatial-domain analysis is shown

in Fig 4.3 This CCD camera has a dynamic range of 8 bit, which is comparable to a conventional optical recording medium This dynamic range can fully satisfy the measurement requirement in this study However, in digital holography, the resolved spatial frequency is still limited due to the small dimension of CCD, and the higher maximum spatial frequency can be obtained with a smaller pixel size of CCD camera

In off-axis digital holography, since the small spatial frequency limits the dimension

of the test object, a high-resolution CCD camera is highly required In this study, the CCD camera as shown in Fig 4.3 can effectively satisfy the above requirement since the resolution of the CCD camera (an array of 1392 1040× pixels and pixel size of 4.65 µm) is high In addition, the camera is a miniature, monochrome and progressive scan CCD camera, which can be applied to many research fields, such as medical imaging and pattern recognition In this static measurement, the dimension of the captured holograms is set as 1024 1024× pixels to facilitate the subsequent image processing, such as fast Fourier transform

In this experimental setup, a plate with a central point load is chosen as a test specimen shown in Fig 4.4(a) The aluminum plate is clamped at the edges, and a point loading is applied from the rear at the center. The thickness t of the plate at the

center is 1.5 mm, and the diameter D of the plate is 50 mm In the experiment, the

point loading is small in order to avoid a pattern recording with too dense fringes White lacquer is sprayed on the surface of the plate to improve the reflectivity of the test specimen The distance between the CCD and the surface of the plate in Fig 4.1

is 92.5 cm to satisfy the minimum distance requirement (dmin =2D x∆ λ, where D is

object size) in digital holography. Figure 4.4(b) shows a photograph of the test plate

Figure 4.3 The recording equipment: a high-resolution CCD camera

(a) (b)

Figure 4.4 (a) A schematic for an aluminum plate with a central point loading at the rear; (b) a photograph of the test plate

4.2 Experimental setup for temporal-domain analysis

4.2.1 Continuous deformation of a cantilevel beam

4.2.1.1 Experimental setup

Figure 4.5 shows a schematic experimental setup for the analysis of continuous deformation of a cantilever beam The basic principle of this experiment is the same

D

Micrometer head

t

as that in Fig 4.1 The wavelength of a He-Ne laser is 632.8 nm, and the laser beam is separated by a variable beam splitter into reference and object beams A microscopic lens and a pinhole consist of a spatial filter, and a polarizer is also placed in the reference wave path to regulate the intensity Another beam splitter cube is located in front of a high-speed CCD camera (KODAK Motion Corder Analyzer, SR-Ultra) to recombine object and reference beams The high-speed CCD camera with 512×480 pixels (pixel size of 7.4 µm) is employed with a sampling rate of 125 frames per second Hence, a total of 546 consecutive holograms are recorded by using the high-speed CCD camera within 4.4 s, and 380 consecutive wrapped phase maps are selected in this case study In this experiment, a cantilever beam is chosen as the specimen, and a piezoelectric transducer (PZT) is used to excite the cantilever beam Note that only continuous deformation is studied, and one of the twin images is selected for the analysis

Figure 4.5 A schematic experimental setup for recording a series of digital holograms: VBS, variable beam splitter; BSC, beam splitter cube; SF, spatial filter;

CB, cantilever beam; CCD, high-speed CCD

CB

VBS

Lens

Mirror

BSC

Mirror Mirror

SF

CCD

Computer Polarizer

PZT

4.2.1.2 Equipment and specimen

The recording rate of a normal CCD camera (as shown in Fig 4.3) is usually below

30 frames per second However, in a dynamic situation, this type of recording rate can not satisfy the Nyquist sampling rate Hence, in the temporal-domain analysis, a high speed CCD camera is essential, and in this study a high speed CCD camera (Kodak Motion Carder Analyzer, Monochrome Model SR-Ultra) is used The photo of this high speed camera is shown in Fig 4.6(a) The sensor of this high speed camera is a CCD with pixel size of 7.4 µm and an array of 658 × 496 pixels In addition, this CCD has a recording rate ranging from 30 to 10000 frames per second, and the size of

a captured digital hologram varies with the recording rate In this study, a recording rate of 125 frames per second is used, and 546 images with 512×480 pixels are recorded This high-speed CCD camera has a standard 8-bit monochrome BMP or TIFF output with 256 gray levels

Figure 4.6 (a) A photo of the high speed CCD camera; (b) the specimen:

a cantilever beam (length 21cm; width 1cm; thickness 0.25cm)

A cantilever beam is used as the specimen for the dynamic study Figure 4.6(b) shows a real image of this cantilever beam Since not whole cantilever beam can be illuminated by the laser in the experiment, an area of interest is processed by the proposed method The cropped dimension is described in Section 5.2.1

4.2.2 Multi-illumination method for object profiling

4.2.2.1 Experimental setup

For surface profiling of a specimen with height steps, a schematic experimental setup for the recording of a series of digital holograms is shown in Fig 4.7 A He-Ne laser beam (wavelength of 632.8 nm) is split into two beams by a fiber coupler One beam serves as a reference wave, and another beam illuminates the specimen as an object wave A beam splitter cube located in front of a high-speed CCD camera (KODAK Motion Corder Analyzer, SR-Ultra) with 512×480 pixels (pixel size of 7.4 µm) combines these beams, and a series of digital holograms is recorded by the high-speed CCD camera The frame rate of the high-speed CCD camera is 125 frames per second, and 546 holograms are captured In this experiment, the multi-illumination method (Quan et al., 2007) is used and realized by translating the optical fiber at a constant speed During the movement of this optical fiber end, a series of digital holograms is recorded

Figure 4.7 A schematic experimental setup for surface profiling of a specimen

1

s

Specimen with height steps

Beam splitter cube

Object beam

He-Ne laser

Computer Optical fiber

Reference beam

High-speed CCD camera

2

s

12

s

L

z

x

θ

As can be seen in Fig 4.7, the tilt between illumination points s1 and s2 can

be assumed to be perpendicular to the illumination direction, and the movement distance between the two points is denoted as s12 The value of s12 is much smaller

than the value of L (see Fig 4.7) As two illumination points are considered, the tilt

angle can be approximately calculated by ∆ ≈θ s12 L

Figure 4.8 shows a schematic arrangement for the multi-illumination method Continuous movement of the illumination point is realized by translating the optical fiber As shown in Fig 4.7, the first hologram is recorded with the fiber end at illumination point s1 After the fiber end is slightly translated to the illumination point

2

s , the second hologram is recorded by the CCD A series of holograms is recorded

by repeating the above procedure with the translation of the optical fiber

Figure 4.8 A schematic setup for the multi-illumination method

If the unit vectors sv1 and sv2 (see Fig 4.8) are replaced by a common unit

vector sv, phase difference between the first and second illuminations is expressed as

2

ps

π

ϕ

λ

∆ = r r (4.1) where ∆ϕ denotes a phase difference map at two positions, and vector pr = AP−BP

Specimen with

height steps

2

pr

Illumination point A

Observation direction

Illumination point B

N

n

s

r

4.2.2.2 Equipment and specimen

In this experiment, the multi-illumination method is used, and a series of digital holograms is recorded by moving the optical fibre end Figure 4.9(a) shows a Newport 150 mm mid-range travel steel linear stage used in this experiment The stage features a built-in tachometer to provide superior speed stability with a travel range of 150 mm Figure 4.9(b) shows a Melles Griot nanostep motor controller which is used to drive and control the linear stage Software is used to control the operation of the stepper actuators and automatically compensate the positioning errors

Figure 4.9 (a) Newport 150 mm mid-range travel steel linear stage; (b) Melles Griot nanostep motor controller

A specimen with height steps is used to verify the proposed method in digital holography, and Fig 4.10 shows a top view of the specimen and an area of interest (inside the black box) The concrete dimension is discussed in Section 5.2.2

Figure 4.10 Top view of the specimen and an area of interest

4.2.3 Detection and compensation of phase-shifting error

A schematic optical system for recording digital holograms in phase-shifting digital holography is shown in Fig 4.11 A linearly polarized laser beam (the wavelength of

635 nm) is collimated by using a spatial filter and a lens, and is then divided into two beams by using a beam splitter cube One beam illuminates the test specimen as an object wave, while another beam is considered a plane reference beam Another beam splitter cube is placed in front of a CCD camera in order to record an on-line digital hologram The image size is 512×512 pixels, and the pixel size of CCD camera

is7.4µm The distance between object plane and hologram (or CCD) plane is 26 cm When the phase-shifting operation (phase shift of π 2 using PZT) is carried out to modify the phase of the plane reference wave, another digital hologram is recorded by the CCD In this case study, two-step phase-shifting digital holography is investigated

To effectively remove zero-order term in the proposed method, two shutters are respectively located in the object and reference wave paths to capture the reference and object intensities, respectively

Figure 4.11 A schematic system for two-step phase-shifting digital holography

Lens

Mirror 2

CCD

PZT

Spatial filter

Test object

Computer

Polarizer Laser

Beam splitter cube

Shutter 1

Shutter 2 Mirror 1

Beam splitter cube

4.3 Extension of the depth of focus

The measurement of a particle field is important in many practical applications However, a problem which exists in the particle field measurement is the small depth

of focus, and micro-objects in the different recording distances can not be in focus simultaneously To verify the feasibility and effectiveness of the proposed method described in Section 3.3, a numerical experiment is carried out Figure 4.12 shows a schematic experimental setup for a particle field measurement A He-Ne laser (wavelength of 632.8 nm) is used as the light source, and is first collimated by using a combination of spatial filter and a lens Then the collimated wave illuminates the particle field Hence, the waves diffracted by the particles are considered as the object wave, while the waves which are not affected by the particles are considered as the reference wave Subsequently, the object and reference waves combine at the image plane of a CCD camera, and a digital hologram is recorded The pixel size of the CCD camera is 4.65µm, and pixel number is 512×512 In addition, the thickness of the whole particle holding volume is 12.76 mm The digital hologram is stored in the computer for post-processing (such as numerical reconstruction) which is described in Section 5.3 If digital holographic microscopy is applied, the recording distance might

be smaller (Cuche et al., 1999b)

Figure 4.12 A schematic experimental arrangement for a particle field measurement

mm 76 2

Particles

CCD camera

Computer Spatial filter

He-Ne laser

Lens

4.4 Complex-valued object reconstruction

In conventional diffraction imaging, a stronger support constraint is usually required for a complex-valued object than that for a real or non-negative object In Section 3.4,

a new simple method is described to recover the complex-valued object Here, a numerical experiment is carried out as shown in Fig 4.13 to demonstrate the feasibility and effectiveness of the proposed method There are three planes in the proposed optical system, i.e., object plane, phase mask plane and CCD plane The light source with a wavelength of 632.8 nm is used to illuminate the object, and the diffracted wave illuminates the phase mask again Subsequently, the diffraction intensity distribution is recorded by a CCD camera The pixel size of CCD camera is

µm,

65

4 and pixel number is 512×512 In addition, the distance between object and CCD camera is 55 mm, and the values of z1, z2, z and 3 z4 are 15 mm, 36 mm, 2

mm and 2 mm, respectively In this case study, a complex-valued object is tested, and the pure phase mask is a map randomly distributed in a range of 0 to 2π When the random-phase mask is translated to positions 2 and 3, another two intensity distributions are recorded by the CCD Note that the proposed method might be considered as the conventional in-line digital holography

Figure 4.13 A schematic experimental setup for recording diffraction intensities

Specimen

Phase mask (Position 1) Position 3

Position 2

1

CCD

Light