Development of spatial and temporal phase evaluation techniques in digital holography 2

Since holographic technique uses a coherent light source to generate an interference pattern, the speckle phenomenon also exists in holography.. 2.2 Optical holography 2.2.1 Hologram re

CHAPTER TWO

LITERATURE REVIEW

2.1 Basic principles of holography

2.1.1 Wave theory of light

Light is an electromagnetic radiation, particularly radiation of a wavelength that is visible to the human eye (about 400 nm – 700 nm) In the field of physics, the term light usually refers to an electromagnetic radiation of any wavelength Light that exists in tiny packets called photons can exhibit properties of both waves and particles This property is referred to as the wave–particle duality In addition, light waves can

be described either by the electrical or by the magnetic field in many applications There are four primary properties of light wave, i.e., intensity, frequency (or wavelength), polarization and phase The study of light (also known as optics) is an important research area in modern physics and various application fields

Diffraction and interference are perfectly described by the wave model, which

is based on the theory of classical electromagnetism Interference and diffraction also form the basis of holographic technique Since electromagnetic waves obey the Maxwell equations (Kreis, 2005; Schnars and Jueptner, 2005), the propagation of a light wave in vacuum can be described by

where E is the electric field strength, ∇2 denotes Laplace operator described by

( , , )x y z denotes the spatial coordinate, t denotes the temporal coordinate, and c is the

propagation speed of the light wave in vacuum (3.0 10 m s× 8 )

The electrical field E is a vector quantity, and can vibrate in any direction

perpendicular to the light wave propagation However, in most real applications, it is not essential to consider the full vector quantity, and vibration in a single plane is usually assumed In this case, the light is called linear polarized light The above scalar wave equation can be rewritten as (Kreis, 2005; Schnars and Jueptner, 2005)

where the propagation of the light is in the z direction

For a linearly polarized and harmonic plane wave, the important solution of Eq (2.3) is described by

The expression of the complex exponential can be written by

0

E x y z t = A j π ft−k rr r+ϕ  (2.5)

where j= −1 In practice, only the real part of this complex exponential represents

the physical wave, and 2 ftπ can be ignored since the spatial part of the electrical field is of the most interest in most cases

The wavelengths of visible light are in the range of 400 nm to 700 nm, and the range of the light frequency is from 4.3 10× 14 Hz to 7.5 10× 14 Hz Hence, commonly-used sensors, such as photodiodes, photographic films and CCD, are not able to detect such high frequencies The only measurable quantity is the intensity, which is defined

by the energy flux through an area per time The intensity distribution I for a plane

wave can be described by (Schnars and Jueptner, 2005)

12

t

I =ε c E = ε c A (2.6)

where t is the time average over the light periods, and ε0 is the vacuum permittivity In many applications, the factor ε0c 2 can be ignored For simplicity, the coordinate in Eq (2.6) is omitted

2.1.2 Interference

Interference is the superposition of two or more waves, which can result in a new wave pattern In holography, interference usually refers to the interaction of monochromatic waves that are correlated or coherent with each other since they may come from the same source or have the same frequencies and wavelengths (Schnars

and Jueptner, 2005) In this study, two monochromatic waves are considered, and the complex amplitudes for these two waves are described by

where the asterisk denotes complex conjugate

It can be seen from Eq (2.9) that a constructive interference is formed when the value of (ϕ ϕ1− 2) is equal to 2nπ (n=0,1, 2 ) Similarly, a destructive

(2n+1)π (n=0,1, 2 ) When the constructive interference is generated, the wavefronts can be considered to be in phase; when the destructive interference is generated, the wavefronts can be considered to be out of phase In digital holography, after the intensity distribution captured by the CCD is multiplied by a numerical reference wave, the first two terms ( 2 2)

E + E on the right-hand side of Eq (2.9)

form zero-order term of the diffraction, and the third (E E ) and fourth terms (1* 2 E E ) 1 2*

form real and virtual images, respectively It is assumed that the light wave E 2

represents the reference wave in digital holography

The visibility or contrast of the interference pattern is defined by

The fringe spacing of the recorded interference pattern can be defined by the distance between two neighboring maxima points P1 and P2 as shown in Fig 2.1 The fringe spacing is described by (Schnars and Jueptner, 2005)

Figure 2.1 Interference between two plane waves E1 and E2

2.1.3 Spatial and temporal coherence

It is well known that interference phenomenon is rarely observed under natural sunlight or lamplight This is mainly due to the lack of sufficient coherence with this type of illumination light Coherence is the measure of the ability of light to interfere The coherence of two waves follows from how well correlated the individual waves are and is derived from the phase relationship between two points which are separated

in either space or time along the wave train For instance, we consider two points along the train that are spatially separated and moving with the train If the phase relationship between the waves at these points remains constant in time, the waves between these points are coherent On the other hand, if the phase relationship is random or rapidly changing, the waves at these two points are incoherent Two aspects of the general coherence are the spatial and temporal coherence

Spatial coherence describes the mutual correlation of different parts of the same wavefront and can be physically explained using Young’s double aperture interferometer experiment arrangement (Kreis, 2005; Schnars and Jueptner, 2005) as shown in Fig 2.2 In the Young’s double aperture interferometry experimental arrangement, an aperture with two transparent holes is mounted between the light source and a screen, and the two holes are separated with a certain distance It was demonstrated in the experiment that only with the distance below the critical limit a , cl

the interference pattern could be observed In addition, the interference fringes also vanish with the decrease of the distance between the light source and the aperture Hence, it can be concluded that the spatial coherence is not related to the spectral width of the light source, but depends on the properties of the light source and the geometry of the interferometer

Figure 2.2 Experimental arrangement for Young’s double aperture interferometer

Temporal coherence describes the correlation of a wave with itself at different instants and is related to the finite bandwidth of the light source The temporal coherence length L is the greatest distance between two points for which there is a

phase difference that still remains constant in time When the points are separated by

a certain distance greater than the temporal coherence length, there is no phase correlation and the interference fringes vanish Typical coherence lengths of commonly-used lasers in digital holography have temporal coherence lengths from a few millimeters to centimeters The temporal-coherence property of a light source can

be investigated using Michelson-interferometer experimental arrangement as shown

in Fig 2.3 As can be seen in Fig 2.3, through translating mirror 2, we can adjust the optical path difference between the two-wave paths Interference fringes can be observed only when the optical path difference is below the temporal coherence length L of the light source

Figure 2.3 Michelson interferometer arrangement

Figure 2.4 Diffraction based on an opaque screen with a transparent hole

With a simple assumption about the amplitude and phase of the secondary waves, Huygens’s principle is able to accurately determine the light distribution of diffraction patterns (Goodman, 1996) Huygens principle was mathematically described by Fresnel who considered the approximation methods and Kirchhoff who figured out all the correct multiplying terms Subsequently, some problems inherent in diffraction principle were solved by Fresnel and Fraunhofer Recently, several effective diffraction theories have been widely applied, such as Kirchhoff theory, first Rayleigh-Sommerfeld solution and second Rayleigh-Sommerfeld solution

The Huygens-Fresnel principle which is predicted by the first Sommerfeld solution can be mathematically described by (Goodman, 1996)

the observation place is a superposition of the diverging spherical waves

[exp(jkρHR) ρHR] originating from the secondary waves located at the point H within the aperture Σ

2.1.5 Speckles

When a surface is illuminated by a light wave, each point on an illuminated surface acts as a source of secondary spherical waves according to the diffraction theory The light at any other places is made up of waves which are scattered from each point of the illuminated surface If the surface is rough enough to create path-length differences exceeding one wavelength, the intensity of the resultant light will vary randomly, which is called speckles A typical speckle pattern is shown in Fig 2.5 However, if the light of low coherence (for instance, with multiple wavelengths) is applied, a speckle pattern is rarely observed The reason is that the speckle pattern produced by individual wavelengths has different dimensions which average one another (Kreis, 2005)

There are two main types of speckle patterns according to the experimental arrangement, i.e., objective and subjective speckle patterns When a laser light that is scattered by a rough surface directly develops on a screen without any intermediate optical imaging optics or system, an objective speckle pattern is formed When the illuminated surface is focused with an imaging optics or system, a subjective speckle

pattern is formed In objective or subjective speckle pattern, speckle size d is

calculated by d =( λz a), where z denotes the distance between the object or the imaging optics and the screen, and a denotes the dimension of the object or the

aperture of an imaging optics

Figure 2.5 A typical speckle pattern

Since holographic technique uses a coherent light source to generate an interference pattern, the speckle phenomenon also exists in holography However, the coherent noise of speckles can disturb the image quality to some degrees, and make the identification of features in the scattering structures highly difficult This has been considered as a major barrier for the widespread application of coherent imaging techniques, such as holography

2.2 Optical holography

2.2.1 Hologram recording

2.2.1.1 In-line optical holographic arrangement

In holography, the first step is to record a hologram based on a preset optical arrangement Different optical holographic arrangements can meet different requirements and also need different processing methods Figure 2.6 shows a typical in-line optical holographic experimental arrangement The in-line optical holographic arrangement was first proposed by Dr Gabor (1948) In Fig 2.6, a light source (such

as a He-Ne laser) with a sufficient coherence length is first split into two waves, i.e., reference and object waves The angle between object and reference waves is small or close to zero (Xu et al., 2002, 2003; Sucerquia et al., 2006a, 2006b) A test object is illuminated, and the diffracted wavefront from the object (called object wave) then propagates to a recording medium The object and reference waves combine in front

of the recording medium by a beam splitter cube, and interfere on the surface of the recording medium The recorded interference pattern is called hologram To ensure the stability of a recorded intensity distribution, phase difference between the above two waves must be stationary In the optical holographic setup, a photographic film is usually used as the recording medium

Figure 2.6 A typical in-line optical holographic experimental arrangement

Complex amplitude of the object wave in the recording plane is described by

Reference wave

where A O( )x y, and ϕO( )x y, denote real amplitude and phase of the object wave The complex amplitude of the reference wave in the recording plane can be described by

E x y = A x y jϕ x y  (2.14)

where A R( )x y, and ϕR( )x y, denote real amplitude and phase of the reference wave

In an in-line optical holographic setup, a plane reference wave is commonly used, so the phase part of Eq (2.14) may be ignored

The intensity I x y( ), which is formed by using the interference principle is recorded by the recording medium, and can be expressed by

2.2.1.2 Off-axis optical holographic arrangement

In the conventional in-line optical holographic setup, the angle between the object and reference waves is small or close to zero This setup induces some problems during holographic reconstruction, such as a superposition of different reconstructed terms mentioned in Section 2.2.2 Leith and Upatnieks (1962, 1963) proposed to apply an off-axis experimental setup to overcome the problems inherent in the on-line holographic setup An appropriate angle between object and reference waves is introduced in order to solve the problem existing in the in-line holographic setup Figure 2.7 shows an off-axis optical holographic experimental arrangement (Schnars

and Jueptner, 2005) The basic principles of wave propagation and interference are the same to those in the in-line optical holographic setup However, in the off-axis optical holographic setup, the angle between object and reference waves is relatively large The introduction of this angle results in a spatial carrier (Takeda et al., 1982), which can greatly facilitate the subsequent analyses

Figure 2.7 An off-axis optical holographic experimental arrangement

2.2.2 Optical reconstruction

To reconstruct the image recorded by the photographic film, the same reference wave used in the recording step should be applied again to illuminate the hologram Figure 2.8 shows an optical reconstruction of the object wave based on the off-axis optical holographic setup (Kreis, 2005) Similarly, the reference wave can also be used to illuminate the recorded hologram in the in-line optical holographic setup As shown in

Beam splitter cube

Mirror Mirror

Lens

Reference wave

Object wave

Fig 2.8, the object wave path is blocked, and the reference wave is used to illuminate the recorded hologram

Figure 2.8 Optical reconstruction based on an off-axis optical holographic setup

When the reference wave illuminates the recorded hologram, the modulation procedure can be described by

where the first term on the right-hand side of Eq (2.16) forms a zero-order term or

DC term, and the second and third terms form real and virtual images, respectively In off-axis optical holographic setup, these three terms can be effectively separated However, in the in-line optical holographic setup, these terms superpose

Beam splitter cube

Mirror Mirror

Recorded hologram

Reconstructed virtual image

Laser

Spatial filter

Lens Beam stop

Lens

2.3 Optical holographic interferometry

In the practical engineering field, not only a single reconstruction is of interest, but the comparison of two or more wavefronts is also required Holographic interferometry is defined as the interferometric comparison of two or more wave fields (Vest, 1979) Holographic interferometry is a non-contact and non-destructive technique which can

be applied to various applications, such as deformation measurement and object surface contouring (Schnars and Jueptner, 2005) There are two main types in holographic interferometry, i.e., double exposure and real time Figures 2.9 and 2.10 show the recording and reconstruction of holographic interferogram based on double exposure method In Fig 2.9, the reference state of the object is first recorded by the recording medium, and then the second (such as deformed) state of the object is recorded by the same recording medium In the optical reconstruction, the same reference wave during the recording is applied to illuminate the hologram, and the superposition result (such as an interference fringe) is visible

Figure 2.9 Optical recording of a holographic interferogram

Beam splitter cube

Mirror Mirror

Hologram

Object under deformation Lens

Figure 2.10 Optical reconstruction of holographic interferogram

In the recording procedure, the reference and deformed states of the object can

of the two reconstructed complex amplitudes

Beam splitter cube

Mirror Mirror

2.4 Digital holography

2.4.1 Digital hologram recording

With the rapid development of computer technique and charged-couple device (CCD), the automation becomes a major theme and the accurate and quantitative measurement of physical quantities becomes feasible When the CCD technology is introduced to the holographic research field, the concept of digital holography was developed (Schnars and Jüptner, 1994a; Cuche et al., 1999a) The experimental arrangements for digital holography are almost the same as those shown in Figs 2.6 and 2.7 Simple schematics for in-line and off-axis digital holographic experimental setups are shown in Figs 2.11(a) and 2.11(b), respectively A CCD shown in Fig 2.11 is used to replace the conventional recording medium of photographic film, and the numerical recording in digital holography brings in many new topics into the realm of research However, basic principles of holographic techniques mentioned above, such as coherence, interference and diffraction, are also valid for digital holography

Digital holography has several advantages (Schnars, 1994) compared with conventional optical holographic technique: (1) The hologram is recorded digitally and no chemical or physical development process is required; (2) the hologram can be recorded in the video frequency; (3) numerical hologram recording, subsequent numerical reconstruction and phase evaluation can be integrated into a single system

Figure 2.11 (a) In-line digital holographic experimental setup; (b) off-axis digital holographic experimental setup

2.4.2 Numerical reconstruction

2.4.2.1 Fresnel approximation method

There are three planes in digital holography, i.e., object plane, hologram (or CCD) plane and reconstruction (or image) plane A coordinate system for the numerical reconstruction of a digital hologram is shown in Fig 2.12 The numerical reconstruction procedure is mainly based on the scalar diffraction principle (Goodman, 1996) The original object wave ( ', ')Γξ η can be reconstructed by a modification of

Eq (2.12), and can be expressed by Fresnel-Kirchhoff integral

Object

Reference wave

CC

CCD

where H x y denotes the hologram function, ( , )( , ) R x y denotes a numerical reference

wave, ρ is the distance from a point P in the hologram (or CCD) plane to a point Q in the image plane ρ= (x−ξ')2 +(y−η')2 +d2 as shown in Fig 2.12, d is the axial

reconstruction distance, and θ is the angle between the normal nr and ρ (see Fig 2.12) In the most practical situations, θ is very small and cosθ ≈1

Figure 2.12 A coordinate system for numerical reconstruction of digital hologram

The expression of ρ can be expanded to a Taylor series (Schnars and Jueptner, 2005)

π λ

π ξ η λ

where Γ(m,n)denotes a matrix of M ×N points, and ∆x and y∆ denote pixel sizes

in the hologram (CCD) plane For the sake of brevity, the pixel in the image plane is

denoted as (m, n) The pixel sizes in the image plane can be calculated by

∆ =

∆ (2.26)

The intensity distribution ( , )I m n and the phase distribution ( , )ϕ m n can be directly extracted from the reconstructed complex amplitude

Hence, numerical reconstruction based on convolution method can be written

as (Kreis et al., 1997; Schnars and Jueptner, 2005)