Speckles are a statistical interference phenomenon which arises when coherent light is scattered by an optically rough surface. The resulting speckle field can be examined using statistical techniques and some results that arise are related to the form of the resulting speckles. Characteristic features are the so-called lateral and longitudinal speckle size as well as the orientation of speckles in the z-direction. It has been shown that the 3D orientation of speckles in free-space and in Fourier transforming systems are different from each other. Here we review some of Sheridan's work in this area. We first examine the speckle field formed by a diffuser in free space over a 3D volume. Next an Optical Fourier Transform (OFT) system is assumed, where the diffuser is placed in the front focal length of a Fourier transforming lens and we examine the resulting speckle field in the back focal plane of this OFT system. We compare and contrast the properties of the speckles from both setups free-space and Fourier.
Digital holographic microscopy (DHM) has a wide variety of applications in biomedical sciences. Here we examine the use of an off-axis holographic system to monitor bacterial growth in commercially available ibidi micro-fluidic chambers. The imaged volume spans roughly 120 x 120 x 200 μm3, at 50x magnification. By use of an off-axis reference wave and some standard numerical filtering operations it is possible to recover the complex amplitude and thus the phase of the object wave. Two strains of bacteria are used in the research study: Escherichia coli and Staphylococcus warneri with dimensions around 1-4 microns. Examining these samples, showed that through the use of phase compensation techniques, the method can sustain an average Signal-to-Noise ratio (SNR) of around 5, allowing for the use of a simple thresholding algorithm for cell counting. In addition, the cell count dynamics determined using this method reflect the exponential increase of dilution steps very well, when compared against hypothetical cell counts. Details of the system operation, including a discussion of the signal processing routines, together with modifications to the general optic design of DHM systems are presented.
It is possible to use of 3D lateral and longitudinal static speckle fields to determine the location of the optical axis in a system. In this manuscript we examine a variation of this idea where we use the 3D temporal correlation properties of multiple speckle fields to determine the axis. Previously it was shown how a spatial averaging approach could be used to estimate the 3D speckle orientation, here we compare the characteristics of this spatial averaging technique to a temporal averaging approach (where multiple speckle fields are used to produce an ensemble), experimental results are compared with the theoretical predictions and we present some conclusions.
The accurate measurement of optical phase has many applications in metrology. For biological samples, which appear transparent, the phase data provides information about the refractive index of the sample. In speckle metrology, the phase can be used to estimate stress and strains of a rough surface with high sensitivity. In this theoretical manuscript we compare and contrast the properties of two techniques for estimating the phase distribution of a wave field under the paraxial approximation: (I) A digital holographic system, and (II) An idealized phase retrieval system. Both systems use a CCD or CMOS array to measure the intensities of the wave fields that are reflected from or transmitted through the sample of interest. This introduces a numerical aspect to the problem. For the two systems above we examine how numerical calculations can limit the performance of these systems leading to a near-infinite number of possible solutions.
A new optical technique for understanding, analyzing and developing optical systems is presented. This approach is statistical in nature, where information about an object under investigation is discovered by examining deviations from a known reference statistical distribution.
A practical technique is presented based on DH, for the reconstruction of a wavefront from three recorded intensity images. Combining the off-axis Fourier spatial filtering (OFSF) technique with iterative phase retrieval algorithms, it is shown how the twin image can be eliminated. The proposed method overcomes system geometry constraints and improves both the flexibility and resolution associated with OFSF-based DH. It also overcomes the cost problem associated with phase-shifting interferometry-based DH. In order to demonstrate the performance of the proposed DH method, both simulation and experiment results for objects having smooth and rough surfaces are presented.
In this paper, a new practical technique is presented based on digital holography, for the reconstruction of a wave front from three intensity recordings. Combining the off-axis Fourier filtering technique with boundary detection and iterative phase retrieval algorithms, it is shown how problems such as elimination of the twin image can be overcome. The proposed methods deal with also the issues of feasibility and accuracy associated with off-axis Fourier spatial filtering (OFSF), and those of cost and alignment associated with phase shifting interferometry (PSI). Problems associated with working with diffuse objects are also overcome.
Measurement of the refractive index of regular shaped glass by speckle correlation is reported. One intensity image in the diffraction field of a speckle-illuminated sample is captured by a CCD before the presence of glass sample and another intensity image is captured after the presence of glass sample. As the position of peak correlation coefficient is quantitatively related to the change in optical path length arising due to the presence of glass, the refractive index of the glass can be evaluated by the correlation of the intensity images before and after the glass insertion. The theoretical correlation function is first derived that describes the relationship between optical path length change and speckle decorrelation. In experiment, various regular shaped glasses are measured to demonstrate the accuracy and robustness of the proposed technique.
In modern optical systems discrete digital devices for measuring intensity distributions play an indispensible role. The intensity incident on the CCD or CMOS array is averaged over the spatial extent of each pixel for a given exposure time. Fluctuations in the power of a laser, vibrations on an optical table, and electronic noise from the digital sensor all contribute to some degree to a base-line noise level for a particular optical system. Hence we expect that the intensity value measured by each pixel will fluctuate over time. In this paper we investigate the effects of noise of cameras over time using a speckle field for a range of different camera parameters such as exposure time, gain factor, and light power. We then examine how this baseline noise level changes when the incident speckle field is mixed with a plane reference wave forming to form a hologram at the camera plane. We comment on our experimental results and how they apply to general optical systems that measure the phase distribution for a complex field.
Optical systems that can recover both the amplitude and phase of a scattered wave eld are important for a range of di erent practical imaging and metrology applications. In this manuscript we examine two di erent techniques: (A) Fresnel based digital holography and (B) Teague's transport of intensity phase retrieval technique, using a special analytical function that serves to act as the scattered wave eld we would like to recover. Nowadays both systems use modern CCD or CMOS arrays to make the necessary intensity measurements. In system (A) an ideal plane wave reference eld is required and should overlap, and interfere, with the scattered eld at at the CCD plane. The resulting intensity distribution recorded by the CCD is a digital hologram. If several captures are recorded, where the phase of the reference has been changed (stepped) between captures, it is possible to recover an approximation to the complex amplitude of the scattered wave eld. In system (B) no reference eld is needed, which is a signi cant advantage from a practical implementation point of view. Rather, the intensity of the scattered wave eld has to be measured at two axially displaced planes. We expect that the performance of both systems will be fundamentally limited by at least three separate factors, (i) the nite extent of CCD array, (ii) the nite extent of the CCD pixels which average the light intensity incident upon them, and (iii) the sampling operation which occurs because the intensity is recorded at a set of uniformly displaced discrete locations. In this manuscript, we examine how factors (i) and (iii), e ect the imaging performance of each system by varying the spatial frequency extent of the scattered wave eld. We nd that system A has superior performance compared to system B.
Speckle fields are formed when quasi-monochromatic light is scattered by an optically rough surface. These fields
are usually described by reference to their first and second order statistical properties. In this paper we review
and extend some of these fundamental properties and propose a novel technique for estimating the refractive
index of a smooth sample. Theoretical and experimental results are presented. Separately, we also report on
a preliminary experiment to determine some characteristics of speckle fields formed in free space by a rotating
compound diffuser. Some initial measurements are made where we examine how the speckle intensity pattern in
the output plane changes as a function of the relative rotation angle.
Quasi-monochromatic light reflected from an optically rough surface produces a complicated 3D speckle field. This
speckle field is often described using a correlation function from which the 3D speckle properties can be examined. The
derivation of the correlation function is based on a physical model where several critical assumptions about the input and
output fields in the model are made. However, experimental works verifying this correlation function are rare and
sometimes produce inconsistent results. In this paper, we examine some practical issues encountered when
experimentally measuring this correlation function, including: The realization of the ensemble average between speckle
fields at two point positions; and, The pixel integrating effect of the recording camera and the implications this has for
the statistics of the measured speckle field. Following verification of the correlation function and examining the speckle
decorrelation properties in 3D space, two practical applications are proposed, one is the aligning of the system optical
axis with the camera center and the other is the measurement of the out-of-plane displacement of an object surface.
Simulation and experimental results that support our analysis are presented.
Iterative Phase Retrieval (PR) techniques represent an alternative means to Digital Holography (DH) for estimating the
complex amplitude of an optical wavefront. To achieve a high-resolution reconstruction from a digital hologram, one
must use Phase-Shifting Interferometric (PSI) techniques to remove the DC and twin image terms that are a feature of
holographic recordings. Unfortunately this approach is not suitable for imaging dynamic events, since a minimum of 3
sequential captures, are typically required and the scene cannot change during this recording process. PR algorithms
may provide a solution to this dynamic imaging problem, however these algorithms provide solutions that are not unique
and hence cannot ensure an accurate solution to the problem.
We examine some fundamental theoretical limits on the ability of practical digital holography (DH) systems to resolve detail in an image. Unlike conventional diffraction-limited imaging systems, where a projected image of the limiting aperture is used to define the system performance, there are at least three major effects that determine the performance of a DH system: (i) The spacing between adjacent pixels on the CCD, (ii) an averaging effect introduced by the finite size of these pixels, and (iii) the finite extent of the camera face itself. Using a theoretical model, we define a single expression that accounts for all these physical effects. With this model, we explore several different DH recording techniques: off-axis and inline, considering both the dc terms, as well as the real and twin images that are features of the holographic recording process. Our analysis shows that the imaging operation is shift variant and we demonstrate this using a simple example. We examine how our theoretical model can be used to optimize CCD design for lensless DH capture. We present a series of experimental results to confirm the validity of our theoretical model, demonstrating recovery of super-Nyquist frequencies for the first time.
We show how the reconstruction of digital holograms can be speeded up on ordinary computers by precomputing
the chirp factor in the Fresnel transform for a given detector array size. The
speedup in time is shown for various hologram sizes. We also run the same algorithm on a Nvidia GPU. The
speedup and the error introduced due to quantizing to different levels is investigated. Additionally a variance based
An optical implementation of the amplitude encoded double random phase encryption/decryption technique is implemented, and both numerical and experimental results are presented. In particular, we examine the effect of quantization in the decryption process due to the discrete values and quantized levels, which a spatial light modulator (SLM) can physically display. To do this, we characterize a transmissive SLM using Jones matrices and then map a complex image to the physically achievable levels of the SLM using the pseudorandom encoding technique. We present both numerical and experimental results that quantify the performance of the system.
The theorems of Nyquist, Shannon and Whittaker have long held true for sampling optical signals. They showed
that a signal (with finite bandwidth) should be sampled at a rate at least as fast as twice the maximum spatial
frequency of the signal. They proceeded to show how the continuous signal could be reconstructed perfectly
from its well sampled counterpart by convolving a Sinc function with the sampled signal. Recent years have
seen the emergence of a new generalized sampling theorem of which Nyquist Shannon is a special case. This
new theorem suggests that it is possible to sample and reconstruct certain signals at rates much slower than
those predicted by Nyquist-Shannon. One application in which this new theorem is of considerable interest is
Fresnel Holography. A number of papers have recently suggested that the sampling rate for the digital recording
of Fresnel holograms can be relaxed considerably. This may allow the positioning of the object closer to the
camera allowing for a greater numerical aperture and thus an improved range of 3D perspective. In this paper
we: (i) Review generalized sampling for Fresnel propagated signals, (ii) Investigate the effect of the twin image,
always present in recording, on the generalized sampling theorem and (iii) Discuss the effect of finite pixel size
for the first time.
The lateral correlation properties of speckle fields have been shown to be useful in aligning multiple optical channels relative to one another. Relative rotational alignment can also be achieved using a sub-sectioning extension of this technique. In this paper, we examine the three dimensional correlation properties of speckle, and by so doing, create a technique that allows for absolute positioning of a single channel free space optical system on the optical axis without the need for markers or gratings.
Latest advances in femtosecond technology have strongly emphasized the control of ultra-short pulses in many
applications where the preservation of the pulse duration is most important. Recently, the delivery of ultra-short pulses
through optical fibers has become possible which opens up remarkable chances for simplifying optical setups or reaching
inaccessible regions. In this study we report on fiber delivery of 2 nJ and sub 65 fs pulses from a Ti:Sapphire laser
through 1.5 m LMA photonic crystal fiber. Application of such a fiber in an all-integrated THz imaging system to obtain
contactless information on the doping concentration of semiconductor wafers is shown.
A numerical method, based on the fast Fourier transform, is proposed that efficiently calculates the 2D (x,y) diffraction
pattern formed when an ultrashort pulse of light is incident upon an aperture. Since ultrashort pulses are becoming
increasingly important in modern optics from THz generation and spectroscopy to confocal microscopy, a fast
numerical technique for calculating typical diffraction patterns is of significant interest. Pulses are not monochromatic
but rather have a finite spectral distribution about some central frequency. Under paraxial conditions, the spatial
diffraction pattern due to an individual spectral component may be calculated using the Fresnel transform. This is
performed for each spectral component giving a spatio-spectral distribution. The diffracted spatio-temporal pulse can
then be calculated by performing an inverse Fourier transform (with respect to the temporal frequency) on this spatio-spectral
distribution. Numerical implementation raises two questions: (a) for a given distance and temporal frequency
what is the minimum number of samples needed to efficiently calculate the corresponding Fresnel diffraction pattern
and (b) for a given temporal pulse profile how many spectral components are required to accurately describe the
diffraction of the pulse? By examining the distribution of the pulses energy in phase space using Wigner diagrams we
identify a simple set of rules for determining these optimal sampling conditions. Then, using these rules we examine
the diffraction patterns from both a square and circular aperture. A discussion of the results and potential THz
applications follows.
Speckle is an inherent characteristic of coherent imaging systems. Often, as in the case of Ultrasound, Synthetic Aperture
Radar, Laser Imaging and Holography, speckle is a source of noise and degrades the reconstructed image. Various
methods exist for the removal of speckle in such images. One method, which has received attention for the removal of
speckle from coherent imaging, is to use a temporally incoherent source. We create a novel digital signal processing
technique for the reduction of speckle from digital holograms by simulating temporal incoherence during the digital
reconstruction process. The method makes use of the discrete implementation of the Fresnel Transform, which calculates
the reconstructed image for a range of different wavelengths. These different spectral components can be weighted to
suit a temporally incoherent source and the intensities from each wavelength are added together. The method is
examined using the speckle index metric.
Complex paraxial optical systems, consisting of multiple lenses and sections of free space propagation, can be described
using the Linear Canonical Transform (LCT). Indeed it can be shown that many well know optical transforms such as
the Optical Fourier Transform (OFT), Optical Fractional Fourier Transform (OFRT), the effect of a lens or Chirp
Modulation Transform (CMT) are all subsets of the more general LCT. Using the ABCD Collins matrix formula it is
possible to represent these integral transforms in a simpler form, which facilitates system analysis and design. Apertures
are necessary in speckle based metrology systems to control the size of the speckle. We examine their effect on LCT
systems and show using the "generalized Yamaguchi correlation factor" that a useful interpretation of the system's behavior, using the LCT, may still be obtained. Furthermore, we experimentally demonstrate our ability to determine
simultaneous tilt and translation motion by capturing two, sequential, mixed doma in images with a single camera. We
also show how localized deformations in an object may be measured using this system.
The numerical calculation of the Fresnel transform (FST) presents significant challenges due to the high sampling rate associated with the chirp function in the kernel. The development of an efficient algorithm is further complicated by the fact that the output extent of the FST is dependent on the propagation distance. In this paper, we implement a recently proposed technique for efficiently calculating the FST in which we apply the Wigner distribution function and the space bandwidth product to identify suitable sampling rates. This method is shown to be suitable for all propagation distances. Our method can also be applied to describe the effect of a thin lens modeled as a chirp modulation transform (CMT). Combining our results for the FST and the CMT, we numerically calculate the light distribution at the output of both Cai-Wang and Lohmann Type-I optical fractional Fourier transform (OFRT) systems. Analytic solutions for the OFRT of rectangular window and circular apertures are presented. The analytical solutions are compared to experimental data and to numerical results for equivalent cases. Finally the numerical method is applied to examine the effect that apertured lenses, in the OFRT system, have on the output distribution.
It has been shown that complex paraxial optical systems, consisting of various lens and distances of free space propagation, can be described using the Linear Canonical Transform (LCT). Indeed it can be shown that many well know optical transforms such as the Optical Fourier Transform (OFT), Optical Fractional Fourier Transform (OFRT), the effect of a lens or Chirp Modulation Transform (CMT) are all subsets of the more general LCT. Using the ABCD Collins matrix formula it is possible to represent these integral transforms in a simpler form, which facilitates system analysis and design. Speckle Photography (SP) in combination with an optical LCT can be used to measure surface motion of an optically rough body. It has previously been shown that Optical FRT's (OFRT) can be used in speckle based metrology systems to vary the range and sensitivity of a metrology system and also to determine both, the magnitude and direction, of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. In this paper we extend the OFRT analysis to more general LCT systems and demonstrate how simultaneous tilt and translation measurements can be discerned from the speckle images captured prior to, and after motion. A spherical wavefront can be conveniently described using the Collin's matrix notation. By changing the wavefront of the illuminating light we show that we effectively change the domain of the LCT system without changing the bulk elements in the optical system. Thus the complete motion (in-plane translation and small surface tilting) of a rigid body can be determined using one optical LCT system and illuminating fields of varying curvature.
We consider a double random phase encoding Encryption/Decryption system in which the image encryption/decryption process is performed numerically. In this paper we look at the effect of quantisation in the decryption process due to the discrete values which a spatial light modulator can display. We look at the characterisation of a transmissive spatial light modulator and we present results from simulations of the system.
Phase retrieval is the recovery of missing phase information in an optical field using measurable quantities. Usually the directly measurable quantity in an optical system is the intensity of the image and a diffraction pattern. The Gerchberg-Saxton (GS) algorithm utilizes the spatial and the spatial frequency intensities and the Fourier Transform (FT) is used to numerically transform between the two. Recently the fractional Fourier transform (FRT) has been proposed in the same context. Studies have shown that the efficiency of this FRT based Gerchberg-Saxton algorithm is dependent upon the discrete FRT used in the procedure. If this discrete FRT is unitary optimum results are obtained. In this paper we investigate the use of a recently proposed set of theoretically unitary discrete FRT algorithms. These algorithms are based on tracking the shape of the signal's Wigner Distribution Function (WDF) and the relationship between this shape and the Space-Bandwidth Product (SBP). We provide numerical simulations and compare these with those obtained using the traditional FT-GS algorithm.
The numerical calculation of the Fresnel Transform (FST) presents significant challenges due to the high sampling rate associated with the chirp function in the kernel. The development of a general algorithm is further complicated due the fact that the output extent of the FST is dependant on the propagation distance. In this paper we implement a novel technique for calculating the FST in which we use the Wigner Distribution Function (WDF) to choose an optimal sampling rate. This method is suitable for all propagation distances. Our method can also be used to describe the effect of a lens in an optical system. A lens in an optical system is modeled as a Chirp Modulation Transform (CMT), which acts on the spatial frequency distribution of a signal in a manner that is analogous to the effect an FST has on the spatial distribution of a signal. Combining our results for the FST and the CMT we numerically calculate the light distribution at the output of both a Cai-Wang and a Lohmann Type-I Optical Fractional Fourier Transform (OFRT) system. Analytic solutions for the OFRT of a square (1-D) and circular (1-D) aperture are presented. The numerical results are compared to the analytically derived solution.
The Linear Canonical Transform (LCT) is a general transform which can be used to describe linear lossless quadratic phase systems (QPS). It can be shown that the Optical Fourier Transform (OFT), Optical Fractional Fourier Transform (OFRT) and the effect of a thin lens or Chirp Modulation Transform (CMT), are all special cases of the more general LCT. Using the Collins formula it is possible to represent these transforms as ABCD matrices. By cascading relevant matrices together, quite complicated bulk optical systems can be described in a compact manner. Digital Speckle Photography (DSP) can be used in the analysis of surface motion in combination with an optical LCT. It has previously been shown that Optical FRT's (OFRT) can be used in speckle based metrology systems to vary the range and sensitivity of a metrology system and also to determine both, the magnitude and direction, of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. In this paper we extend this analysis to more general LCT systems. We demonstrate that a spherical illuminating wavefront can be conveniently described using matrix notation. We show that by changing the sphericity of wavefront we can change the domain of the LCT system. Hence by illuminating a target with a plane wavefront and then a spherical wavefront, we capture the motion in two separate LCT domains and we are thus in a position to fully determine the motion of a rigid body without a priori knowledge.
In Optical Signal Processing (OSP) one of the most important operations is the Optical Fourier Transform (OFT). Many different optical arrangements exist which allow implementation of the OFT, however one of the more popular is the Scaled Optical Fourier Transform (SOFT) because, as the name implies, this arrangement allows control over the scale of the output Fourier distribution. Using cylindrical coordinates we examine some of the practical limits introduced by the use of an illuminating spherical lens of finite aperture in the implementation of the SOFT. By deriving simple rules of thumb, based on examining phase and intensity deviations from the ideal unapertured case, we define a volume inside the geometric shadow, which we refer to as a sub-geometric shadow. Inside this sub-geometric shadow we show that the worst case errors in the resulting SOFT, arising due to diffraction, can be quantified and avoided.
KEYWORDS: Spatial light modulators, Image encryption, Quantization, Optical signal processing, Transform theory, Wigner distribution functions, Signal processing, Digital holography, Data processing, Free space
Coherent optical signal processors, due to their ability to process and relay information in two dimensions, have been receiving increasing attention in recent years. These systems involve a coherent field being propagated through some bulk optical system consisting of thin lenses and sections of free space (such paraxial systems being described mathematically using the Linear Canonical Transformation). A Spatial Light Modulator (SLM) may be used to modulate the input digital data onto a coherent wave-field as well as to modulate the amplitude and/or phase of the complex wave-field at any desired plane. The complex field (amplitude and phase) at any desired plane may be recorded quantitatively using a CCD camera, using digital holographic techniques allowing the further processing of data digitally. Such hybrid optoelectronic systems have applications for 2D and 3D data processing covering fields as diverse as data storage, data security and pattern recognition. But devices such as SLMs and CCD cameras can represent only discrete levels of data necessitating a quantisation of continuous valued analog information. In this paper, we take the example of an optical system used to encrypt 2D and 3D data and evaluate the effect of the finite discrete levels of an SLM on the encryption/decryption process.
A Linear Canonical Transform (LCT) is a general integral transform which can be used to describe a whole host of complex paraxial optical systems. It can be shown that Fourier Transform (FT), Fractional Fourier Transform (FRT), Chirp Multiplication Function (CMT), (which is used as a model for a thin lens), and the Fresnel Transform (FST) are all specific cases of LCT's and are particularly important in optics. Using the Collins ABCD matrix formula it is possible to represent the above integral transforms in matrix notation. Furthermore since most bulk optical systems can be built using lenses of different curvatures (CMT) and free space propagation (FST) it becomes straight forward, to describe optical systems using matrix notation, (which is interchangeable with LCT integral notation). Speckle Photography (SP) can be used in the analysis of surface motion in combination with an optical LCT. It has previously been shown that Optical FRT's (OFRT) can be used in speckle based metrology systems to vary the range and sensitivity of the metrology system. Using a novel correlation technique it is possible to determine both, the magnitude and direction, of tilting (rotation) and translation motion simultaneously. In this paper these ideas are extended to more general LCT's, which allow the consideration of more complicated bulk optical systems. Combined with correlation techniques we experimentally demonstrate our ability to determine both, the magnitude and direction, of tilting (rotation) and translation motion of a surface over a greater range and sensitivity than previous OFRT techniques allowed.
The Fractional Fourier Transform (FRT) in combination with speckle photography has previously been used to measure surface tilting and translation. Previous Optical Fractional Fourier Transform (OFRT) based techniques used to determine such motion, have not been able to discern the direction of the tilt/translation. A simple new approach involving the use of correlation is presented to overcome this limitation. This is combined with an OFRT system, and controlled variation of the minimum resolution and dynamical range of measurement is demonstrated. It is then confirmed that if a rigid body’s motion is captured using two separate OFRT systems, the direction of both the tilting and in plane translation motion of the body can be determined without apriori knowledge. Experimental results are presented which confirm the validity of theoretical predictions.
Analogue Optical Signal Processing is a very important area in optics and central to it, is the implementation of the Optical Fourier Transform (OFT). There are many bulk optical arrangements for implementing the OFT however one which is particularly popular is the Scaled Optical Fourier Transform (SOFT) because it offers the user to the ability to change the size or scale of the output Fourier distribution. In this paper we examine the some of the practical limits introduced by the use of a square and circular apertured spherical lens. We examine the phase deviations and provide a simple rule of thumb, which allows the cautious user to avoid some of the worst case errors arising due to diffraction.
Analogue Optical Signal Processing is a very important area in optics and central to it, is the implementation of the Optical Fourier Transform (OFT). There are many bulk optical arrangements for implementing the OFT however one which is particularly popular is the Scaled Optical Fourier Transform (SOFT) because it offers the user to the ability to change the size or scale of the output Fourier distribution. In this paper we examine the some of the practical limits introduced by the use of a cylindrical lens of finite aperture. We examine the phase and intensity deviations and provide simple rules of thumb, which allows the cautious user to, not only avoid, but also to quantify the worst case errors arising due to diffraction.
The Optical Fourier Transform (OFT) is the most fundamental operation in analogue Optical Signal Processing (OSP). The Scaled Optical Fourier Transform (SOFT) is widely used as it provides a great deal of flexibility, which is invaluable in real implantations. In this paper we study some of the practical limits introduced by the use of an illuminating lens of finite aperture. We show, by deriving simple rules of thumb based on examining phase and intensity deviations, that the worst case diffraction errors can be avoided.
We discuss the significance of the Wigner Optics (WiO) and the Wigner Distribution Function (WDF), in understanding problems usually dealt with using Wave Optics (WaO). We first present a derivation of the WDF transport equation, equivalent to the Helmhotz equation. We then discuss the corresponding first order WDF differential equation and the 'paraxial’ equation. The Fresnel transform can be derived from the Paraxial Approximation (PA). However another integral transform is required to describe our first order equation. This equation is not a paraxial transform but does take account of the characteristics of the field. We clarify how these transforms are linked to electromagnetic theory. Finally we justify the practical significance we attach to WiO by describe how a new way of understanding motion measurement systems can be achieved using these ideas.
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