Collins formulae in both space and frequency domains for ABCD optical systems with small deformations

Abstract

Generalized Collins formulae for arbitrary imperfect ABCD optical systems with small deformations (misalignments and/or deviations from ideal optical operations) are obtained in both the space domain and the frequency domain. These formulae can provide a unified way to analyse the performance of a practical optical system in both domains, including the ideal as its special case. Particularly, it shows that a reciprocally symmetrical ABCD system with small deformations can implement so-called almost-fractional Fourier transformation (FRT) simultaneously in both domains with the same order. Some other applications in practice are also discussed to verify the effectiveness of our proposed method.     

Comment on ‘The Fractional Fourier Transform in Optical Propagation Problems’

Although their mathematical forms apparently resemble each other, the diffraction integral and fractional Fourier transformation (FRT) have completely different physical meanings. We point out that an interpretation of the FRT given recently in a paper by Alieva et al. is not physically appropriate. We then show how those integral transformations can be treated in a unified way within the framework of the special affine Fourier transformation. Finally the multidimensional FRT presented in the above paper is further generalized to allow n independent fractional degrees.     

Fractional Fourier transform for partially coherent off-axis Gaussian Schell-model beam

The fractional Fourier transform (FRT) is applied to a partially coherent off-axis Gaussian Schell-model (GSM) beam, and an analytical formula is derived for the FRT of a partially coherent off-axis GSM beam. The corresponding tensor ABCD law for performing the FRT of a partially coherent off-axis GSM beam is also obtained. As an application example, the FRT of a partially coherent linear laser array that is expanded as a sum of off-axis GSM beams is studied. The derived formulas are used to provide numerical examples. The formulas provide a convenient way to analyze and calculate the FRT of a partially coherent off-axis GSM beam.     

Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane

An analytical and concise formula is derived for the fractional Fourier transform (FRT) of partially coherent beams that is based on the tensorial propagation formula of the cross-spectral density of partially coherent twisted anisotropic Gaussian-Schell-model (GSM) beams. The corresponding tensor ABCD law performing the FRT is obtained. The connections between the FRT formula and the generalized diffraction integral formulas for partially coherent beams passing through aligned optical systems and misaligned optical systems are discussed. With use of the derived formula, the transformation and spectrum properties of partially coherent GSM beams in the FRT plane are studied in detail. The results show that the fractional order of the FRT has strong effects on the transformation properties and the spectrum properties of partially coherent GSM beams. Our method provides a simple and convenient way to study the FRT of twisted anisotropic GSM beams.     

The Wigner distribution function of self-Fourier functions

Abstract

Particular forms of the Wigner distribution function and its moments of arbitrary order are derived for a generalized definition of the self-Fourier functions. The general form of the optical system matrix for which an input self-Fourier function transforms to an output self-Fourier function is found. The bright soliton solution of the electromagnetic wave is treated as an example of a generalized self-Fourier function.     

White-light optical implementation of the fractional fourier transform with adjustable order control

An optical implementation of the fractional Fourier transform (FRT) with broadband illumination is proposed by use of a single imaging element, namely, a blazed diffractive lens. The setup displays an achromatized version of the FRT of order P of any two-dimensional input function. This fractional order can be tuned continuously by shifting of the input along the optical axis. Our compact and flexible configuration is tested with a chirplike input signal, and the good experimental results obtained support the theory.     

Chirp filtering in the fractional Fourier domain

In the Wigner domain of a one-dimensional function, a certain chirp term represents a rotated line delta function. On the other hand, a fractional Fourier transform (FRT) can be associated with a rotation of the Wigner-distribution function by an angle connected with the FRT order. Thus with the FRT tool a chirp and a delta function can be transformed one into the other. Taking the chirp as additive noise, the FRT is used for filtering the line delta function in the appropriate fractional Fourier domain. Experimental filtering results for a Gaussian input function, which is modulated by an additive chirp noise, are shown. Excellent agreement between experiments and computer simulations is achieved.     

Approximate propagation equations of flattened gaussian beams passing through a paraxial ABCD system with hard-edge aperture

Abstract

The propagation of flattened Gaussian beams through a paraxial optical ABCD system with hard-edge aperture is studied. Approximate closed-form equations of FGBs for the apertured case are derived by means of the expansion of the circ function into a finite sum of complex Gaussian functions. The advantages of the method as compared with the straightforward integration of the Collins formula are discussed and illustrated with numerical examples.     

Beam Quality Dependence on the Coherence Length of Gaussian Schell-model fields Propagating Through ABCD Optical Systems

Abstract

The quality behaviour of Gaussian Schell-model fields propagating through general (non-orthogonal) ABCD optical systems has been characterized by introducing a number of parameters. Such parameters have been given in terms of the Wigner distribution function of the field, closely connected with its second-order coherence features. Dependence of the focusing capabilities, transversal widths and beam asymmetry on the coherence length of the field has been investigated.     

Generation and Statistical Properties of Optical Dead-time Effects

Abstract

An optoelectronic device that can generate either the non-paralysable or the paralysable dead-time effect has been constructed using a feedback scheme in conjunction with the acousto-optic deflector. ‘Semi-classical’ experiments show good agreement with the theory, which predicts antibunching and sub-Poissonian statistics. An expression for the correlation function of an arbitrary order is obtained for a general renewal process. From this general result, exact expressions for autocorrelation functions for both the non-paralysable and the paralysable counters are derived when the dead-time is greater than twice the sample time. The Fano factor is considered for both the non-paralysable and the paralysable counters. It is seen that the Fano factor for the paralysable counter can be smaller than that for the non-paralysable counter for a certain range of the input count rate.     

Effect of nish rolling temperature on static recrystallisation in hot bands of electrical steel containing 1·3% silicon

Abstract

The effect of the finish rolling temperature (FRT) on recrystallisation behaviour in hot bands of an electrical steel containing 1·3%Si was investigated. Four sequential passes of hot rolling were carried out on the 1·3%Si electrical steel, with finish rolling temperatures ranging from 980 to 700°C, followed by isothermal annealing at 720°C. The experimental results showed that when Ar ₁ <FRT <Ar ₃, fine equiaxed subgrains formed at the boundaries between deformed and non-deformed grains in a necklacelike arrangement, and strain induced boundary migration (SIBM) was the main mechanism corresponding to the formation of recrystallisation nuclei for steels finish rolled below Ar ₁. However, the study also demonstrated that when FRT <(Ar ₁­100 K), a second nucleation mechanism, i.e. subgrain growth, became active in recrystallisation, this resulted in an increase of nucleus density. Steels in which SIBM was the dominant mechanism of recrystallisation possessed the largest grain size, and strongest textures with major component {100}〈110〉.     

Fast numerical algorithm for the linear canonical transform

The linear canonical transform (LCT) describes the effect of any quadratic phase system (QPS) on an input optical wave field. Special cases of the LCT include the fractional Fourier transform (FRT), the Fourier transform (FT), and the Fresnel transform (FST) describing free-space propagation. Currently there are numerous efficient algorithms used (for purposes of numerical simulation in the area of optical signal processing) to calculate the discrete FT, FRT, and FST. All of these algorithms are based on the use of the fast Fourier transform (FFT). In this paper we develop theory for the discrete linear canonical transform (DLCT), which is to the LCT what the discrete Fourier transform (DFT) is to the FT. We then derive the fast linear canonical transform (FLCT), an N log N algorithm for its numerical implementation by an approach similar to that used in deriving the FFT from the DFT. Our algorithm is significantly different from the FFT, is based purely on the properties of the LCT, and can be used for FFT, FRT, and FST calculations and, in the most general case, for the rapid calculation of the effect of any QPS.     

A further comparative study of flattened gaussian beams and super-Gaussian beams

Abstract

A further comparison between flattened Gaussian beams (FGBs) and super-Gaussian beams (SGBs) is performed. It is shown that the two FGB and SGB having the same beam propagation factor (M²?factor) but different waist widths demonstrate a similar irradiance profile at the position of the equal generalized Fresnel number, while they propagate through a paraxial ABCD system. Furthermore, the focusing by a spherically aberrated lens is studied, and nearly the same propagation characteristics are achieved for the two FGB and SGB with the same M²?factor and waist width.     

Quantum Optics of Linear Systems

Abstract

The classical Jones calculus for optical networks is utilized to formulate a general theory of transfer of quantum fields through a linear quantum optics device. For input fields having Gaussian quasi-distribution functions, the output distributions can be calculated explicitly. The formalism is applied to the simple beam splitter as an illustration.     

Investigation of dry ice blockage in an ultra-low temperature cascade refrigeration system using CO2 as a working fluid

Dry ice blockage in a CO₂-solid-gas-flow-based ultra-low temperature cascade refrigeration system is investigated experimentally by a visualization test and a system study of the liquid CO₂ blew into an expansion tube through a Throttle needle valve. The visualization test shows that dry ice sedimentation occurs in low flow velocity and the dry ice formation makes the heat transfer behavior of CO₂ complicated. The sedimentation also occurs at low condensation temperature and low heating power input. Based on the present investigation, it is found that the present ultra-low temperature cascade refrigeration system is better to work at heating power input above 900 W and condensation temperature above −20 °C. At suitable operating condition, the present ultra-low temperature cascade refrigeration system has been shown the capability of achieving ultra-low temperature −62 °C continuously and stably.     

Space-variant Characteristics in Multiple Object Recognition with a Coherent Optical Processor

Abstract

When the input plane of a Van der Lugt correlator contains a composite object, the output plane will in general contain several correlation peaks. When these correlation peaks overlap, the total intensity will be given by the coherent addition of the amplitudes of the individual peaks. It is shown in this paper that the total intensity depends on the phase difference between the correlation signals. The phase differences are dependent both on the displacement of the input object and any aberration in the correlator. A theoretical analysis of the system is given, showing that the interference between the correlation signals causes the total correlation intensity to change with object displacement in an oscillatory fashion rather than in a monotonic way. Some experimental verification of the theoretical predictions is also presented.     

Optimization techniques: Studies in cell culture

Abstract

With the present large worldwide demand for vaccines and cellular by products, a number of anchorage-dependent cell mass culture systems have been developed. To obtain the maximum yield of product per run, the number of cells per system volume must be maximized and yet maintain efficiency. There is a large number of independent variables that could affect the cell yield. During the course of an experimental program to increase the cell yield, the usefulness of an existing computer optimization program was evaluated.

A series of nine statistically designed experiments was done using multi-disk propagators for the mass culture of primary chick embryo cells. Cell input and time between medium refeeds were chosen as the independent variables from the many variables that could affect cell yield. The primary dependent variable was cell yield. The results indicated a substantial increase in cell yield as cell input was increased (57 to 85 percent increase by doubling the cell input) but a lesser increase in cell yield with an increasing number of refeeds.

The data were subjected to regression analysis by the existing computer program. The second order polynomial equation (cell output as a function of cell input and time between refeeds) aptly described the experimental data. Three additional experiments were done with 50 percent more cell input than the maximum cell input in the experimental design. The predicted cell yields from the polynomial equation were approximately 10% greater than the cell yields obtained from the three experiments.     

Parameter identification of a class of nonlinear systems with numerical optimization technique

Abstract

A time domain technique for the identification of unknown parameters in the nonlinear system of some given structures is presented. The nonlinearities considered in this paper are in the forms of saturation, deadzone and backlash. The steps of the identification process are first to set the initial or guessed system parameters in a learning model, then apply the recorded input signal of the actual system to the mathematical learning model. By comparing the output errors between the model's and the system's responses, an optimization algorithm can be applied to search for the true parameters of the nonlinear system in order to meet a least mean square error criterion. The computer simulations have demonstrated the feasibility of this numerical technique.     

Propagation of hollow Gaussian beams through apertured paraxial optical systems

On the basis of the generalized Collins formula and the expansion of the hard-aperture function into a finite sum of complex Gaussian functions, an approximate analytical formula for a hollow Gaussian beam propagating through an apertured paraxial stigmatic (ST) ABCD optical system is derived. Some numerical examples are given. Furthermore, by using a tensor method, we derive approximate analytical formulas for a hollow elliptical Gaussian beam propagating through an apertured paraxial general astigmatic ABCD optical system and an apertured paraxial misaligned ST ABCD optical system. Our results provide a convenient way for studying the propagation and transformation of a hollow Gaussian beam and a hollow elliptical Gaussian beam through an apertured general optical system.