Complex Fractional Fourier Transformation Research Articles

On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations

By a quantum mechanical analysis of the additive rule Fα[Fβ[f]]=Fα+β[f], which the fractional Fourier transformation (FrFT) Fα[f] should satisfy, we reveal that the position-momentum mutual-transformation operator is the core element for constructing the integration kernel of FrFT. Based on this observation and the two mutually conjugate entangled-state representations, we then derive a core operator for enabling a complex fractional Fourier transformation (CFrFT), which also obeys the additive rule. In a similar manner, we also reveal the fractional transformation property for a type of Fresnel operator.

New fractional entangling transform and its quantum mechanical correspondence

In this Letter, a new fractional entangling transformation (FrET) is proposed, which is generated in the entangled state representation by a unitary operator exp{iθ(ab†+a†b)} where a(b) is the Bosonic annihilate operator. The operator is actually an entangled one in quantum optics and differs evidently from the separable operator, exp{iθ(a†a+b†b)}, of complex fractional Fourier transformation. The additivity property is proved by employing the entangled state representation and quantum mechanical version of the FrET. As an application, the FrET of a two-mode number state is derived directly by using the quantum version of the FrET, which is related to Hermite polynomials.

New complex integration transformation and its compatibility with complex Weyl-Wigner transformation and entangled state representation

Enlightened by the special transformation in our preceding paper [J. Mod. Opt. 56, 1227 (2009)], we propose a new complex integration transformation corresponding to two mutually conjugate two-mode entangled states <eta| and <xi| that is compatible with eta-xi phase space quantum mechanics and can be used to obtain the complex fractional Fourier transformation kernel from the chirplet function. This transformation obeys the Parseval theorem and is invertible.

Two-mode SU(2) quadratic canonical operators and bipartite-entangled state representations for deriving complex fractional Fourier transformation

We demonstrate that the complex fractional Fourier transformation (CFRFT) can be deduced by two-mode SU(2) quadratic canonical operators and bipartite-entangled state representations. It turns out that the CFRFT is a special case of the two-dimensional (2D) complex Fresnel transformation, just as the 1D fractional Fourier transformation is adapted to the 1D Fresnel transformation.

Classical versus complex fractional Fourier transformation

The quantum optical complex fractional Fourier transform (FRFT) has been related to the classical FRFT using both classical and quantum formalisms. In particular, it was shown that the kernel of the complex FRFT can be classically produced with rotated astigmatic optical systems that mimic the quantum entanglement property.

Quantum Optical Squeezing Transform for Generalizing Fractional Fourier Transform

By establishing the relation between the optical scaled fractional Fourier transform (FFT) and quantum mechanical squeezing-rotating operator transform, we employ the bipartite entangled state representation of two-mode squeezing operator to extend the scaled FFT to more general cases, such as scaled complex FFT and entangled scaled FFT. The additivity and eigenmodes are presented in quantum version. The relation between the scaled FFT and squeezing-rotating Wigner operator is studied.

Convolution Theorem of Fractional Fourier Transformation Derived by Representation Transformation in Quantum Mechancis

Based on our previous paper (Commun. Theor. Phys. 39 (2003) 417) we derive the convolution theorem of fractional Fourier transformation in the context of quantum mechanics, which seems a convenient and neat way. Generalization of this method to the complex fractional Fourier transformation case is also possible.

Eigenfunctions of the complex fractional Fourier transform obtained in the context of quantum optics

We employ the recently established basis (the two-variable Hermite-Gaussian function) of the generalized Bargmann space (BGBS) [Phys. Lett. A303, 311 (2002)] to study the generalized form of the fractional Fourier transform (FRFT). By using the technique of integration within an ordered product of operators and the bipartite entangled-state representations, we derive the generalized generating function of the BGBS with which the undecomposable kernel of the two-dimensional FRFT [also named complex fractional Fourier transform (CFRFT)] is obtained. This approach naturally shows that the BGBS is just the eigenfunction of the CFRFT.

CONVOLUTION THEOREM OF THE COMPLEX FRACTIONAL FOURIER TRANSFORMATION DERIVED BY TRANSFORMATION OF ENTANGLED STATES

Based on our previous paper [H.-Y. Fan and H.-L. Lu, Int. J. Mod. Phys.19, 799 (2005)] and the two mutually-conjugate entangled state representations, we derive the convolution theorem of complex fractional Fourier transformation in the context of quantum mechanics. This approach seems convenient and neat.

From Complex Fractional Fourier Transform to Complex Fractional Radon Transform

We show that for n-dimensional complex fractional Fourier transform the corresponding complex fractional Radon transform can also be derived, however, it is different from the direct product of two n-dimensional real fractional Radon transforms. The complex fractional Radon transform of two-mode Wigner operator is calculated.

Fractional Hankel transform studied by charge-amplitude state representations and complex fractional Fourier transformation.

Starting from a complex fractional Fourier transformation [Opt. Lett. 28, 680 (2003)], it is shown that the integral kernel of a fractional Hankel transformation is equivalent to the matrix element of an appropriate operator in the charge-amplitude state representations; i.e., the fractional Hankel transformation is endowed with a definite physical meaning (definite quantum-mechanical representation transform).

New Eigenmodes of Propagation in Quadratic Graded Index Media and Complex Fractional Fourier Transform**The project supported by National Natural Science Foundation of China under Grant No. 10175057

By introducing a convenient complex form of the α-th 2-dimensional fractional Fourier transform (CFFT) operation we find that it possesses new eigenmodes which are two-mode Hermite polynomials. We prove the eigenvalues of propagation in quadratic graded-index medium over a definite distance are the same as the eigenvalues of the α-th CFFT, which means that our definition of the α-th CFFT is physically meaningful.

EPR entangled states and complex fractional Fourier transformation

Starting from the Einstein-Podolsky-Rosen entangled state representations of continuous variables we derive a new formulation of complex fractional Fourier transformation (CFFT). We find that two-variable Hermite polynomials are just the eigenmodes of the CFFT. In this way the CFFT is linked to the appropriate operator transformation between two kinds of entangled states in the context of quantum mechanics. In so doing, the CFFT of quantum mechanical wave functions can be derived more directly and concisely.

Extended fractional Fourier transforms

The concept of an extended fractional Fourier transform (FRT) is suggested. Previous FRT’s and complex FRT’s are only its subclasses. Then, through this concept and its method, we explain the physical meaning of any optical Fresnel diffraction through a lens: It is just an extended FRT; a lens-cascaded system can equivalently be simplified to a simple analyzer of the FRT; the two-independent-parameter FRT of an object illuminated with a plane wave can be readily implemented by a lens of arbitrary focal length; when cascading, the function of each lens unit and the relationship between the adjacent ones are clear and simple; and more parameters and fewer restrictions on cascading make the optical design easy.