Lü Bai-Da

Vectorial Nonparaxial Four-Petal Gaussian Beams and Their Propagation in Free Space

The vectorial nonparaxial four-petal Gaussian beam (FPGB) is introduced. The closed-form propagation expressions for the free-space propagation of FPGBs are derived and their more general applicable advantages are illustrated analytically and numerically. Some special interesting cases, in particular the paraxial one, are discussed. It is found that the parameter f = 1/kw0 with the k being the wave number and w0 being the waist width plays a crucial role in determining the nonparaxiallity of FPGBs. For small values of the f parameter the paraxial approximation is allowable. In the nonparaxial regime the beam order n additionally affects the vectorial and nonparaxial behaviour of FPGBs.

Nonparaxial Dark-Hollow Gaussian Beams

The concept of nonparaxial dark-hollow Gaussian beams (DHGBs) is introduced. By using the Rayleigh–Sommerfeld diffraction integral, the analytical propagation equation of DHGBs in free space is derived. The on-axis intensity, far-field equation and, in particular, paraxial expressions are given and treated as special cases of our result. It is shown that the parameter f = 1/kw0 with k being the wave number and w0 being the waist width determines the nonparaxiality of DHGBs. However, the parameter range, within which the paraxial approach is valid, depends on the propagation distance. The beam order affects the beam profile and position of maximum on-axis intensity.

Fractional Fourier transform of flat-topped multi-Gaussian beams based on the Wigner distribution function

By using the Wigner distribution function, the fractional Fourier transform (FRFT) of flat-topped multi-Gaussian (FTMG) beams is studied. Analytical expressions for the intensity distribution, beam width, far-field divergence angle, M2 factor and K parameter of FTMG beams are derived. The influence of fractional order on transformation properties of FTMG beams in the FRFT plane is illustrated with numerical examples.

Propagation of the off-axis superposition of partially coherent beams through atmospheric turbulence

The propagation properties of the off-axis superposition of partially coherent beams through atmospheric turbulence and their beam quality in terms of the mean-squared beam width w(z) and the power in the bucket (PIB) are studied in detail, where the effects of partial coherence, off-axis beam superposition and atmospheric turbulence are considered. The analytical expressions for the intensity, the beam width and the PIB are derived, and illustrative examples are given numerically. It is shown that the maximum intensity I max and the PIB decrease and w(z) increases as the refraction index structure constant C 2 n increases. Therefore, the turbulence results in a degradation of the beam quality. However, the resulting partially coherent beam with a smaller value of spatial correlation parameter γ and larger values of separate distance x d and beam number M is less affected by the turbulence than that with a larger value of γ and smaller values of x d and M. The main results obtained in this paper are explained physically.

Characterization of Nonparaxial Truncated Cosine-Gaussian Beams and the Beam Quality in the Far Field

The analytical expression characterizing the propagation of nonparaxial truncated cosine-Gaussian (CoG) beams in free space is derived, and some special cases are discussed. The extended power in the bucket (PIB) is proposed to characterize the beam quality of nonparaxial truncated beams in the far field. It is shown that the extended PIB is applicable to nonparaxial truncated beams, and the PIB of nonparaxial truncated CoG beams depends on the decentred parameter, waist-width-to-wavelength ratio, truncation parameter, and bucket size chosen.

Partially coherent nonparaxial modified Bessel–Gauss beams

The concept of partially coherent nonparaxial modified Bessel–Gauss (MBG) beams is proposed. Based on the generalized Rayleigh-Sommerfeld diffraction integral, the analytical propagation equations of nonparaxial MBG beams in free space are derived and analysed and some special cases are discussed. In particular, under the paraxial approximation our results reduce to the corresponding paraxial ones. Numerical calculation examples are given to illustrate the dependence of intensity and spectral degree of coherence on the beam order m, ξ and f parameters and to compare the difference between the paraxial and nonparaxial results.

Effect of astigmatism on spectral switches of partially coherent beams

A detailed study of the spectrum of partially coherent beams diffracted at an astigmatic aperture lens is performed. Considerable attention is paid to the effect of astigmatism on spectral switches of polychromatic Gaussian Schell-model beams. It is shown that the spectral switch can also take place in the vicinity of intensity minimum in a geometrical focal plane for the astigmatic case, but the astigmatism of the lens and the spatial correlation of the beam affect the critical position u c , spectral minimum S min and transition height Δ of spectral switches.

Off-Axis Astigmatic Gaussian Beam Combination Beyond the Paraxial Approximation

Taking the off-axis astigmatic Gaussian beam combination as an example, the beam-combination concept is extended to the nonparaxial regime. The closed-form propagation expressions for coherent and incoherent combinations of nonparaxial off-axis astigmatic Gaussian beams with rectangular geometry are derived and illustrated with numerical examples. It is shown that the intensity distributions of the resulting beam depend on the combination scheme and beam parameters in general, and in the paraxial approximation (i.e., for the small f-parameter) our results reduce to the paraxial ones.

Coherence vortices of partially coherent beams in the far field

Based on the propagation law of cross-spectral density function, studied in this paper are the coherence vortices of partially coherent, quasi-monochromatic singular beams with Gaussian envelope and Schell-model correlator in the far field, where our main attention is paid to the evolution of far-field coherence vortices into intensity vortices of fully coherent beams. The results show that, although there are usually no zeros of intensity in partially coherent beams with Gaussian envelope and Schell-model correlator, zeros of spectral degree of coherence exist. The coherence vortices of spectral degree of coherence depend on the relative coherence length, mode index and positions of pairs of points. If a point and mode index are kept fixed, the position of coherence vortices changes with the increase of the relative coherence length. For the low coherent case there is a circular phase dislocation. In the coherent limit coherence vortices become intensity vortices of fully coherent Laguerre?Gaussian beams.

Composite optical vortices in noncollinear Laguerre--Gaussian beams and their propagation in free space

Taking two Laguerre—Gaussian beams with topological charge l = ± 1 as an example, this paper studies the composite optical vortices formed by two noncollinear Laguerre—Gaussian beams with different phases, amplitudes, waist widths, off-axis distances, and their propagation in free space. It is shown by detailed numerical illustrative examples that the number and location of composite vortices at the waist plane are variable by varying the relative phase β, amplitude ratio η, waist width ratio ζ, or off-axis distance ratio μ. The net topological charge lnet is not always equal to the sum lsum of charges of the two component beams. The motion, creation and annihilation of composite vortices take place in the free-space propagation, and the net charge during the propagation remains unchanged and equals to the net charge at the waist plane.