Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system

https://doi.org/10.1016/S0030-4018(03)01729-2Get rights and content

Abstract

By introducing the hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression for the one-dimensional off-axial Hermite–cosine–Gaussian beams passing through an apertured and misaligned paraxially ABCD optical system has been derived. As special cases, the corresponding closed-forms for the off-axial or non-off-axial Hermite–cosine–Gaussian beams passing through apertured or unapertured and misaligned or aligned paraxially ABCD optical systems have also been given. The results provide more convenient for studying their propagation and transformation than the usual way by using diffraction integral directly, which can be straightforward to the two-dimensional case. Some numerical examples are also illustrated.

Introduction

In recent years, the Hermite–sinusoidal–Gaussian beams are one of the solutions of the paraxial wave equation in the rectangular coordinate system shown by Casperson and Tovar [1], [2], and a comprehensive study of their theory including the field distribution, production and propagation has been given. In fact, the Hermite–sinusoidal–Gaussian beams represent the more general beams, and the Gaussian, Hermite–Gaussian, cosine (sine)–Gaussian, cosh (sinh) and cosine (sine) beams could be regarded as their special cases. The cosh-Gaussian beams and cosine-Gaussian beams passing through a paraxial ABCD optical system had been extensively studied [3], [4], [5]. Up to now, the off-axial Hermite–cosine–Gaussian beams passing through an apertured and misaligned paraxially ABCD optical system, to our knowledge, have not been studied elsewhere. Practically, misaligned optical systems always exist and exhibit small deviations from perfect alignment [5]. The tolerance in designing and manufacturing optical elements, the error in locating and aligning optical systems, and the thermal displacements or thermal deformations of optical elements are common characteristics of real devices. In this paper, we will study their propagation or transformation based on the expansion of the hard aperture function into a finite sum of complex Gaussian functions.

The paper is organized as follows. The matrix analyses of the misaligned optical systems are simply given in Section 2. The propagation of one-dimensional off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned paraxially ABCD optical system is analysed and some discussions are given in Section 3. Some numerical simulations are illustrated in Section 4. Finally, a simple conclusion is outlined in Section 5.

Section snippets

Generalized diffraction integral formulae for misaligned optical systems

For simplicity, let us consider the one-dimensional case. In this case, a generalized form of the diffraction integral for misaligned optical systems in spatial-domain could be written in terms of transfer matrix elements [5], [6]E2(x2)=iλB−∞+∞E1(x1)expik2BAx12−2x1x2+Dx22+Ex1+Gx2dx1,where a constant phase term exp(−ikL0) has been omitted, andE=2(αTε+βTε),G=2(BγT−DαT)ε+2(BδT−DβT.Note that the transfer matrix elements A,B,D and the misalignment matrix elements αT, βT, γT, δT are not only

Propagation of one-dimensional off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned paraxially ABCD optical system

It is known that the field distribution of the one-dimensional off-axial Hermite–cosine–Gaussian beams at the input plane is characterized by [2]E1(x1)=E0expik2qx12+sx1+pHm2w(x1−δ)cos(ux1),where E0 is the constant amplitude, q represents the complex curvature radius of Gaussian beam, s denotes the displacement parameter associated with the Gaussian part, p means the constant phase factor, w is the spot size of Gaussian beam, δ indicates the displacement parameter associated with the Hermite

Numerical simulations

First, let us examine what extent the expansion of the Gaussian function represented by Eq. (11) matches the hard aperture function. Fig. 1(a) shows the real and imaginary parts of Eq. (11), evaluated by the coefficients An and Bn listed in Table 1 which are given by Wen an Breazeale [7]. Fig. 1(b) also gives the magnitude and phase of this expansion. One can see at this figure that the errors reach ∼5–6% and even 12% near the aperture edge. However, it should be noted that Eq. (11) is only an

Conclusions

In conclusion, by using the expansion of the hard aperture function into a finite sum of complex Gaussian functions, the closed-form propagation expression for the one-dimensional off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned paraxially ABCD optical system is given in this paper, which could easily reduce to the cases of unapertured or non-off-axial or aligned. The obtained results provide more convenient for treating their propagation or transformation, for

Acknowledgements

This work is supported by National Natural Science Foundation of China (NSAF United Foundation), Grant No. 10276034.

References (9)

  • L.W. Casperson et al.

    J. Opt. Soc. Am. A

    (1998)
  • A.A. Tovar et al.

    J. Opt. Soc. Am. A

    (1998)
  • B. Lu et al.

    Opt. Lett.

    (1999)
  • S. Wang et al.

    Optik

    (1995)
There are more references available in the full text version of this article.

Cited by (61)

View all citing articles on Scopus
View full text