廣義裂片的繞射理論分析兼論向量光束

Abstract

對切透鏡有比勒(Billet)横向及梅斯林(Meslin)縱向切割兩種型式。比勒對切透鏡在遠場會產生雙曲面及等間距直線這兩種干涉條紋，而梅斯林對切透鏡在兩透鏡焦點中間附近會產生半圓型的干涉條紋。本論文延伸對切透鏡成裂解透鏡，並討論比勒裂解透鏡之焦點形成一個圓形時的遠場干涉條紋分佈情形及其相對於光軸的對稱性；在梅斯林裂解透鏡中，我們將一個圓透鏡切割成2N塊等角度透鏡，編號為單數與雙數分別放置於不同焦點處，並討論其光場分佈之對稱性。相對於通過兩焦點之中心點的垂直面；N為單數時，振幅及相位分別為鏡面反射對稱及反對稱(扣除π相位角)，N為偶數時，在鏡面反射對稱與反對稱外要再加上額外的2π/N的旋轉角。此外，從光場的漸近表示式可以得知比勒裂解透鏡可以用來產生貝索光束(Bessel beam)，並可再經由另一透鏡消除發散相位成一無繞射貝索光束(non-diffracting Bessel beam)。相對於傳統的環形孔徑，使用此透鏡所產生的貝索光束可以攜帶更多能量。有趣的是在漸近表示式中，貝索函數之參數與數值孔徑無關，但是數值孔徑卻決定了漸近表示式的適用範圍。因此，數值孔徑會決定貝索光束的發散情形與其所適用的漸近表示式範圍。 另一方面，在完美透鏡(perfect lens)成像系統中，焦點位移(Focal shift)效應發生於菲涅耳數(Fresnel number)小於10的情形之下，若使用線偏振光(linearly polarized)的入射光學系統中，則焦點位移效應與菲涅耳數及數值孔徑皆成反比。本論文討論在徑向偏振光(radially polarized)及方位偏振光(azimuthally polarized)入射情形下，焦點位移效應與菲涅耳數及數值孔徑並不會只有單純反比關係。此外，三種不同偏振光在同一個系統參數下(亦即相同菲涅耳數和數值孔徑)，方位偏振光所造成的焦點位移效應最嚴重，徑向偏振光次之，線偏振光最輕微。

This study examines the diffraction properties of the generalized split N-sector lens originating from the configuration of Meslin’s experiment and the Billet’s split bi-sector lens. In Billet’s N-split lens, the type of lens splitting selected causes the interference pattern of equidistant straight lines in the original Billet’s lens to form an N-fold angularly distributed pattern with an angle difference of 2π/N. For an odd number of splitting N, there is an additional angle shift of π/N for the azimuthally distributed patterns of equidistant straight lines. In other words, there are two kinds of symmetry even for simple splitting operations. On the other hand, the peak intensity distribution in the central portion resembles a concentric-circle-like pattern, when N is large as a result of N-beam interference. As to the Meslin’s N-split lens, the amplitude and the phase follow and respectively when the splitting is with double of an even number. On the other hand, for the case of double of an odd number, the relation changes to hold with and , where the optical units u and v are used to denote the z- and the radial coordinates respectively and the azimuthal angle is ψ. Additional symmetry properties are also explored and identified, particularly for the distributions on the focal plane. Moreover, the Bessel beam is studied and by the use of the Billet’s N-split lens distributing the focal points circularly on the focal plane. This study explores the characteristics of beam propagation and analytically derives the asymptotic characteristics of beam propagation based on the stationary phase approximation and the moment-free Filon-type method. Results show that the unique Billet’s N-split lens can generate a quasi-Bessel beam if the number of splitting N is large enough, e.g., N≧24. This study also explores the diffraction efficiency of corresponding quasi-Bessel beam and the influence of aperture size. The potential advantage of proposed split-lens approach is that, unlike the classical means of annual aperture, this simple lens approach allows a much large throughput in creating the Bessel beam and hence the Bessel beam could have more optical energy. The diffraction behaviors of cylindrical vector beam, particularly the focal shifts further caused by different polarizations, namely linear, radial and azimuthal, are also investigated. The variation of focal shifts associated with numerical aperture and the Fresnel number is also explored. It is found that with a low numerical aperture, e.g., 0.1, the focal shifts associated by the radially and azimuthally polarized illuminations are nearly the same, while they are about 1.65 times as large as that of linearly polarized illumination. As the system is of high numerical aperture, e.g., 0.9, the focal shifts associated by the radially and azimuthally polarized illuminations have ~10% difference and their ratios with that of linearly polarized illumination become double in comparing with the case of low numerical aperture. In general, azimuthally polarized illumination has the largest power in shifting the focal point.