廣義裂片的繞射理論分析兼論向量光束

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國立交通大學

光電工程研究所

博士論文

廣義裂片的繞射理論分析兼論向量光束

Diffraction Theoretic Study on Vectorial Beams

and Split Lenses

研究生：鄭介任

指導教授：陳志隆

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廣義裂片的繞射理論分析兼論向量光束

Diffraction Theoretic Study on Vectorial Beams and Split

Lenses

研究生：鄭介任 Student : Chieh-Jen Cheng

指導教授：陳志隆 Advisor : Jyh-Long Chern

國立交通大學

光電工程研究所

博士論文

A Dissertation

Submitted to Department of Photonics and Institute of Electro-Optical

Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Electro-Optical Engineering

August 2010

Hsinchu, Taiwan

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廣義裂片的繞射理論分析兼論向量光束

博士研究生：鄭介任指導教授：陳志隆教授

國立交通大學光電工程研究所

摘要

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。

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Diffraction Theoretic Study on

Vectorial Beams and Split Lenses

Doctoral Student: Chieh-Jen Cheng Advisor: Dr. Jyh-Long Chern

Institute of Electro-Optical Engineering National Chiao Tung University

Abstract

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 ( , ,ψ 2π) U(u,v,ψ)

N v u U − + = and π ψ π ψ + =−Φ − − Φ( , , 2 ) (u,v, ) N v

u 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

) , , ( ) , , ( u vψ U u vψ

U − = and Φ(−u,v,ψ)=−Φ(u,v,ψ)−π , 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.

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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.

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誌謝

從大學畢業後就來到風城求學，經過這九個年頭的努力後，終於順利拿到光電工程博士學位，感謝上天！感謝博士論文指導教授陳志隆老師多年來的指導與提攜。陳老師傳道、授業、解惑的恩惠，學生沒齒難忘。也感謝博士論文考試委員們施宙聰老師、許瑞榮老師、郭浩中老師、賴暎杰老師及謝文峰老師，給予學生許多寶貴的建議。感謝鄭伊凱學長、朱淑君同學、曹兆璽同學及鄭竹明同學在求學過程中的諸多幫忙及鼓勵。也感謝學弟妹們家佑、志雲、建成、夢華、玫君、家瑜、奎佐、燃宏、志傑、冠廷、健榮、璧滎、建德、志明、肇佑、保嘉、怡文、偉傑、宏智、偉宏、柏宇、意雯的協助，有了你們的加入，實驗室更顯朝氣。感謝偉宏、柏宇、意雯三位學弟妹們於己丑年孟秋幫忙進行實驗室搬遷，讓實驗室能順利從電資大樓搬到田家炳光電大樓。最後，感謝父母親對我的栽培，也感謝家人的支持、鼓勵與包容；同時也感謝芳能一直陪在我身邊；感謝每一位曾經幫助我的人，謝謝。－庚寅年申月風城交大

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List of Figures

Fig. 1-1 Schematic diagram of diffraction by a circular aperture...3 Fig. 1-2 Diffraction by a circular aperture with radius a=20λ. Top: Geometry and the

intensity line scans at various distances z=0.5, 90, 210, 350, 500, and 650λ.

Middle: The intensity distribution in the meridional plane. Bottom: The on-axis

intensity. ...6 Fig. 1-3 Geometry for focusing through a circular aperture. The focal point lies on the

origin of the XYZ Cartesian coordinate and the diffracted field is observed at

P(x,y,z). ...7

Fig. 1-4 Density plot of intensity distribution in the focal plane and the Airy pattern is clearly seen. The intensity is normalized to unity at focus. ...9 Fig. 1-5 Density and contour plot of intensity distribution in the meridional plane near

focus of a converging spherical wave diffracted at a circular aperture. The intensity is normalized to unity at focus. The dashed lines represent the

boundary of the geometrical shadow. ...9 Fig. 1-6 Schematic diagram of ray-tracing of a lens in the presence of primary

spherical aberration. The caustic curve can be easily seen and the wavefront is also seen by connecting the arrows in all of the rays...11 Fig. 1-7 Density and contour plot of intensity distribution in the meridional plane near

focus diffracted at a circular aperture in the aberration-free (W040=0) and in the presence of primary spherical aberration ( 4

040ρ

W

=

Φ ) of half a wavelength (W040=-0.5λ) and one wavelength (W040=-1λ). The dashed lines indicate the geometrical caustics and red vertical line denotes the diffraction focus. (a)

W040=0; (b) W040=-0.5λ and (c) W040=-1λ...12 Fig. 1-8 Schematic diagram of ray-tracing through a lens in the presence of primary

comatic aberration. The geometrical confusion figures are also shown. ...13 Fig. 1-9 Intensity distribution at the geometrical focal plane in the presence of primary

comatic aberration. The boundary of geometrical confusion figures is also shown. (a) W031=0.5λ; (b) W031=1λ and (c) W031=3λ...15 Fig. 1-10 Schematic diagram of a focusing lens in the presence of primary

astigmatism. ...17 Fig. 1-11 Intensity distribution in the presence of primary astigmatism W221=0.64 (a)

at the sagittal focal plane; (b) u=kW221 and (c) at tangential focal plane

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Fig. 1-12 Intensity distribution at the focal plane of an aplanatic lens with NA=0.866 and focal length f=30,000λ, illuminated by a linearly polarized plane wave.

Frames (a)-(c) display the intensity plot with linear scaling, while frames (d)-(f) show the intensity distribution with logarithm scaling. The peak intensities in (a), (b), (c) are in the ratios 1.0:0.0036:0.13, respectively...22 Fig. 1-13 The electric energy density near the focal region of an aplanatic lens with

NA=0.866 and focal length f=30,000λ, illuminated by a linearly polarized

plane wave. The logarithmic scaling is used here...23 Fig. 1-14 Intensity distribution at the focal plane of an aplanatic lens with NA=0.866

and focal length f=30,000λ, illuminated by a radially polarized plane wave.

Frames (a)-(b) display the intensity plot with linear scaling, while frames (c)-(d) show the intensity distribution with logarithm scaling. The peak

intensities in (a), (b) are in the ratios 0.73:1.0, respectively. ...25 Fig. 1-15 The electric energy density near the focal region of an aplanatic lens with

NA=0.866 and focal length f=30,000λ, illuminated by a radially polarized

Bessel-Gauss wave. The logarithmic scaling is used here...26 Fig. 1-16 Intensity distribution at the focal plane of an aplanatic lens with NA=0.866

and focal length f=30,000λ, illuminated by an azimuthally polarized plane

wave. Frame (a) displays the intensity plot with linear scaling, while frame (b) shows the intensity distribution with logarithm scaling. ...27 Fig. 1-17 The electric energy density near the focal region of an aplanatic lens with

NA=0.866 and focal length f=30,000λ, illuminated by an azimuthally

polarized Bessel-Gauss wave. The logarithmic scaling is used here...28 Fig. 2-1 Top: Front view from the left side, showing the arrangements of sectors when

N=2, 3, 4, 5, and 6, where N is the number of sectors. Bottom: Schematic

diagram of Billet split bi-sector lens. F1 and F2 are the first focus and second focus, respectively, and 2d is the separation distance between the foci of the

two sectors. ...34 Fig. 2-2 Notation representation of the coordinate system of beam propagation...35 Fig. 2-3 Density plot of normalized intensity distribution of the generalized N-split

lens in the XY-plane at z=5000λ where (a) N=2, (b) N=3, (c) N=4, (d) N=5, (e) N=6, (f) N=7, (g) N=8, (h) N=9, and (i) N=10. ...38

Fig. 2-4 Normalized intensity distribution of the generalized N-split lens in the

XY-plane at z=5000λ where (a) N=4, (b) N=12, and (c) Enlargement of (b). ..42

Fig. 3-1 (a) Notation of the coordinate system for beam propagation. (b) Schematic diagram of a split lens where N is the number of sectors. F1 and F2 are the first focus and second focus respectively and Δz is the separation between the two

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shown. ...48 Fig. 3-2 Summary of symmetry relations with respect to the XY-plane of z=0: (left)

classical form for perfect lens; (right) the top shows the case of double of odd number (N=2, 6, and 10) while the bottom shows the case of double of even

number (N=4, 8, and 12)...56

Fig. 3-3 Normalized intensity distribution and the corresponding phase one in the

XY-plane through the mid-point between two foci where the symbol (a) is for

intensity and the symbol (b) is for phase, while (1) for N=2, (2) for N=4, (3)

for N=6, (4) for N=8, (5) for N=10, and (6) for N=12. The separation distance

along the z-axis Δz=300λ...62

Fig. 3-4 Normalized intensity distribution and the corresponding phase one in the

XY-plane through the mid-point between two foci where the symbol (a) is for

intensity and the symbol (b) is for phase, while (1) for N=2, (2) for N=4, (3)

for N=6, (4) for N=8, (5) for N=10, and (6) for N=12. The separation distance

along the z-axis Δz=400λ...62

Fig. 3-5 Normalized intensity distribution for N=2 at various XY-planes along z-axis.

(a) z=-200λ, (b) z=-150λ, (c) z=-100λ, (d) z=-50λ, (e) z=0λ, (f) z=50λ, (g) z=100λ, (h) z=150λ, (i) z=200λ. ...63

Fig. 3-6 The corresponding phase structure for N=2 at various XY-planes along z-axis.

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z=100λ, (h) z=150λ, (i) z=200λ. ...66

Fig. 3-13 The on-axis intensity and the intensity by two half-lenses are also plotted for comparison. The separation distance along the z-axis Δz=400λ. ...67

Fig. 3-14 Intensity on mid-point of two foci varies with respect to the separation distance Δz...68

Fig. 3-15 Normalized intensity distribution near focus in the meridional plane. (a)

N=2 and ψ=0, (b) N=6 and ψ=0, (c) N=2 and ψ=π , (d) N=6 and ψ=2 π . .70 2

Fig. 3-16 Normalized intensity distribution near focus in the meridional plane. (a)

N=4 and ψ=π , (b) N=8 and ψ=4 π , (c) N=4 and ψ=8 3π , (d) N=8 and 4

ψ=3π ...70 8 Fig. 4-1 (a) Schematic diagram of the Billet split bi-sector lens. F1 and F2 are the first

focus and second focus, respectively, and 2d is the separation distance

between the foci of the two sectors. A front view on the left side shows the arrangement of sectors with N=2, 10, and 24, where N is the number of sectors.

(b) Notation representation of the coordinate system of beam propagation....76 Fig. 4-2 Normalized intensity distribution of the generalized N-split lens in the focal

plane, where (a) N=10 and (b) N=24. ...79

Fig. 4-3 Phase distribution of the generalized N-split lens in the focal plane, where (a) N=10 and (b) N=24. ...79

Fig. 4-4 The intensity distributions in the meridional plane with ψ =0 (XZ-plane) for

different number of split sectors, (a) N=10 and (b), N=24, where the intensity

is normalized to 100. Plots with enlarged scale are shown in (c), N=10, and (d), N=24, where the first three dark rings of the J0 are illustrated at the bottom. The on-axis intensity of asymptotic approximations is denoted with solid lines. The intensity within z=d/NA has been multiplied by 100 as denoted by a circle

in the plots (see text)...81 Fig. 4-5 The intensity disturbances in the meridional plane with the number of split

sectors N=24 and ψ =0 (XZ-plane) for the aperture radius at a=40000λ,

24000λ and 16000λ which corresponds to NA=0.5, 0.3 and 0.2, respectively.

The intensity is normalized by the maximum intensity of the case with

a=40000λ. The logarithmic scale is used here. The solid lines also illustrate the

dark rings of the J0 Bessel beam. ...84

Fig. 4-6 The intensity disturbances in the meridional plane with the number of split sectors N=24. The parameters were the same as in Figure 4 except that the

azimuthal angleψ =π/24...85

Fig. 4-7 The on-axis intensity for the aperture radius at a=40000λ, 32000λ, 24000λ,

16000λ and 8000λ which corresponds to NA=0.5, 0.4, 0.3, 0.2 and 0.1,

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the case with a=40000λ. The inset displays the logarithmic scaling for the

on-axis intensity. ...86 Fig. 5-1 Schematic diagrams of the geometry of a focusing system where a vector

diffraction theory is applied over a spherical wavefront surface with a circular aperture of radius a. (see text) (a) three dimensional schematic plot and (b)

schematic diagram of aperture. (c) Schematic illustration of radially polarized (RP) and azimuthal polarized (AP) beams. The corresponding arrows are used to denote the polarization...95 Fig. 5-2 Density plot of the normalized time-averaged electric energy density for the

radially polarized beams in the ρ–z plane with a numercial aperture NA=0.7,

where the dashed line is used to indicate the geometrical focal plane. The rows from top to down are with the Fresnel number N=1, N=2, N=5 and N=10

correspondingly. The radial components are shown the first colum and the corresponding phase structures are shown in the second column. The

longitudinal components are shown in the third column and the corresponding phase structures are shown in the fourth column, respectively. The density plots of the normalized total time-averaged electric energy density are shown in the fifth column...105 Fig. 5-3 Density plot of the normalized time-averaged electric energy density for the

azimuthally polarized beams in the ρ–z plane with a numerical aperture

NA=0.7, where the dashed line indicates the geometrical focal plane. The rows from top to down is with the Fresnel numbers N=1, N=2, N=5, and N=10. The

first column is the energy density where the corresponding phase structures are shown in the second column respectively. ...107 Fig. 5-4 Fractional focal shift Δf/f versus the Fresnel number N for the radially

polarized illumination (RPI). ...110 Fig. 5-5 Fractional focal shift Δf/f versus the Fresnel number N for the azimuthally

polarized illumination (API)...112 Fig. 5-6 Fractional focal shift Δf/f versus the Fresnel number N for the linearly

polarized illumination (LPI). ...113 Fig. 5-7 The variation of the ratios of (fractional) focal shifts of the radially polarized

(RP) and the azimuthally polarized (AP) to the linearly polarized illuminations (LPI). The former is denoted by a solid line while the latter is with a dash line. ...114

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List of Tables

Table 4-1 Fractions of energy at z=2d/NA for N=10, 24 and ∞, where the aperture

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1 Introduction

It is a classical topic to study field propagation and its associated diffraction behavior in the image space of an optical system. The resultant electric fields in the image space are determined crucially by the optical system. In existing literature, the optical systems with perfect lenses are classical platforms for exploring diffraction behavior. The current study considers a different approach that may be able to provide an additional basic reference for diffraction study, namely the generalized form of a split lens. There are many ways to achieve lens splitting; for example, in a configuration of Meslin’s experiment or using Billet’s split lens. Once a lens is split in multiple pieces, the resulting interference involves multiple beams and the configuration of multiple paths. This creates complicated beam propagation and interference. Nevertheless, if this generalization is implemented symmetrically, the field distribution exhibits an embedded symmetry, which reduces and simplifies the complexity of analysis and calculation. Thus, exploring the diffraction behavior with such a generalization, particularly the symmetry properties, is worthy of further research. Therefore, this study presents such a generalization of Billet’s split lens and Meslin’s split lens and

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therefore the propagation characteristics of the beams generated by Billet’s N-split

lens manipulating the resultant electromagnetic fields.

The exploration of the beam propagation characteristics of a polarized beam is not only important for fundamental understanding but also to provide a useful mean in exploring and probing the other systems as polarized illumination. In paraxial optics, the numerical aperture of the lens is small and the polarization characteristic properties of light sources are ignored. Nonetheless the polarization effect can not be ignored when the numerical aperture of lens becomes extremely large. When vector nature is included in consideration, the performance will be influenced where the focal property is always the first item to be checked because of its importance to application. The polarization properties of incident wave, and therefore, are important when we study the electromagnetic fields in an optical system having a high numerical aperture. For a light source with a special polarization, the polarization properties are crucial to the diffracted electromagnetic fields. Here some polarizations are listed below.

1. Uniform polarizations; linearly polarization, circularly polarization.

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1.1 Incident beams – scalar field

Fig. 1-1 Schematic diagram of diffraction by a circular aperture.

Fig. 1-1 shows the schematic diagram for the diffraction by an aperture. By the application of the first Rayleight-Sommerfeld diffraction formula, the diffracted field

U(P) by an aperture, having the field u(ξ,η,0), can be written as [1]

(

)

( )

∫ ∫

−∞∞ ∞ ∞ − ∂ ∂ − = ξ η ξ η π R d d ikR z u P U , ,0 exp 2 1 ) ( , (1-1)

where k=2π/λ is the wavenumber of the incident wave, λ is the wavelength and

(

) (

)

[

2 2 2

]

12. z y x R= −ξ + −η +

(

)

( )

∫ ∫

−∞∞ ∞ ∞ − − − = ξ η ξ η π R d d ikR R ikR u z P U , ,0 1exp ₂ 2 ) ( . (1-2) Q y x z a P z=0 η ξ ρ r

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When kR >> 1, the diffraction field can be expressed as

(

) ( )

∫ ∫

−∞∞ ∞ ∞ − = ξ η ξ η λ R d d ikR u i z P U( ) , ,0 exp ₂ , (1-3)

By use of Taylor series expansion of the square root

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ≈ 1 2 1 2 1 2 2 z y z x z

R ξ η and keep only the first two terms the diffraction

field now is

(

)

[

(

) (

)

]

∫ ∫

−∞∞ ∞ ∞ − _⎭⎬ ⎫ ⎩ ⎨ ⎧ ₋ ₊ ₋ = ξ η ξ η ξ η λ z x y d d k i u z i e P U ikz 2 2 2 exp 0 , , ) ( , (1-4)

which is the Fresnel diffraction integral or the near field diffraction of the aperture. To meet sufficient criteria for accuracy, the maximum phase changed by the leading term of dropped series has much less than 1 radian. Thus z has to satisfy

(

) (

)

[

]

2 max 2 2 3 4λ ξ η π ₋ ₊ ₋ >> x y z . (1-5) If

(

)

2 max 2 2 _η ξ + >> k

z is applicable the diffracted field can be further simplified as

( )

(

)

(

)

∫ ∫

−∞∞ ∞ ∞ − + ⎥⎦ ⎤ ⎢⎣ ⎡₋ ₊ = ξ η ξ η λ π η ξ λz u i z x y d d i e e P U y x z k i ikz ₂ exp 0 , , ) ( 2 2 2 , (1-6)

which is the Fraunhofer diffraction integral or the far field diffraction of the aperture. When a circular aperture with radius a illuminated by a unit-amplitude plane-wave, i.e., u(ξ,η,0)=1, the on-axis diffracted field by the first Rayleigh-Sommerfeld theory can be expressed as [2]

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( )

(

)

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + + − = 2 2 2 2 exp exp ) ( a z a z ik z ikz z z U . (1-7)

The Fresnel diffraction integral can be written as [1]

∫

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = a r z ik ikz d z ik z kr J z i e e P U 0 2 0 2 2 exp 2 ) ( 2 ρ ρ ρ ρ λ π , (1-8)

The Fraunhofer diffraction integral now is [1]

z r ka z r ka J i e e a P U r z k i ikz ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 2 2 1 2 ) ( 2 π . (1-9)

This is the so-called Airy pattern.

Fig. 1-2 shows the diffracted intensity U

( )

P 2 by a circular aperture with radius a=20λ. In the top of Fig. 1-2, the geometry and the intensity line scans through the

radial direction at various distances from the aperture z=0.5, 90, 210, 350, 500, and

650λ are illustrated. The near field of the aperture when z=0.5λ is clearly shown and

the Fresnel region can be recognized when z is beyond the near field of the aperture to z~400λ. The Fraunhofer region is approximately beyond z=400λ as the last two curves

shown. The middle of Fig. 1-2 displays the intensity distribution in the meridional plane and the on-axis intensity is shown in the bottom of Fig. 1-2.

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Fig. 1-2 Diffraction by a circular aperture with radius a=20λ. Top: Geometry and the

intensity line scans at various distances z=0.5, 90, 210, 350, 500, and 650λ. Middle:

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1.2 Optics

Assume we have an aberrated wavefront and pass through a circular aperture. The aberrated wavefront is a deformation of a spherical wave and the spherical wave converges towards an axial focal point F(0,0,0). According to the classical scalar

Debye diffraction theory the field at an observation point P(x,y,z) is then given as [3]

( )

∫ ∫

⎢⎣⎡Φ − − − ⎥⎦⎤ − = 1 0 2 0 2 1 ) cos( , ) ( 2 2 2 2 ) ( π ρθ ρ θ ψ ρ ρ ρ θ λ f e e d d A a i P U a u i k v u f i , (1-10)

where the optical units u, v, and ψ represent the Cartesian coordinate positions of P (x,

y, z). These values are z

f a u 2 2 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = λ π , 2 2 _x2 _y2 f a r f a v _⎟⎟ + ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = λ π λ π , where ψ cos r

x= and y=rsinψ . Φ

( )

ρ,θ is the aberration function. Fig. 1-3 shows the coordinate system.

Fig. 1-3 Geometry for focusing through a circular aperture. The focal point lies on the origin of the XYZ Cartesian coordinate and the diffracted field is observed at P(x,y,z).

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1.2.1 Perfect lens

In the absence of aberrations, i.e. when Φ

( )

ρ,θ =0, the diffracted field of the double integral in Eq. (1-10) can be simplified to one integral by carrying out the integral

with respect to θ using α

( )

α

π π _α _α n in ix n J e e i =

∫

− ₂ 0 cos _d

2 where Jn(α) is Bessel function

of first kind.

( )

∫

− − = 1 0 2 1 0 ) ( 2 2 2 2 2 ) ( ρ ρ ρ λ π ρ d e v J e f A a i P U a u iu f i , (1-11)

This integral can be further carried out when u=0 i.e., the field in the focal plane is

( )

v v J e f A a i P U a u f i 1 ) ( 2 2 2 2 ) ( λ π − = , (1-12)

The intensity I=|U|2₌

( )

0 2 1 2 I v v J ⎥⎦ ⎤ ⎢⎣

⎡ _{is the Airy formula for Fraunhofer diffraction at a}

circular aperture where

2 2 2 0 ⎟⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = f A a I λ π .

The intensity distribution in the focal plane is characterized by the Airy function shown in Fig. 1-4. The bottom of Fig. 1-4 shows the line scan through the focus and the Airy function can be observed clearly. Fig. 1-5 shows the intensity distribution in the meridional plane. The dashed lines indicate the boundary of the geometrical shadow. The Airy pattern can be also observed in the radial directionFig. 1-4.

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Fig. 1-4 Density plot of intensity distribution in the focal plane and the Airy pattern is clearly seen. The intensity is normalized to unity at focus.

Fig. 1-5 Density and contour plot of intensity distribution in the meridional plane near focus of a converging spherical wave diffracted at a circular aperture. The intensity is normalized to unity at focus. The dashed lines represent the boundary of the geometrical shadow.

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1.2.2 A lens with aberration

1.2.2.1 Primary spherical aberration

In the presence of primary spherical aberration only, the aberration function is

( )

4 040

,θ ρ

ρ =W

Φ , (1-13)

where W040 is the amount of wavefront deformation at the aperture edge, measured in units of wavelengths. On substituting Eq. (1-13) and carrying out the integral carrying out the integral with respect to θ by the Bessel function, the Eq. (1-10) yields

( )

∫

⎟⎠ ⎞ ⎜ ⎝ ⎛ ₋ − = 1 0 2 1 0 ) ( 2 2 4 2 040 2 2 ) ( ρ ρ ρ λ π ρ ρ d e v J e f A a i P U a u i kW u f i . (1-14)

From Eq. (1-14) the intensity is rotationally symmetry about the optical axis in the primary spherical aberration.

∫

− ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ₋ ′ − = 12 2 1 2 2 1 ) ( 2 2 2 2 2 ) ( ρ λ π ρ d e e f A a i P U F uF u u i u a f i , (1-15)

where u'=u-uF and uF=4πW040.

The axial intensity from Eq (1-15) is symmetric about the point u=uF=4πW040 and the diffraction foci, a unique position (uF,vF,ψF) has the maximum intensity, is (4πW040,0,0). The schematic diagram for a lens having primary spherical aberration

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with ray-tracing is shown in Fig. 1-6 where the caustic curves can be readily observed. The intensity distribution in the meridional plane for the case of the aberration-free, in the presence of primary spherical aberration W040=-0.5λ and W040=-1λ are shown in Fig. 1-7. The dashed lines indicate the geometrical caustics.

Fig. 1-6 Schematic diagram of ray-tracing of a lens in the presence of primary spherical aberration. The caustic curve can be easily seen and the wavefront is also seen by connecting the arrows in all of the rays.

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Fig. 1-7 Density and contour plot of intensity distribution in the meridional plane near focus diffracted at a circular aperture in the aberration-free (W040=0) and in the presence of primary spherical aberration ( 4

040ρ

W

=

Φ ) of half a wavelength

(W040=-0.5λ) and one wavelength (W040=-1λ). The dashed lines indicate the geometrical caustics and red vertical line denotes the diffraction focus. (a) W040=0; (b)

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1.2.2.2 Primary comatic aberration

In the presence of primary comatic aberration only, the aberration function is

( )

ρ,θ ρ3cosθ

031

W

=

Φ , (1-16)

and therefore, the Eq. (1-10) gives rise to

∫ ∫

⎢⎣⎡ − − − ⎥⎦⎤ − = 1 0 2 0 2 1 ) cos( cos ) ( 2 2 3 2 031 2 ) ( π ρ θ ρ θ ψ ρ ρ ρ θ λ f e e d d A a i P U a u i kW v u f i . (1-17)

Schematic diagram of ray-tracing through a lens in the presence of primary comatic aberration is shown in Fig. 1-8 and the geometrical confusion figures are also shown on the right side.

Fig. 1-8 Schematic diagram of ray-tracing through a lens in the presence of primary comatic aberration. The geometrical confusion figures are also shown.

Geometrical image

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The intensity distribution at the geometrical focal plane in the presence of primary comatic aberration W031=0.5λ, W031=1λ and W031=3λ are shown in Fig. 1-9. The diffraction focus lies in the geometrical focal plane in the presence of primary comatic aberration. When the W031 is small, the location of diffraction focus is given by [3]

vF=2_3kW031 and ψF=0. As expected, it agrees well with the diffraction foci in Fig. 1-9 (a) and (b). However, the diffraction foci in Fig. 1-9 (c) is vF~5.4 instead of vF=4π because a large comatic aberration W031=3λ is used here.

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Fig. 1-9 Intensity distribution at the geometrical focal plane in the presence of primary comatic aberration. The boundary of geometrical confusion figures is also shown. (a)

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1.2.2.3 Primary astigmatism

In the presence of primary astigmatism only, the aberration function can be written as

( )

ρ θ ρ2 2θ

022 cos

, =W

Φ , (1-18)

and therefore, the Eq. (1-10) yields

∫ ∫

⎢⎣⎡ − − − ⎥⎦⎤ − = 1 0 2 0 2 1 ) cos( cos ) ( 2 2 2 2 2 022 2 ) ( π ρ θ ρ θ ψ ρ ρ ρ θ λ f e e d d A a i P U a u i kW v u f i . (1-19)

From Eq. (1-19) the intensity distribution has reflection symmetry about both x and y axis. Fig. 1-10 shows the schematic diagram of a lens having primary astigmatism. In the sagittal focal plane u=uS=0, the Eq. (1-19) gives rise to

[ ]

∫ ∫

− − − = 1 0 2 0 ) cos( cos ) ( 2 2 2 2 022 2 ) ( π ρ θ ρ θ ψ ρ ρ θ λ f e e d d A a i P U i _af u ikW v _{. (1-20)}

In the tangential focal plane u=uT=2kW022, the Eq. (1-19) is

[ ]

∫ ∫

− + − − = 1 0 2 0 ) cos( sin ) ( 2 2 2 2 022 2 ) ( π ρ θ ρ θ ψ ρ ρ θ λ f e e d d A a i P U _a u ikW v f i . (1-21)

The diffraction pattern in the sagittal focal plane is the same as in the tangential focal plane, except for a rotation of 90°. The diffraction focus in the presence of a small amount of primary astigmatism situates in the central plane uF=kW022, midway between the sagittal and the tangential focal lines [3]. Fig. 1-11 shows the intensity

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distribution in the presence of primary astigmatism W022=0.64λ at the geometrical focal plane, at u=k W022 and u=2kW022. As expected, the intensity patterns in Fig. 1-11(a) and Fig. 1-11(c) show the rotation of 90° with respect to the optical axis.

Fig. 1-10 Schematic diagram of a focusing lens in the presence of primary astigmatism. [5]

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Fig. 1-11 Intensity distribution in the presence of primary astigmatism W221=0.64 (a) at the sagittal focal plane; (b) u=kW221 and (c) at tangential focal plane u=2kW221.

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1.2.3 Split lens

A split lens is proposed and used to generate desired focus shape near the focal region and therefore to manipulate a beam having particular propagation characteristic in the entire image space. Lens splitting can be implemented in many different ways, such as a configuration of Meslin’s experiment or Billet’s split lens [3]. Once a lens is split in multiple pieces, the resulting interference will involve multiple beams and the configuration of multiple paths, creating a relatively complex situation for beam propagation and interference.

1.3 Revisit on incident beams – vector fields in

an optical system

In an optical system with a large relative aperture, the polarization effect has been considered. For an aplanatic system illuminated by a linearly polarized incident beam, the diffracted electromagnetic fields by vector Debye theory are [4]

(

)

(

)

(

2 cos

)

, , 2 sin , 2 cos 1 2 2 0 ψ ψ ψ L iB E L iB E L L iB E z y x − = − = + − = (1-22) The factor B is

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λ πE f

B₌ 0 _{, (1-23)}

and the L functions are defined as

(

1 cos

) (

sin

)

exp

[

cos

]

d , sin cos 0 0 0 θ θ θ ρ θ θ θ α

∫

+ = J k ikz L (1-24a)

(

sin

)

exp

[

cos

]

d , sin cos 0 1 2 1 θ θ ρ θ θ θ α

∫

= J k ikz L (1-24b)

(

θ

) (

ρ θ

)

[

θ

]

θ θ θ α d cos exp sin cos 1 sin cos 0 2 2 =

∫

− J k ikz L (1-24c) where _α ₌_sin−1

(

a f

)

_.

For radially polarized illumination, the diffracted electromagnetic fields by vector Debye theory are [6]

( ) ( ) (

)

[

]

( ) (

sin

)

exp

[

cos

]

d . sin cos 2 , 0 , d cos exp sin 2 sin cos 0 0 2 0 1 θ θ θ ρ θ θ θ θ θ θ ρ θ θ θ α φ α ρ

∫

= = = ikz k J l iB E E ikz k J l B E z (1-25)

For azimuthally polarized illumination, the diffracted electromagnetic fields by vector Debye theory are [6]

( ) (

)

[

]

, 0 , d cos exp sin sin cos 2 , 0 0 1 = = =

∫

z E ikz k J l B E E θ θ θ ρ θ θ θ α φ ρ (1-26) where

( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = α θ β α θ β θ sin sin 2 sin sin exp ₁ ₀ 2 2 0 J

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radius and the beam waist. It is taken as 3/2 in the numerical simulations.

The time-averaged electric and magnetic energy densities for the diffracted fields are

defined as * 16 1 E E⋅ ≡ π e W .

Fig. 1-12 shows the intensity distribution of three components of electric fields at the focal plane by linearly polarized (along the x-axis) illumination with NA=0.866 and the focal length f=30,000λ. The intensity distribution in the left column shows by linear scaling and in the right column displays with the logarithmic scaling. The z component shows clearly an oscillating electric dipole.

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Fig. 1-12 Intensity distribution at the focal plane of an aplanatic lens with NA=0.866 and focal length f=30,000λ, illuminated by a linearly polarized plane wave. Frames (a)-(c) display the intensity plot with linear scaling, while frames (d)-(f) show the intensity distribution with logarithm scaling. The peak intensities in (a), (b), (c) are in the ratios 1.0:0.0036:0.13, respectively.

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Fig. 1-13 The electric energy density near the focal region of an aplanatic lens with NA=0.866 and focal length f=30,000λ, illuminated by a linearly polarized plane wave. The logarithmic scaling is used here.

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Fig. 1-13 shows the electric energy density in the image space near the focus by slicing the XZ- and YZ-planes together with the XY-plane at the focus and z=12λ. The logarithmic scaling is used here. The electric energy density is not rotationally symmetric about the optical axis because of the bending of the polarization is clearly observed. The diverging characteristic beyond the focus can also be seen.

Fig. 1-14 shows the intensity distribution of two components of electric fields at the focal plane with radially polarized illumination with NA=0.866 and the focal length

f=30,000λ. The intensity distribution in the left column shows by linear scaling and in

the right column displays with the logarithmic scaling. Fig. 1-15 shows the electric energy density in the image space near the focus by slicing the XZ- and YZ-planes together with the XY-plane at the focus and z=18λ. The logarithmic scaling is used here. The electric energy density is rotationally symmetric about the optical axis can be clearly observed. The diverging characteristic beyond the focus can also be seen. Fig. 1-16 shows the intensity distribution of two components of electric fields at the focal plane with azimuthally polarized illumination with NA=0.866 and the focal length f=30,000λ. The intensity distribution in the left column shows by linear scaling and in the right column displays with the logarithmic scaling. Fig. 1-17 shows the electric energy density in the image space near the focus by slicing the XZ- and

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is used here. The electric energy density is rotationally symmetric about the optical axis can be clearly observed. The diverging characteristic beyond the focus can also be seen.

Fig. 1-14 Intensity distribution at the focal plane of an aplanatic lens with NA=0.866 and focal length f=30,000λ, illuminated by a radially polarized plane wave. Frames (a)-(b) display the intensity plot with linear scaling, while frames (c)-(d) show the intensity distribution with logarithm scaling. The peak intensities in (a), (b) are in the ratios 0.73:1.0, respectively.

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Fig. 1-15 The electric energy density near the focal region of an aplanatic lens with NA=0.866 and focal length f=30,000λ, illuminated by a radially polarized Bessel-Gauss wave. The logarithmic scaling is used here.

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Fig. 1-16 Intensity distribution at the focal plane of an aplanatic lens with NA=0.866 and focal length f=30,000λ, illuminated by an azimuthally polarized plane wave. Frame (a) displays the intensity plot with linear scaling, while frame (b) shows the intensity distribution with logarithm scaling.

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Fig. 1-17 The electric energy density near the focal region of an aplanatic lens with NA=0.866 and focal length f=30,000λ, illuminated by an azimuthally polarized Bessel-Gauss wave. The logarithmic scaling is used here.

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1.4 Organization of this dissertation

This dissertation is organized as follow. The split lenses are studied in Chapter 2 and Chapter 3 in terms of the transversal and longitudinal arrangement of foci, respectively. In Chapter 2, the transversal arrangement i.e., Billet’s N-split lens, is discussed and focused on the symmetry properties of the interference patterns. The longitudinal arrangement of foci i.e., Meslin’s N-split lens is studied in Chapter 3 for the symmetry properties of field distribution. The Quasi J0 Bessel beam generated by Billet’s N-split lens is investigated in Chapter 4 by the numerical simulations and the asymptotic solution. In Chapter 5, The focal shifts has been investigated by the application of the vector Kirchhoff diffraction theory on vector beams including linearly, radially and azimuthally polarization. A comparison has also made among these three vector beams. Finally we draw our conclusions and future works in Chapter 6.

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2 Transversal foci: Billet’s N-split lens

2.1 Introduction

This chapter discusses a split having transversal foci arrangement where the focal points of sectors locating in the same plane, the original focal plane and therefore the interference pattern varies with the number of sectors of the split lens. First, we revisit the original Billet’s split lens where there are two foci located either in the X- or Y- axes and there are two kinds of interference pattern in the XY-plane can be observed in the far field away from the focal plane. One is the straight line and the other is the hyperbolas. The interference pattern of equidistant straight lines are running perpendicular to the connection line of two foci and the interference pattern of hyperbolas are the cross section of the hyperboloids of revolution having the two focal points as common foci. In the Billet’s N-split lens, we cut a conventional lens into N sectors and placing the focal points of sectors on a circle. Note that the arrangement of sectors foci is not restricted to a circle only. The interference pattern of hyperbolas lies between two adjacent equidistant straight lines and having a radian of π/N. This type of lens splitting selected causes both the interference patterns of

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equidistant straight lines and of hyperbolas 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 and hyperbolas. Moreover, there is an

N-peak interference pattern near the optical axis, resembling a concentric-circle-like

interference, can be readily observed when N is large as a result of N-beam interference.

The study of field propagation and its associated diffraction behavior is a classical topic in optics research [7]. Previous studies show that this topic has many important applications in optical testing [8] and the development of new optical devices using nano-technology [9]. The current design approach for creating optical products is still primarily based on ray optics, while diffraction-based theory generally provides a reference and base line of resolution and performance limitations. Nevertheless, the diffraction theory of optical fields remains an important research topic. Researchers continue to make active progress in this area, as indicated by the selected works of E. Wolf [10]. In viewing the demands of technology development and academic interest, Chu and Chern are dedicated to exploring far-field behavior with sub-wavelength variations, where aperture (stop) plays a key role in information retrieval [11]. In existing literature, the aperture stop (circular and rectangular) and

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perfect lens are classical platforms for exploring diffraction behavior. Studies on this topic generally fall into one of two categories:

(1) Light sources could be different, e.g., a cylindrical beam or a vector polarized beam.

(2) The lens can have aberrations, e.g., defocus, spherical aberration, coma and astigmatism.

The current study considers a different approach that may be able to provide an additional basic reference for diffraction study, namely the generalized form of a split lens.

There are many ways to achieve lens splitting; for example, in a configuration of Meslin’s experiment or using Billet’s split lens [3]. Once a lens is split in multiple pieces, the resulting interference involves multiple beams and the configuration of multiple paths. This creates complicated beam propagation and interference. Nevertheless, if this generalization is implemented symmetrically, the field distribution exhibits an embedded symmetry, which reduces and simplifies the complexity of analysis and calculation. Thus, exploring the diffraction behavior with such a generalization, particularly the symmetry properties, is worthy of further research. Therefore, this study presents such a generalization of Billet’s split lens.

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Note that previous research has developed such a Billet’s split lens for multiple imaging and multichannel optical processing [12].

2.2 Symmetry properties

The bottom of Fig. 2-1 shows a schematic diagram of Billet’s split bi-sector lens, where a conventional focusing lens is split into two identical halves (two sectors). The upper half and the lower half are then moved a distance d up and down the Y-axis, respectively. This split lens creates a collimated uniform monochromatic wave with a wavelength of λ for two different foci, F1 and F2, in the focal plane. The diffraction theory applied here assumes that the aperture radius a >> λ, the focal length f >> a >>

λ, and the Fresnel number a2/λf is much larger than unity. When the (half) translation length d is zero, the two foci will coincide and the integral representation of the disturbance U(P) at a point P(x,y,z) in the image space is [3]

∫ ∫

−⎢⎣⎡ − + ⎥⎦⎤ − = 1 0 2 0 2 1 ) cos( ) ( 2 2 2 2 ) ( π ρ θ ψ ρ ρ ρ θ λ f e e d d A a i P U a u iv u f i , (2-1)

where the optical units u, v, and ψ represent the Cartesian coordinate positions of P (x,

y, z). These values are z

f a u 2 2 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = λ π , 2 2 _x2 _y2 f a r f a v _⎟⎟ + ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = λ π λ π , where

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ψ

cos

r

x= and y=rsinψ . Fig. 2-2 shows the coordinate system. The disturbance

U(P) is [ ]

∫

⎥⎦ ⎤ ⎢ ⎣ ⎡ ₊ − − = 1 0 2 0 sin cos 2 2 1 2 ) ( π λ ρ θ θ π ρ ρ θρd d e e C P U f x y a i u i , (2-2) where a u f i e f A a i C 2 ) ( 2 2 λ − = .

Fig. 2-1 Top: Front view from the left side, showing the arrangements of sectors when

N=2, 3, 4, 5, and 6, where N is the number of sectors. Bottom: Schematic diagram of

Billet split bi-sector lens. F1 and F2 are the first focus and second focus, respectively, and 2d is the separation distance between the foci of the two sectors.

To generalize lens splitting, a focusing lens is divided into N equiangular sectors. Each sector is exploded and translated a distance d in the r direction along the

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perpendicular bisector of the angle. Fig. 2-1 shows the schematic layouts of the simplest cases of N=2, 3, 4, 5, and 6. Ray-based analysis shows that the foci of all sectors constitute a regular N-sided polygon in the focal plane. Therefore, the focal point of each sector is

(

1 2

)

, 0,1,.., 1 2 , sin , cos − = + = = = N m m N d y d x m m m m m π ψ ψ ψ (2-3)

Fig. 2-2 Notation representation of the coordinate system of beam propagation.

By applying coordinate translation and summing the contributions from all sectors, the disturbance U(P) is

( ) ( ) [ ] ( ) , ) ( 1 0 1 0 2 1 2 sin cos 2 2 1 2

∫

∑∫

− = + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ₋ ₊ ₋ − − = _π π λ ρ θ θ θρ ρ π ρ d d e e C P U N m N m N m y y x x f a i u i m m (2-4)

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Substituting Eq. (2-3) into Eq. (2-4) leads to ( ) ( ) [ ] ( )

∫

∑∫

₌− + ⎭⎬ ⎫ ⎩ ⎨ ⎧ ₋ ₋ ₋ − − = 1 0 1 0 2 1 2 cos cos 2 2 1 2 ) ( θρ ρ π π ψ θ ψ θ ρ λ π ρ d d e e C P U N m N m N m d r f a i u i m (2-5)

We can change the interval of integration for each segment to be the same value from 0 to 2π/N and the disturbance is

∫

∑∫

₌− ⎭⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ₊ ₋ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ₊ ₋ − − = 1 0 1 0 2 0 2 cos 2 cos 2 2 1 2 ) ( θρ ρ π _ρ _θ π _ψ _θ π _ψ λ π ρ d d e e C P U N m N N m d N m r f a i u i m (2-6)

After substituting Eq. (2-3) into Eq. (2-6), the d term in the brackets of exponential function is no longer a function of m. The summation can be put into the integrand of

r term only, i.e.,

∫

∑

₌− ⎟⎠ ⎞ ⎜ ⎝ ⎛ ₋ ₊ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − × = 1 0 2 0 1 0 2 cos cos 2 1 2 ) ( θρ ρ π _ρ _θ π _ρ _θ _ψ π ρ d d e e e C P U N N m N m iv N iv u i d , (2-7) where d f a v_d _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = λ π 2

. Now, the azimuthally symmetrical property at a specific z plane with respect to the optical axis is

), , , ( ) 2 , , ( ψ π U u vψ N v u U − = (2-8)

This shows that a disturbance on a specified z plane is rotationally symmetrical with an angle of 2π/N. In other words, the disturbance repeats itself every 2π/N along the

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azimuthal direction, and hence, an N-fold rotational symmetry originates from the symmetry in the split form in the lens.

2.3 Billet’s N-split lens

The light distributions of Billet’s N-split lens are explored by using numerical simulations. To fulfill the condition of diffraction beam, i.e., the numerical aperture should be around 0.1, we used a typical lens with a focal length of a few ten mm, e.g.,

f=30,000λ, to create an aperture radius a=3,000λ. This lens also has the (half)

separation distance d=100λ. For the numerical example of λ=630 nm, these settings lead to f=18.9mm, a=1.89mm, and d=0.063mm. The plot of intensity distributions was normalized to 100. As a base reference, we revisited the classical Billet spilt lens, i.e., N=2. After the focal plane, the interference pattern in the XY-plane formed within the overlap region lit by two sectors. The intensity distribution in Fig. 2-3(a) reflects this result, clearly showing a two-fold symmetry [3]. The diffraction pattern contributed from each sector beyond the focal plane is similar to the original half sector, but rotated π radians around the new translated axis. This new translated axis is parallel to the optical axis throughout the focus of each sector. The lights from the two semicircles form an overlapped region near the optical axis and create interference. The interference patterns, therefore, are equidistant straight lines parallel to the lines

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that cut the spherical wavefront into two hemispheres, and are perpendicular to the line through two focal points F1 and F2. On the other hand, the diffraction patterns outside the overlapped region lit by two sectors are hyperbolas, which are sections of hyperboloids of revolution about the F1F2 axis and have F1 and F2 as common foci [3].

Fig. 2-3 Density plot of normalized intensity distribution of the generalized N-split lens in the XY-plane at z=5000λ where (a) N=2, (b) N=3, (c) N=4, (d) N=5, (e) N=6, (f)

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2.3.1 Interference pattern of straight lines

To identify how splitting a lens changes the intensity distribution, this study evaluates the distributions for N=2 to N=10 in the far field (here, z=5000λ=3.15mm, if

λ=630nm). Fig. 2-3(b) shows that a 3-fold symmetry is readily apparent when N=3.

However, the fringes of hyperbolas must have near 2π/3 radians, instead of π radians in the case of N=2. In the case of N=2, they are in the opposite direction of the equidistant straight lines, and the fringes of equidistant straight lines in the XY-plane appear inside the overlap region lit by each two sectors. On the other hand, for the case of N=3, there are three distinct straight-line fringes that are parallel to three lines located at the angles of π/3, π, and 5π/3, respectively. Nevertheless, these three lines do not coincide with the lines of sector division, but rotate an additional angle of π/3 around the optical axis. This is because the diffraction pattern of each sector (beyond the focal plane) rotates π radians around each translated axis parallel to optical axis through the focus of each sector. In addition, the two straight cutting edges of each sector also rotate by π radians, and therefore interfere with each other after a rotation of the azimuthal angle of π. When N is odd, the rotation prevents the interference pattern of each straight line from coinciding with the original cutting edge of each sector, and all fringes of straight lines resemble an angle of rotation of π/N-radian (mod 2π) around the optical axis. When N is even, the rotation of the azimuthal angle

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of π also shifts the interference pattern of straight line π radians, but the rotated interference pattern of straight lines coincides with the original cutting edge of other sectors due to reflection symmetry between the X- and Y-axes. Under these circumstances, all straight line fringes look like they have not rotated, and remain located within the cutting edge of each sector. Similarly, the interference pattern of equidistant straight lines by an N-split lens is oriented at the angle of m·2π/N when N is an even number but for the case of odd-number lens splitting, the interference pattern is at the angle of π/N + m·2π/N, which requires an additional angle shift of

π/N.

2.3.2 Concentric-circle like interference pattern

Next, consider the central region of intensity distribution near the optical axis. Fig. 2-3 indicates that as N becomes larger, the intensity distribution centered on a specified location along the optical axis begins to resemble a concentric-circle-like interference pattern, while for a small N, the distribution is more like a regular

N-sided polygon. This polygon basically represents the split distance d, i.e., the

circumradius is limited by d. The intensity distribution in the central region involves multiple-beam interference, i.e., all sectors contribute to the total field. However, only the beam interference of two sectors, i.e., the overlap of the field from two sectors

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causes the straight-line-like interference pattern in the outward azimuthal regime. For example, consider the case of N=12, where Fig. 2-4(b) plots the far-field intensity distribution (z=5000λ for simplification). The central regime contains concentric-circle-like interference patterns. The second bright ring actually has twelve peaks, and each of which is located along the azimuthal direction with an angle difference of π/6. However, the difference between the minimum and maximum intensity of the second bright ring is too small to identify in Fig. 2-4(b). Therefore, Fig. 2-4(c) presents an enlargement of Fig. 2-4(b). Here, the peak distributions are readily apparent in the inner rings, and twelve peaks clearly appear in the second and fourth bright rings. From the symmetrical properties deduced above, numerical simulation reveal that the distance between two successive peaks is ~dcos(π 12) ~0.122mm, if λ=630nm, where the peak location is defined according to the maximum, even at such a far-field distance. Numerical simulations also show that as N becomes larger, the concentric-circle-like interference patterns inside the region of circumscribed circle become significant at various XY-planes, i.e., at any specified locations along the optical axis, provided that the interference occurs after the focal plane. However, the symmetry properties deduced above remain the same.

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(a)

(b)

(c)

Fig. 2-4 Normalized intensity distribution of the generalized N-split lens in the

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2.4 Summary

In summary, the diffraction behavior of a generalized Billet’s N-split lens has been derived based on symmetry consideration. The intensity distributions vary significantly as the number of split sectors increases, particularly compared with the original Billet’s split lens. Nevertheless, there is a symmetry relationship embedded in this class of split lenses. Due to lens-splitting form adopted in this study, the intensity distribution has an N-fold rotational symmetry with respect to the optical axis in the

XY-plane. The interference patterns of equidistant straight lines are orientated at the

angle of m⋅2π N when N is even, but at the angle of π N+m⋅2π N when N is odd, where m=0, 1, 2,…, N-1. In other words, there are two kinds of symmetry even though the corresponding splitting operation is simple. The interference of the disturbance by two adjacent sectors of the split lens is the physical origin of the fringes of equidistant straight lines. In addition, this symmetrical property is physically traceable based on the symmetry embedded in the splitting form of lens.

A concentric-circle-like interference pattern near the optical axis appears when N is larger than 10. This feature is primarily due to multiple-beam interference. The multiple-beam interference inside the inner regime forms a polygon boundary of intensity distribution in which the distance between two successive maximum peaks

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is~dcos(π N). When the number of sectors in split lens becomes very large, the polygon nearly becomes a circle.

Note that the symmetry embedded within the generalized split lens and the straight-line provide two basic guidelines for forming the azimuthal light distribution while the central regime hosts a concentric-like distribution. Practically, the proposed approach to the generalized split lens provides more means of controlling light beams. Though this study is limited to Billet’s split lens, different symmetrical forms in lens splitting will lead to different kinds of light distribution. It is also possible to implement this generalized Billet’s N-split lens with liquid crystal, i.e., a segmented-aperture optical system in which phase-shifting material, here liquid crystal, fills each segmented region [13-14].

廣義裂片的繞射理論分析兼論向量光束

國立交通大學

光電工程研究所

博士論文

廣義裂片的繞射理論分析兼論向量光束

Diffraction Theoretic Study on Vectorial Beams

and Split Lenses

研 究 生：鄭介任

指導教授：陳志隆

廣義裂片的繞射理論分析兼論向量光束

Diffraction Theoretic Study on Vectorial Beams and Split

Lenses

研究生：鄭介任 Student : Chieh-Jen Cheng

指導教授：陳志隆 Advisor : Jyh-Long Chern

國立交通大學

光電工程研究所

博士論文

A Dissertation

Submitted to Department of Photonics and Institute of Electro-Optical

Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Electro-Optical Engineering

August 2010

Hsinchu, Taiwan

廣義裂片的繞射理論分析兼論向量光束

博士研究生：鄭介任 指導教授：陳志隆教授

國立交通大學 光電工程研究所

摘 要

Diffraction Theoretic Study on

Vectorial Beams and Split Lenses

Doctoral Student: Chieh-Jen Cheng Advisor: Dr. Jyh-Long Chern

Abstract

誌 謝

Contents

List of Figures

List of Tables

1

Introduction

1.1 Incident beams – scalar field

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研究生：鄭介任

博士研究生：鄭介任指導教授：陳志隆教授

國立交通大學光電工程研究所

摘要

誌謝