Summary - 廣義裂片的繞射理論分析兼論向量光束

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⋅2

π

N when N is even, but at the angle of

π

N+m⋅2

π

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

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].

3 Longitudinal foci: Meslin’s N-split lens

3.1 Introduction

In previous chapter, we have studied the generalization in Billet’s configureation of split lens where the focal points are distributed in a XY-plane especially in a circular shape. Here we would like to consider a different approach which should be able to provide additional reference for the diffraction study of a split lens. In this chapter, we present our result of such a generalization in Meslin’s configuration of split lens where the focal points are distributed along the optical axis. Typically, the lens splitting could be implemented in many different ways as may be referenced in the literatures of multiple-beam interference and interferometry [6,7]. We, however, focus on the characteristics of focal point which is the key of identification in considering beam propagation. It is worthwhile to note that the corresponding distributions of focal points in classical Meslin’s experiment and Billet’s split lens [6]. For Meslin’s experiment, the two focal points are along the optical axis, while for Billet’s split lens, the two focal points are located vertically, i.e., on a plane normal to the optical axis.

In other words, it is possible that by successive lens splitting, the lens becomes a

special lenslet array and the distribution of focal points becomes a line either along the optical axis or on a plane that is normal to the optical axis.

It could be understood that once lens is split in multiple pieces, the incident beam will be separated into multiple beams in a multiple configurations of path, which result in a quite complicated situation in beam propagation and interference pattern.

Nevertheless, if the generalization is implemented with symmetry, the field distribution is expecting to exhibit the embedded symmetry, and hence the complexity of analysis may be reduced and the calculation could be simplified. As an academic exploration, it should be worthwhile to investigate the diffraction behavior with such a generalization, particularly the symmetrical property. In this chapter, we present our result of such a generalization in Meslin’s configuration of split lens where the focal points are distributed along the optical axis, the longitudinal arrangement of foci.

3.2 Theoretical formalism

Referring to Fig. 3-1(a), the notation of the coordinate system for the beam propagation with a perfect lens is provided. The perfect lens, having focal length f and aperture radius a, brings a collimated uniform monochromatic wave of wavelength λ to the image space. If f >> a >> λ, and if, in addition, the Fresnel number a²/λf is

much larger than unity, the Debye integral, will give a good approximation of the disturbance U(P) at the point P(x,y,z) in image space which follows [3]

∫ ∫

⁻^⎢⎣^⎡ ⁻ ⁺ ^⎥⎦^⎤

where A is the amplitude of incident beam, and the optical units u and v together with azimuthal angle ψ are used to specify the location, i.e., _z

π . The disturbance U(P) has a symmetric property,

i.e.,U(−u,v,

ψ

π

)= − U(u,v,

[ ψ

)

]

^∗ or U(−u,v,

ψ

)= − U(u,v,

[ ψ

)

]

^∗ because of rotational symmetry, where * is the complex conjugate as shown by Collett and Wolf [15]. It can be readily shown that the symmetric properties of amplitude and the phase Φ are

The amplitude (intensity) has a symmetry of reflection about the focal plane z=0, while the phase has reflection anti-symmetry, apart from an additive factor π.

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 lenses. On the top, the sector arrangements for N=2, 4, 6, 8, 10, and 12 are shown.

Referring to Fig. 3-1(b), when the lens is split (the distance Δz is finite), a conventional focusing lens with a circular aperture becomes a split lens and two different foci, F1 and F2, appear along the Z-axis. The major concern of this chapter is to explore the interference pattern in the XY-plane and together with the embedded

symmetry properties could be changed by the lens splitting. The motivation is to keep

the focal points still along the optical axis. Hence, we follow the well-known Meslin’s experiment configuration to split lens in N equiangular sectors, i.e., the sectors are alternatively shifted and un-shifted as illustrated on the top of Fig. 3-1(b). For simplification, we only consider the case of even-number N. The new origin of coordinate system (z=0) is set at the mid-point between two foci.

In the following we will deduce the disturbance first. There are mainly two kinds of operation in exploring the symmetry.

(1) u→ −u, which is to identify the reflection symmetry with respect to the mid-focus point at z=0.

(2) u= u *, which is a fixed value to explore the rotational symmetry on the XY-plane.

To evaluate the disturbance we consider the contributions from the shifted sectors and the un-shifted ones. The optical units for these two kind of sectors are now

written as _⎟

point P follows

[ ]

′ . The first part of integrand is for the shifted sectors and the second

part is for the un-shifted ones. As to be shown below, it is not straightforward that the symmetry properties could be categorized in two kinds following the splitting number is double of an even number (e.g., N=4, 8, 12, …, defined as “double-even”) or double of an odd number (e.g., N=2, 6, 10, …, defined as “double-odd”).

3.2.1 N is double of even number

We first discuss the number of sectors N is double-even. Based on Eq. (3-3), we separate the set of angular integration into two sets of equal length. The former one is with the index k from 0 to ^N_{4 −}¹ and the latter one is from ^N₄ to ^N_{2 −}¹, i.e.,

[ ] set is also shifted to have the same interval with the former set.

[ ]

The additional angle of π in the cosine makes the former and latter sets to be complex conjugates to each other. Therefore, the imaginary part of the integrand of polar angle can be canceled out. By changing the integration interval of each term to be the same, the disturbance becomes:

⎪⎭

We introduce a factor

words, the amplitude and the phase with respect to the XY-plane of z=0 follow

U(−u,v,

ψ

π

Next let us explore the symmetry property on the XY plane, particularly for z=0. Also with Eq. (3-7), when u ~=u , a fixed value, it could be shown that

Φ . In addition, the disturbance in the XY-plane

at mid-focus z=0 can be further reduced as 2 )

[

(0, , )

]

the rotational symmetry of intensity distribution in this plane is

)

I + = , i.e., an N-fold rotational symmetry. The rotational symmetry of phase distribution in this plane is Φ

ψ

+ 2

π

)=−Φ(0, ,

ψ

)−

π

an N-fold rotational antisymmetry, apart from a constant π. In other words, there is an transition of symmetry, the symmetry changed from N/2-fold to N-fold and back to N/2-fold as cross over z=0 plane.

3.2.2 N is double of odd number

Next, we derive the disturbance for the case of double-odd. The disturbance is

( )

By changing the order of later set and splitting into two sets with equal length, U(P) becomes

We further shift the angle of integration of former set by π and by –π in later set such that

(3-11)

By adding a negative sign and remove the π in the cosine in the later set, the disturbance becomes

( )

After reuniting the later set, Eq. (3-12) becomes

( )

The disturbance could be further simplified by changing the integration interval of each term to be the same, i.e., the former and latter sets are complex conjugates to each other. Then, the imaginary part of integrand of polar angle can be canceled out and the disturbance follows

( )

With Eq. (3-14), it could be seen that the disturbance has a symmetrical property with

respect to the XY-plane of z=0, i.e., U(−u,v,

ψ

)= − U(u,v,

[ ψ

)

]

^∗. As a result, the symmetrical properties of amplitude and phase can be expressed as

U(−u,v,

ψ

) = U(u,v,

ψ

) and Φ(−u,v,

ψ

)= −Φ(u,v,

ψ

)−

π

. (3-15)

Next let us explore the rotational symmetry about the optical axis on various XY-planes, where the location on the z-axis is fixed ( u ~=u ), follows

) ,

~, ( 4 )

(

ψ π

U u u v

ψ

v N u u

U = + = = . In other words, the amplitude and the phase with respect to the optical axis in various XY-planes follow

) ,

~, ( 4 )

(

ψ π

U u u v

ψ

v N u u

U = + = = and 4 ) ( ~, , )

(

ψ π

u u v

ψ

v N u

u= + =Φ =

respectively. Both the amplitude and phase have N/2-fold rotational symmetry about the optical axis in various XY-planes. In addition, from Eq. (3-14) the disturbance in the XY-plane of z=0 (u=0) is purely imaginary. Unlike that case of double even, there is no symmetry transition for double-odd.

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).

A comparison of symmetry properties associated with the operation (1), u→ −u, is provided in Fig. 3-2. On the left, the classical result of symmetry property deduced by Collett and Wolf [15] is indicated; on the right-top the split case of double-odd is summarized while on the right-bottom, it contains the cases of double-even. The major difference is the appearance of angle shift, 2π/N in the double-even case, while in the case of even-odd, the reflection symmetry is kept in intensity distribution and phase distribution is still has reflection antisymmetry apart from an additive factor π.

For both of two cases, the axial disturbance can be expressed as

( ) ( )

And hence, the disturbance at origin is

( )

with the Euler’s formula:

⎥⎦⎤

For example, if the focal length f=30000λ, a=3000λ and Δz=300λ, so that the disturbance at origin is

λ

) 150 0

( iA

U = .

3.3 Numerical explorations

In this section, we provide the result of numerical exploration. Without loss of generality, the focal length is taken as f=30,000λ, aperture radius a=3,000λ, and the separation distance along the z axis Δz=300λ or Δz=400λ. The plots of intensity distribution are normalized to 100. If with a He-Ne laser, the wavelength λ=632.8nm, then f=18.984 mm which is a typical lens, and the aperture a=1.90 mm, while Δz=0.18984 or =0.25312 mm are generally available. The observation plane is set at z=0 where the interference pattern can be clearly seen as shown in Fig. 3 where the

intensity distribution and the phase distribution for the case of N=2 (i.e., 2X1) are denoted with Fig. 3-3(1a) and Fig. 3-3(1b) respectively. The result (1a) is well known in literature [3,7]. The corresponding phase Fig. 3-3(1b) shows winding broken strips in distribution where the strip boundary indicates phase jumps as noted by the color changes in the plot. The connection between intensity and phase could be identified by the similarity sharing in the forms of distribution.

Next let us see the intensity distributions of N=6 (i.e., 2X3) and N=10 (i.e., 2X5), which are denoted by Fig. 3-3(3a) and Fig. 3-3(5a), while the corresponding phase distributions are shown in Fig. 3-3(3b) and Fig. 3-3(5b) respectively. Now, the N/2-fold rotational symmetry about the optical axis is clearly observed whereas there

is no rotational symmetry for N=2, though it is still 2/2-fold, i.e., 1-fold. On the other hand, the phase distribution displays the phase changes by π abruptly and as numerically identified that there are only two kinds of value in phase, i.e., ±

π

2. This is because the disturbance in this XY-plane is purely imaginary and this feature was also observed in the focal plane focused by a conventional focusing lens [3]. The origin is mainly the inversion symmetry in imaging for a conventional lens (singlet);

this also leads the same feature to the double-odd case.

Now we change to look on the cases of N=4 (i.e., 2X2), N=8 (i.e., 2X4), and N=12 (i.e., 2X6). The results are dramatically different and they are denoted with Fig.

3-3(2a), (4a), and (6a) for the intensity distribution respectively, while the corresponding phase distributions are labeled with Fig. 3-3(2b), (4b), and (6b) respectively. The phase has N-fold rotational anti-symmetry apart from a factor π about the optical axis in the XY-planes at z=0 as shown in Fig. 3-3 with the labels of Fig. 3-3(2b), (4b), (6b) and the intensity pattern in this plane has N-fold rotational symmetry as denoted by Fig. 3-3(2a), (4a), and (6a). The variations in phase distribution are much wild; the phase values are no more kept with only two values because of the disturbance is not purely imaginary.

In short, one could numerically identify that although the splitting operation is simply with an even number, there are two kinds of distribution and they could be further classified according the number of splitting, i.e., either double-even or double-odd.

Fig. 3-4 plots the intensity and phase distribution in the XY-plane having the same condition with Fig. 3-3 but the separation distance along the z-axis Δz now is 400λ. The embedded symmetry can still be observed but the intensity in the vicinity of the optical axis is faint. This destructive interference is caused by the nearly -180°

Gouy phase shift of between two beams focused by the “shifted” and “un-shifted”

half-lenses.

Fig. 3-5 and Fig. 3-6 show the intensity and phase distribution for N=2 in various XY-planes along the optical axis. From (a) to (i) the observation plane moves in a step

of 50λ from the first focus at F1, to the second focus, at F2. The separation distance along the z-axis Δz now is 400λ. As expected, there is no rotational symmetry property with respect to the optical axis.

The intensity and phase distribution for N=6 are plotted in Fig. 3-7 and Fig. 3-8 having the same condition with Fig. 3-5 and Fig. 3-6, respectively. The symmetry properties around the XY-plane passing through the mid-point of two foci can be readily observed from these figures. The 3-fold symmetry properties with respect to the optical axis are clearly shown in the intensity and phase distribution.

Fig. 3-9 and Fig. 3-10 show the intensity and phase distribution for N=4 and the intensity distribution has 2-fold symmetry and phase has 2-fold anti-symmetry can be readily observed. In the case of N=8, the intensity distribution has 4-fold symmetry and the phase distribution has 4-fold anti-symmetry are shown in Fig. 3-11 and Fig.

3-12, respectively.

Fig. 3-13 shows the on-axis intensity with the separation distance along the z-axis Δz=400λ. The maximum intensity for the two beams are located at F1 and F2 as

shown in Fig. 3-13 but the total maximum intensities are not located at F1 and F2

because of constructive and deconstructive interference. As expected, the intensity in the vicinity of the mid-point of two foci, z=0, are small due to the destructive interference caused by the nearly -180° Gouy phase shift.

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.

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.

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