- Email: [email protected]

S0030-4026(16)30838-5 http://dx.doi.org/doi:10.1016/j.ijleo.2016.07.071 IJLEO 57991

To appear in: Received date: Accepted date:

26-5-2016 25-7-2016

Please cite this article as: Sameen Ahmed Khan, Quantum Methodologies in Helmholtz Optics, (2016), http://dx.doi.org/10.1016/j.ijleo.2016.07.071 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Manuscript

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Sameen Ahmed Khan1 Department of Mathematics and Sciences College of Arts and Applied Sciences (CAAS) Dhofar University Post Box No. 2509, Postal Code: 211 Salalah, Sultanate of Oman.

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Quantum Methodologies in Helmholtz Optics

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Abstract It is well-known that the Helmholtz equation describing scalar optics has a striking mathematical similarity with the Klein-Gordon equation of relativistic quantum mechanics. Exploiting this similarity, quantum methodologies are applied to the Helmholtz equation leading to a new formalism of scalar optics. Paraxial and aberrating Hamiltonians are derived for a general spatially varying refractive index. The beam-optical Hamiltonians thus derived give rise to wavelengthdependent contributions. Example of the graded-index medium is covered in detail. Explicit formulae relating the incident and emergent light rays for the general case of a thick lens are also presented.

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PACS: 02, 42.15.-i, 42.90.+m Keywords: Scalar optics, Helmholtz equation, Hamiltonian description, beam propagation, aberrations, wavelength-dependent effects, graded-index fiber, Feshbach-Villars linearization procedure, Foldy-Wouthuysen transformation technique.

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1

Introduction

It is well-known that the Helmholtz equation describing scalar optics has a striking algebraic similarity with the Klein-Gordon equation for a spin-0 relativistic particle. A new formalism of scalar optics is being developed exploiting this similarity. Before going into the technical details, let us briefly recall the relevant aspects of the traditional approaches to scalar optics. Historically, the scalar wave theory of optics is based on a beam-optical Hamiltonian obtained from the Fermat’s principle of least time [1]. This geometric E-mail address: r[email protected] URL: http://orcid.org/0000-0003-1264-2302. 1

1

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approach has been successful in understanding diverse phenomena including aberrations. Now, we know that the entire optics is governed by the Maxwell’s equations of electromagnetism. The most widely used method for deriving a scalar theory of optics from the Maxwell’s equations is through the Helmholtz equation. Helmholtz equation forms the basis of scalar optics. The required beam-optical Hamiltonian is obtained by taking the square-root of the Helmholtz operator [2]-[5]. The resulting beam-optical Hamiltonian is identical to the one derived using the Fermat’s principle of least time. This points to the great universality of the variational principles and their major role in describing natural phenomena. Here, we emphasize that the square-root method replaces the original boundary value problem with a firstorder initial value problem. This transition has significant applications, as it leads to the powerful Fourier optic approach [6]. With all its plus points the square-root approach has been under question! Mathematically, the reduction process is done by taking the “square-root” of the Helmholtz operator. This can never be claimed to be exact or even rigorous enough. Hence, there have been attempts to do the reduction in other ways and several reduction schemes are discussed in [7, 8]. There has been a wide interest in the square-root operators and they have been extensively studied (see [9] for details). By exploiting the mathematical similarities between the Helmholtz equation governing scalar optics and the Klein-Gordon equation for a spin-0 relativistic particle, it is possible to avoid the issue of the reduction process through the square-root. Consequently, we develop a formalism of Helmholtz optics, utilizing the powerful machinery of relativistic quantum mechanics. From the inception, we borne in mind that the electromagnetic field and the Klein-Gordon field are two exclusively different objects. But the similitude in the mathematical structures of the two systems can be utilized to work out some relevant calculations leading to well-established results. The algebraic similarities between the Helmholtz equation governing scalar optics and the Klein-Gordon equation for the relativistic spin-0 particle are well known. In relativistic quantum mechanics, one tries to understand the behaviour of the system as the nonrelativistic limit along with the relativistic corrections in the quasirelativistic regime, order-by-order. In optics, the situation is analogous as the beam propagation is to be understood as the ideal paraxial behaviour along with the aberrations in the quasiparaxial regime, again order-by-order. For the Klein-Gordon this is done using the two-step algorithm described below. The Klein-Gordon equation is second-order in time. It is first linearized using a procedure due to Feshbach and Villars [10].

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Quantum Methodologies in Helmholtz Optics

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The resulting equation is first-order in time and is mathematically very similar to the Dirac equation [11]. The second step of the algorithm is to obtain a series expansion by applying the Foldy-Wouthuysen transformation procedure extensively used in the Dirac theory of the electron. The two-step algorithm is also applicable to the Helmholtz equation as it is mathematically very similar to the Klein-Gordon equation. The proposal to use the Foldy-Wouthuysen procedure for the Helmholtz equation was first mentioned as a comment by Fishman and McCoy [12]. The same idea was independently outlined by Jagannathan and Khan (see, p. 277 in [13]). Here, we shall exploit this idea in detail following the work initiated in [7, 8]. The approach presented here provides an elegant alternative to the customary square-root approach. A few comments on the Foldy-Wouthuysen are essential. The FoldyWouthuysen transformation was initially developed for the Dirac equation, which is for the spin-1/2 particles [14]-[23]. Seeing its utility, the FoldyWouthuysen transformation technique was extended to other spins [24, 25]. Later, it was realized that the Foldy-Wouthuysen technique can also be applied to a class of equations satisfying a specific mathematical structure and the equations need not be even quantum mechanical! Consequently, the Foldy-Wouthuysen technique has found useful applications in diverse systems such as atomic systems [26, 27]; synchrotron radiation [28]; derivation of the Bloch equation for polarized beams [29]; among others. The relevant examples also include acoustics [30]-[35] and ocean acoustics [36]. A detailed account of the Foldy-Wouthuysen transform and its applications in optics is available in [37, 38]. The Foldy-Wouthuysen transform is the key technique in the development of the matrix-formulation of Maxwell optics [39, 40]. In this approach the Maxwell’s equations are first written in a matrix form. The exact representation Maxwell’s equations in a medium with spatially and temporally varying permittivity and permeability necessarily requires 8 × 8 matrices [41]. We are utilizing the Foldy-Wouthuysen transformation as a powerful calculation tool to develop a non-traditional formalism of Helmholtz optics, bearing in mind that the Dirac equation for the electron and Helmholtz equation governing scalar optics are two exclusively different objects. The Foldy-Wouthuysen transformation leads to what we call as the non-traditional prescription of Helmholtz optics [8]. The mathematical machinery of the non-traditional formalism is adopted from the machinery used in the quantum theory of charged-particle beam optics, based on the Dirac equation [42]-[50] and the Klein-Gordon equation [13, 50]. A unified treat-

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Traditional Prescriptions

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ment of the beam-optics and the spin dynamics of a Dirac particle beam, considering the anomalous magnetic moment is to be found in [51]-[57]. A diffractive quantum limit has been also considered for particle beams [58]. The emerging field of QABP, the quantum aspects of beam physics, has been recognized by a series of international conferences carrying the same name [59]-[61]. The aim of this article is to make a detailed presentation of a formalism of Helmholtz optics by employing the quantum methodologies. We shall derive the general Hamiltonians for a spatially varying refractive index. The Hamiltonians thus derived contain wavelength-dependent parts, the key outcome of our formalism. In the limit of low wavelength, our approach reproduces the Lie algebraic formalism of light beam optics (for Lie methods in optics, see [2, 4, 5] and [62]-[65]). Section 2 has an outline of the traditional prescriptions. Section 3 has the formalism in detail. Section 4 has the applications. The example of an axially symmetric graded-index medium is covered in detail. Next, we examine the general case of thick and thin lenses. Explicit formulae relating the incident and emergent light rays for the general case of a thick lens are presented for this system. Section 5 has our concluding remarks.

In this section, we shall briefly review the traditional prescription of Helmholtz Optics. The beam is assumed to be monochromatic and quasiparaxial. The optic-axis is along the z-axis. Furthermore, it is assumed that a Schr¨odingerlike equation given below describes the z-evolution of the corresponding optical wavefunction ψ(r)

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∂ c i–λ ψ(r) = Hψ(r) , (1) ∂z where –λ = λ/2π = c/ω is the reduced wavelength and the analogue of h ¯ , the c Planck’s constant in quantum mechanics. The beam-optical Hamiltonian H is

1/2

c = − n2 (r) − p b 2⊥ H , (2) – ⊥ . It is assumed that the light rays are b = −i– b ⊥ = −iλ∇ where p λ∇ and p propagating very close to the optic-axis (chosen to be the z-axis). Matheb ⊥ | ≪ pz ≈ 1. The only condition on the refractive index is that matically, |p

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Quantum Methodologies in Helmholtz Optics

!

n2 (r) ψ(r) = 0 . ∇ − –2 λ 2

(3)

b 2⊥ ψ(r) . ψ(r) = n2 (r) − p

an

!2

Then, one takes the square-root [2] as

1/2 ∂ b 2⊥ ψ(r) = − n2 (r) − p ψ(r) , ∂z

M

i–λ

us

The terms in the Helmholtz equation are rearranged as ∂ i–λ ∂z

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ip t

it varies smoothly around a constant value n0 , satisfying |n(r) − n0 | ≪ n0 . The beam-optical Hamiltonian in (2) is traditionally obtained by two distinct procedures. The first procedure predating Maxwell is to derive it from the Fermat’s principle of least time [1]. The starting point for the second procedure is the Helmholtz equation

(4)

(5)

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corresponding to the requirement that the propagation be entirely in the positive z-direction; if the propagation is in the negative z-direction, the right hand side of (5) will have the opposite sign. It is to be noted that the passage from (3) to (5) replaces the original boundary value problem with a first-order initial value problem in z. It is interesting to note that the purely geometric approach of the Fermat’s principle of least time and the Helmholtz equation based on the Maxwell’s equations of electromagnetism lead to the same beam-optical Hamiltonian. The Hamiltonian in (2) or equivalently in (5) is expanded in a power b ⊥ respectively. In order to series in the transverse components, r ⊥ and p c have an Hermitian H, it is required to incorporate a suitable ordering and symmetrization, of the polynomials resulting from the expansion [2]-[5]. For a homogeneous medium, the refractive index, n(r) reduces to a constant value n0 . Then, the Hamiltonian has the simplified Taylor expansion

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c = − n2 (r) − p b 2⊥ H (

1/2

)1/2

1 2 b = −n0 1 − 2 p n0 ⊥ ( 1 2 1 4 1 6 b⊥ + b + b = n0 −1 + 2 p p p 4 ⊥ 2n0 8n0 16n60 ⊥

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Sameen Ahmed Khan )

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5 7 b8 + b 10 + p p ⊥ +··· 8 ⊥ 128n0 256n10 0 ∞ X (2m − 3)!! 1 m b , p = n0 m ⊥ (2m)!! n 0 m=0

(6)

Quantum Methodologies

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3

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where we recall that 1!! = 1, 0!! = 1, (−1)!! = 1, and (−3)!! = −1. In practice, the refractive index does not have a constant value. In such situations, one makes a Taylor expansion of the varying refractive index n(r) = n(rb ⊥ , z). Then the radical is expanded to the required degree of accuracy in powers of b 2⊥ /n20 , consistently taking into account the terms arising from the expansion p of n(r). The aforementioned expansion of the Hamiltonian has quadratic terms and higher-order terms. The quadratic terms govern the paraxial or ideal behaviour. The higher-order terms describe the aberrations of corresponding order, order-by-order [2].

The wave equation for a spatially varying refractive index n(r) is

d

n2 (r) ∂ 2 Ψ(r, t) . v 2 ∂t2 We shall focus on a monochromatic system,

te

∇2 Ψ(r, t) =

Ψ(r, t) = ψ(r)e−iωt ,

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ω > 0.

(7)

(8)

Rearranging the terms in (8) leads to (

)

n2 (r) 2 ω ψ(r) = 0 . ∇ + v2 2

(9)

Now, we introduce the procedure of ‘wavization’ through the familiar Schr¨odinger replacement b⊥ , −i–λ∇⊥ −→ p

−i–λ

∂ −→ pz , ∂z

(10)

where the reduced wavelength –λ = λ/2π = c/ω is the analogue of Planck’s constant. Then, we obtain

∂ −i–λ ∂z

!2

b 2⊥ − n2 (r) − p

ψ(r) = 0 .

(11)

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Quantum Methodologies in Helmholtz Optics

ψ1 (r) ψ2 (r)

!

=

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Before proceeding further, we note that [p, q] = pq − qp = −i–λ, which is the analogue of the fundamental commutation relation [p, q] = −i¯h in quantum mechanics. It is to be borne in mind that the refractive index n(r) and b ⊥ = −i∇⊥ do not commute. Thus, the formalism the transverse momenta p presented here naturally leads to the wavelength-dependent contributions, absent in the traditional approaches of beam-optics. In the limit of low wavelength (–λ −→ 0), the formalism presented here leads to the Lie algebraic formalism of light beam optics (see [2, 4, 5] and [62]-[65]). The FoldyWouthuysen transformation technique is ideally suited for the Lie algebraic approach to optics. As stated earlier, we shall follow the Feshbach-Villars-like procedure to linearize Eq. (11). We introduce two functions ψ1 (r) and ψ2 (r) ψ(r) –λ ∂ −i n0 ∂z ψ(r)

!

,

(12)

ψ1 (r) ψ2 (r)

!

=

1 n20

0 2 b 2⊥ n (r) − p

d

–λ ∂ −i n0 ∂z

M

enabling us to write the Helmholtz equation in Eq. (11) as

1 0

ψ1 (r) ψ2 (r)

!

.

(13)

!

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It is to be noted that Eq. (13) has the form of a first-order system. Following the first step of our algorithm, we have to write Eq. (13) in a Dirac-like form. This requires the following transformation of Eq. (12)

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ψ1 (r) ψ2 (r)

−→

ψ+ (r) ψ− (r)

where

M =M

−1

!

= M

ψ1 (r) ψ2 (r)

!

– ∂ ψ(r) 1 ψ(r) − i nλ0 ∂z √ = , –λ ∂ 2 ψ(r) + i n ∂z ψ(r)

(14)

0

1 =√ 2

1 1 1 −1

!

.

(15)

To understand the significance of this transformation, let us first consider a monochromatic quasiparaxial beam moving in the +z-direction having the principal z-dependence (

n(r) ψ(r) ∼ exp i – z λ

)

,

(16)

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Sameen Ahmed Khan

then

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1 n(r) ψ(r) , ψ+ (r) ∼ √ 1 + n0 2 ! 1 n(r) ψ− (r) ∼ √ 1 − ψ(r) . n0 2

cr

(17)

ψ+ (r) ψ− (r)

!

=

b H

ψ+ (r) ψ− (r)

!

an

∂ i–λ ∂z

us

Recalling that n(r) varies smoothly about a constant value n0 such that |n(r) − n0 | ≪ n0 , which leads to the important inequality |ψ+ (r)| ≫ |ψ− (r)|. This inequality plays a crucial role in the Foldy-Wouthuysen technique, central to our formalism. Consequently, Eq. (11) can be written in a Dirac-like form as

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b = −n σ + Eb + O b H 0 z o n 1 b 2⊥ + n20 − n2 (r) σz p Eb = 2n0 o 1 n 2 2 b = b ⊥ + n0 − n2 (r) (iσy ) , O p 2n0

(18)

where σy and σz are from the triplets of the Pauli matrices, #

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σ =

"

σx =

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0 1 1 0

, σy =

"

0 −i i 0

#

, σz =

"

1 0 0 −1

#!

.

(19)

The beam-optical Hamiltonian, we obtained in Eq. (18), using the twostep algorithm is exact. This is evident, from the square of the beam-optical 2 b b 2⊥ . Notably, the exact beam-optical Hamiltonian, i.e., H = n2 (r) − p Hamiltonian has a one-to-one algebraic correspondence with the Dirac equation, enriched with appropriate physical interpretations. These aspects are summarized in the table below Standard Dirac Equation

Light Beam-Optical Form

b m0 c2 β + EbD + O D 2 m0 c Positive Energy b ≪ m0 c Nonrelativistic, |π| Non relativistic Motion + Relativistic Corrections

b −n0 σz + Eb + O −n0 Forward Propagating Beam b ⊥ | ≪ n0 Paraxial Beam, |p Paraxial Behavior + Aberration Corrections

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Quantum Methodologies in Helmholtz Optics

∂ c(2) |ψi , i–λ |ψi = H ∂z

1 b2 . σz O 2n0

an

c(2) = −n σ + Eb − H 0 z

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The algebraic correspondence between the two systems facilitate the usage of the rich mathematical machinery of the relativistic quantum mechanics. In the present context, it is the Foldy-Wouthuysen transformation technique. We have established the mathematical and physical correspondence between the Dirac-like form of the Helmholtz equation in Eq. (18) via the FeshbachVillars procedure and the original Dirac equation (see the Table). This enables us to apply the Foldy-Wouthuysen procedure. The first iteration of the Foldy-Wouthuysen procedure, gives the formal Hamiltonian to leading order b 2⊥ /n20 ) as (p

(20)

o2 1 n 2 2 b ⊥ + n0 − n2 (r) p . 8n30

(21)

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+

o 1 n 2 2 b ⊥ + n0 − n2 (r) p 2n0

d

c(2) = −n + H 0

M

Note, that the square of each of the Pauli matrices is a unit matrix. Since, we are essentially interested in a beam propagating in the forward direction, we omit σz . This simplifies the formal Hamiltonian to the more familiar form

The second iteration of the Foldy-Wouthuysen procedure is formally expressed as

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i–λ

∂ c(4) |ψi , |ψi = H ∂z

c(4) = −n σ + Eb − H 0 z "

1 b2 σz O 2n0

h i 1 ∂ b b b Eb + i– − 2 O, O, λ O 8n0 ∂z

!#

h i 1 ∂ b b4 + b Eb + i– + 3 σz O O, λ O 8n0 ∂z

!2

.

(22)

We again omit the σz and express the formal Hamiltonian in (22) in a more familiar form n o c(4) = −n + 1 b 2⊥ + n20 − n2 (r) H p 0 2n0

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Sameen Ahmed Khan o2 1 n 2 2 b ⊥ + n0 − n2 (r) p 8n30 # " i–λ ∂ 2 2 b , − n (r) p 32n40 ⊥ ∂z !2 –λ2 ∂ 2 n (r) + 32n50 ∂z o3 1 n 2 2 b ⊥ + n0 − n2 (r) p + 16n50 o4 5 n 2 2 b ⊥ + n0 − n2 (r) p . + 128n70

(23)

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+

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It is to be noted that the imaginary unit, ‘i’ in the Hamiltonian disappears once the commutator is executed. The Hamiltonian in (23) has all the trab 2⊥ /n20 )4 as it should. The terms arising from the ditional terms to order (p Taylor expansion of n(r) are retained to the same order. The Hamiltonian in (23) also has the wavelength-dependent terms, which are an outcome of the non-traditional approach using quantum methodologies. These terms persist whenever the refractive index has any inhomogeneities. There are no such terms in the traditional square-root approach. It is straightforward to note that the wavelength-dependent terms are present at each order including the paraxial Hamiltonian and continuing to the aberrating Hamiltonians of each order. This will be seen in substantial detail, when we do the applications in Section 4. The Foldy-Wouthuysen scheme works to all orders, and we can obtain the higher order Hamiltonians to a desired degree of accuracy.

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4

Applications

In Section 2, we presented an outline of the traditional square-root approach. In Section 3, we obtained an alternate method to the traditional square-root approach by exploiting the mathematical similarities between the Helmholtz equation and the Klein-Gordon equation. The resulting beam-optical Hamiltonian was seen to contain the traditional terms modified by wavelengthdependent contributions. Now, we shall consider the application of the formalism to the specific systems.

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Quantum Methodologies in Helmholtz Optics

4.1

Medium with a Constant Refractive Index

(24)

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b H c = −n0 σz + Dσz + D (iσy ) o 1 n 2 2 b ⊥ + n0 − n2c D = p , 2n0

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As a first example, let us consider a medium with a constant or uniform refractive index. This system constitutes the ideal or the simplest conceivable system. Using our method, it can be exactly diagonalized. In relativistic quantum mechanics too, very few systems can be exactly diagonalized and all others require the Foldy-Wouthuysen iterative procedure. When the refractive index n(r) = nc , the Hamiltonian in (18) is

n

M

which can be exactly diagonalized. This is done in the same manner in which the free particle Dirac Hamiltonian is exactly diagonalized (see [11, 13] for details). The resulting diagonalized Hamiltonian is b diagonal = − n2 − p b 2⊥ H c c

o1

2

σz .

(25)

c

= −n0

n

≈ −n0 1 − P 2 n

b 2⊥ = − n2c − p

n

1 1 1 5 4 1 − P − P2 − P3 − P σz 2 8 16 128

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c(4) H

d

The Foldy-Wouthuysen iterative procedure gives

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b diagonal . = H c

o1

2

o1 2

σz

σz (26)

o1

b 2⊥ + (n20 − n2c ) 2 . This is consistent with the traditional where P = n12 p 0 prescriptions. The transfer operator relating any two points {(z ′′ , z ′ )|z ′′ > z ′ } on the optic-axis (z-axis) has the formal expression

with

|ψ(z ′′ , z ′ )i = Tb (z ′′ , z ′ ) |ψ(z ′′ , z ′ )i , ∂ c Tb (z ′′ , z ′ ) , i–λ Tb (z ′′ , z ′ ) = H ∂z

(27)

Tb (z ′′ , z ′ ) = Ib ,

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Sameen Ahmed Khan

(

"

#)

Z

(28)

us

cr

ip t

i z ′′ c dz H(z) = ℘ exp − – λ z′ Z i z ′′ c b dz H(z) = I−– λ z′ Z Z z i 2 z ′′ c H(z c ′) dz + −– dz ′ H(z) λ z′ z′ +... ,

Tb (z ′′ , z ′ )

an

where ℘ signifies the path-ordered exponential and Ib is the identity operator. For a very general choice of the refractive index n(r), there are no closed form expressions for the transfer operators, Tb (z ′′ , z ′ ). In such situations, the zevolution operator Tb (z ′′ , z ′ ) (also called the z-propagator), is expressed as

with

Z

z ′′

z′

c dz H(z)

d

Tˆ (z ′′ , z ′ ) =

M

i Tb (z ′′ , z ′ ) = exp − – Tb (z ′′ , z ′ ) , λ

i 1 + −– 2 λ +... ,

te Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Z

z ′′

z′

dz

Z

z

z′

(29)

h

c c ′) dz ′ H(z) , H(z

i

(30)

coming from the Magnus formula [66]-[69]. Now, the transfer operator is

i c (zout , zin ) = exp − – ∆zHc λ i 1 pb2⊥ 1 = exp + – nc ∆z 1 − − λ 2 n2c 8

Ub

pb2⊥ n2c

!2

− · · · ,

(31)

where ∆z = (zout − zin ). Equation (31) leads to the transfer maps hr ⊥ i hp⊥ i

!

out

=

1 0

√

1 n2c −

p 1

2 ⊥

∆z

hr ⊥ i hp⊥ i

!

.

(32)

in

It is evident that the beam-optical Hamiltonian is fundamentally aberrating. Even the ideal system of a constant or uniform refractive index has aberrations to all orders.

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Quantum Methodologies in Helmholtz Optics

4.2

Axially symmetric graded-index medium

cr

ip t

The first example was the ideal system of constant refractive index. The second example is the realistic case of the axially symmetric graded-index medium. In general, the refractive index for such a system is expressed as an infinite series (see, pp. 117 in [2]): n(r) = n0 + α2 (z)r 2⊥ + α4 (z)r 4⊥ + α6 (z)r 6⊥ + α8 (z)r 8⊥ + · · · .

(33)

us

We note that the optic-axis is along the z-axis. The other powers of r ⊥ are absent due to the axial symmetry. We shall express the beam-optical Hamiltonian using the notation indicated below

(34)

te

d

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an

b ⊥ · r⊥ + r⊥ · p b ⊥) Tb = (p d {2n0 α2 (z)} w1 (z) = dz o d n 2 w2 (z) = α2 (z) + 2n0 α4 (z) dz d {2n0 α6 (z) + 2α2 (z)α4 (z)} w3 (z) = dz o d n 2 α4 (z) + 2α2 (z)α6 (z) + 2n0 α8 (z) . w4 (z) = dz

The paraxial and the aberrating Hamiltonians up to order-eight (describing seventh-order aberrations) are c = H c +H c c c H 0 ,p 0 ,(4) + H0 ,(6) + H0 ,(8) – – – – c(λ) + H c(λ) + H c(λ) + H c(λ) +H 0 ,(2) 0 ,(4) 0 ,(6) 0 ,(8) 1 2 c b − α2 (z)r 2⊥ H p 0 ,p = −n0 + 2n0 ⊥ 1 4 α2 (z) 2 2 c b⊥ − b ⊥ r ⊥ + r 2⊥ p b 2⊥ − α4 (z)r 4⊥ H p p 0 ,(4) = 8n30 4n20 o α2 (z) n 4 2 1 6 c b − b ⊥ r ⊥ + r 2⊥ p b 4⊥ + p b 2⊥ r2⊥ p b 2⊥ p p H 0 ,(6) = 5 ⊥ 4 16n0 8n0 o 1 n 2 b 2⊥ r 4⊥ + r 4⊥ p b 2⊥ + 2α22 (z)r 2⊥ p b 2⊥ r 2⊥ + 3 α2 (z) − 2n0 α4 (z) p 8n0 −α6 (z)r 6⊥ 5 5α2 (z) 4 h 2 2 i 8 c b b b H = p − p , p , r 0 ,(8) ⊥ ⊥ ⊥ + 128n70 ⊥ 64n60 +

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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Sameen Ahmed Khan i i h h 1 4 2 4 2 2 2 2 b b p , r p , r 3α (z) − 4n α (z) + 5α (z) + 0 4 ⊥ ⊥ + ⊥ ⊥ + 2 2 32n50

ip t

b 2⊥ r 4⊥ p b 2⊥ − 2α22 (z) + 4n0 α4 (z) p

−

5α23 (z)

r 4⊥ ,

h

i

cr

i h 1 3 2 b 2⊥ , r6⊥ + p 4 α (z) + n α (z)α (z) + n α (z) 0 2 4 6 2 0 + 16n40 b 2⊥ , r 2⊥ p + +

−α8 (z)r 8⊥ –2

h

b 2⊥ r2⊥ r 2⊥ , r 2⊥ p

us

+ 2α23 (z) + 4n0 α2 (z)α4 (z)

+

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an

– λ c(λ) w1 (z)Tb H 0 ,(2) = − 32n40 –λ2 –λ2 (– λ) 2 b 2 c b H0 ,(4) = − w (z) r ⊥ T + T r⊥ + w 2 (z)r 4⊥ 4 2 5 1 32n0 32n0 2 –2 – – 3λ 4 b b r4 + λ w (z)w (z)r 6 c(λ) w (z) r T + T H = − 3 1 2 ⊥ ⊥ ⊥ 0 ,(6) 32n40 16n50 –λ2 – 6 b c(λ) b r6 H = − w (z) r T + T 4 ⊥ ⊥ 0 ,(8) 8n40 o –λ2 n 2 + w (z) + 2w (z)w (z) r 8⊥ . 1 3 2 5 32n0

i

(35)

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Note, that [A, B]+ = (AB + BA) is the anticommutator. There are reasons c in this manner. The ideal for partitioning the beam-optical Hamiltonian H c . The Hamiltonibehaviour is described by the paraxial Hamiltonian H 0 ,p c c c ans, H0 ,(4) , H0 ,(6) and H0 ,(8) are responsible for the third-order, fifth-order and seventh-order aberrations respectively. All the four Hamiltonians get – c(–λ) c(–λ) c(λ) , H modified by the wavelength-dependent terms present in H 0 ,(2) 0 ,(4) , H0 ,(6) – c(λ) respectively. The wavelength-dependent Hamiltonians are aband H 0 ,(8) c sent in the traditional approaches. The Hamiltonian H 0 ,(4) is responsible for the six third-order aberrations namely: (1) Spherical Aberration; (2) Coma; (3) Astigmatism; (4) Curvature of Field; (5) Distortion; and (6) Nameless – c(λ) is responsible for the wavelengthor POCUS [2]. The Hamiltonian H 0 ,(4) dependent modifications of all the aforementioned six third-order aberrations. Likewise, the higher-order Hamiltonians describe the higher-order

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4.3

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aberrations, which in turn get modified by the corresponding wavelengthdependent Hamiltonians respectively.

General Thick Lens

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cr

Let us consider a thick lens of thickness t = t1 + t2 as shown in Figure 1. Let the refractive index of the lens be n and that of the surrounding medium to be unity without loss of generality. The optic-axis is along z-axis; x and y are along the face of the lens. The following equations of the curved surfaces of the lens describe the lens system

an

z1 (r ⊥ ) = γ2 (r 2⊥ ) + γ4 (r 4⊥ ) + γ6 (r 6⊥ ) + · · · , z2 (r ⊥ ) = β2 (r 2⊥ ) + β4 (r 4⊥ ) + β6 (r 6⊥ ) + · · · .

(36)

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M

The other powers of r ⊥ are absent due to the assumed axial symmetry.

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Figure 1: Lens of thickness t = t1 + t2 and uniform refractive index n. The transfer operator for this lens is given by the product of three transfer operators corresponding to the two surfaces and the transit through the lens respectively. The transfer operator within the paraxial approximation leads to the transfer map hr ⊥ i hp⊥ i

where S =

"

1 0 2β2 (1 − n) 1

#"

!

= S

out

1 t/n 0 1

#"

hr⊥ i hp⊥ i

!

,

(37)

in

1 0 2γ2 (1 − n) 1

#

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Sameen Ahmed Khan #

1 + 2γ2 (1 − n)t/n t/n (38) . 2(1 − n) [(β2 + γ2 ) + 2(1 − n)γ2 β2 t/n] 1 + 2β2 (1 − n)t/n

ip t

=

"

1+ t/n 1 i h(n − 1)t/nR (n−1)t 1 1 (1 − n) R1 − R2 + nR1 R2 1 − (n − 1)t/nR2

us

S =

"

cr

Let the spherical lens have surfaces with radii of curvature R1 and R2 respectively. Then, the constants can be identified as γ2 = 1/2R1 and β2 = −1/2R2 . Then the transfer map S simplifies to

"

"

#

.

(39)

also known as the Ko#

(n − 1)t 1 1 − + . R1 R2 nR1 R2

M

1 F = −C = = (n − 1) f

an

A B This is the well known ABCD matrix or C D gelnik’s ABCD-law [70]. The dioptric power F is

#

(40)

5

te

d

This is the familiar relation giving the power of a general thick lens along with the Lensmaker’s equation. The example of the thick lens has been done in detail using Lie algebraic methods in [65].

Concluding Remarks

In summary, we have exploited the striking algebraic similarities between the Helmholtz equation and the Klein-Gordon equation, to develop a new formalism of Helmholtz optics. This formalism provides an alternate method for the aberration expansion. The beam-optical Hamiltonian was derived from the Helmholtz equation by adopting the quantum methodologies used in the case of the Klein-Gordon equation. This enabled us to rewrite the Helmholtz equation in a linear form using a Feshbach-Villars-like procedure widely used for linearizing the Klein-Gordon equation. The linearized form thus obtained was shown to have the mathematical structure of the Dirac equation. Consequently, it was seen that we can adopt the powerful machinery of quantum mechanics, to the beam-optics formalism. In our formalism, we used the Foldy-Wouthuysen transformation technique, to get an aberration expansion of the beam-optical Hamiltonian. We could obtain formal expressions for the ideal/paraxial Hamiltonian along with the aberrating Hamiltonians to eighth-order, for a general spatially varying refractive index. It was further

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d

M

an

us

cr

ip t

shown that both the paraxial and aberrating Hamiltonians are accompanied with wavelength-dependent Hamiltonians. It was seen that these wavelengthdependent Hamiltonians modify the paraxial and aberrating behaviour of the system. In the limit of low wavelength, we obtain the Lie algebraic formalism of light beam optics [2, 4, 63, 65]. To illustrate the general theory, we considered the example of the constant refractive index. This is exactly solvable and we obtained the transfer maps. Next, we considered the axially symmetric graded-index medium in detail. For this system, we obtained the paraxial and aberrating Hamiltonians up to eighth-order, governing up to seventh-order aberrations respectively. Each of these Hamiltonians is modified by wavelength-dependent terms. Such wavelength-dependent contributions are completely absent in the traditional prescriptions and are a result of the quantum methodologies, we have used. The traditional Hamiltonians are recovered from our formalism, when we take the limit, –λ −→ 0. We call this as the traditional limit of our formalism. This is very similar to the situation of obtaining classical limit from quantum prescriptions by taking h ¯ −→ 0. In passing, we note that the mathematical machinery of the non-traditional prescription of Helmholtz optics presented here is very similar to the one used in the quantum theory of charged-particle beam optics (see [13] and [42]-[57]). In the latter, the limit –λ0 −→ 0, leads to the Lie algebraic formalism of charged-particle beam optics [71]-[81]. Note, that the –λ0 = h ¯ /p0 is the de Broglie wavelength and p0 is the average or design momentum of the system under study. Unlike the first two examples, the lens constitutes a complex system. The transfer map for a complex system in our formalism is a product of the transfer maps corresponding to the sub-systems constituting the complex system. The transfer map for a thick lens is the product of three transfer maps corresponding to the two surfaces of the lens and the transit through the lens. This is very similar to the situation in the Lie algebraic formalism of optics [65]. The extra terms arising in our formalism by employing quantum methodologies require a closer examination. The i source of these extra terms is the h h i 1 ∂ b b b b – commutator, − 8n2 O, O, E + iλ ∂z O in the formal expression in (22). 0 i – h 2 ∂ 2 b , (n (r)) On simplifying this formal expressions translates to − iλ p

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32n40

⊥ ∂z

in (23). The corresponding commutator terms in the Foldy-Wouthuysen formalism of relativistic quantum mechanics lead to a correct explanation of several phenomena such as the spin-orbit energy (see [7, 11] for details).

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Sameen Ahmed Khan

So, it would be meaningful to explore the additional wavelength-dependent contributions experimentally, however tiny they may be.

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Quantum methodologies have been also exploited to study multi-particle effects in charged-particle beam optics. Here, it is relevant to note the models developed by Fedele et al. (the thermal wave model [82]-[83]) and Cufaro Petroni et al. (stochastic collective dynamical model [84]) for treating the beam phenomenologically as a quasiclassical many-body system. In the aforementioned approaches, the basic equation is a Schr¨odinger-like equation in which the beam emittance plays the role of h ¯ , the Planck’s constant. Recently the quantum-like approach has been applied to construct a Diffraction Model for the beam halo [85]-[88]. The analogy between geometrical optics and the classical theories of charged-particle beam optics are well-known. It is seen that the Hamilton’s analogy persists even in the wavelength-dependent regime [89]-[91]. Let us recall that the Medieval Arab scholars had actively worked in optics. The opto-mechanical analogy can now be traced to Ibn alHaytham (0965-1037) [92]-[98]. But it was Hamilton, who in 1833, examined the trajectories of material particles in various potential fields and compared them with the paths of light rays in a media with varying refractive index. Thus it was Hamilton, who laid a rigorous mathematical foundation to the opto-mechanical analogy. This analogy played an important role in the early years of the classical theories of electron optics. In 1926, Hans Busch used the analogy to develop an electromagnetic lens, leading to the invention of the electron microscope by Ernst Ruska in 1931. In the 1920’s the optomechanical analogy influence the development of quantum mechanics. These historical aspects of the analogy were widely covered in the just concluded International Year of Light and Light-based Technologies [99]-[102].

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6

Acknowledgments

I am indebted to Professor Ramaswamy Jagannathan (IMSC: Institute of Mathematical Sciences, Chennai, India) for all my training in the field of quantum mechanics of beam optics, which was the topic of my doctoral thesis, which he so elegantly supervised. I am thankful to him for suggesting the novel project of investigating the scalar optics (Helmholtz optics) using the FoldyWouthuysen transformation.

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References Thya2002),

ip t

[1] V. Lakshminarayanan, Ajoy Ghatak and K. garajan, Lagrangian Optics, (Springer, http://dx.doi.org/10.1007/978-1-4615-1711-5.

us

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[2] A.J. Dragt, R. Forest, K.B. Wolf, Foundations of a Lie algebraic theory of geometrical optics. in: Lie Methods in Optics, Lecture Notes in Physics, 250, J. S´anchez Mondrag´on and K. B. Wolf (Eds.), Springer Verlag, Berlin, Germany, (1986), pp. 105-157, http://dx.doi.org/10.1007/3-540-16471-5_4.

an

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[7] Sameen Ahmed Khan, Ramaswamy Jagannathan, and Rajiah Simon, Foldy-Wouthuysen transformation and a quasiparaxial approximation scheme for the scalar wave theory of light beams, E-Print arXiv: arXiv:physics/0209082 [physics.optics]; http://arXiv.org/abs/physics/0209082/ (2002). (communicated). [8] Sameen Ahmed Khan, Wavelength-dependent modifications in Helmholtz Optics, International Journal of Theoretical Physics, 44(1) (2005) 95-125, http://dx.doi.org/10.1007/s10773-005-1488-0; E-Print arXiv: arXiv:physics/0210001 [physics.optics]; http://arxiv.org/abs/physics/0210001. [9] T.L. Gill and W.W. Zachary, Analytic representation of the square-root operator, Journal of Physics A: Mathematical and General, 38 (2005) 2479-2496, http://dx.doi.org/10.1088/0305-4470/38/11/010.

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[32] L. Fishman, One-way wave equation modeling in two-way wave propagation problems, in: B. Nilsson, L. Fishman (Eds.), Mathematical Modelling of Wave Phenomena 2002, Mathematical Modelling in Physics, Engineering and Cognitive Sciences, vol. 7, V¨axj¨o University Press, V¨axj¨o Sweden, (2004), pp. 91-111. [33] L. Leviandier, The one-way wave equation and its invariance properties, Journal of Physics A: Mathematical and Theoretical, 42 (26) (2009) 265402, http://dx.doi.org/10.1088/1751-8113/42/26/265402.

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[34] D. Wurmser, A new strategy for applying the parabolic equation to a penetrable rough surface, J. Acoust. Soc. Am. 109(5) (2001) 2300, http://dx.doi.org/10.1121/1.4744070. [35] D. Wurmser, A parabolic equation for penetrable rough surfaces: using the Foldy-Wouthuysen transformation to buffer density jumps, Annals of Phys. 311 (2004) 53-80, http://dx.doi.org/10.1016/j.aop.2003.11.006. [36] R.S. Patton, in: Path Integrals from meV to MeV, (M. Gutzwiller, A. Inomata, J.R. Klauder, L. Streit, Eds.), pp. 98-115. World Scientific, Singapore, (1986).

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[37] Sameen Ahmed Khan, The Foldy-Wouthuysen Transformation Technique in Optics, Optik-International Journal for Light and Electron Optics, 117(10) (2006) 481-488, http://dx.doi.org/10.1016/j.ijleo.2005.11.010.

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[38] Sameen Ahmed Khan, The Foldy-Wouthuysen Transformation Technique in Optics, in: P.W. Hawkes (Ed.), Advances in Imaging and Electron Physics, Vol. 152, Elsevier, Amsterdam, (2008) pp. 49-78, http://dx.doi.org/10.1016/S1076-5670(08)00602-2.

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[39] Sameen Ahmed Khan, Maxwell Optics of Quasiparaxial Beams, Optik-International Journal for Light and Electron Optics, 121(5) (2010) 408-416. http://www.elsevier-deutschland.de/ijleo/, http://dx.doi.org/10.1016/j.ijleo.2008.07.027. EPrint arXiv: arXiv:physics/0205084 [physics.optics]; http://arxiv.org/abs/physics/0205084.

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[40] Sameen Ahmed Khan, Aberrations in Maxwell Optics, OptikInternational Journal for Light and Electron Optics, 125(3) (2014) 968-978. http://www.elsevier-deutschland.de/ijleo/, http://dx.doi.org/10.1016/j.ijleo.2013.07.097. EPrint arXiv: arXiv:physics/0205085 [physics.optics]; http://arxiv.org/abs/physics/0205085.

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[41] Sameen Ahmed Khan, An Exact Matrix Representation of Maxwell’s Equations, Physica Scripta, 71(5) (2005) 440-442. http://dx.doi.org/10.1238/Physica.Regular.071a00440; E-Print arXiv: arXiv:physics/0205083 [physics.optics]; http://arXiv.org/abs/physics/0205083/. [42] R. Jagannathan, R. Simon, E.C.G. Sudarshan, N. Mukunda, Quantum theory of magnetic electron lenses based on the Dirac equation, Phys. Lett. A 134 (1989) 457-464. http://dx.doi.org/10.1016/0375-9601(89)90685-3. [43] R. Jagannathan, Quantum theory of electron lenses based on the Dirac equation, Phys. Rev. A 42 (1990) 6674-6689. http://dx.doi.org/10.1103/PhysRevA.42.6674.

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[44] R. Jagannathan, Dirac equation and electron optics, in: R. Dutt, A.K. Ray (Eds.), Dirac and Feynman: Pioneers in Quantum Mechanics, Wiley Eastern, New Delhi, India 1993, pp. 75-82.

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[46] S.A. Khan and R. Jagannathan, Theory of relativistic electron beam transport based on the Dirac equation, pp. 102-107. In Proceedings of the 3rd National Seminar on Physics and Technology of Particle Accelerators and their Applications, PATPAA-93, (S. N. Chintalapudi, ed) November 25-27, 1993, Kolkata (Calcutta), IUC-DAEF, Kolkata (Calcutta), India, (1993).

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[47] S.A. Khan and R. Jagannathan, Quantum mechanics of charged-particle beam optics: An operator approach, Presented at the JSPS-KEK International Spring School on High Energy Ion Beams - Novel Beam Techniques and their Applications, March 1994, Japan, (1994).

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[84] Nicola Cufaro Petroni, Salvatore De Martino, Silvio De Siena and Fabrizio Illuminati Stochastic collective dynamics of charged-particle beams in the stability regime, Phys. Rev. E 63, 016501 (2000), http://dx.doi.org/10.1103/PhysRevE.63.016501.

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[85] Sameen A. Khan and Modesto Pusterla, “Quantum mechanical aspects of the halo puzzle,” in Proceedings of the 1999 Particle Accelerator Conference, Editors: A. Luccio and W. MacKay, (New York City, NY, 1999), Vol. 5, pp. 3280-3281. http://dx.doi.org/10.1109/PAC.1999.792276; E-Print arXiv: arXiv:physics/9904064 [physics.acc-ph]; http://arxiv.org/abs/physics/9904064.

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[86] S.A. Khan and M. Pusterla, Quantum-like approach to the transversal and longitudinal beam dynamics. The halo problem, European Physical Journal A 7(4), 583587 (2000). http://dx.doi.org/10.1007/s100500050430. E-Print arXiv: arXiv:physics/9910026 [physics.acc-ph]; http://arxiv.org/abs/physics/9910026.

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[87] Sameen A. Khan and Modesto Pusterla, “Quantum-like approaches to the beam halo problem,” in Proceedings of the 6th International Conference on Squeezed States and Uncertainty Relations, Editors: D. Han, Y.S Kim, and S. Solimeno, (NASA Conference Publication Series, 2000-209899, 2000), pp. 438-441. E-Print arXiv: arXiv:physics/9905034 [physics.acc-ph]; http://arxiv.org/abs/physics/9905034. [88] S.A. Khan and M. Pusterla, Quantum approach to the halo formation in high current beams, Nuclear Instruments and Methods in Physics Research Section A, 464(1-3), 461-464 (2001). http://dx.doi.org/10.1016/S0168-9002(01)00108-5;

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[91] Sameen Ahmed Khan, Passage from scalar to vector optics and the Mukunda-Simon-Sudarshan theory for paraxial systems, Journal of Modern Optics 63 (2016) 1-9, http://dx.doi.org/10.1080/09500340.2016.1164257. (Taylor & Francis, 2016).

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[93] K. B. Wolf and G. Kr¨otzsch. Geometry and dynamics in refracting systems, European Journal of Physics, 16(1) (1995) 14-20, http://dx.doi.org/10.1088/0143-0807/16/1/003.

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[94] K. B. Wolf, Geometric Optics on Phase Springer, Berlin, Germany, (2004) pp. http://www.springer.com/us/book/9783540220398.

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[95] Sameen Ahmed Khan, Medieval Arab Understanding of the Rainbow Formation, Europhysics News, 37(3) (2006) 10, http://www.europhysicsnews.org/articles/epn/pdf/2006/03/epn2006-37-3.pdf.

[96] Sameen Ahmed Khan, Arab Origins of the Discovery of the Refraction of Light; Roshdi Hifni Rashed Awarded the 2007 King Faisal International Prize, Optics & Photonics News (OPN), 18(10), 22-23 (October 2007). http://www.osa-opn.org/Content/ViewFile.aspx?id=10890.

[97] Sameen Ahmed in Optics, Il

Khan, Nuovo

Medieval Saggiatore,

Islamic 31(1-2),

Achievements pp. 36-45

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(January-February 2015). (Publication of SIF: the Societ`a Italiana di Fisica, the Italian Physical Society). http://prometeo.sif.it/papers/online/sag/031/01-02/pdf/06-percorsi.pdf.

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[98] Sameen Ahmed Khan, Medieval Arab Contributions to Optics, Digest of Middle East Studies (DOMES), 25(1), 19-35 (Spring 2016). http://dx.doi.org/10.1111/dome.12065.

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[99] Sameen Ahmed Khan, 2015 declared the International Year of Light and Light-based Technologies. Current Science 106(4), 501 (25 February 2014). http://www.currentscience.ac.in/Volumes/106/04/0501.pdf.

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[100] Sameen Ahmed Khan, International Year of Light and Light-based Technologies, LAP LAMBERT Academic Publishing, Germany (2015). http://isbn.nu/9783659764820/.

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[101] Sameen Ahmed Khan, International Year of Light and History of Optics, Chapter-1 in: Advances in Photonics Engineering, Nanophotonics and Biophotonics, Editor: Tanya Scott, (Nova Science Publishers, New York, 2016, http://www.novapublishers.com/). pp. 1-56 (April 2016). http://isbn.nu/978-1-63484-498-7. [102] Sameen Ahmed Khan, Reflecting on the International Year of Light and Light-based Technologies, (communicated).

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