Aberrations in Helmholtz optics

Optik 153 (2018) 164–181

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Optik journal homepage: www.elsevier.de/ijleo

Original research article

Aberrations in Helmholtz optics Sameen Ahmed Khan Department of Mathematics and Sciences, College of Arts and Applied Sciences (CAAS), Dhofar University, Post Box No. 2509, Postal Code: 211 Salalah, Oman

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 21 June 2017 Accepted 2 October 2017 Keywords: Scalar optics Helmholtz optics Helmholtz equation Aberrations Wavelength-dependent effects Graded-index fiber Graded index slab Feshbach–Villars linearization procedure Foldy–Wouthuysen transformation

The scalar optics is based on a Hamiltonian derived using the Fermat’s principle of least time. The same Hamiltonian can now be derived from the Maxwell equations. The Helmholtz equations has a striking mathematical similarity to the Klein–Gordon equation for the relativistic spin-0 particle. It is possible to make use of this strong similarity through quantum techniques and develop an alternative to the traditional approaches. The non-traditional formalism of Helmholtz optics reproduces the traditional results as expected. Moreover, it leads to the wavelength-dependent modifications of the paraxial as well as the aberrating behavior. To illustrate the formalism, we consider the propagation of light in optical fibers in substantial detail. We also consider the propagation of light through a graded index slab. © 2017 Elsevier GmbH. All rights reserved.

1. Introduction We consider a monochromatic quasiparaxial light beam propagating along a straight optic-axis chosen to be the positive z-axis. Such a beam is governed by

i ¯

∂  (r), (r) = H ∂z

(1)

where  ¯ = /2 = c/ω has the status of  = h/2, the reduced Planck’s constant in quantum theory. The beam-optical Hamiltonian is



 = − n2 (r) −  H p⊥ 2

1/2 ,

(2)

where  p⊥ = −i p⊥ |  ¯ ∇ ⊥ . The light rays are assumed to be propagating in a close proximity to the optic-axis. Algebraically, | pz ≈ 1. It is further assumed that the refractive index varies smoothly around a constant value n0 , satisfying |n(r) − n0 |  n0 . The beam-optical Hamiltonian in Eq. (2) can be derived using two very independent approaches. Historically, the first procedure is based on the Fermat’s principle of least time [1] and precedes Maxwell’s theory of electromagnetism by two

E-mail address: [email protected] https://doi.org/10.1016/j.ijleo.2017.10.006 0030-4026/© 2017 Elsevier GmbH. All rights reserved.

S.A. Khan / Optik 153 (2018) 164–181

165

centuries. The second procedure is based on the Helmholtz equation derived from Maxwell’s equations. The Helmholtz equation is



∇ − 2

n2 (r)



(r) = 0.

2

 ¯

(3)

After rearranging the terms in the Helmholtz equation, we obtain



∂ i ¯ ∂z

2

(r) =



2

n2 (r) −  p⊥

 (r).

(4)

At this stage, one resorts to the square-root [2] in the following manner



i ¯

∂ 2 p⊥ (r) = − n2 (r) −  ∂z

1/2

(r).

(5)

The negative/positive sign indicates that the light ray is propagating in the positive/negative z-direction. We have chosen the negative sign and consequently the propagation in the positive z-direction in our treatment. It is interesting to note, that the beam-optical Hamiltonian in the square-root approach is identical to the one obtained using the Fermat’s principle. The two equivalent Hamiltonians in Eqs. (2) and (5) are to be expanded in a meaningful power series. This is best done  . An appropriate ordering p⊥ . The resulting expansion does not lead to an Hermitian H in the transverse components, r⊥ and  and symmetrization is applied to the polynomials in order to ensure this [2–5]. For a medium with a constant refractive index, n(r) = n0 , the Hamiltonian has the Taylor expansion

 H

=



2

− n2 (r) −  p⊥

 =

−n0

=

n0

1

1− −1 + 5

1/2

2 p⊥ n20



+

1/2



1 2n20

8  p + 8 ⊥

128n0

2  p⊥ +

7

1 8n40

4  p⊥ +

1 16n60

6  p⊥

(6)



10  p + ··· 10 ⊥

256n0

(2m − 3)!! 1 ∞

=

n0

(2m)!! m=0

nm 0

m  p⊥ .

We recall that the double factorial leads to 1!! = 1, 0!! = 1, (−1)!! = 1, and (−3)!! = −1. In most practical situations, the refractive index need not have a constant value. In all such situations, the infinite Taylor expansion of the spatially varying refractive index n(r) = n( r ⊥ , z) is inevitable. From this infinite expansion of the radical, one keeps the terms to a desired degree of 2

p⊥ /n20 . This expansion is accompanied with the Taylor expansion of n(r) to the same degree of accuaccuracy in powers of  racy. The power series expansion of the Hamiltonian thus obtained has quadratic terms along with the higher-order terms. The paraxial or ideal behavior is governed by the quadratic terms in the Hamiltonian. The higher-order terms are responsible for the deviations from the ideal behavior and are called as the aberrating terms. These terms govern the aberrations of the corresponding order [2]. It should be noted, that the transition from Eqs.(3) to (5) reduces the original boundary value problem to a first order initial value problem in z. This reduction is of great practical value, since it leads to the powerful system or the Fourier optic approach [6–8]. However, the above reduction process is not mathematically rigorous or exact! Hence, there is room for alternative procedures for the reduction. Of course, any such reduction scheme is bound to lack in rigor to some extent, and the ultimate justification lies only in the results the scheme leads to. The square-root operators have drawn a wide interest as they occur in diverse situations (see [9] for details). Helmholtz equation of scalar optics and the Klein–Gordon equation of relativistic quantum mechanics have a striking algebraic similarity. It is possible to make use of this strong similarity through quantum techniques and develop an alternative to the widely-used square-root approach. The resulting approach, utilizing the experience of quantum mechanics leads to a non-traditional formalism of Helmholtz optics [10–12]. The initial results in this direction were reported in this journal [12] and the present article is a logical continuation with a focus on the aberrations. From the very beginning the reader is reminded that the electromagnetic field and the Klein–Gordon field are two entirely different entities. But the underlying similarity in the mathematical structure of the two systems, enables us to use quantum methodologies in Helmholtz optics. This approach is justified as it leads to well-established results. In Section 2, we shall recount briefly some of the other attempts to go beyond the paraxial regime. Section 3 has the formalism based on the quantum methodologies. Section 4 is all about aberrations. Section 5 has the applications. The propagation in optical fibers is done in detail. Propagation in a graded-index medium slab is also covered in detail. Section 6 has our concluding remarks.

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2. Beyond the paraxial approximation There have been several notable attempts to go beyond the paraxial approximation and, in that process, to obtain a precise knowledge of the meaning and accuracy of paraxial wave optics itself. We highlight here only some of these, and the reader may consult these works for further references. A significant early attempt in this regard is due to Lax et al. [13]. These authors pointed out that the process of neglecting ∇ (∇ · E) and seeking a solution that is plane polarized in the same sense everywhere is simply incompatible with the exact Maxwell equations. They developed an expansion procedure in powers of the beam parameter (w0 /), where w0 is the waist size and  = w02 / ¯ is the diffraction length or (twice the) Rayleigh range of the beam under consideration. In addition to showing that the zero-order field obeyed the Maxwell system of equations, they developed the equations obeyed by the higher-order corrections. The first-order correction was shown to be longitudinal. Agrawal and Pattanayak [14] studied the propagation of Gaussian beam in a simple (linear, homogeneous, and isotropic) dielectric using the angular spectrum representation for the electric field, and showed that the exact solution consisted of the paraxial result plus higher-order non-Gaussian correction terms. They demonstrated, in particular, that the second-order correction term satisfied an equation consistent with the work of Lax et al. [13]. In another paper, Agarwal and Lax [15] examined the role of the boundary condition in respect of the corrections to the paraxial approximation of Gaussian beams. This work resolved the controversy between the work of Couture and Belanger [16] and that of Agarwal and Pattanayak [14], by tracing it to the fact that the two works had used qualitatively different boundary conditions for the correction terms: while Agarwal and Pattanayak had made the more natural demand that the field distribution in the waist plane be strictly Gaussian in the exact treatment, as in the paraxial case, Couture and Belanger had demanded the on-axis field to be the same in both treatments. A major step in directly connecting solutions of the paraxial equation to those of the exact treatment was taken by Wünsche [17], who showed that it is possible to construct a linear transition operator, which transforms arbitrary solutions of the paraxial equation into exact (monochromatic) solutions of the scalar wave equation (Helmholtz equation). Indeed, Wünsche constructed two such operators T1 , T2 corresponding to two different boundary conditions and noted, moreover, that the transition operator method is equivalent to the complete integration of the system of coupled differential equations of Lax et al., restricted to the scalar case. Cao and Deng [18] derived a simpler transform operator under the condition that the evanescent waves can be ignored, and used this transform to study the corrections to the paraxial approximation of arbitrary freely propagating beam. They verified the consistency of their conclusions with those of the perturbation approach of Lax et al. Subsequently, Cao [19] applied the method of Lax et al. to non-paraxial light beam propagating in a transversely nonuniform refractive index medium, computed the correction terms in detail, and specialized the results to the case of Gaussian beam propagating in transversely quadratic refractive index media. The transition operator method has been further extended by Alonso et al. [20]. The uncertainty product has played a fundamental role in the analysis of paraxial (both coherent and partially coherent) beams. Alonso and Forbes [21] have recently generalized the uncertainty product to the case of non-paraxial fields. There are two aspects to the issue of going beyond paraxial optics. The first one is to do with the spatial dependence alone, and hence applies to ‘scalar’ optics. The second one is to do with the vectorial nature of the light field and, more specifically, with the fact that Maxwell’s is a constrained system of equations. The restriction ∇ · E = 0 (in free space and in homogeneous dielectrics) demands that the spatial dependence of the field, even in a transverse plane like the waist plane, cannot be chosen independent of polarization. Thus, an input plane-polarized plane wave going through a thin spherical lens results, in the exit plane of the lens, in a wave which is spherical and hence necessarily has spatially-varying polarization so that the Poynting vector at all points in the exit plane points to the focus of the lens. Though the work of Lax et al. pointed to both these aspects, the subsequent ones noted above largely concentrated on the first. Examining the fundamental Poincaré symmetry of the Maxwell system in the front-form of Dirac, Mukunda et al. [22,23] developed a formalism which converts a solution of the scalar wave equation into a corresponding (approximate) solution of the Maxwell system, resulting in a generalization of Fourier optics for vector electromagnetic beams [24]. This formalism leads to simple-looking electromagnetic Gaussian beams [25,26], and predicts a cross-polarization term [27] which is consistent with experimental observations [28]. Further analysis of electromagnetic Gaussian beams has been presented by Sheppard and Saghafi [29], and Chen et al. [30]. There are two primary reasons for our believing that this approach may have advantage over the ones noted above. First, the Foldy–Wouthuysen transformation iteratively transforms the field to a new representation, where the forward propagating components get progressively decoupled from the backward propagating components [10–12]. It enables us to study non-paraxial beams and the passage through optical systems, which are not necessarily paraxial (Gaussian). Secondly, the Foldy–Wouthuysen method appears ideally suited for the Lie algebraic approach. Finally, the Foldy–Wouthuysen technique generalizes to the vector case with very little extra effort [31,32]. In 1984, Fishman and McCoy had made a suggestion to use the Foldy–Wouthuysen transformation in the case of the Helmholtz equation [33]. In 1996, Jagannathan and Khan independently chartered the same idea, while working out the quantum theory of charged-particle beam optics (see, p. 277 in [34] for details). It was only in 2002, when Jagannathan et al. executed this idea [10]. This article would be technically incomplete without a note on the Foldy–Wouthuysen transformation technique. We shall just note the background leading to the Foldy–Wouthuysen transformation. The technical details will follow as we use it in our formalism. Historically, the Foldy–Wouthuysen transformation was developed to take the non-relativistic

S.A. Khan / Optik 153 (2018) 164–181

167

limit of the Dirac equation (the relativistic equation for the spin-1/2 particles [35–44]). The success of the spin-1/2 case, prompted the extension of the Foldy–Wouthuysen transformation to arbitrary spins [45,46]. A closer mathematical analysis lead to the important realization that the Foldy–Wouthuysen technique is applicable to a class of matrix-equations with a particular mathematical structure. The class of equations is much larger and includes equations outside the domain of quantum mechanics! This universality of the Foldy–Wouthuysen transformation has led to numerous applications: (a) atomic systems [47,48]; (b) derivation of the Bloch equation for polarized beams [49]; (c) synchrotron radiation [50]; (d) acoustics [51–56]; (e) ocean acoustics [57]. A comprehensive account of the power of the Foldy–Wouthuysen transform in the context of optics is to be found in [31,32]. The Foldy–Wouthuysen technique is central to the non-traditional formalism of Helmholtz optics-[10–12] and the matrix-formulation of Maxwell optics [58,59]. In the latter formalism, one requires an exact representation of Maxwell’s equations in a medium with varying refractive index [60]. The algebraic tools of the nontraditional formalism of Helmholtz optics and the non-traditional formalism of Maxwell optics were decisively influenced by the quantum theory of charged-particle beam optics, based on the Dirac equation [61–69] and the Klein–Gordon equation [34,69]. The quantum theory of charged-particle beam optics leads to a unified formalism of beam-optics and spin dynamics of a beam of Dirac particles with due consideration to the anomalous magnetic moment [70–79]. Christopher Hill explored the diffractive limit for charged-particle beams [80]. The frontier field of QABP, the quantum aspects of beam physics has held several meetings, whose proceedings are an invaluable source [81–83]. 3. Quantum methodologies in Helmholtz optics The Helmholtz wave equation with varying refractive index n(r) is

∇ 2  (r, t) =

n2 (r) ∂

v2

2

∂t 2

 (r, t).

(7)

For a monochromatic light beam, we have (r)e−iωt , ω > 0.

 (r, t) =

(8)

We rearrange the terms and rewrite Eq. (8) as

 ∇2 +

n2 (r)

v2

 ω2

(r) = 0.

(9)

Now, we incorporate the process of wavization [3] through the very familiar Schrödinger replacement −i p⊥ , −i ¯ ∇ ⊥ −→  ¯

∂ −→ pz . ∂z

This leads to



∂ −i ¯ ∂z

2



2

− n

2 (r) − p⊥





(10)

 (r) = 0.

(11)

In beam-optics, we are primarily interest in the ‘profile of the light beam’ and not the ‘time of flight’. Hence, the spatial variable ‘z’ along the optic axis (z-axis) is preferred over the variable time ‘t’. In beam-optics, the commutation relation, [ p⊥ ,  r ⊥ ] = −i p⊥ = ¯ is analogous to the commutation relation [p, q] =− i in quantum mechanics. The transverse momenta  −i ¯ ∇ ⊥ and the spatially varying refractive index n(r) need not commute. It is this non-commutativity, which generates the wavelength-dependent terms. In the limit −→0 (low wavelength), the non-traditional formalism of Helmholtz optics very ¯ naturally leads to the Lie algebraic formalism of light beam optics (see [2,4,5]). Unlike in the traditional approaches, we shall linearize Eq. (11) using the Feshbach–Villars-like procedure [84,85]. To this end, we introduce two functions 1 (r) and 2 (r)



1 (r)

=

2 (r)

(r)  ∂ −i ¯ n0 ∂z

(r)

.

(12)

Then, the time-independent Helmholtz equation in Eq. (11) is written as

 ∂ −i ¯ n0 ∂z

1 (r) 2 (r)

⎛  =⎝ 1 n20

0 2

n2 (r) −  p⊥



1 0

⎞ ⎠

1 (r) 2 (r)

.

(13)

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S.A. Khan / Optik 153 (2018) 164–181

Table 1 The correspondence between particle dynamics and paraxial optics Standard Dirac equation

Light beam-optical form

m0 c ˇ +  ED +  OD m0 c2 Positive energy

−n0 z +  E+ O −n0 Forward propagating beam

2

|  m0 c Nonrelativistic, | Non-relativistic motion + relativistic corrections

p⊥ |  n0 Paraxial beam, | Paraxial behavior + aberration corrections

Eq. (13) thus obtained has the structure of a first-order system. Next, we rewrite Eq. (13) in a Dirac-like form, using the following transform



1 (r)



+ (r)



−→ − (r)

⎛ 2 (r) 1 = √ 2

1 M= √ 2



=M

2 (r)

where

1 (r)

1

1

⎜ ⎜ ⎝

⎞

 ∂ (r) − i ¯ n0 ∂z

(r)

 ∂ (r) + i ¯ n0 ∂z

(r)

⎟ ⎟, ⎠

(14)

 = M −1 .

1 −1

(15)

The significance of the aforementioned transformation is evident, when we apply it to a monochromatic quasiparaxial beam propagating in the positive z-direction with the z-dependence



(r)∼ exp

i

n(r) z  ¯

then + (r)

− (r)

∼

1 √ 2

∼

1 √ 2

 



,

1+

n(r) n0

n(r) 1− n0

(16)

 (r),



(17) (r).

Let us recall the basic assumption that the refractive index varies smoothly around a constant value n0 , satisfying |n(r) − n0 |  n0 . This basic assumption leads to the fundamental inequality | + (r)| | − (r)|, central to the Foldy–Wouthuysen transformation technique. This enables us to rewrite Eq. (11) readily in a Dirac-like form as i ¯

∂ ( ∂z

+ (r)

+ (r)

( )=H

)

− (r)

− (r)

 = −n0 z + E + O  H

(18)

2 E = 1 { p⊥ + (n20 − n2 (r))}z

2n0

= O

1 2 { p + (n20 − n2 (r))}(iy ), 2n0 ⊥

where  y and  z are from the triplets of the Pauli matrices,



=



x =

0

1

1

0





, y =

0

−i

i

0





, z =

1

0

0

−1

 .

(19)

The beam-optical Hamiltonian in Eq. (18) is exact, which can be verified by squaring it 2

 = (n2 (r) −  H p⊥ ). 2

(20)

The beam-optical Hamiltonian is exact and furthermore has a one-to-one algebraic and physical correspondence with the Dirac equation. This correspondence is concisely illustrated in Table 1. Having established the mathematical and physical correspondence, it is now possible to apply the powerful tools of relativistic quantum mechanics to the beam-optical system

S.A. Khan / Optik 153 (2018) 164–181

169

in Eq. (18). The foremost technique to be incorporated is the Foldy–Wouthuysen transformation. The first iteration (that is 2

p⊥ /n20 )) of the Foldy–Wouthuysen transformation leads to the formal Hamiltonian to order ( i ¯

∂  (2) | , | =H ∂z

(21)

 (2) = −n0 z + E − 1 z O 2. H 2n0

It is straightforward to see, that the square of the Pauli matrices is a unit matrix. Furthermore, we are interested in a light beam propagating in the positive z-direction. So, we omit  z . This leads to the very familiar form of the Hamiltonian (2)

 H

=

1 2n0

−n0 + +

1

   2  p⊥ + n20 − n2 (r)

  2 2 2 2  p + n − n . (r) ⊥ 0 3

(22)

8n0

The formal expression for the Hamiltonian after the second iteration of the Foldy–Wouthuysen procedure is i ¯

∂  (4) | , | = H ∂z (4)

 H

=

−n0 z +  E−

1 2 z O 2n0 (23)

∂

1

 , ([O  , E] + i¯ O  )] [O ∂z 8n20

−

1

+

4

8n30

 + ([O  , E] + i¯ z {O

2

∂  O) }. ∂z

After some algebra and omitting  z , the Hamiltonian in Eq. (23) simplifies to (4)

 H

=

−n0 + + −

1

1 2 { p + (n20 − n2 (r))} 2n0 ⊥ 2

8n30

{ p⊥ + (n20 − n2 (r))}

i ¯

2

32n40

[ p⊥ ,

2

∂ 2 (n (r))] ∂z

(24)

2

2

 ∂ + ¯ 5 ( (n2 (r))) ∂ 32n0 z 3 1 2 + { p⊥ + (n20 − n2 (r))} 16n50 4 5 2 + { p⊥ + (n20 − n2 (r))} . 128n70

The imaginary unit, ‘i’ present in the Hamiltonian is nullified by the unit ‘i’ coming from execution of the commutator. 2

4

p⊥ /n20 ) . For consistency, we also The Hamiltonian in Eq. (24) has all the terms from the traditional prescriptions to order ( have the terms to the same order coming from the Taylor expansion of the refractive index. Significantly, the Hamiltonian has the wavelength-dependent terms, which is an outcome of the quantum methodologies. Such terms are absent in the traditional approaches. We further note, that the wavelength-dependent terms arise at each order starting with the paraxial order. In the next section, we shall examine the implications of such extra terms on the aberrations. The Foldy–Wouthuysen procedure can be carried out to still higher orders to a desired degree of accuracy. Before addressing the aberrations, we make a few remarks on the paraxial Hamiltonian. (p)

 H

=

−n0 + −

i ¯

1 2 { p + (n20 − n2 (r))} 2n0 ⊥ 2

[ p⊥ , 4

32n0

∂ 2 (n (r))]. ∂z

(25)

The extra wavelength-dependent term in the paraxial Hamiltonian in Eq. (25) manifests itself, whenever the refractive index of the medium possesses either longitudinal or transverse inhomogeneities. The source of this term is the commutator term



− 1/8n20



, O





 /∂z  , E + i¯ ∂O O



(4)

 in H

in Eq. (23). For comparison, the corresponding term in the Foldy–Wouthuysen

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S.A. Khan / Optik 153 (2018) 164–181

representation of the Dirac equation leads to the accurate explanation of the spin-orbit term (including the Thomas precession effect) and the Darwin term (ascribed to the zitterbewegung) (see Section 4.3 of [86]). The analogous term in the Foldy–Wouthuysen representation of the Klein–Gordon equation corresponds the correction term to the classical electrostatic interaction of a point charge (see Section 9.7 of [86]). In the quantum theory of charged-particle beam optics (based on the Klein–Gordon and Dirac equations respectively) the corresponding terms are quantum corrections to the classical paraxial and aberrating behavior. Guided by analogy, it would be interesting to study the influence of the commutator term on the propagation of Gaussian beams in a parabolic index medium, whose focusing strength is modulated in the axial variable z, however tiny it may be compared to the classical terms. In the first-order optical systems, the ray parameters in the input and output planes are related as



r ⊥

p⊥



 =

P

Q

R

S



r ⊥

,

p⊥

out

(26)

in

where P, Q, R and S are the formal solutions of the paraxial Hamiltonian in Eq. (25). These are the formal paraxial transfer maps. This is known as the Kogelnik’s ABCD-law [87] (even though, we have used the functions P, Q, R and S). From the symplecticity conditions, it follows that P S − Q R = 1. The structure of the paraxial equations further leads to the relations R = P and S = Q , where () denotes the z-derivative. We shall shortly see that the aberration coefficients to all orders can be expressed in a closed form in terms of the four paraxial solutions (P, Q, R and S). We shall do so explicitly in the case of the third-order aberrations. 4. Aberrations In this section, we shall focus on the aberrations. The formal expression of the transfer operator connecting any pair of points {(z , z )|z > z } on the optic axis is | (z , z ) =  T(z , z )| (z , z ) ,

(27)

with i ¯

∂  T(z , z ), T(z , z ) = I, T(z , z ) = H ∂z

 z T(z , z ) = ℘{exp[− i  (z)]} dz H  ¯ z  z i   (z) dz H = I−  ¯

z

i 2 + (− )  ¯





z

dz z

z

(28)

 (z)H  (z ) dz H

z

+· · ·, where ℘ denotes the path-ordered exponential and  I is the identity operator. In general,  T(z , z ) does not have a closed form expression. Following is the most useful way to express the z-evolution operator  T(z , z ), or the z-propagator

T(z , z ) = exp [− i  T (z , z )],

(29)

 ¯

with

 T (z , z ) =



z

 (z) dz H

z

+

1 i (− ) 2  ¯





z

dz z

z

 (z), H  (z )] dz [H

(30)

z

+· · ·. The series expression in Eq. (30) is the Magnus formula [88–91]. The beam-optical Hamiltonian in Eq. (24) can be partitioned as

 = H

0,p + H 0,(4) + H  0,(6) + H  0,(8) H ) ) ) ) ¯ +H ¯ +H ¯ +H ¯ .  (0,(2)  (0,(4)  (0,(6)  (0,(8) +H

(31)

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171

Table 2 The Table of Aberrations Symbol

Polynomial

Name

p⊥  2  p⊥ , p⊥ · r ⊥ + r ⊥ ·p⊥ +  2  p⊥ · r ⊥ + r ⊥ ·  p⊥ 2  2 p⊥ r 2⊥ + r 2⊥p⊥ 2   r⊥ ,  p⊥ · r ⊥ + r ⊥ ·  p⊥ + 4

C K A F D

Spherical aberration Coma Astigmatism Curvature of field Distortion

r 4⊥

E

Pocus

0,(4) is responsible for the third-order aberrations. The Hamiltonians, H 0,(6) and H 0,(8) lead to the fifth-order The Hamiltonian, H and seventh-order aberrations respectively. All these Hamiltonians invariably get modified by the wavelength-dependent ¯ ,H ¯ and H ¯ respectively. Here, we shall examine the third-order aberrations in detail.  ()  ()  () contributions coming from H 0,(4)

0,(6)

0,(8)

The most accurate and compact way to express the transfer operator in terms of the formal paraxial solutions namely, (P, Q, R, S), is through the interaction picture developed in the ‘Lie algebraic treatment’ of both light-beam optics and charged-particle beam optics.

  T(z, z0 ) = exp − i  T (z, z0 ) ,  ¯



i 4 p⊥ = exp − {C(z , z )  ¯ 2

p⊥ , ( p⊥ · r ⊥ + r ⊥ ·  p⊥ )]+ +K(z , z )[

⊥ · r ⊥ + r ⊥ ·  p⊥ )

+A(z , z )(p

2

(32)

2

2

+F(z , z )( p⊥ r 2⊥ + r 2⊥ p⊥ ) +D(z , z )[r 2⊥ , ( p⊥ · r ⊥ + r ⊥ ·  p⊥ )]+ +E(z , z )r 4⊥ }]. The quantity [A, B]+ = (AB + BA) is called the anticommutator. The six functions (C, K, A, F, D, E) have one-to-one correspondence with the classical Seidel third-order monochromatic aberrations (see [1,4] for details). This correspondence is illustrated in Table 2. The name Pocus is adopted from page 137 in [2]. The effect of the six aberrations on the beam are enumerated below

1. Spherical aberration: 4  T = C (z , z )  p⊥ 2

r ⊥ out = r ⊥ in − 4C (z , z )  p⊥  p⊥ in

(33)

 p⊥ out =  p⊥ in 4

The effect of the term C (z , z )  p⊥ is that the deviations in the actual arrival point of a light beam at the image plane depends on its Gaussian arrival direction r⊥ by an amount given in Eq. (33) and does not depend on the its Gaussian arrival point r⊥ . This effect is known as the spherical aberration. 2. Coma:

   2  T = K (z , z )  p⊥ ,  p⊥ · r ⊥ + r ⊥ ·  p⊥   +   2 r ⊥ out = r ⊥ in − 2K (z , z )   r⊥,  p⊥ +  p⊥ ,  p⊥ · r ⊥ + r ⊥ ·  p⊥

+ in 2

 p⊥ out =  p⊥ in + 4K (z , z )  p⊥  p⊥ in

+

(34)

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S.A. Khan / Optik 153 (2018) 164–181

3. Astigmatism:

 2  T = A (z , z )  p⊥ · r ⊥ + r ⊥ ·  p⊥    r ⊥ out =  r ⊥ in − 2A (z , z )   r⊥,  p⊥ · r ⊥ + r ⊥ ·  p⊥

+ in     p⊥ out =  p⊥ in + 2A (z , z )   p⊥ ,  p⊥ · r ⊥ + r ⊥ ·  p⊥

+ in

(35)

4. Curvature of field:

  2 2  T = F (z , z )  p⊥ r 2⊥ + r 2⊥ p⊥   2 r ⊥ out = r ⊥ in − 2F (z , z )   p⊥ ,  r ⊥ in  + 2  p⊥ out =  p⊥ in + 2F (z , z )   r⊥,  p⊥ in

(36)

+

5. Distortion:

    T = D (z , z ) r 2⊥ ,  p⊥ · r ⊥ + r ⊥ ·  p⊥ +

r ⊥ out  p⊥ out

2

r ⊥ in − 4D (z , z )  r ⊥ r ⊥ in

=

     2 =  p⊥ in + 2D (z , z )   r⊥,  p⊥ +  r⊥,  p⊥ · r ⊥ + r ⊥ ·  p⊥

+ in

(37)

+

6. Pocus:

 T = E (z , z ) r 4⊥ r ⊥ out = r ⊥ in  p⊥ out =

(38)

2 p⊥ in + 4E (z , z ) r ⊥ r ⊥ in



 

This term has no effect on the quality of an image. However, it does effect the arrival direction  p⊥ of a ray at the image plane and is of significance if the optical system is part of a larger optical system. The terms C, K, A, F D and E describe the well-known Seidel aberrations. In the Lie algebraic approach, each monomial corresponds to a particular aberration. Hence, it has been possible to classify the aberrations using Lie methods [92]. The classification also works for the asymmetric systems [93–95]. Matrix methods have been used in various other areas of optics and have significant advantages for simplifying and presenting the equations of optics [96]. Same is true for other algebraic techniques [97,98]. 5. Applications In Section 1, we outlined the traditional approaches using the Fermat’s principle of least time and the square-root of the Helmholtz operator. In Section 2, we presented the issues related to the square-root approach along with the numerous ways to go beyond the paraxial approximation. In Section 3, we obtained an alternative the customary square-root approach by exploiting the algebraic similarities between the Helmholtz equation and the Klein–Gordon equation. In that section, we developed the use of quantum methodologies in Helmholtz optics. The resulting beam-optical Hamiltonian derived using the quantum methodologies contains the traditional terms accompanied with wavelength-dependent contributions. In Section 4, we discussed the aberrations without assuming any specific form for the refractive index. Now, we shall apply the formalism to some specific systems. 5.1. Medium of constant refractive index Let us first consider the ideal system described by a constant refractive index, n(r) = nc . For this system the Hamiltonian in Eq. (18) is



 c = −n0 z + Dz + D iy H D=

1 2n0



   2  p⊥ + n20 − n2c .

(39)

S.A. Khan / Optik 153 (2018) 164–181

173

The constant refractive index beam-optical Hamiltonian is mathematically similar to the free-particle Dirac Hamiltonian. We recall that the free-particle Dirac Hamiltonian is exactly diagonalizable (see [34,86] for details). So, we shall follow the same recipe, which leads to diagonalized beam-optical Hamiltonian diagonal

c H



2

= − n2c −  p⊥

1/2



z .

Using the notation, P = 1/n20 (4)

c H

= −n0



1−

(40)

1/2  2  , the Foldy–Wouthuysen iterative procedure gives p⊥ + (n20 − n2c )

1 1 1 3 5 4 P − P2 − P − P 2 8 16 128

 z

1  ≈ −n0 1 − P 2 2 z (41)



2

= − n2c −  p⊥ diagonal

c =H

1

2

z

.

This result is consistent with the traditional prescriptions. The transfer operator is also obtained exactly in a closed form

 c (zout , zin ) = exp[− i zHc ] U  ¯ 2

p2⊥ p2 i 1 1  = exp[+ nc z{1 − − ( ⊥ ) − · · ·}], 2 2 nc 8 n2c  ¯

(42)

where z = (zout − zin ). Eq. (42) gives the exact transfer maps



r ⊥

⎛

p⊥

=⎝

1 0

out

1

!

n2c − p2⊥ 1

z

⎞ ⎠

r ⊥



p⊥

.

(43)

in

It is apparent that the beam-optical Hamiltonian is intrinsically aberrating. Even the simplest conceivable system of constant refractive index possesses aberrations to infinite orders! 5.2. Propagation in optical fibers An optical fiber can be modeled using an axially symmetric graded-index medium with the following profile (see p. 117 in [2]): n(r ⊥ , z) = n0 + ˛2 (z)r 2⊥ + ˛4 (z)r 4⊥ + ˛6 (z)r 6⊥ + ˛8 (z)r 8⊥ + · · ·.

(44)

The axial symmetry forbids the other powers of r⊥ . For this medium to function as an optical fiber, the refractive index n(r⊥ , z) should have the largest value at r⊥ = 0 and lower values as r⊥ increases. This implies that n0 ≥ 1 and ˛2 (z) < 0. The beam-optical Hamiltonian shall be expressed in terms of

   T =  p⊥ · r ⊥ + r ⊥ ·  p⊥ w1 (z) =

d dz

w2 (z) =

d dz

w3 (z) =

d dz

w4 (z) =

d dz

   

2n0 ˛2 (z) ˛22 (z) + 2n0 ˛4 (z) 2n0 ˛6 (z) + 2˛2 (z)˛4 (z) ˛24 (z) + 2˛2 (z)˛6 (z) + 2n0 ˛8 (z) .

(45)

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S.A. Khan / Optik 153 (2018) 164–181

The paraxial and the aberrating Hamiltonians up to eighth-order (governing seventh-order aberrations) are

 = H

0,p + H 0,(4) + H 0,(6) + H 0,(8) H ( ) ( ) ( ) ( ) ¯ +H ¯ +H ¯ +H ¯ 0,(2) 0,(4) 0,(6) 0,(8) +H

0,p H

=

0,(4) H

=

0,(6) H

=

−n0 + 1

1 2  p − ˛2 (z)r 2⊥ 2n0 ⊥ ˛2 (z)

4  p − 3 ⊥

4n20

8n0 1

˛2 (z)

6  p − 5 ⊥

8n40

16n0 +

 1

8n30

  2 2  p⊥ r 2⊥ + r 2⊥ p⊥ − ˛4 (z)r 4⊥    4 4 2 2  p⊥ r 2⊥ + r 2⊥ p⊥ +  p⊥ r 2⊥ p⊥

−˛6 (z)r 6⊥

0,(8) H

=

5

+

5˛2 (z)

8  p − 7 ⊥

128n0





1

   2 4 2 2  p⊥ r ⊥ + r 4⊥ p⊥ + 2˛22 (z)r 2⊥ p⊥ r 2⊥

˛22 (z) − 2n0 ˛4 (z)

32n50

64n60

    4 2  p⊥ ,  p⊥ , r 2⊥

+ +

 2  4 4 2 3˛22 (z) − 4n0 ˛4 (z)  p⊥ , r ⊥ + 5˛22 (z)  p⊥ , r 2⊥





2

2

− 2˛22 (z) + 4n0 ˛4 (z)  p⊥ r 4⊥ p⊥ +

+

+



 

1

 2 6  p⊥ , r ⊥

4 ˛32 (z) + n0 ˛2 (z)˛4 (z) + n20 ˛6 (z)

16n40



   2 −5˛32 (z) r 4⊥ ,  p⊥ , r 2⊥

+ +





+ 2˛32 (z) + 4n0 ˛2 (z)˛4 (z)

2

r 2⊥ , r 2⊥ p⊥ r 2⊥

(46) +

  +

−˛8 (z)r 8⊥ 2

( ) ¯ 0,(2) H

=

 − ¯ 4 w1 (z) T 32n0

( ) ¯ 0,(4) H

=

 − ¯ 4 w2 (z) r 2⊥  T + T r 2⊥ 32n0

( ) ¯ 0,(6) H

=

−

( ) ¯ 0,(8) H

=

 − ¯ 4 w4 (z) r 6⊥  T + T r 6⊥ 8n0

2

3 ¯

2

32n40

2







w3 (z) r 4⊥  T + T r 4⊥



2

 + ¯ 5 32n0







2

+

 ¯ w12 (z)r 4⊥ 32n50

+

 ¯ w1 (z)w2 (z)r 6⊥ 16n50

2

w22 (z) + 2w1 (z)w3 (z) r 8⊥ .

0,p . The Hamiltonians, H 0,(4) , H 0,(6) and H 0,(8) are responsible for The ideal behavior is governed by the paraxial Hamiltonian H the third-order, fifth-order and seventh-order aberrations respectively. All the four traditional Hamiltonians are accompa¯ ,H ¯ ,H ¯ and H ¯ respectively. These wavelength-dependent  ()  ()  ()  () nied by the wavelength-dependent counterparts, H 0,(2)

0,(4)

0,(6)

0,(8)

0,(4) is responsible for all the six third-order aberHamiltonians are absent in the traditional prescriptions. The Hamiltonian H rations namely: (1) Spherical Aberration; (2) Coma; (3) Astigmatism; (4) Curvature of Field; (5) Distortion; and (6) Pocus [2]. ¯ modifies all the six aberration coefficients with wavelength-dependent components. Similarly, the  () The Hamiltonian H 0,(4)

higher-order Hamiltonians are responsible for the higher-order aberrations, which in turn get modified by the corresponding wavelength-dependent Hamiltonians respectively. 5.2.1. The paraxial Hamiltonian For this system the commutator term modifies the paraxial Hamiltonian as

S.A. Khan / Optik 153 (2018) 164–181

 H

(2)

=

( ) ¯  H0,(2) H 0,p + 

=

−n0 +

1 2 p − ˛2 (z)r 2⊥ 2n0 ⊥

175

(47)

2

 d − ¯ 3 { ˛2 (z)}( p⊥ · r ⊥ + r ⊥ ·  p⊥ ). 16n0 dz

This extra commutator term is bound to have a bearing on the beam optics of the system. The paraxial transfer maps are formally given by Eq. (26). 5.2.2. Aberrations The six aberration coefficients including the wavelength-dependent parts are



C (z , z )



z

=

dz

1 8n30

z

S4 −

˛2 (z) 2n20

2

 K (z

, z

)

Q 2 S 2 − ˛4 (z)Q 4

2

  − ¯ 4 w2 (z)Q 3 S + ¯ 5 w12 (z)Q 4 32n0 8n0



z

=

dz

1 8n30

z

˛2 (z)

RS 3 −

4n20

 ,

QS(PS + QR) − ˛4 (z)PQ 3

   − ¯ 4 w2 (z) Q 2 (PS + QR) + 2PQ 2 S 32n0 2

2

 A (z , z )

 + ¯ 5 w12 (z)PQ 3 32n0



z

=

dz

1 8n30

z



,

R2 S 2 −

˛2 (z) 2n20

PQRS − ˛4 (z)P 2 Q 2

2

 − ¯ 4 w2 (z) (PQ (PS + QR)) 16n0 2

 F (z

, z

)

 + ¯ 5 w12 (z)P 2 Q 2 32n0



z

=

dz

1 8n30

z



,

R2 S 2 −

˛2 (z) 4n20

(P 2 S 2 + Q 2 R2 ) − ˛4 (z)P 2 Q 2

2

 − ¯ 4 w2 (z) (PQ (PS + QR)) 16n0 2

 D (z

, z

)

 + ¯ 5 w12 (z)P 2 Q 2 32n0



z

=

dz

1 8n30

z



, ˛2 (z)

R3 S −

4n20

PR(PS + QR) − ˛4 (z)P 3 Q

   − ¯ 4 w2 (z) P 2 (PS + QR) + 2P 2 QR 32n0 2

2

 E (z , z )

 + ¯ 5 w12 (z)P 3 Q 32n0



z

=

dz z

1 8n30



R4 −

, ˛2 (z) 2n20

2

 − ¯ 4 w2 (z)P 3 R 8n0 2

 + ¯ 5 w12 (z)P 4 32n0

 .

P 2 R2 − ˛4 (z)P 4

(48)

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S.A. Khan / Optik 153 (2018) 164–181

Thus we see that the transfer operator and the aberration coefficients are modified by -dependent contributions. The ¯ traditional beam optics is obtained in the limit −→0. The sixth and eighth order Hamiltonians are modified by the presence ¯ of wavelength-dependent terms. These will in turn modify the fifth and seventh order aberrations respectively. 5.3. Propagation in a graded-index slab Let us consider one more example, that is the propagation through a graded-index slab of thickness ‘t’ comprising of a z-independent graded index medium. In such a medium n = n(r⊥ ). Consequently, the quantities ˛m in Eq. (44) are constants. So, the paraxial solutions in Eq. (26) turn out to be





P(z) = cos (z − z )

  1 sin (z − z ) 

Q (z) =



(49)



R(z) = − sin (z − z )





S(z) = cos (z − z ) , where

"

=

−

2˛2 . n0

(50)

It is straightforward to see the underlying symplectic structure of the paraxial solutions in Eq. (49). In this example, the aberration coefficients in Eq. (48) can be evaluated exactly as C =

  1 (−2n0 ˛2 )−1 (s4 − 4s2 ) − z 8

K =

1 (−2n0 ˛2 )−1 [ (−c4 + 2c2 )] 2









A=−

1 1 1 s4 + z − ˛2 z 2 3 3n20

F =−

1 1 1 s4 + z + ˛2 z 4 3 3n20

D=

1 1 (−2n0 ˛2 ) 2 [ (c4 + 2c2 )] 2

E =

  1 (−2n0 ˛2 ) (s4 + 4s2 ) − z , 8

where



sm =

cm =

sin mz m





1 − cos mz

=−

=−



m ˛2 4n20 ˛2 4n20

(51)



1+2

 5−6

n0 ˛4 ˛22 n0 ˛4 ˛22



(52)

 .

6. Concluding remarks We have developed a non-traditional formalism of Helmholtz optics using quantum methodologies. This was possible due to the mathematical similarities between the Helmholtz equation and the Klein–Gordon equation. We adopted two techniques from the relativistic quantum mechanics, which are the Feshbach–Villars linearization procedure and the Foldy–Wouthuysen transformation technique. The beam-optical Hamiltonians were derived to eighth-order without making

S.A. Khan / Optik 153 (2018) 164–181

177

any assumptions on the form of the refractive index. These general Hamiltonians have the parts obtained using traditional approaches accompanied with wavelength-dependent parts. It is shown that the formalism works very well in the wavelength-dependent regime. The new formalism was demonstrated by working out specific examples. We considered the exactly solvable example of a medium with constant refractive index. Then, we considered the propagation of light beam through an optical fiber. For this system, we derived the six third-order aberration coefficients. It was found that the aforementioned six aberration coefficients have wavelength-dependent components. In the limit, −→0 (low wavelength), our formalism leads to the Lie ¯ algebraic formalism of light beam-optics (see [2,4,99–102]). We have named this as the traditional limit of our formalism. This is analogous to the classical limit of quantum prescriptions. We note, that the mathematical machinery of the non-traditional formalism of Helmholtz optics presented here is very similar to the one used in the quantum theory of charged-particle beam optics (see [34,61–79]). In the latter, the classical limit is obtained by taking  ¯ 0 −→ 0, where  ¯ 0 = /p0 is the de Broglie wavelength and p0 is the average or design momentum of the system under study. The resulting classical limit is the Lie algebraic formalism of charged-particle beam optics [103–113]. Finally, we considered the propagation of light in a z-independent graded index slab. In this example, the six third-order aberration coefficients are derived exactly [2]. Quantum methodologies have diverse applications. For instance, it is possibly to apply quantum methodologies to multiparticle effects in charged-particle beam optics. Models developed using such approaches include: thermal wave model [114,115]); stochastic collective dynamical model [116]; and Diffraction Model for the beam halo [117–120]. In these models beam emittance plays the role of  the Planck’s constant and the analogous . ¯ These models are phenomenological and have successfully explained some aspects of beam physics. The close analogy between geometrical optics and the classical theories of charged-particle beam optics need no introduction. It is fascinating to note that the analogy between these two systems operates in the wavelength-dependent regime [121–126]. The Medieval Arab scholars had actively worked in optics using the corpuscular theories of light. A closer examination of the Medieval Arab texts on optics points to the fact opto-mechanical analogy can now be traced to Ibn al-Haytham (0965-1037) [127–134]. But it was only in 1833, when Hamilton rigorously 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 gave a firm mathematical foundation to the opto-mechanical analogy. This analogy played a pivotal role in the early years of electron optics (in the classical regime). In 1926, Hans Busch exploited the analogy to advance the electromagnetic lens. In 1931, Ernst Ruska used the magnetic lens to invent the electron microscope. It is to be noted that the opto-mechanical analogy had a decisive influence on the initial development of quantum mechanics in the 1920’s. Some aspects of the analogy received a wider attention during the United Nations designated 2015 International Year of Light and Light-based Technologies [135–141]. The extra wavelength-dependent terms arising in our formalism by employing quantum methodologies require a closer examination. It would be meaningful to explore the influence of these extra wavelength-dependent contributions experimentally, however tiny they may be. Acknowledgments I am grateful to Professor Ramaswamy Jagannathan (Institute of Mathematical Sciences, Chennai, India) for my training in the field of quantum mechanics of beam-optics, which was the topic of my doctoral thesis. He gave the brilliant suggestion to employ the Foldy–Wouthuysen transformation technique to investigate the scalar optics (Helmholtz optics). The new formalism of optics using quantum methodologies described in this article was selected for the Year in Optics 2016 by the Optical Society [of America]. The summary of the two selected papers [12,123] is available in the special issue of the Optics & Photonics News [125]. It is to be noted that Ref. [12] was published in this journal! References [1] V. Lakshminarayanan, A. Ghatak, K. Thyagarajan, Lagrangian Optics, Springer, 2002, http://dx.doi.org/10.1007/978-1-4615-1711-5. 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