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The Foldy–Wouthuysen transformation technique in optics
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Optics
Optik 117 (2006) 481–488 www.elsevier.de/ijleo
The Foldy–Wouthuysen transformation technique in optics Sameen Ahmed Khan Department of Mathematics & Applied Sciences, Middle East College of Information Technology (MECIT), Technowledge Corridor, Knowledge Oasis Muscat, P.B. No. 79, Al Rusayl 124, Muscat, Sultanate of Oman Received 14 July 2005; accepted 12 November 2005
Abstract We describe the standard Foldy–Wouthuysen transformation technique along with its applications in optics and the new results it leads to. The use of the Foldy–Wouthuysen transformation technique is central in the non-traditional prescriptions of Helmholtz optics and Maxwell optics, respectively, resulting in the wavelength-dependent modifications of light beam optics, even at the paraxial level. It is further shown that the wavelength-dependent modifications in the newly developed formalisms of light beam optics are in close anology with the quantum theory of charged-particle beam optics. r 2006 Elsevier GmbH. All rights reserved. PACS: 02; 41.75.i; 41.85.p; 42.15.i; 42.25.p Keywords: Helmholtz optics; Maxwell optics; Maxwell’s equations; Vector optics; Charged-particle optics; Wavelength-dependent effects; Beam propagation; Aberrations; Graded-index fiber; Mathematical methods of optics; Foldy–Wouthuysen transformation
1. Introduction The Foldy–Wouthuysen transform is widely used in high-energy physics. It was historically formulated by Leslie Lawrence Foldy and Siegfried Adolf Wouthuysen in 1949 to understand the non-relativistic limit of the Dirac equation, the equation for the spin-12 particles [1]. The approach of the Foldy and Wouthuysen made use of a canonical transform that has now come to be known as the Foldy–Wouthuysen transformation [2,3]. Before their work, there was some difficulty in understanding and gathering all the interaction terms of a Fax: +968 24446028.
E-mail addresses: [email protected], [email protected]. URL: http://www.pd.infn.it/khan/. 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2005.11.010
given order, such as those for a Dirac particle immersed in an external field. With their procedure the physical interpretation of the terms was clear, and it became possible to apply their work in a systematic way to a number of problems that had previously defied solution [4]. The Foldy–Wouthuysen transform was extended to the physically important cases of the spin-0 and the spin1 particles [5], and even generalized to the case of arbitrary spins [6]. The powerful machinery of the Foldy–Wouthuysen transform has found applications in diverse areas such as atomic systems [7] and acoustics [8]. Though the suggestion to employ the Foldy– Wouthuysen technique in the case of the Helmholtz equation existed in the literature as a remark [9], it has been exploited only now, to analyze the quasiparaxial approximations [10]. There is a close algebraic analogy
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between the Helmholtz equation (governing scalar optics) and the Klein–Gordon equation; and the matrix-form of the Maxwell’s equations (governing vector optics) and the Dirac equation. So, it is natural to use the powerful machinery of standard quantum mechanics (particularly, the Foldy–Wouthuysen transform) in analyzing these systems. This results in what we call as the non-traditional prescriptions of Helmholtz optics [11] and Maxwell optics [12], respectively. The non-traditional approaches give rise to very interesting wavelength-dependent modifications of the paraxial and aberrating behavior. The non-traditional formalism of Maxwell optics provides a unified framework of light beam optics and polarization. The non-traditional prescriptions of light optics are in close analogy with the quantum theory of charged-particle beam optics [13–21]. In the following sections we shall look at the standard Foldy–Wouthuysen transform in detail. A brief outline of the quantum theory of charged-particle beam optics and the non-traditional prescriptions of light optics is also presented.
2. The Foldy–Wouthuysen transformation Let us describe here briefly the standard Foldy– Wouthuysen theory so that the way it has been adopted for the purposes of the above studies in optics will be clear. Let us consider a charged-particle of rest-mass m0 , charge q in the presence of an electromagnetic field characterized by E ¼ =f ðq=qtÞA and B ¼ = A. Then the Dirac equation is i_
q b D Cðr; tÞ, Cðr; tÞ ¼ H qt
(1)
b D ¼ m0 c2 b þ qf þ ca b H p 2 b þ O, b ¼ m0 c b þ E b ¼ qf, E b ¼ ca b O p, where " a¼ "
0 r
ð2Þ
r 0
# ; "
r ¼ sx ¼ " sz ¼
"
1
0
0
1
b¼
0 1 1 0 ## .
# ;
1 0
0
1 "
sy ¼
"
# ;
1¼
0
i
#
i
0
1 0
0 1
# ,
;
ð3Þ
p2x þ b p2y þ b p2z Þ. with b p¼b p qA, b p ¼ i_=, and b p2 ¼ ðb In the non-relativistic situation the upper pair of components of the Dirac Spinor C are large compared b which to the lower pair of components. The operator E
does not couple the large and small components of C is b is called an ‘odd’ operator which called ‘even’ and O couples the large to the small components. Note that b ¼ Ob; b bO
b ¼ Eb. b bE
(4)
Now, the search is for a unitary transformation, b such that the equation for C0 does not C0 ¼ C!UC, contain any odd operator. In the free particle case (with f ¼ 0 and b p¼b p) such a Foldy–Wouthuysen transformation is given by b F C, C!C0 ¼ U b F ¼ eiSb ¼ eba^py ; U
tan 2j^pjy ¼
jb pj . m0 c
ð5Þ
This transformation eliminates the odd part completely from the free particle Dirac Hamiltonian reducing it to the diagonal form: i_
q 0 ^ ^ C ¼ eiS ðm0 c2 b þ ca b pÞeiS C0 qt ba b p sin jb pjy ðm0 c2 b þ ca b ¼ cos jb pjy þ pÞ jb pj ba b p sin jb pjy C0 cos jb pjy jb pj ¼ ðm0 c2 cos 2jb pjy þ cjb pj sin 2jb pjyÞbC0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ m20 c4 þ c2 pb2 bC0 .
ð6Þ
In the general case, when the electron is in a timedependent electromagnetic field it is not possible to b which removes the odd operators construct an exp ðiSÞ from the transformed Hamiltonian completely. This necessitates a non-relativistic expansion of the transformed Hamiltonian in a power series in 1=m0 c2 keeping through any desired order. Note that in the nonrelativistic case, when jpj5m0 c, the transformation 2 b with Sb ibO=2m b b F ¼ exp ðiSÞ operator U 0 c , where b O ¼ ca b p is the odd part of the free Hamiltonian. So, in the general case we can start with the transformation b Cð1Þ ¼ eiS1 C;
b ibO iba b p . Sb1 ¼ ¼ 2 2m0 c 2m0 c
(7)
Then, the equation for Cð1Þ is q q q q b b b eiS1 C ¼ i_ eiS1 C þ eiS1 i_ C i_ Cð1Þ ¼ i_ qt qt qt qt q b b b ¼ i_ ei S 1 þ ei S 1 H D C qt q b b b b iSb1 ð1Þ eiS1 eiS1 þ eiS1 H e C ¼ i_ D qt b b iSb1 b1 q b1 iS iS e ¼ eiS1 H e i_e Cð1Þ D qt b ð1Þ Cð1Þ , ¼H D
ð8Þ
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b b where we have used the identity ðq=qtÞ eA eA þ b b eA ðq=qtÞ eA ¼ ðq=qtÞIb ¼ 0. Now, using the two identities b b Ab b b ½A; b B b B b b þ 1 ½A; ¼ B þ ½A; eA Be 2! 1 b b b b þ ½A; ½A; ½A; B þ . 3! 1 b 2 1 b 3 bðtÞ q bðtÞ A A b e e ¼ 1 þ AðtÞ þ AðtÞ þ AðtÞ qt 2! 3!
(
b b b qAðtÞ 1 qAðtÞ b þ AðtÞ b qAðtÞ þ AðtÞ qt 2! qt qt
)
b b 1 qAðtÞ b qAðtÞ AðtÞ b 2 þ AðtÞ b AðtÞ 3! qt qt ) ! b q AðtÞ b 2 þAðtÞ ... qt " # b b qAðtÞ 1 b qAðtÞ AðtÞ; qt 2! qt " " ## b 1 b q AðtÞ b AðtÞ; AðtÞ; 3! qt " " " ### b 1 b qAðtÞ b b AðtÞ; AðtÞ; AðtÞ; ð9Þ 4! qt b ¼ iSb1 , we find with A
" !# b 2 1 1 q O b1 E bþ b b ½O; b E b þ i_ E bO O; 2m0 c2 qt 8m20 c4
1 b4, bO 3 6 8m0 c
(11)
b1 ibO Sb2 ¼ 2m0 c2
! " # h i b 3 ib b q O 1 b E b þ i_ b . ¼ O; 2 4O 2m0 c2 2m0 c2 qt 3m0 c ð12Þ
After this transformation, q ð2Þ b2 þ O b2 b ð2Þ Cð2Þ ; H b ð2Þ ¼ m0 c2 b þ E C ¼H D D qt ! h i b1 b qO b b b b b E2 E1 ; O2 O1 ; E1 þ i_ , 2m0 c2 qt
i_
ð13Þ
b 2 ¼ Oðð1=m0 c2 Þ2 Þ. After the third transwhere, now, O formation b2 ibO Sb3 ¼ 2m0 c2
(14)
we have i_
ð10Þ
b þ O, b simplifying b D ¼ m 0 c2 b þ E Substituting in (10), H b ¼ Ob b and the right-hand side using the relations bO b b bE ¼ Eb and collecting everything together, we have b1 þ O b1, b ð1Þ m0 c2 b þ E H D
1 b3 O 3m20 c4
b Cð2Þ ¼ eiS2 Cð1Þ ,
b Cð3Þ ¼ eiS3 Cð2Þ ;
#
b b b D _ qS1 þ i Sb1 ; H b D _ qS 1 H 2 qt qt " " ## 1 b b b _ qSb1 S1 ; S1 ; H D 2! 3 qt " " " ### i b b b b _ qSb1 S1 ; S1 ; S1 ; H D . 3! 4 qt
qt
b 1 and O b 1 obeying the relations bO b 1 ¼ O b 1 b and with E b1 ¼ E b 1 b exactly like E b and O. b It is seen that while the bE b b term O in H D is of order zero with respect to the b ¼ Oðð1=m0 c2 Þ0 Þ the expansion parameter 1=m0 c2 (i.e., O
b ð1Þ H D
b E b þ i_ ½O;
ð1Þ
(
"
b 2m0 c2
b 1 , contains only terms of b , namely O odd part of H D 2 order 1=m0 c and higher powers of 1=m0 c2 (i.e., b 1 ¼ Oðð1=m0 c2 ÞÞ). O To reduce the strength of the odd terms further in the transformed Hamiltonian a second Foldy–Wouthuysen transformation is applied with the same prescription:
q b þ 1 AðtÞ b 2 1 AðtÞ b 3 1 AðtÞ qt 2! 3!
1 b 2 1 b 3 b ¼ 1 þ AðtÞ þ AðtÞ þ AðtÞ 2! 3!
b1 O
483
! b qO
q ð3Þ b ð3Þ Cð3Þ ; C ¼H D qt
b3 þ O b3 b ð3Þ ¼ m0 c2 b þ E H D
! h i b2 b q O b2 E b1; O b3 b 2 þ i_ b3 E b2; E E O , ð15Þ 2m0 c2 qt b 3 ¼ O 1=m0 c2 3 . So, neglecting O b3, where O b þ 1 bO b2 b ð3Þ m0 c2 b þ E H D 2 2m0 c " !# h i b 1 q O b b E b þ i_ O; O; qt 8m20 c4 8 !2 9 < = h i b 1 b 4 þ O; b E b þ i_ qO . b O 3 6 : qt ; 8m0 c
ð16Þ
It may be noted that starting with the second b OÞ b pairs can be obtained transformation successive ðE;
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recursively using the rule bj ¼ E b 1 ðE b!E b j1 ; O b!O b j1 Þ E bj ¼ O b 1 ðE b!E b j1 ; O b!O b j1 Þ; O
j41,
ð17Þ
and retaining only the relevant terms of desired order at each step. b ¼ qf and O b ¼ ca b With E p, the final reduced Hamiltonian (16) is, to the order calculated, ! 4 2 ð3Þ b b q_ p p b ¼ b m 0 c2 þ H þ qf bR B D 2m0 c 2m0 8m30 c6
iq_2 q_ R curl E 2 2 R E b p 2 2 8m0 c 4m0 c
q_2 div E, 8m20 c2
ð18Þ
with the individual terms having direct physical interffi parenthesis result from qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pretations. The terms in the first the expansion of m20 c4 þ c2 b p2 showing the effect of the relativistic mass increase. The second and third terms are the electrostatic and magnetic dipole energies. The next two terms, taken together (for hermiticity), contain the spin–orbit interaction. The last term, the so-called Darwin term, is attributed to the zitterbewegung (trembling motion) of the Dirac particle: because of the rapid coordinate fluctuations over distances of the order of the Compton wavelength (2p_=m0 c) the particle sees a somewhat smeared out electric potential. It is clear that the Foldy–Wouthuysen transformation technique expands the Dirac Hamiltonian as a power series in the parameter 1=m0 c2 enabling the use of a systematic approximation procedure for studying the deviations from the non-relativistic situation.
3. Quantum formalism of charged-particle beam optics The classical treatment of charged-particle beam optics has been extremely successful in the designing and working of numerous optical devices, from electron microscopes to very large particle accelerators. It is natural, however to look for a prescription based on the quantum theory, since any physical system is quantum mechanical at the fundamental level! Such a prescription is sure to explain the grand success of the classical theories. It is sure to help us in getting a deeper understanding and lead to better designing of chargedparticle beam devices. The starting point of the quantum prescription of charged particle beam optics is to build a theory based on the basic equations of quantum mechanics (Schro¨dinger, Klein–Gordon, Dirac) appropriate to the situation under study. In order to analyze the evolution of
the beam parameters of the various individual beam optical elements (quadrupoles, bending magnets, etc.) along the optic axis of the system, the first step is to start with the basic time-dependent equations of quantum mechanics and then obtain an equation of the form i_
q c y; sÞcðx; y; sÞ, cðx; y; sÞ ¼ Hðx; qs
(19)
where ðx; y; sÞ constitute a curvilinear coordinate system, adapted to the geometry of the system. Eq. (19) is the basic equation in the quantum formalism, called as the beam-optical equation; H and c as the beam-optical Hamiltonian and the beam wavefunction, respectively. The second step requires obtaining a relationship between any relevant observable fhOiðsÞg at the transverse-plane at s and the observable fhOiðsin Þg at the transverse plane at sin , where sin is some input reference point. This is achieved by the integration of the beamoptical equation in (19) b sin Þcðx; y; sin Þ, cðx; y; sÞ ¼ Uðs;
(20)
which gives the required transfer maps hOiðsin Þ!hOiðsÞ ¼ hcðx; y; sÞjOjcðx; y; sÞi, b y OUjcðx; b y; sin Þi. ¼ hcðx; y; sin ÞjU
ð21Þ
The two-step algorithm stated above gives an oversimplified picture of the quantum formalism. There are several crucial points to be noted. The first step in the algorithm of obtaining the beam-optical equation is not to be treated as a mere transformation which eliminates t in preference to a variable s along the optic axis. A clever set of transforms are required which not only eliminate the variable t in preference to s but also give us the s-dependent equation which has a close physical and mathematical correspondence with the original t-dependent equation of standard time-dependent quantum mechanics. The imposition of this stringent requirement on the construction of the beam-optical equation ensures the execution of the second-step of the algorithm. The beam-optical equation is such that all the required rich machinery of quantum mechanics becomes applicable to the computation of the transfer maps that characterize the optical system. This describes the essential scheme of obtaining the quantum formalism. The rest is mostly a mathematical detail which is inbuilt in the powerful algebraic machinery of the algorithm, accompanied with some reasonable assumptions and approximations dictated by the physical considerations. The nature of these approximations can be best summarized in the optical terminology as a systematic procedure of expanding the beam optical Hamiltonian in a power series of jb p? =p0 j, where p0 is the design (or average) momentum of beam particles moving predominantly along the direction of the optic axis and b p? is the small transverse kinetic
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momentum. The required expansion is obtained using the ambiguity-free procedure of the Foldy–Wouthuysen transformation. The Feshbach–Villars procedure [22] brings the Schro¨dinger and the Klein–Gordon equations to a two-component form facilitating the application of the Foldy–Wouthuysen expansion. The leading-order approximation along with jb p? =p0 j51, constitutes the paraxial or ideal behavior and higher order terms in the expansion give rise to the nonlinear or aberrating behavior. It is seen that the paraxial and aberrating behavior get modified by the quantum contributions which are in powers of the de Broglie wavelength (|0 ¼ _=p0 ). The classical limit of the quantum formalism reproduces the well known Lie algebraic formalism of charged-particle beam optics [23]. A complete coverage to the new field of Quantum Aspects of Beam Physics (QABP), can be found in the proceedings of the series of meetings under the same name [24].
4. Light optics: various prescriptions Historically, the scalar wave theory of optics (including aberrations to all orders) is based on the Fermat’s principle of least time. In this approach the beam-optical Hamiltonian is derived using the Fermat’s principle. This approach is purely geometrical and works adequately in the scalar regime. All the laws of geometrical optics can be deduced from the Maxwell’s equations [25]. This deduction is traditionally done using the Helmholtz equation, which is derived from the Maxwell’s equation. In this approach one takes the squareroot of the Helmholtz operator followed by an expansion of the radical [23,26]. It should be noted that the square-root approach reduces the original boundary value problem to a first-order initial value problem. This reduction is of great practical value, since it leads to the powerful system or the Fourier optic approach [27]. However, the beam-optical Hamiltonian in the squareroot approach is no different from the geometrical approach of the Fermat’s principle. Moreover, the reduction process itself can never be claimed to be rigorous or exact! The Helmholtz equation governing scalar optics is algebraically very similar to the Klein–Gordon equation for a spin-0 particle. Exploiting this similarity the Helmholtz equation is linearized in a procedure very similar to the one due to Feshbach–Villars [22], for linearizing the Klein–Gordon equation. This brings the Helmholtz equation to a Dirac-like form enabling the procedure of the Foldy–Wouthuysen expansion used in the Dirac electron theory. This formalism gives rise to wavelength-dependent contributions modifying the paraxial behavior [10] and the aberration coefficients
485
[11]. This is the non-traditional prescription of scalar optics. As for the polarization: a systematic procedure for the passage from scalar to vector wave optics to handle paraxial beam propagation problems, completely taking into account the way in which the Maxwell’s equations couple the spatial variation and polarization of light waves, has been formulated by analyzing the basic Poincare´ invariance of the system, and this procedure has been successfully used to clarify several issues in Maxwell optics [28–31]. In all the aforementioned approaches, the beamoptics and the polarization are studied separately, using very different machineries. The derivation of the Helmholtz equation from the Maxwell’s equations is an approximation as one neglects the spatial and temporal derivatives of the permittivity and permeability of the medium. It is very natural to look for a prescription based fully on the Maxwell’s equations. The starting point for such a prescription is the exact matrix representation of the Maxwell’s equations, taking into account the spatial and temporal variations of the permittivity and permeability [32]. The derived representation using 8 8 matrices has a close algebraic analogy with the Dirac equation, enabling the use of the rich machinery of the Foldy–Wouthuysen transform. The beam optical Hamiltonian derived from this representation reproduces the Hamiltonians obtained in the traditional prescription along with wavelengthdependent matrix terms, which we have named as the polarization terms. These polarization terms are very similar to the spin terms in the Dirac electron theory and the spin-precession terms in the beam-optical version of the Thomas-BMT equation [18]. The matrix formulation provides a unified treatment of beam optics and light polarization. Some well-known results of light polarization are obtained as the paraxial limit of the matrix formulation [28–31]. Results from the specific example of the graded-index medium considered in the non-traditional prescription of Maxwell optics [12] are worth noting. Firstly, it predicts an image rotation (proportional to the wavelength) and its magnitude is explicitly given. Secondly, we obtain all the nine aberrations permitted by the axial symmetry. The traditional approaches give six aberrations. The exact treatment of Maxwell optics modifies the six aberration coefficients by wavelength-dependent contributions and also gives rise to the remaining three permitted by the axial symmetry. The existence of the nine aberrations and image rotation are well-known in axially symmetric magnetic lenses, even when treated classically. The quantum treatment of the same system leads to the wavelength-dependent modifications [17]. The alternate procedure for the Helmholtz optics in [11] gives the usual six aberrations (though modified by the wavelength-dependent contributions) and does not give
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any image rotation. These extra aberrations and the image rotation are the exclusive outcome of the fact that the formalism is based on a treatment starting with an exact matrix representation of the Maxwell’s equations. The traditional beam-optics (in particular, the Lie algebraic formalism of light beam optics [23,26] is completely obtained from the non-traditional prescriptions in the limit wavelength, |!0, which we call as the traditional limit of our formalism. This is analogous to the classical limit obtained by taking _!0 in the quantum prescriptions. The scheme of using the Foldy–Wouthuysen machinery in the non-traditional prescriptions of Helmholtz optics and Maxwell optics is very similar to the one used in the quantum theory of charged-particle beam optics [13–21]. There too one recovers the classical prescriptions (Lie algebraic formalism of charged-particle beam optics [33,34] in the limit |0 !0, where |0 ¼ _=p0 is the de Broglie wavelength and p0 is the design momentum of the system under study. The Foldy–Wouthuysen transformation has enabled entirely new approaches to light optics and charged-particle optics, respectively.
5. Concluding remarks The use of the Foldy–Wouthuysen transformation technique in optics has been able to shed light on the deeper connections in the wavelength-dependent regime
between the light optics and charged-particle optics. The enclosed Table 1 summarizes the Hamiltonians in the different prescriptions of light beam optics and chargedparticle beam optics for magnetic systems respectively. b 0;p are the paraxial Hamiltonians, with lowest order H wavelength-dependent contributions. From the Hamiltonians in the Table 1 we make the following observations: The classical/traditional Hamiltonians of particle/ light optics are modified by wavelength-dependent contributions in the quantum/non-traditional prescriptions, respectively. The algebraic forms of these modifications in each row is very similar. The starting equations have one-to-one algebraic correspondence: Helmholtz 2 Klein–Gordon; Matrix form of Maxwell 2 Dirac equation. Lastly, the de Broglie wavelength, |0 , and | have an analogous status, and the classical/ traditional limit is obtained by taking |0 !0 and |!0, respectively. The parallel of the analogies between the two systems is sure to provide us with more insights. If not for the Foldy–Wouthuysen transformation, it would not have been possible to see the new aspects of the analogies between the light optics and the charged-particle optics [35]. An historical account of the analogies is available in [25,36] Notation: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Refractive index, nðrÞ ¼ c ðrÞmðrÞ. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Resistance, hðrÞ ¼ mðrÞ=ðrÞ. 1 uðrÞ ¼ =nðrÞ. 2nðrÞ
Hamiltonians in different prescriptions
Table 1.
Light beam optics
Charged-particle beam optics
Fermat’s principle
Maupertuis’ principle
2
H ¼ fn ðrÞ
p2? g1=2
H ¼ fp20 p2? g1=2 qAz
Non-traditional Helmholtz
Klein–Gordon formalism
b 0;p H
b 0;p H
1 2 b ¼ nðrÞ þ p 2n0 ? i| 2 q b ; p nðrÞ 16n30 ? qz
1 2 b p ¼ p0 qAz þ 2p0 ? i_ q 2 b p2? ; b þ p? 4 qz 16p0
Maxwell, Matrix
Dirac formalism
b 0;p H
b 0;p H 1 2 b p 2n0 ? i|bR u 1 2 2 | w b þ 2n0
¼ nðrÞ þ
¼ p0 qAz þ
1 2 b p 2p0 ?
_ fmgR? B ? þ ðq þ mÞSz Bz g 2p0 _ Bz þi m0 c
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1 =hðrÞ. 2hðrÞ R and b are the Dirac matrices. b p? ¼ b p? qA? . anomalous magnetic moment, ma . anomalous electric moment, a . m ¼ 2m0 ma =_, ¼ 2m0 a =_. g ¼ E=m0 c2 .
wðrÞ ¼
References [1] L.L. Foldy, S.A. Wouthuysen, On the Dirac theory of Spin 1/2 particles and its non-relativistic limit, Phys. Rev. 78 (1950) 29–36. [2] A brief historical account of the Foldy–Wouthuysen transformation is available in H. Leopold, Obituary of Siegfried A Wouthuysen, Phys. Today 50 (11) (1997) 89; R.W. Brown, L.M. Krauss, P.L. Taylor, Obituary of Leslie Lawrence Foldy, Phys. Today 54 (12) (2001) 75. [3] J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York, San Francisco, 1964. [4] J.P. Costella, B.H.J. McKellar, The Foldy–Wouthuysen transformation, arXiv: hep-ph/9503416, 1995. [5] K.M. Case, Some generalizations of the Foldy–Wouthuysen transformation, Phys. Rev. 95 (1954) 1323–1328. [6] J. Jayaraman, A note on the recent Foldy–Wouthuysen transformations for particles of arbitrary spin, J. Phys. A: Math. Gen. 8 (1975) L1–L4. [7] T. Asaga, T. Fujita, M. Hiramoto, Foldy–Wouthuysen transformed EDM operator in atomic system, arXiv: hepph/0005314, 2000. [8] D. Wurmser, A new strategy for applying the parabolic equation to a penetrable rough surface, J. Acoust. Soc. Am. 109 (5) (2001) 2300. [9] L. Fishman, J.J. McCoy, Derivation and application of extended parabolic wave theories, Part I: the factored Helmholtz equation, J. Math. Phys. 25 (1984) 285–296. [10] S.A. Khan, R. Jagannathan, R. Simon, Foldy–Wouthuysen transformation and a quasiparaxial approximation scheme for the scalar wave theory of light beams, arXiv: physics/0209082, 2002, communicated. [11] S.A. Khan, Wavelength-dependent modifications in Helmholtz Optics, Int. J. Theor. Phys. 44 (1) (2005) 95–125 (Kluwer Academic Publishers, https://www. editorialmanager.com/ijtp/). [12] S.A. Khan, Wavelength-dependent modifications in Maxwell optics, arXiv: physics/0210027, 2002, pp. 59, communicated. [13] 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. [14] R. Jagannathan, Quantum theory of electron lenses based on the Dirac equation, Phys. Rev. A 42 (1990) 6674–6689. [15] 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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[16] S.A. Khan, R. Jagannathan, On the quantum mechanics of charged particle beam transport through magnetic lenses, Phys. Rev. E 51 (1995) 2510–2515. [17] R. Jagannathan, S.A. Khan, Quantum theory of the optics of charged particles, in: P.W. Hawkes (Ed.), Advances in Imaging and Electron Physics, vol. 97, Academic Press, San Diego, 1996, pp. 257–358. [18] M. Conte, R. Jagannathan, S.A. Khan, M. Pusterla, Beam optics of the Dirac particle with anomalous magnetic moment, Part. Accel. 56 (1996) 99–126. [19] R. Jagannathan, S.A. Khan, Quantum mechanics of accelerator optics, ICFA Beam Dynam. Newslett. 13 (1997) 21–27 (ICFA: International Committee for Future Accelerators). [20] S.A. Khan, Quantum theory of charged-particle beam optics, Ph.D. Thesis, University of Madras, Chennai, India, 1997. [21] R. Jagannathan, S.A. Khan, Several articles in Proceedings/E-Prints (http://arxiv.org/) on the Quantum theory of charged-particle beam optics, arXiv: physics/9803042; arXiv: physics/0101060; arXiv: physics/9809032; arXiv: physics/9904063; arXiv: physics/ 0112085; arXiv: physics/0112086 and arXiv: physics/ 0304099, 1998. [22] H. Feshbach, F.M.H. Villars, Elementary relativistic wave mechanics of spin 0 and spin 1/2 particles, Rev. Mod. Phys. 30 (1958) 24–45. [23] A.J. Dragt, R. Forest, K.B. Wolf, Foundations of a Lie algebraic theory of geometrical optics, in: J. Sa´ndez Mondrago´n, K.B. Wolf (Eds.), Lie Methods in Optics, Lecture Notes in Physics, vol. 250, Springer, Berlin, 1986, pp. 105–157. [24] P. Chen (Ed.), in: Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics, 4–9 January 1998, Monterrey, California, USA, World Scientific, Singapore, 1999, http:// www.slac.stanford.edu/grp/ara/qabp/qabp.html; P. Chen (Ed.), in: Proceedings of the 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics, 15–20 October 2000, Capri, Italy, World Scientific, Singapore, 2002, http://qabp2k.sa.infn.it/; P. Chen (Ed.), in: Proceedings of the Joint 28th ICFA Advanced Beam Dynamics and Advanced & Novel Accelerators Workshop on Quantum Aspects Of Beam Physics and Other Critical Issues of Beams in Physics and Astrophysics, 7–11 January 2003, Hiroshima University, Japan, World Scientific, Singapore, 2003, http://home. hiroshima-u.ac.jp/ogata/qabp/home.html, http://www.slac. stanford.edu/pubs/slacreports/slac-r-630.html; P. Chen (Ed.), Workshop Reports, ICFA Beam Dynamics Newslett. 16 (1988) 22–25; P. Chen (Ed.), Workshop Reports, ICFA Beam Dynamics Newslett. 23 (2000) 13–14; P. Chen (Ed.), Workshop Reports, ICFA Beam Dynamics Newslett. 30 (2003) 72–75; Bull. Assoc. Asia Pacific Phys. Soc. 13(1) (2003) 34–37. [25] M. Born, E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, UK, 1999. [26] A.J. Dragt, Lie Algebraic Method for Ray and Wave Optics, University of Maryland Physics Department Report, 1988; A.J. Dragt, A Lie Algebraic theory of
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[29]
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geometrical optics and optical aberrations, J. Opt. Soc. Am. 72 (1982) 372. J.W. Goodman, Introduction to Fourier Optics, second ed., McGraw-Hill, New York, 1996. N. Mukunda, R. Simon, E.C.G. Sudarshan, Paraxialwave optics and relativistic front description I: the scalar theory, Phys. Rev. A 28 (1983) 2921–2932; N. Mukunda, R. Simon, E.C.G. Sudarshan, Paraxialwave optics and relativistic front description, II: the vector theory, Phys. Rev. A 28 (1983) 2933–2942. N. Mukunda, R. Simon, E.C.G. Sudarshan, Fourier optics for the Maxwell field: formalism and applications, J. Opt. Soc. Am. A 2 (3) (1985) 416–426. R. Simon, E.C.G. Sudarshan, N. Mukunda, Gaussian– Maxwell beams, J. Opt. Soc. Am. A 3 (4) (1986) 536–540. R. Simon, E.C.G. Sudarshan, N. Mukunda, Cross polarization in laser beams, Appl. Opt. 26 (9) (1987) 1589–1593. S.A. Khan, An exact matrix representation of Maxwell’s equations, Phys. Scr. 71 (5) (2005) 440–442 http:// www.physica.org/.
[33] E. Todesco, Overview of single-particle nonlinear dynamics, CERN-LHC-99-1-MMS, 1999. 16pp; E. Todesco, Talk given at 16th ICFA Beam Dynamics Workshop on Nonlinear and Collective Phenomena in Beam Physics, Arcidosso, Italy, 1–5 September 1998, AIP Conf. Proc. 468 (1999) 157–172. [34] G. Turchetti, A. Bazzani, M. Giovannozzi, G. Servizi, E. Todesco, Normal forms for symplectic maps and stability of beams in particle accelerators, in: Proceedings of the Dynamical Symmetries and Chaotic Behaviour in Physical Systems, Bologna, Italy, 1989, pp. 203–231. [35] S.A. Khan, Analogies between light optics and chargedparticle optics, ICFA Beam Dynamics Newsletter 27 (2002) 42–48 arXiv: physics/0210028 (ICFA: International Committee for Future Accelerators). [36] P.W. Hawkes, E. Kasper, Principles of Electron Optics, vols. I and II, Academic Press, London, 1989 P.W. Hawkes, E. Kasper, Principles of Electron Optics, vol. 3, Wave Optics, Academic Press, London, San Diego, 1994.
BOOK REVIEW W.M. Steen, Laser Material Processing, third ed., Springer, Heidelberg, ISBN 1-85233-698-6, 2003 (XV/ 408pp., 257, illust., EUR69.95, $59.95, Softcover). Material processing is one of the most important applications of lasers. It is an already well-established field in many fields of industry and was modernising industrial production methods drastically. There exist a lot of books for the education of students in this field, but only a few of them will help students and engineers to specialise their knowledge. Laser Material Processing gives a compact survey and can be used as university or industrial course material for senior undergraduate, graduate and nondegree technical training in optoelectronics, laser processing and advanced manufacturing. The book will guide the reader smoothly from the basics of laser physics to the detailed treatment of all major material processing techniques for which lasers are essential. In its third edition this book will help to understand how the laser works and to decide which laser is best for
your purposes. Some new chapters on bending and cleaning reflect the changes in the field since the last 10 years completing the range of practical knowledge about the processes possible with lasers already familiar to users of this textbook. Instead of a number of cartoons which are not fully connected to the content of the book, the authors should introduce some more attractive applications: laser supported cutting tools, laser polishing and some others which are not yet described in the present edition. In some chapters it would also be helpful to introduce some more physics to describe the effects and to explain the results. But even with such small deficiencies, I can recommend this book as an authoritative source of information on the rapidly expanding use of industrial lasers in material processing.
Roland Baur
Optik
Optics
Optik 117 (2006) 481–488 www.elsevier.de/ijleo
The Foldy–Wouthuysen transformation technique in optics Sameen Ahmed Khan Department of Mathematics & Applied Sciences, Middle East College of Information Technology (MECIT), Technowledge Corridor, Knowledge Oasis Muscat, P.B. No. 79, Al Rusayl 124, Muscat, Sultanate of Oman Received 14 July 2005; accepted 12 November 2005
Abstract We describe the standard Foldy–Wouthuysen transformation technique along with its applications in optics and the new results it leads to. The use of the Foldy–Wouthuysen transformation technique is central in the non-traditional prescriptions of Helmholtz optics and Maxwell optics, respectively, resulting in the wavelength-dependent modifications of light beam optics, even at the paraxial level. It is further shown that the wavelength-dependent modifications in the newly developed formalisms of light beam optics are in close anology with the quantum theory of charged-particle beam optics. r 2006 Elsevier GmbH. All rights reserved. PACS: 02; 41.75.i; 41.85.p; 42.15.i; 42.25.p Keywords: Helmholtz optics; Maxwell optics; Maxwell’s equations; Vector optics; Charged-particle optics; Wavelength-dependent effects; Beam propagation; Aberrations; Graded-index fiber; Mathematical methods of optics; Foldy–Wouthuysen transformation
1. Introduction The Foldy–Wouthuysen transform is widely used in high-energy physics. It was historically formulated by Leslie Lawrence Foldy and Siegfried Adolf Wouthuysen in 1949 to understand the non-relativistic limit of the Dirac equation, the equation for the spin-12 particles [1]. The approach of the Foldy and Wouthuysen made use of a canonical transform that has now come to be known as the Foldy–Wouthuysen transformation [2,3]. Before their work, there was some difficulty in understanding and gathering all the interaction terms of a Fax: +968 24446028.
E-mail addresses: [email protected], [email protected]. URL: http://www.pd.infn.it/khan/. 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2005.11.010
given order, such as those for a Dirac particle immersed in an external field. With their procedure the physical interpretation of the terms was clear, and it became possible to apply their work in a systematic way to a number of problems that had previously defied solution [4]. The Foldy–Wouthuysen transform was extended to the physically important cases of the spin-0 and the spin1 particles [5], and even generalized to the case of arbitrary spins [6]. The powerful machinery of the Foldy–Wouthuysen transform has found applications in diverse areas such as atomic systems [7] and acoustics [8]. Though the suggestion to employ the Foldy– Wouthuysen technique in the case of the Helmholtz equation existed in the literature as a remark [9], it has been exploited only now, to analyze the quasiparaxial approximations [10]. There is a close algebraic analogy
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between the Helmholtz equation (governing scalar optics) and the Klein–Gordon equation; and the matrix-form of the Maxwell’s equations (governing vector optics) and the Dirac equation. So, it is natural to use the powerful machinery of standard quantum mechanics (particularly, the Foldy–Wouthuysen transform) in analyzing these systems. This results in what we call as the non-traditional prescriptions of Helmholtz optics [11] and Maxwell optics [12], respectively. The non-traditional approaches give rise to very interesting wavelength-dependent modifications of the paraxial and aberrating behavior. The non-traditional formalism of Maxwell optics provides a unified framework of light beam optics and polarization. The non-traditional prescriptions of light optics are in close analogy with the quantum theory of charged-particle beam optics [13–21]. In the following sections we shall look at the standard Foldy–Wouthuysen transform in detail. A brief outline of the quantum theory of charged-particle beam optics and the non-traditional prescriptions of light optics is also presented.
2. The Foldy–Wouthuysen transformation Let us describe here briefly the standard Foldy– Wouthuysen theory so that the way it has been adopted for the purposes of the above studies in optics will be clear. Let us consider a charged-particle of rest-mass m0 , charge q in the presence of an electromagnetic field characterized by E ¼ =f ðq=qtÞA and B ¼ = A. Then the Dirac equation is i_
q b D Cðr; tÞ, Cðr; tÞ ¼ H qt
(1)
b D ¼ m0 c2 b þ qf þ ca b H p 2 b þ O, b ¼ m0 c b þ E b ¼ qf, E b ¼ ca b O p, where " a¼ "
0 r
ð2Þ
r 0
# ; "
r ¼ sx ¼ " sz ¼
"
1
0
0
1
b¼
0 1 1 0 ## .
# ;
1 0
0
1 "
sy ¼
"
# ;
1¼
0
i
#
i
0
1 0
0 1
# ,
;
ð3Þ
p2x þ b p2y þ b p2z Þ. with b p¼b p qA, b p ¼ i_=, and b p2 ¼ ðb In the non-relativistic situation the upper pair of components of the Dirac Spinor C are large compared b which to the lower pair of components. The operator E
does not couple the large and small components of C is b is called an ‘odd’ operator which called ‘even’ and O couples the large to the small components. Note that b ¼ Ob; b bO
b ¼ Eb. b bE
(4)
Now, the search is for a unitary transformation, b such that the equation for C0 does not C0 ¼ C!UC, contain any odd operator. In the free particle case (with f ¼ 0 and b p¼b p) such a Foldy–Wouthuysen transformation is given by b F C, C!C0 ¼ U b F ¼ eiSb ¼ eba^py ; U
tan 2j^pjy ¼
jb pj . m0 c
ð5Þ
This transformation eliminates the odd part completely from the free particle Dirac Hamiltonian reducing it to the diagonal form: i_
q 0 ^ ^ C ¼ eiS ðm0 c2 b þ ca b pÞeiS C0 qt ba b p sin jb pjy ðm0 c2 b þ ca b ¼ cos jb pjy þ pÞ jb pj ba b p sin jb pjy C0 cos jb pjy jb pj ¼ ðm0 c2 cos 2jb pjy þ cjb pj sin 2jb pjyÞbC0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ m20 c4 þ c2 pb2 bC0 .
ð6Þ
In the general case, when the electron is in a timedependent electromagnetic field it is not possible to b which removes the odd operators construct an exp ðiSÞ from the transformed Hamiltonian completely. This necessitates a non-relativistic expansion of the transformed Hamiltonian in a power series in 1=m0 c2 keeping through any desired order. Note that in the nonrelativistic case, when jpj5m0 c, the transformation 2 b with Sb ibO=2m b b F ¼ exp ðiSÞ operator U 0 c , where b O ¼ ca b p is the odd part of the free Hamiltonian. So, in the general case we can start with the transformation b Cð1Þ ¼ eiS1 C;
b ibO iba b p . Sb1 ¼ ¼ 2 2m0 c 2m0 c
(7)
Then, the equation for Cð1Þ is q q q q b b b eiS1 C ¼ i_ eiS1 C þ eiS1 i_ C i_ Cð1Þ ¼ i_ qt qt qt qt q b b b ¼ i_ ei S 1 þ ei S 1 H D C qt q b b b b iSb1 ð1Þ eiS1 eiS1 þ eiS1 H e C ¼ i_ D qt b b iSb1 b1 q b1 iS iS e ¼ eiS1 H e i_e Cð1Þ D qt b ð1Þ Cð1Þ , ¼H D
ð8Þ
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b b where we have used the identity ðq=qtÞ eA eA þ b b eA ðq=qtÞ eA ¼ ðq=qtÞIb ¼ 0. Now, using the two identities b b Ab b b ½A; b B b B b b þ 1 ½A; ¼ B þ ½A; eA Be 2! 1 b b b b þ ½A; ½A; ½A; B þ . 3! 1 b 2 1 b 3 bðtÞ q bðtÞ A A b e e ¼ 1 þ AðtÞ þ AðtÞ þ AðtÞ qt 2! 3!
(
b b b qAðtÞ 1 qAðtÞ b þ AðtÞ b qAðtÞ þ AðtÞ qt 2! qt qt
)
b b 1 qAðtÞ b qAðtÞ AðtÞ b 2 þ AðtÞ b AðtÞ 3! qt qt ) ! b q AðtÞ b 2 þAðtÞ ... qt " # b b qAðtÞ 1 b qAðtÞ AðtÞ; qt 2! qt " " ## b 1 b q AðtÞ b AðtÞ; AðtÞ; 3! qt " " " ### b 1 b qAðtÞ b b AðtÞ; AðtÞ; AðtÞ; ð9Þ 4! qt b ¼ iSb1 , we find with A
" !# b 2 1 1 q O b1 E bþ b b ½O; b E b þ i_ E bO O; 2m0 c2 qt 8m20 c4
1 b4, bO 3 6 8m0 c
(11)
b1 ibO Sb2 ¼ 2m0 c2
! " # h i b 3 ib b q O 1 b E b þ i_ b . ¼ O; 2 4O 2m0 c2 2m0 c2 qt 3m0 c ð12Þ
After this transformation, q ð2Þ b2 þ O b2 b ð2Þ Cð2Þ ; H b ð2Þ ¼ m0 c2 b þ E C ¼H D D qt ! h i b1 b qO b b b b b E2 E1 ; O2 O1 ; E1 þ i_ , 2m0 c2 qt
i_
ð13Þ
b 2 ¼ Oðð1=m0 c2 Þ2 Þ. After the third transwhere, now, O formation b2 ibO Sb3 ¼ 2m0 c2
(14)
we have i_
ð10Þ
b þ O, b simplifying b D ¼ m 0 c2 b þ E Substituting in (10), H b ¼ Ob b and the right-hand side using the relations bO b b bE ¼ Eb and collecting everything together, we have b1 þ O b1, b ð1Þ m0 c2 b þ E H D
1 b3 O 3m20 c4
b Cð2Þ ¼ eiS2 Cð1Þ ,
b Cð3Þ ¼ eiS3 Cð2Þ ;
#
b b b D _ qS1 þ i Sb1 ; H b D _ qS 1 H 2 qt qt " " ## 1 b b b _ qSb1 S1 ; S1 ; H D 2! 3 qt " " " ### i b b b b _ qSb1 S1 ; S1 ; S1 ; H D . 3! 4 qt
qt
b 1 and O b 1 obeying the relations bO b 1 ¼ O b 1 b and with E b1 ¼ E b 1 b exactly like E b and O. b It is seen that while the bE b b term O in H D is of order zero with respect to the b ¼ Oðð1=m0 c2 Þ0 Þ the expansion parameter 1=m0 c2 (i.e., O
b ð1Þ H D
b E b þ i_ ½O;
ð1Þ
(
"
b 2m0 c2
b 1 , contains only terms of b , namely O odd part of H D 2 order 1=m0 c and higher powers of 1=m0 c2 (i.e., b 1 ¼ Oðð1=m0 c2 ÞÞ). O To reduce the strength of the odd terms further in the transformed Hamiltonian a second Foldy–Wouthuysen transformation is applied with the same prescription:
q b þ 1 AðtÞ b 2 1 AðtÞ b 3 1 AðtÞ qt 2! 3!
1 b 2 1 b 3 b ¼ 1 þ AðtÞ þ AðtÞ þ AðtÞ 2! 3!
b1 O
483
! b qO
q ð3Þ b ð3Þ Cð3Þ ; C ¼H D qt
b3 þ O b3 b ð3Þ ¼ m0 c2 b þ E H D
! h i b2 b q O b2 E b1; O b3 b 2 þ i_ b3 E b2; E E O , ð15Þ 2m0 c2 qt b 3 ¼ O 1=m0 c2 3 . So, neglecting O b3, where O b þ 1 bO b2 b ð3Þ m0 c2 b þ E H D 2 2m0 c " !# h i b 1 q O b b E b þ i_ O; O; qt 8m20 c4 8 !2 9 < = h i b 1 b 4 þ O; b E b þ i_ qO . b O 3 6 : qt ; 8m0 c
ð16Þ
It may be noted that starting with the second b OÞ b pairs can be obtained transformation successive ðE;
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recursively using the rule bj ¼ E b 1 ðE b!E b j1 ; O b!O b j1 Þ E bj ¼ O b 1 ðE b!E b j1 ; O b!O b j1 Þ; O
j41,
ð17Þ
and retaining only the relevant terms of desired order at each step. b ¼ qf and O b ¼ ca b With E p, the final reduced Hamiltonian (16) is, to the order calculated, ! 4 2 ð3Þ b b q_ p p b ¼ b m 0 c2 þ H þ qf bR B D 2m0 c 2m0 8m30 c6
iq_2 q_ R curl E 2 2 R E b p 2 2 8m0 c 4m0 c
q_2 div E, 8m20 c2
ð18Þ
with the individual terms having direct physical interffi parenthesis result from qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pretations. The terms in the first the expansion of m20 c4 þ c2 b p2 showing the effect of the relativistic mass increase. The second and third terms are the electrostatic and magnetic dipole energies. The next two terms, taken together (for hermiticity), contain the spin–orbit interaction. The last term, the so-called Darwin term, is attributed to the zitterbewegung (trembling motion) of the Dirac particle: because of the rapid coordinate fluctuations over distances of the order of the Compton wavelength (2p_=m0 c) the particle sees a somewhat smeared out electric potential. It is clear that the Foldy–Wouthuysen transformation technique expands the Dirac Hamiltonian as a power series in the parameter 1=m0 c2 enabling the use of a systematic approximation procedure for studying the deviations from the non-relativistic situation.
3. Quantum formalism of charged-particle beam optics The classical treatment of charged-particle beam optics has been extremely successful in the designing and working of numerous optical devices, from electron microscopes to very large particle accelerators. It is natural, however to look for a prescription based on the quantum theory, since any physical system is quantum mechanical at the fundamental level! Such a prescription is sure to explain the grand success of the classical theories. It is sure to help us in getting a deeper understanding and lead to better designing of chargedparticle beam devices. The starting point of the quantum prescription of charged particle beam optics is to build a theory based on the basic equations of quantum mechanics (Schro¨dinger, Klein–Gordon, Dirac) appropriate to the situation under study. In order to analyze the evolution of
the beam parameters of the various individual beam optical elements (quadrupoles, bending magnets, etc.) along the optic axis of the system, the first step is to start with the basic time-dependent equations of quantum mechanics and then obtain an equation of the form i_
q c y; sÞcðx; y; sÞ, cðx; y; sÞ ¼ Hðx; qs
(19)
where ðx; y; sÞ constitute a curvilinear coordinate system, adapted to the geometry of the system. Eq. (19) is the basic equation in the quantum formalism, called as the beam-optical equation; H and c as the beam-optical Hamiltonian and the beam wavefunction, respectively. The second step requires obtaining a relationship between any relevant observable fhOiðsÞg at the transverse-plane at s and the observable fhOiðsin Þg at the transverse plane at sin , where sin is some input reference point. This is achieved by the integration of the beamoptical equation in (19) b sin Þcðx; y; sin Þ, cðx; y; sÞ ¼ Uðs;
(20)
which gives the required transfer maps hOiðsin Þ!hOiðsÞ ¼ hcðx; y; sÞjOjcðx; y; sÞi, b y OUjcðx; b y; sin Þi. ¼ hcðx; y; sin ÞjU
ð21Þ
The two-step algorithm stated above gives an oversimplified picture of the quantum formalism. There are several crucial points to be noted. The first step in the algorithm of obtaining the beam-optical equation is not to be treated as a mere transformation which eliminates t in preference to a variable s along the optic axis. A clever set of transforms are required which not only eliminate the variable t in preference to s but also give us the s-dependent equation which has a close physical and mathematical correspondence with the original t-dependent equation of standard time-dependent quantum mechanics. The imposition of this stringent requirement on the construction of the beam-optical equation ensures the execution of the second-step of the algorithm. The beam-optical equation is such that all the required rich machinery of quantum mechanics becomes applicable to the computation of the transfer maps that characterize the optical system. This describes the essential scheme of obtaining the quantum formalism. The rest is mostly a mathematical detail which is inbuilt in the powerful algebraic machinery of the algorithm, accompanied with some reasonable assumptions and approximations dictated by the physical considerations. The nature of these approximations can be best summarized in the optical terminology as a systematic procedure of expanding the beam optical Hamiltonian in a power series of jb p? =p0 j, where p0 is the design (or average) momentum of beam particles moving predominantly along the direction of the optic axis and b p? is the small transverse kinetic
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momentum. The required expansion is obtained using the ambiguity-free procedure of the Foldy–Wouthuysen transformation. The Feshbach–Villars procedure [22] brings the Schro¨dinger and the Klein–Gordon equations to a two-component form facilitating the application of the Foldy–Wouthuysen expansion. The leading-order approximation along with jb p? =p0 j51, constitutes the paraxial or ideal behavior and higher order terms in the expansion give rise to the nonlinear or aberrating behavior. It is seen that the paraxial and aberrating behavior get modified by the quantum contributions which are in powers of the de Broglie wavelength (|0 ¼ _=p0 ). The classical limit of the quantum formalism reproduces the well known Lie algebraic formalism of charged-particle beam optics [23]. A complete coverage to the new field of Quantum Aspects of Beam Physics (QABP), can be found in the proceedings of the series of meetings under the same name [24].
4. Light optics: various prescriptions Historically, the scalar wave theory of optics (including aberrations to all orders) is based on the Fermat’s principle of least time. In this approach the beam-optical Hamiltonian is derived using the Fermat’s principle. This approach is purely geometrical and works adequately in the scalar regime. All the laws of geometrical optics can be deduced from the Maxwell’s equations [25]. This deduction is traditionally done using the Helmholtz equation, which is derived from the Maxwell’s equation. In this approach one takes the squareroot of the Helmholtz operator followed by an expansion of the radical [23,26]. It should be noted that the square-root approach reduces the original boundary value problem to a first-order initial value problem. This reduction is of great practical value, since it leads to the powerful system or the Fourier optic approach [27]. However, the beam-optical Hamiltonian in the squareroot approach is no different from the geometrical approach of the Fermat’s principle. Moreover, the reduction process itself can never be claimed to be rigorous or exact! The Helmholtz equation governing scalar optics is algebraically very similar to the Klein–Gordon equation for a spin-0 particle. Exploiting this similarity the Helmholtz equation is linearized in a procedure very similar to the one due to Feshbach–Villars [22], for linearizing the Klein–Gordon equation. This brings the Helmholtz equation to a Dirac-like form enabling the procedure of the Foldy–Wouthuysen expansion used in the Dirac electron theory. This formalism gives rise to wavelength-dependent contributions modifying the paraxial behavior [10] and the aberration coefficients
485
[11]. This is the non-traditional prescription of scalar optics. As for the polarization: a systematic procedure for the passage from scalar to vector wave optics to handle paraxial beam propagation problems, completely taking into account the way in which the Maxwell’s equations couple the spatial variation and polarization of light waves, has been formulated by analyzing the basic Poincare´ invariance of the system, and this procedure has been successfully used to clarify several issues in Maxwell optics [28–31]. In all the aforementioned approaches, the beamoptics and the polarization are studied separately, using very different machineries. The derivation of the Helmholtz equation from the Maxwell’s equations is an approximation as one neglects the spatial and temporal derivatives of the permittivity and permeability of the medium. It is very natural to look for a prescription based fully on the Maxwell’s equations. The starting point for such a prescription is the exact matrix representation of the Maxwell’s equations, taking into account the spatial and temporal variations of the permittivity and permeability [32]. The derived representation using 8 8 matrices has a close algebraic analogy with the Dirac equation, enabling the use of the rich machinery of the Foldy–Wouthuysen transform. The beam optical Hamiltonian derived from this representation reproduces the Hamiltonians obtained in the traditional prescription along with wavelengthdependent matrix terms, which we have named as the polarization terms. These polarization terms are very similar to the spin terms in the Dirac electron theory and the spin-precession terms in the beam-optical version of the Thomas-BMT equation [18]. The matrix formulation provides a unified treatment of beam optics and light polarization. Some well-known results of light polarization are obtained as the paraxial limit of the matrix formulation [28–31]. Results from the specific example of the graded-index medium considered in the non-traditional prescription of Maxwell optics [12] are worth noting. Firstly, it predicts an image rotation (proportional to the wavelength) and its magnitude is explicitly given. Secondly, we obtain all the nine aberrations permitted by the axial symmetry. The traditional approaches give six aberrations. The exact treatment of Maxwell optics modifies the six aberration coefficients by wavelength-dependent contributions and also gives rise to the remaining three permitted by the axial symmetry. The existence of the nine aberrations and image rotation are well-known in axially symmetric magnetic lenses, even when treated classically. The quantum treatment of the same system leads to the wavelength-dependent modifications [17]. The alternate procedure for the Helmholtz optics in [11] gives the usual six aberrations (though modified by the wavelength-dependent contributions) and does not give
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any image rotation. These extra aberrations and the image rotation are the exclusive outcome of the fact that the formalism is based on a treatment starting with an exact matrix representation of the Maxwell’s equations. The traditional beam-optics (in particular, the Lie algebraic formalism of light beam optics [23,26] is completely obtained from the non-traditional prescriptions in the limit wavelength, |!0, which we call as the traditional limit of our formalism. This is analogous to the classical limit obtained by taking _!0 in the quantum prescriptions. The scheme of using the Foldy–Wouthuysen machinery in the non-traditional prescriptions of Helmholtz optics and Maxwell optics is very similar to the one used in the quantum theory of charged-particle beam optics [13–21]. There too one recovers the classical prescriptions (Lie algebraic formalism of charged-particle beam optics [33,34] in the limit |0 !0, where |0 ¼ _=p0 is the de Broglie wavelength and p0 is the design momentum of the system under study. The Foldy–Wouthuysen transformation has enabled entirely new approaches to light optics and charged-particle optics, respectively.
5. Concluding remarks The use of the Foldy–Wouthuysen transformation technique in optics has been able to shed light on the deeper connections in the wavelength-dependent regime
between the light optics and charged-particle optics. The enclosed Table 1 summarizes the Hamiltonians in the different prescriptions of light beam optics and chargedparticle beam optics for magnetic systems respectively. b 0;p are the paraxial Hamiltonians, with lowest order H wavelength-dependent contributions. From the Hamiltonians in the Table 1 we make the following observations: The classical/traditional Hamiltonians of particle/ light optics are modified by wavelength-dependent contributions in the quantum/non-traditional prescriptions, respectively. The algebraic forms of these modifications in each row is very similar. The starting equations have one-to-one algebraic correspondence: Helmholtz 2 Klein–Gordon; Matrix form of Maxwell 2 Dirac equation. Lastly, the de Broglie wavelength, |0 , and | have an analogous status, and the classical/ traditional limit is obtained by taking |0 !0 and |!0, respectively. The parallel of the analogies between the two systems is sure to provide us with more insights. If not for the Foldy–Wouthuysen transformation, it would not have been possible to see the new aspects of the analogies between the light optics and the charged-particle optics [35]. An historical account of the analogies is available in [25,36] Notation: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Refractive index, nðrÞ ¼ c ðrÞmðrÞ. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Resistance, hðrÞ ¼ mðrÞ=ðrÞ. 1 uðrÞ ¼ =nðrÞ. 2nðrÞ
Hamiltonians in different prescriptions
Table 1.
Light beam optics
Charged-particle beam optics
Fermat’s principle
Maupertuis’ principle
2
H ¼ fn ðrÞ
p2? g1=2
H ¼ fp20 p2? g1=2 qAz
Non-traditional Helmholtz
Klein–Gordon formalism
b 0;p H
b 0;p H
1 2 b ¼ nðrÞ þ p 2n0 ? i| 2 q b ; p nðrÞ 16n30 ? qz
1 2 b p ¼ p0 qAz þ 2p0 ? i_ q 2 b p2? ; b þ p? 4 qz 16p0
Maxwell, Matrix
Dirac formalism
b 0;p H
b 0;p H 1 2 b p 2n0 ? i|bR u 1 2 2 | w b þ 2n0
¼ nðrÞ þ
¼ p0 qAz þ
1 2 b p 2p0 ?
_ fmgR? B ? þ ðq þ mÞSz Bz g 2p0 _ Bz þi m0 c
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1 =hðrÞ. 2hðrÞ R and b are the Dirac matrices. b p? ¼ b p? qA? . anomalous magnetic moment, ma . anomalous electric moment, a . m ¼ 2m0 ma =_, ¼ 2m0 a =_. g ¼ E=m0 c2 .
wðrÞ ¼
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BOOK REVIEW W.M. Steen, Laser Material Processing, third ed., Springer, Heidelberg, ISBN 1-85233-698-6, 2003 (XV/ 408pp., 257, illust., EUR69.95, $59.95, Softcover). Material processing is one of the most important applications of lasers. It is an already well-established field in many fields of industry and was modernising industrial production methods drastically. There exist a lot of books for the education of students in this field, but only a few of them will help students and engineers to specialise their knowledge. Laser Material Processing gives a compact survey and can be used as university or industrial course material for senior undergraduate, graduate and nondegree technical training in optoelectronics, laser processing and advanced manufacturing. The book will guide the reader smoothly from the basics of laser physics to the detailed treatment of all major material processing techniques for which lasers are essential. In its third edition this book will help to understand how the laser works and to decide which laser is best for
your purposes. Some new chapters on bending and cleaning reflect the changes in the field since the last 10 years completing the range of practical knowledge about the processes possible with lasers already familiar to users of this textbook. Instead of a number of cartoons which are not fully connected to the content of the book, the authors should introduce some more attractive applications: laser supported cutting tools, laser polishing and some others which are not yet described in the present edition. In some chapters it would also be helpful to introduce some more physics to describe the effects and to explain the results. But even with such small deficiencies, I can recommend this book as an authoritative source of information on the rapidly expanding use of industrial lasers in material processing.
Roland Baur