arXiv:physics/0210001v1 [physics.optics] 30 Sep 2002

An alternate way to obtain theaberration expansion in Helmholtz

OpticsSameen Ahmed KHAN

khan@fis.unam.mx,rohelakhan@yahoo.comhttp://www.pd.infn.it/∼khan/

http://www.imsc.ernet.in/∼jagan/khan-cv.htmlCentro de Ciencias Fisicas,

Universidad Nacional Aut´onoma de M´exico (UNAM)Apartado Postal 48-3, Cuernavaca 62251, Morelos,

M EXICO

AbstractExploiting the similarities between the Helmholtz wave equation

and the Klein-Gordon equation,the former is linearized using theFeschbach-Villars procedure used for linearizing the Klein-Gordon equa-tion.Then the Foldy-Wouthuysen iterative diagonalization techniqueis applied to obtain a Hamiltonian description for a system with vary-ing refractive index.Besides reproducing allthe traditionalquasi-paraxialterms,this method leads to additionalterms,which are de-pendent on the wavelength,in the opticalHamiltonian.This alter-nate prescription to obtain the aberration expansion is applied to theaxially symmetric graded index fiber.This results in the wavelength-dependent modification of the paraxial behaviour and the aberrationcoefficients.Explicit expression for the modified coefficients oftheaberration to third-order are presented.Sixth and eighth order Hamil-tonians are derived for this system.

Contents1 Introduction 3

2 Traditional Prescriptions 4

1

3 Applications 113.1 Medium with Constant Refractive Index............ 113.2 Axially Symmetric Graded Index Medium ........... 14

4 Concluding Remarks 17

Appendix A.The Feshbach-Villars Form of the Klein-Gordon Equation19

Appendix B.The Foldy-Wouthusysen Representation of the Dirac Equa-tion 21

Appendix C.The Magnus Formula 28

Appendix D.Analogies between light optics and charged-particle optics:Recent Developments 31D.1 Quantum Formalism of Charged-Particle Beam Optics.... 32D.2 Light Optics:Various Prescriptions............... 34

Table A.Hamiltonians in Different Prescriptions 36

Bibliography 37

2

1 IntroductionThe traditionalscalar wave theory ofoptics (including aberrations to allorders) is based on the beam-optical Hamiltonian derived using the Fermat’sprinciple.This approach is purely geometrical and works adequately in thescalar regime.The other approach is based on the Helmholtz equation whichis derived from the Maxwellequations;Then one makes the square-root ofthe Helmholtz operator followed by an expansion of the radical [1, 2].Thisapproach works to all orders and the resulting expansion is no different fromthe one obtained using the geometrical approach of the Fermat’s principle.

Another way ofobtaining the aberration expansion is based on the al-gebraic similarities between the Helmholtz equation and the Klein-Gordonequation.Exploiting this algebraic similarity the Helmholtz equation is lin-earized in a procedure very similar to the one due to Feschbach-Villars,forlinearizing the Klein-Gordon equation.This brings the Helmholtz equationto a Dirac-like form and then follows the procedure of the Foldy-Wouthuysenexpansion used in the Dirac electron theory.This approach, which uses thealgebraic machinery of quantum mechanics, was developed recently [3], pro-viding an alternative to the traditionalsquare-root procedure.This scalarformalism gives rise to wavelength-dependent contributions modifying theaberration coefficients.The algebraic machinery ofthis formalism is verysimilar to the one used in the quantum theory of charged-particle beam optics,based on the Dirac [4]-[6]and the Klein-Gordon [7]equations respectively.The detailed account for both ofthese is available in [8].A treatment ofbeam optics taking into account the anomalous magnetic moment is avail-able in [9]-[12].

Generalexpressions for the Hamiltonians are derived without assumingany specific form for the refractive index.These Hamiltonians are shown tocontain the extra wavelength-dependent contributions which arise very nat-urally in our approach.We apply the general formalism to the specific exam-ples:A. Medium with Constant Refractive Index.This example is essentiallyfor illustrating some ofthe details ofthe machinery used.The Feschbach-Villars technique for linearizing the Klein-Gordon equation is summarized inAppendix-A. The Foldy-Wouthuysen transformation technique is outlined inAppendix-B.

The other application,B. Axially Symmetric Graded Index Medium isused to demonstrate the power of the formalism.The traditional approaches

3

give six aberrations.Our formalism modifies these six aberration coefficientsby wavelength-dependent contributions.

The traditional beam-optics is completely obtained from our approach inthe limit wavelength,–λ −→ 0,which we callas the traditionallimit of ourformalism.This is analogous to the classical limit obtained by taking ¯h −→ 0in the quantum prescriptions.The scheme of using the Foldy-Wouthuysenmachinery in this formalism is very similar to the one used in the quantumtheory of charged-particle beam optics [4]-[12].There too one recovers theclassical prescriptions in the limit λ0 −→ 0 where λ0 = h/p0 is the de Brogliewavelength and p0 is the design momentum of the system under study.

2 Traditional PrescriptionsRecalling,that in the traditional scalar wave theory for treating monochro-matic quasiparaxial light beam propagating along the positive z-axis, the z-evolution of the optical wave function ψ(r) is taken to obey the Schr¨odinger-like equation

i–λ ∂∂zψ(r) =

cHψ(r) , (1)

where the optical HamiltoniancH is formally given by the radical

cH = − n2(r) −bp2⊥1/2 , (2)

and the refractive index,n(r)= n(x, y, z).In beam optics the rays areassumed to propagate almostparallelto the optic-axis,chosen to be z-axis,here.That is,|bp⊥ |≪ pz ≈ 1 and |n(r) − n0|≪ n0. The refractiveindex is the order ofunity.Let us further assume that the refractive in-dex varies smoothly around the constant background value n0 without anyabrupt jumps or discontinuities.For a medium with uniform refractive index,n(r) = n0 and the Taylor expansion of the radical is

n2(r) −bp2⊥1/2 = n 0

(1 − 1n20

bp2⊥) 1/2

= n 0

(1 − 1

2n20bp2⊥ −

18n40

bp4⊥ −1

16n60bp6⊥

4

− 5128n80

bp8⊥ −7

256n100bp10⊥ − · · ·

). (3)

In the above expansion one retains terms to any desired degree of accuracyin powers of1n20 bp

2⊥ . In generalthe refractive index is not a constant and

varies.The variation of the refractive index n(r),is expressed as a Taylorexpansion in the spatial variables x, y with z-dependent coefficients.To getthe beam opticalHamiltonian one makes the expansion ofthe radicalasbefore,and retains terms to the desired order of accuracy in1

n20bp2⊥ along

with all the other terms (coming from the expansion of the refractive indexn(r)) in the phase-space components up to the same order.In this expansionprocedure the problem is partitioned into paraxial behaviour + aberrations,order-by-order.

In relativistic quantum mechanics too,one has the problem ofunder-standing the behaviour in terms of nonrelativistic limit + relativistic correc-tions,order-by-order.In the Dirac theory of the electron this is done mostconveniently through the Foldy-Wouthuysen transformation.

Here,we follow a procedure similar to the one used for linearizing theKlein-Gordon equation via the Feshbach-Villars linearizing procedure [13].The resulting Feshbach-Villars-like form has an algebraic structure very sim-ilar to the Dirac equation.This enables us to make an expansion using theFoldy-Wouthuysen transformation technique well-known in the Dirac elec-tron theory [14, 15].The resulting expansion reproduces the above expansionin (3) as it should.Furthermore it gives rise to a set of wavelength-dependentcontributions.The formalism presented here is an elaboration of the recentwork which provides an alternative to the traditional square-root techniqueof obtaining the optical Hamiltonian [3].

Let us start with the wave-equation in the rectilinear coordinate system.(∇ 2 − n2(r)

v2∂2∂t2

)Ψ = 0 . (4)

LetΨ = ψ(r)e−iωt, ω > 0 , (5)

then(∇ 2 + n2(r)

v2 ω2)ψ(r) = 0 . (6)

5

At this stage we introduce the wavization,

−i–λ∇ ⊥ −→ bp⊥ , −i–λ ∂∂z −→ pz, (7)

where–λ is the reduced wavelength given by–λ = λ/2π,c =–λω and n(r) =c/v(r).It is to be noted that pq − qp = −i–λ. This is similar to the com-mutation relation in quantum mechanics.In our formalism–λ plays the samerole which is played by the Planck constant,h in quantum mechanics.Thetraditional beam-optics formalism is completely obtained from our formalismin the limit–λ −→ 0.Then, we get,

−i–λ ∂

∂z

! 2+ bp2⊥ − n2(r)

ψ(r) = 0 . (8)

Next, we linearize Eq. (8) following a procedure similar to, the one whichgives the Feshbach-Villars [13]form of the Klein-Gordon equation.To thisend, let

ψ1(r)ψ2(r)

!=

ψ(r)−i–λn0

∂∂zψ(r)

!. (9)

Then, Eq. (8) is equivalent to

−i–λn0

∂∂z

ψ1(r)ψ2(r)

!=

0 1

1n20

n2(r) −bp2⊥ 0

ψ1(r)ψ2(r)

!(10)

Next, we make the transformation, ψ1(r)ψ2(r)

!−→ Ψ (1) =

ψ+ (r)ψ− (r)

!= M

ψ1(r)ψ2(r)

!

= 1√2

ψ1(r) + ψ2(r)ψ1(r) − ψ2(r)

!

= 1√2

ψ(r) − i–λn0∂∂zψ(r)

ψ(r) + i–λn0∂∂zψ(r)

(11)

where

M = M −1 = 1√2

1 11 −1

!, det M = −1 . (12)

6

It is to be noted that the transformation matrix M is independent of z.Fora monochromatic quasiparaxialbeam (in forward direction),with leadingz-dependence ψ(r) ∼ exp in(r)z/–λ.Then

ψ+ ∼ 1√2

(1 +n(r)n0

)ψ(r)

ψ− ∼ 1√2

(1 −n(r)n0

)ψ(r) (13)

Since, |n(r) − n0| ≪ n0, we have ψ+ ≫ ψ− .Consequently,Eq. (8) can be written as

i–λ ∂∂z

ψ+ (r)ψ− (r)

!= bH

ψ+ (r)ψ− (r)

!,

bH = −n 0σz + bE + bObE = 1

2n0nbp2⊥ + n20 − n2 (r)

oσz

bO = 12n0

nbp2⊥ + n20 − n2(r)

o(iσy) , (14)

where σy and σz are,respectively,the y and z components of the triplet ofPauli matrices,

σ = σx =

0 11 0

!, σy =

0 −ii 0

!, σz =

1 00 −1

!!. (15)

It is to be noted that the even-part and odd-part in Hamiltonian (14) differonly by a Paulimatrix.This simplifies the commutations a lot as we shallsee, shortly.The details of the Feshbach-Villars linearizing procedure for theKlein-Gordon equation are available in Appendix-A.

The square of the Hamiltonian isbH 2 =

nn2(r) −bp2⊥

o, (16)

as expected.Thus we have have taken the square-root is a different way.This has certain distinct advantages over the traditional procedure of directlytaking the square-root.

7

The purpose ofcasting Eq.(8) in the form ofEq.(14) willbe obviousnow, when we compare the latter with the form of the Dirac equation

ih ∂∂t

Ψ uΨ l

!= cH D

Ψ uΨ l

!

cH D = m 0c2β + bED + bO DbED = qφbO D = cα ·bπ , (17)

where u and l stand for the upper and lower components respectively and

α ="0 σσ 0

#, β =

"1l 00 −1l

#, 1l =

"1 00 1

#. (18)

To proceed further, we note the striking similarities between Eq. (14) andEq.(17).In the nonrelativistic positive energy case,the upper componentsΨ u are large compared to the lower components Ψl. The odd (bO) part of(cH D − m0c2β), anticommuting with β couples the large Ψu to Ψl while theeven (bE) part commuting with β,does not couple them.Using this fact,the wellknown Foldy-Wouthuysen formalism ofthe Dirac electron theory(see,e.g.,[15]) employs a series oftransformations on Eq.(17) to reach arepresentation in which the Hamiltonian is a sum of the nonrelativistic partand a series of relativistic correction terms; |cbπ|/m0c2 serves as the expansionparameter and the nonrelativistic part corresponds to an approximation oforder up to |cbπ|/m0c2. The terms ofhigher order in |cbπ|/m0c2 constitutethe relativistic corrections.Examining Eq.(14) we conclude ψ+ ≫ ψ− ,and the odd operatorbO, anticommuting with σz, couples the large ψ+ withthe smallψ− ,while the even operatorbE does not make such a coupling.This spontaneously suggests that a Foldy-Wouthuysen-like technique can beused to transform Eq. (14) into a representation in which the correspondingbeam optical Hamiltonian is a series with expansion parameter |bp⊥ |/n0.Thecorrespondence between the beam opticalHamiltonian (14) and the Diracelectron theory is summarized in the following table:

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The AnalogyStandard Dirac Equation Beam Optical Formm 0c2β + bED + bO D −n 0σz + bE + bOm 0c2 −n 0Positive Energy Forward PropagationNonrelativistic, |bπ| ≪ m0c Paraxial Beam, |bp⊥ | ≪ n0Non relativistic Motion Paraxial Behavior+ Relativistic Corrections + Aberration Corrections

Application of the Foldy-Wouthuysen-like technique to Eq. (14) involvesa series of transformations on it and after the required number of transfor-mations,depending on the degree of accuracy,Eq.(14) is transformed intoa form in which the residual odd part can be neglected and hence the upperand lower components (ψ+ and ψ− ) are effectively decoupled.In this repre-sentation the larger component (ψ+ ) corresponds to the beam moving in the+z-direction and the smaller component (ψ− ) corresponds to the backwardmoving component of the beam.

Using the correspondence between Eq.(14)and Eq.(17)the Foldy-Wouthuysen expansion given formally in terms ofbE and bO leads to theHamiltonian

i–λ ∂∂z|ψi = cH (2)|ψi ,

cH (2) = −n 0σz + bE − 12n0

σz bO 2, (19)

To simplify the formal Hamiltonian we use,bO 2 = − 14n20

nbp2⊥ + (n20 − n2(r))

o2

and recall thatbE = 12n0

nbp2⊥ + (n20 − n2(r))

oσz.Dropping the σz the formal

Hamiltonian in in (19) is expressed in terms of the phase-space variables as:

cH (2) = −n 0 +12n0

nbp2⊥ + n20 − n2(r)

o

+ 18n30

nbp2⊥ + n20 − n2(r)

o 2 . (20)

The details of the Foldy-Wouthuysen iterative procedure are described in de-tain in Appendix-B. The lowest order Hamiltonian obtained in this procedureagrees with the traditional approaches.

9

To go beyond the expansions in (20) one goes a step further in the Foldy-Wouthuysen iterative procedure.To next-to-leading order the Hamiltonianis formally given by

i–λ ∂∂z|ψi = cH (4)|ψi ,cH (4) = −n 0σz + bE − 1

2n0σz bO 2

− 18n20

"bO, h bO, bE

i+ i–λ ∂

∂zbO!#

+ 18n30

σz

bO 4 +

h bO, bEi+ i–λ ∂

∂zbO! 2

. (21)

As before we drop the σz and the resulting Hamiltonian in the phase-spacevariable is

cH (4) = −n 0 +12n0

nbp2⊥ + n20 − n2(r)

o

+ 18n30

nbp2⊥ + n20 − n2(r)

o 2

− i–λ32n40

"bp2⊥ ,

∂∂z n2(r)

#

+–λ232n50

∂∂z n2(r)

! 2

+ 116n50

nbp2⊥ + n20 − n2(r)

o 3

+ 5128n70

nbp2⊥ + n20 − n2(r)

o 4 . (22)

The Hamiltonian thus derived has allthe terms which one gets in the tra-ditionalsquare-root approaches.In addition we also get the wavelength-dependent contributions.

The details ofthe various transforms and the beam opticalformalismbeing discussed here turns out to be a simplified analog of the more generalformalism recently developed for the quantum theory of charged-particle beamoptics [4]-[12],both in the scalar and the spinor cases,respectively.A verydetailed description of these transforms and techniques is available in [8].

10

Now, we can compare the above Hamiltonianswith the conventionalHamiltonian given by the square-root approach [2].The square-root ap-proach does not give all the terms, such as the one involving the commutatorof p2⊥ with∂

∂z (n2(r)).Our procedure of linearization and expansion in pow-ers of |bp⊥ |/n0 gives all the terms which one gets by the square-root expansionof (3) and some additional terms, which are the wavelength-dependent terms.Such, wavelength-dependent terms can in no way be obtained by any of theconventional prescriptions, starting with the Helmholtz equation (6).

3 ApplicationsIn the previous sections we presented an alternative to the square-root ex-pansion and and we obtained an expansion for the beam-opticalHamilto-nian which works to allorders.Formalexpressions were obtained for theparaxialHamiltonian and the leading order aberrating Hamiltonian,with-out assuming any form for the refractive index.Even at the paraxiallevelthe wavelength-dependent effects manifest by the presence of a commutatorterm, which does not vanish for a varying refractive index.

Now, we apply the formalism to specific examples.Firstone isthemedium with constant refractive index.This is perhaps the only problemwhich can be solved exactly in a closed form expression.This is just to illus-trate how the aberration expansion in our formalism can be summed to givethe familiar exact result.

The next example is that of the axially symmetric graded index medium.This example enables us to demonstrate the power of the formalism, repro-ducing the familiar results from the traditional approaches and further givingrise to new results, dependent on the wavelength.

3.1 Medium with Constant Refractive IndexFor a medium with constant refractive index, n(r) = nc, we have,

bH c = −n 0σz + Dσz + D (iσy)D = 1

2n0nbp2⊥ + n20 − n2c

o. (23)

11

The Hamiltonian in (23) can be exactly diagonalized by the following trans-form,

T± = exphi (±iσz) bOθ

i

= exp [∓σxDθ]= cosh (Dθ) ∓ σx sinh (Dθ) . (24)

We choose,

tanh (2Dθ) =Dn0 − D=n20 − n2c − bp2⊥n20 + n2c + bp2⊥

< 1 , (25)

then we obtain,bH diagonalc = T + bH cT−

= T + −n0σz + Dσz + D (iσy) T−

= −σ znn20 − 2n0D

o 12

= −σ znn2c − bp2⊥

o 12 (26)

We next, compare the exact result thus obtained with the approximate one,obtained through the systematic series procedure we have developed.Wedefine P =1n20

nbp2⊥ + (n20 − n2c)

o.Then,

cH (4)c = −n 0 1 −12P −

18P

2 − 116P

3 − 5128P

4 σz

≈ −n 0n1 − P2

o 12

= −nn2c − bp2⊥

o 12

= bH diagonalc . (27)

Knowing the Hamiltonian, we can compute the transfer maps.The trans-fer operator between any pair ofpoints (z′′, z′) |z′′> z′ on the z-axis,isformally given by

|ψ(z′′, z′)| =bT (z′′, z′) |ψ(z′′, z′)i , (28)

12

with

i–λ ∂∂z

bT (z′′, z′) =cH bT (z′′, z′) , bT (z′′, z′) =bI ,

bT (z′′, z′) = ℘(exp

"− i–λZ z′′

z′dzcH(z)

#)

= bI − i–λZ z′′

z′dzcH(z)

+ − i–λ

2 Z z′′

z′dz

Z z

z′dz′cH(z)cH(z′)

+ . . . , (29)

wherebI is the identity operator and ℘ denotes the path-ordered exponential.There is no closed form expression forbT (z′′, z′) for an arbitrary choice of therefractive index n(r).In such a situation the most convenient form of theexpression for the z-evolution operatorbT (z′′, z′), or the z-propagator, is

bT (z′′, z′) = exp− i–λbT (z′′, z′) , (30)

with

T (z′′, z′) =Z z′′

z′dzcH(z)

+ 12 − i–λ

Z z′′

z′dz

Z z

z′dz′

hcH(z) ,cH(z′)i

+ . . . , (31)

as given by the Magnus formula [16] which is described in Appendix-C. Weshallbe needing these expressions in the next example where the refractiveindex is not a constant.

Using the procedure outlined above we compute the transfer operator,

bUc(zout, zin) = exp−i–λ∆zH c

= exp

+ i–λnc∆z

1 −12bp2⊥n2c

− 18

bp2⊥n2c

! 2− · · ·

,

∆z = zout− zin, (32)

13

Using (32), we compute the transfer maps hr⊥ ihp⊥ i

!

out=

1 1√

n2c−p2⊥∆z

0 1

hr⊥ ihp⊥ i

!

in. (33)

The beam-optical Hamiltonian is intrinsically aberrating.Even for simplestsituation of a constant refractive index, we have aberrations to all orders!

3.2 Axially Symmetric Graded Index MediumThe refractive index of an axially symmetric graded-index materialcan bemost generally described by the following polynomial (see, pp. 117 in [1])

n(r) = n0 + α2(z)r2⊥ + α4(z)r4⊥ + α6(z)r6⊥ + α8(z)r8⊥ + · · · ,(34)

where, we have assumed the axis of symmetry to coincide with the optic-axis,namely the z-axis without any loss of generality.To write the beam-opticalHamiltonians we introduce the following notation

bT = ( bp⊥ · r⊥ + r⊥ ·bp⊥ )w1(z) =

ddz2n0α2(z)

w2(z) =ddz

nα22(z) + 2n0α4(z)

o

w3(z) =ddz2n0α6(z) + 2α2(z)α4(z)

w4(z) =ddz

nα24(z) + 2α2(z)α6(z) + 2n0α8(z)

o(35)

We also use, [A, B]+ = (AB + BA).The beam-optical Hamiltonian is

cH = cH 0 ,p+ cH 0 ,(4)+ cH 0 ,(6)+ cH 0 ,(8)

+ cH (–λ)0 ,(2)+ cH (–λ)

0 ,(4)+ cH (–λ)0 ,(6)+ cH (–λ)

0 ,(8)

cH 0 ,p = −n 0 +12n0

bp2⊥ − α2(z)r2⊥cH 0 ,(4)=

18n30

bp4⊥ −α2(z)4n20

bp2⊥ r2⊥ + r2⊥ bp2⊥ − α4(z)r4⊥

14

cH 0 ,(6)=1

16n50bp6⊥ −

α2(z)8n40

nbp4⊥ r2⊥ + r2⊥ bp4⊥ + bp2⊥ r2⊥ bp2⊥

o

+ 18n30

nα22(z) − 2n0α4(z)

bp2⊥ r4⊥ + r4⊥ bp2⊥ + 2α22(z)r2⊥ bp2⊥ r2⊥

o

−α 6(z)r6⊥cH 0 ,(8)=

5128n70

bp8⊥ −5α2(z)64n60

bp4⊥ ,hbp2⊥ r2⊥

i+ +

+ 132n50

3α22(z) − 4n0α4(z) h

bp4⊥ , r4⊥i++ 5α22(z)

hbp2⊥ , r2⊥

i2+

− 2α22(z) + 4n0α4(z) bp2⊥ r4⊥ bp2⊥

+ 116n40

4 α32(z) + n0α2(z)α4(z) + n20α6(z) h

bp2⊥ , r6⊥i+

− 5α32(z)r4⊥ ,hbp2⊥ , r2⊥

i+ +

+ 2α32(z) + 4n0α2(z)α4(z) hr2⊥ , r2⊥ bp2⊥ r2⊥

i+

−α 8(z)r8⊥cH (–λ)0 ,(2)= −

–λ216n40

( ddz(n0α2(z))

)bT

cH (–λ)0 ,(4)= −

–λ232n40

w2(z)r2⊥ bT + bT r2⊥ +–λ232n50

w 21(z)r4⊥

cH (–λ)0 ,(6)= −

3–λ232n40

w3(z)r4⊥ bT + bT r4⊥ +–λ216n50

w1(z)w2(z)r6⊥

cH (–λ)0 ,(8)= −

–λ28n40

w4(z)r6⊥ bT + bT r6⊥ +–λ232n50

nw 22(z) + 2w1(z)w3(z)

or8⊥ (36)

The reason for partitioningcH in the above manner will be clear as we proceed.The paraxial transfer maps are formally given by

hr⊥ ihp⊥ i

!

out=

P QR S

! hr⊥ ihp⊥ i

!

in, (37)

where P , Q, R and S are the solutions of the paraxial Hamiltonian in (36).The transfer operator is most accurately expressed in terms ofthe the

15

paraxial solutions, P , Q, R and S, via the interaction picture [17].

bT (z , z0) = exp − i–λbT (z , z0) ,

= exp − i–λ C (z′′, z′)bp4⊥

+K (z′′, z′)hbp2⊥ , (bp⊥ · r⊥ + r⊥ ·bp⊥ )

i+

+A (z′′, z′) (bp⊥ · r⊥ + r⊥ ·bp⊥ )2

+F (z′′, z′) bp2⊥ r2⊥ + r2⊥ bp2⊥+D (z′′, z′)

hr2⊥ , (bp⊥ · r⊥ + r⊥ ·bp⊥ )

i+

+E (z′′, z′) r4⊥ , (38)

The six aberration coefficients are given by,

C (z′′, z′) =Z z′′

z′dz

( 18n30

S4 − α2(z)2n20

Q 2S2 − α4(z)Q4

−–λ28n40

w2(z)Q3S +–λ232n50

w 21(z)Q4

)

K (z′′, z′) =Z z′′

z′dz

( 18n30

RS 3 − α2(z)4n20

QS(P S + QR) − α4(z)P Q3

−–λ232n40

w2(z)Q 2(P S + QR) + 2P Q2S

+–λ232n50

w 21(z)P Q3

)

A (z′′, z′) =Z z′′

z′dz

( 18n30

R 2S2 − α2(z)2n20

P QRS − α4(z)P2Q 2

−–λ216n40

w2(z) (P Q(P S + QR))

+–λ232n50

w 21(z)P2Q 2

)

F (z′′, z′) =Z z′′

z′dz

( 18n30

R 2S2 − α2(z)4n20

(P2S2 + Q2R 2) − α4(z)P2Q 2

16

−–λ216n40

w2(z) (P Q(P S + QR))

+–λ232n50

w 21(z)P2Q 2

)

D (z′′, z′) =Z z′′

z′dz

( 18n30

R 3S −α2(z)4n20P R(P S + QR) − α4(z)P3Q

−–λ232n40

w2(z)P 2(P S + QR) + 2P2QR

+–λ232n50

w 21(z)P3Q

)

E (z′′, z′) =Z z′′

z′dz

( 18n30

R 4 − α2(z)2n20

P 2R 2 − α4(z)P4

−–λ28n40

w2(z)P 3R

+–λ232n50

w 21(z)P4

). (39)

Thus we see that the transfer operator and the aberration coefficients aremodified by–λ-dependent contributions.

The sixth and eighth order Hamiltonians are modified by the presenceof wavelength-dependent terms.These will in turn modify the the fifth andseventh order aberrations respectively [18]-[21].

4 Concluding RemarksWe exploited the similarities between the Helmholtz equation and the Klein-Gordon equation to obtain an alternate prescription for the aberration expan-sion.In this prescription we followed a procedure due to Feschbach-Villarsfor linearizing the Klein-Gordon equation.After casting the Helmholtz equa-tion to this linear form, it was further possible to use the Foldy-Wouthuysentransformation technique ofthe Dirac electron theory.This enabled us toobtain the beam-optical Hamiltonian to any desired degree of accuracy.Wefurther get the wavelength-dependent contributions to at each order,start-ing with the lowest-order paraxial paraxial Hamiltonian.Formal expressions

17

were obtained for the paraxialand leading order aberrating Hamiltonians,without making any assumption on the form of the refractive index.

As an example we considered the medium with a constant refractive index.This is perhaps the only problem which can be solved exactly,in a closedform expression.This example was primarily for illustrating certain aspectsof the machinery we have used.

The second,and the more interesting example isthatofthe axiallysymmetric graded index medium.For this system we derived the beam-optical Hamiltonians to eighth order.At each order we find the wavelength-dependent contributions.The fourth order Hamiltonian was used to ob-tain the six,third order aberrations coefficients which get modified by thewavelength-dependent contributions.Explicit relations for these coefficientswere presented.In the limit–λ −→ 0, the alternate prescription here, repro-duces the very well known Lie Algebraic Formalism of Light Optics.It wouldbe worthwhile to look for the extra wavelength-dependent contributions ex-perimentally.

The close analogy between geometricaloptics and charged-particle hasbeen known for too long a time.Until recently it was possible to see this anal-ogy only between the geometrical optics and classical prescriptions of charge-particle optics.A quantum theory of charged-particle optics was presentedin recent years [4]-[12].With the current development of the non-traditionalprescriptions of Helmholtz optics [3] and the matrix formulation of Maxwelloptics,using the rich algebraic machinery of quantum mechanics,it is nowpossible to see a parallelofthe analogy at each level.The non-traditionalprescription ofthe Helmholtz optics is in close analogy with the quantumtheory of charged-particles based on the Klein-Gordon equation.The matrixformulation ofMaxwelloptics presented here is in close analogy with thequantum theory of charged-particles based on the Dirac equation [22].Weshall examine the parallel of these analogies in Appendix-D and summarizethe Hamiltonians in the various prescriptions in Table-A.

18

Appendix A.The Feshbach-Villars Form of the

Klein-Gordon EquationThe method we have followed to cast the time-independent Klein-Gordon

equation into a beam optical form linear in∂∂z, suitable for a systematic study,

through successive approximations,using the Foldy-Wouthuysen-like trans-formation technique borrowed from the Dirac theory,is similar to the waythe time-dependent Klein-Gordon equation is transformed (Feshbach and Vil-lars,[13]) to the Schr¨o-dinger form,containing only first-order time deriva-tive,in order to study its nonrelativistic limit using the Foldy-Wouthuysentechnique (see, e.g., Bjorken and Drell, [15]).

DefiningΦ = ∂

∂tΨ , (A.1)the free particle Klein-Gordon equation is written as

∂∂tΦ =

c2∇ 2 − m 02c4

h2!Ψ . (A.2)

Introducing the linear combinations

Ψ + = 12

Ψ + ih

m 0c2Φ!, Ψ − = 1

2

Ψ − ih

m 0c2Φ!

(A.3)

the Klein-Gordon equation is seen to be equivalent to a pair ofofcoupleddifferential equations:

ih ∂∂tΨ + = − h2∇ 2

2m 0(Ψ+ + Ψ− ) + m0c2Ψ +

ih ∂∂tΨ − = h2∇ 2

2m 0(Ψ+ + Ψ− ) − m0c2Ψ − . (A.4)

Equation (A.4) can be written in a two-component language as

ih ∂∂t

Ψ +Ψ −

!= cH F V

0

Ψ +Ψ −

!, (A.5)

19

with the Feshbach-Villars Hamiltonian for the free particle,cH F V0 , given by

cH F V0 =

m 0c2 + bp22m 0

bp22m 0

− bp22m 0

−m 0c2 − bp22m 0

= m 0c2σz +bp22m 0

σz + ibp22m 0

σy . (A.6)

For a free nonrelativistic particle with kinetic energy ≪ m0c2, it is seen thatΨ + is large compared to Ψ− .

In presence ofan electromagnetic field,the interaction isintroducedthrough the minimal coupling

bp −→ bπ = bp − qA , ih ∂∂t−→ ih∂∂t− qφ. (A.7)

The corresponding Feshbach-Villars form of the Klein-Gordon equation be-comes

ih ∂∂t

Ψ +Ψ −

!= cH F V

Ψ +Ψ −

!

Ψ +

Ψ −

= 1

2

Ψ + 1m 0c2 ih ∂

∂t − qφ Ψ

Ψ − 1m 0c2 ih ∂

∂t − qφ Ψ

cH F V = m 0c2σz + bE + bObE = qφ + bπ2

2m 0σz, bO = ibπ

2

2m 0σy . (A.8)

As in the free-particle case, in the nonrelativistic situation Ψ+ is large com-pared to Ψ− .The even termbE does not couple Ψ+ and Ψ− whereasbO is oddwhich couples Ψ+ and Ψ− . Starting from (A.8),the nonrelativistic limit ofthe Klein-Gordon equation, with various correction terms, can be understoodusing the Foldy-Wouthuysen technique (see, e.g., Bjorken and Drell, [15]).

It is clear from the above that we have just adopted the above techniquefor studying the z-evolution of the Klein-Gordon wavefunction of a charged-particle beam in an optical system comprising a static electromagnetic field.The additional feature of our formalism is the extra approximation of drop-ping σz in an intermediate stage to take into account the fact that we areinterested only in the forward-propagating beam along the z-direction.

20

Appendix-B.Foldy-Wouthuysen Transformation

In the traditional scheme the purpose of expanding the light optics Hamil-toniancH = − n2(r) −bp2⊥

1/2 in a series using1n20bp2⊥ as the expansion

parameter is to understand the propagation ofthe quasiparaxialbeam interms of a series of approximations (paraxial + nonparaxial).Similar is thesituation in the case of the charged-particle optics.Let us recall that in rela-tivistic quantum mechanics too one has a similar problem of understandingthe relativistic wave equations as the nonrelativistic approximation plus therelativistic correction terms in the quasirelativistic regime.For the Diracequation (which is first order in time) this is done most conveniently usingthe Foldy-Wouthuysen transformation leading to an iterative diagonalizationtechnique.

The main framework ofthe formalism ofoptics,used here (and in thecharged-particle optics)isbased on the transformation technique oftheFoldy-Wouthuysen theory which casts the Dirac equation in a form display-ing the different interaction terms between the Dirac particle and and an ap-plied electromagnetic field in a nonrelativistic and easily interpretable form(see,[14,23],for a generaldiscussion of the role of the Foldy-Wouthuysen-type transformations in particle interpretation of relativistic wave equations).In the Foldy-Wouthuysen theory the Dirac equation is decoupled through acanonical transformation into two two-component equations:one reduces tothe Pauliequation in the nonrelativistic limit and the other describes thenegative-energy states.

Let us describe here briefly the standard Foldy-Wouthuysen theory so thatthe way it has been adopted for the purposes of the above studies in opticswill be clear.Let us consider a charged-particle of rest-mass m0, charge q inthe presence of an electromagnetic field characterized by E = −∇φ −∂

∂tAand B = ∇ × A.Then the Dirac equation is

ih ∂∂tΨ(r, t)=cH D Ψ(r, t) (B.1)

cH D = m 0c2β + qφ + cα ·bπ

21

= m 0c2β + bE + bObE = qφbO = cα · bπ , (B.2)

where

α ="0 σσ 0

#, β =

"1l 00 −1l

#, 1l =

"1 00 1

#,

σ ="σx =

"0 11 0

#,σy =

"0 −ii 0

#,σz =

"1 00 −1

##. (B.3)

withbπ = bp − qA,bp = −i¯h∇, andbπ2 = bπ2x + bπ2y + bπ2z .In the nonrelativistic situation the upper pair of components of the Dirac

Spinor Ψ are large compared to the lower pair of components.The opera-torbE which does not couple the large and smallcomponents of Ψ is called‘even’ andbO is called an ‘odd’ operator which couples the large to the smallcomponents.Note that

β bO = − bOβ , β bE = bE β . (B.4)

Now, the search is for a unitary transformation, Ψ′= Ψ −→ bUΨ, such thatthe equation for Ψ′does not contain any odd operator.

In the free particle case (with φ = 0 andbπ = bp) such a Foldy-Wouthuysentransformation is given by

Ψ −→ Ψ′ = bUF ΨbUF = e ibS = eβα ·bpθ, tan 2|bp|θ =|bp|m 0c

. (B.5)

This transformation eliminates the odd part completely from the free particleDirac Hamiltonian reducing it to the diagonal form:

ih ∂∂tΨ′ = e ibS m 0c2β + cα ·bp e−ibSΨ ′

= cos |bp|θ +βα ·bp|bp| sin |bp|θ

!m 0c2β + cα ·bp

× cos |bp|θ −βα ·bp|bp| sin |bp|θ

!Ψ ′

22

= m 0c2cos 2|bp|θ + c|bp| sin 2|bp|θ βΨ ′

=qm 2

0c4 + c2bp2 β Ψ′. (B.6)

In the general case, when the electron is in a time-dependent electromag-netic field it is not possible to construct an exp(ibS) which removes the oddoperators from the transformed Hamiltonian completely.Therefore, one hasto be content with a nonrelativistic expansion of the transformed Hamilto-nian in a power series in 1/m0c2 keeping through any desired order.Notethat in the nonrelativistic case,when |p|≪ m0c,the transformation oper-atorbUF = exp(ibS) withbS ≈ −iβbO/2m 0c2,where bO = cα ·bp is the oddpart of the free Hamiltonian.So,in the generalcase we can start with thetransformation

Ψ (1)= eibS1Ψ, bS1 = −iβbO2m 0c2

= −iβα ·bπ2m 0c. (B.7)

Then, the equation for Ψ(1)is

ih ∂∂tΨ(1) = ih ∂∂t eibS1Ψ = ih ∂∂t eibS1 Ψ + eibS1

ih ∂∂tΨ

!

="ih ∂∂t eibS1 + eibS1 cH D

#Ψ

="ih ∂∂t eibS1 e−ibS1 + eibS1 cH D e−ibS1

#Ψ (1)

="eibS1 cH D e−ibS1 − i¯heibS1

∂∂t e−ibS1

#Ψ (1)

= cH (1)D Ψ (1) (B.8)

where we have used the identity∂∂t ebA e− bA + ebA ∂

∂t e− bA = ∂∂tbI = 0.

Now, using the identities

ebA bBe− bA = bB + [bA, bB] +12![bA, [bA, bB]] +13![

bA, [bA, [bA, bB]]] + . . .

ebA(t)∂∂t e− bA(t)

= 1 + bA(t) +12!bA(t)2 + 1

3!bA(t)3· · ·

23

× ∂∂t 1 − bA(t) +12!

bA(t)2 − 13!

bA(t)3· · ·

= 1 + bA(t) +12!bA(t)2 + 1

3!bA(t)3· · ·

× − ∂

bA(t)∂t + 1

2!

( ∂ bA(t)∂t

bA(t) +bA(t)∂bA(t)∂t

)

− 13!

( ∂ bA(t)∂t

bA(t)2 + bA(t)∂bA(t)∂t

bA(t)

+ bA(t)2∂bA(t)∂t

). . .

!

≈ −∂bA(t)∂t − 1

2!

"bA(t),∂

bA(t)∂t

#

− 13!

"bA(t),

"bA(t),∂

bA(t)∂t

##

− 14!

"bA(t),

"bA(t),

"bA(t),∂

bA(t)∂t

###, (B.9)

withbA = ibS1, we find

cH (1)D ≈ cH D − h∂

bS1∂t + i

"bS1,cH D − h

2∂ bS1∂t

#

− 12!

"bS1,

"bS1,cH D − h

3∂bS1∂t

##

− i3!

"bS1,

"bS1,

"bS1,cH D − h

4∂ bS1∂t

###. (B.10)

Substituting in (B.10),cH D = m 0c2β + bE + bO, simplifying the right handside using the relations βbO = − bOβ and βbE = bE β and collecting everythingtogether, we have

cH (1)D ≈ m 0c2β + bE1 + bO 1

bE1 ≈ bE + 12m 0c2

β bO 2 − 18m 2

0c4"bO, h bO, bE

i+ i¯h∂

bO∂t

!#

− 18m 3

0c6β bO 4

24

bO 1 ≈ β2m 0c2

h bO, bEi+ i¯h∂

bO∂t

!− 13m 2

0c4bO 3, (B.11)

withbE1 and bO 1 obeying the relations βbO 1 = − bO 1β and βbE1 = bE1β exactlylikebE and bO. It is seen that while the termbO incH D is of order zero withrespect to the expansion parameter 1/m0c2 (i.e., bO = O (1/m0c2)0 theodd part ofcH (1)

D , namelybO 1, contains only terms of order 1/m0c2 and higherpowers of 1/m0c2 (i.e.,bO 1 = O ((1/m0c2))).

To reduce the strength of the odd terms further in the transformed Hamil-tonian a second Foldy-Wouthuysen transformation is applied with the sameprescription:

Ψ (2) = e ibS2Ψ (1),bS2 = − iβbO 1

2m 0c2

= − iβ2m 0c2

" β2m 0c2

h bO, bEi+ i¯h∂

bO∂t

!− 13m 2

0c4bO 3#. (B.12)

After this transformation,

ih ∂∂tΨ(2) = cH (2)

D Ψ (2), cH (2)D = m0c2β + bE2 + bO 2

bE2 ≈ bE1, bO 2 ≈β

2m 0c2 h bO 1,bE1

i+ i¯h∂

bO 1∂t

!, (B.13)

where, now,bO 2 = O (1/m0c2)2 .After the third transformation

Ψ (3)= eibS3 Ψ (2), bS3 = −iβbO 22m 0c2

, (B.14)

we have

ih ∂∂tΨ(3) = cH (3)

D Ψ (3), cH (3)D = m0c2β + bE3 + bO 3

bE3 ≈ bE2 ≈ bE1, bO 3 ≈β

2m 0c2 h bO 2,bE2

i+ i¯h∂

bO 2∂t

!, (B.15)

25

wherebO 3 = O (1/m0c2)3 .So, neglectingbO 3,

cH (3)D ≈ m 0c2β + bE + 1

2m 0c2β bO 2

− 18m 2

0c4"bO, h bO, bE

i+ i¯h∂

bO∂t

!#

− 18m 3

0c6β

bO 4 +

h bO, bEi+ i¯h∂

bO∂t

! 2

(B.16)

It may be noted that starting with the second transformation successive(bE ,bO) pairs can be obtained recursively using the rule

bEj = bE1 bE → bEj−1,bO → bO j−1

bO j = bO 1 bE → bEj−1,bO → bO j−1 , j > 1 , (B.17)and retaining only the relevant terms of desired order at each step.

With bE = qφ andbO = cα ·bπ, the final reduced Hamiltonian (B.16) is, tothe order calculated,

cH (3)D = β

m 0c2 +

bπ22m 0

− bp48m 3

0c6!+ qφ − qh

2m 0cβΣ · B

− iqh28m 2

0c2Σ · curl E −qh4m 2

0c2Σ · E ×bp

− qh28m 2

0c2divE , (B.18)

with the individualterms having direct physicalinterpretations.The termsin the first parenthesis result from the expansion of

qm 2

0c4 + c2bπ2 showingthe effect of the relativistic mass increase.The second and third terms arethe electrostatic and magnetic dipole energies.The next two terms,takentogether (for hermiticity), contain the spin-orbit interaction.The last term,the so-called Darwin term,is attributed to the zitterbewegung (tremblingmotion) of the Dirac particle:because of the rapid coordinate fluctuationsover distances of the order of the Compton wavelength (2π¯h/m0c) the particlesees a somewhat smeared out electric potential.

It is clear that the Foldy-Wouthuysen transformation technique expandsthe Dirac Hamiltonian as a power series in the parameter 1/m0c2 enabling the

26

use of a systematic approximation procedure for studying the deviations fromthe nonrelativistic situation.We note the analogy between the nonrelativisticparticle dynamics and paraxial optics:

The Analogy

Standard Dirac Equation Beam Optical Formm 0c2β + bED + bO D −n 0σz + bE + bOm 0c2 −n 0Positive Energy Forward PropagationNonrelativistic, |bπ| ≪ m0c Paraxial Beam, |bp⊥ | ≪ n0Non relativistic Motion Paraxial Behavior+ Relativistic Corrections + Aberration Corrections

Noting the above analogy, the idea of Foldy-Wouthuysen form of the Diractheory has been adopted to study the paraxial optics and deviations from itby first casting the Maxwell equations in a spinor form resembling exactly theDirac equation (B.1,B.2) in allrespects:i.e.,a multicomponent Ψ havingthe upper halfofits components large compared to the lower componentsand the Hamiltonian having an even part (bE ),an odd part (bO),a suitableexpansion parameter,(|bp⊥ |/n0 ≪ 1) characterizing the dominant forwardpropagation and a leading term with a β coefficient commuting withbE andanticommuting withbO. The additional feature of our formalism is to returnfinally to the originalrepresentation after making an extra approximation,dropping β from the final reduced optical Hamiltonian, taking into accountthe fact that we are primarily interested only in the forward-propagatingbeam.

27

Appendix-CThe Magnus Formula

The Magnus formula is the continuous analogue ofthe famous Baker-Campbell-Hausdorff (BCH) formula

eAeB = eA+ B+ 12[A,B]+ 1

12[[A,A],B]+[[A,B],B]+.... (C.1)

Let it be required to solve the differential equation∂∂tu(t) =A(t)u(t) (C.2)

to get u(T ) at T > t0, given the value of u(t0); the operatorA can representany linear operation.For an infinitesimal ∆t, we can write

u(t0 + ∆t) = e∆tA(t0)u(t0). (C.3)

Iterating this solution we have

u(t0 + 2∆t) = e ∆tA(t0+∆t)e∆tA(t0)u(t0)u(t0 + 3∆t) = e ∆tA(t0+2∆t)e∆tA(t0+∆t)e∆tA(t0)u(t0)

. . .and so on. (C.4)

If T = t0 + N∆t we would have

u(T ) =( N−1Y

n=0e∆tA(t0+n∆t)

)u(t0) . (C.5)

Thus,u(T ) is given by computing the product in (C.5) using successivelythe BCH-formula (C.1) and considering the limit ∆t −→ 0, N −→ ∞ suchthat N∆t = T − t0.The resulting expression is the Magnus formula (Mag-nus, [16]) :

u(T )= bT (T, t0)u(t0)

T (T, t0) = exp( Z T

t0dt1 A(t1)

28

+ 12Z T

t0dt2

Z t2

t0dt1

h bA(t2),A(t1)i

+ 16Z T

t0dt3

Z t3

t0dt2

Z t2

t0dt1

hhA(t3),A(t2)

i,A(t1)

i

+hhA(t1),A(t2)

i,A(t3)

i+ . . .

). (C.6)

To see how the equation (C.6) is obtained let us substitute the assumedform ofthe solution,u(t) =bT (t, t0) u (t0),in (C.2).Then,it is seen thatbT (t, t0) obeys the equation

∂∂t

bT (t, t0) = bA(t)T (t, t0), bT (t0, t0) =bI . (C.7)

Introducing an iteration parameter λ in (C.7), let∂∂t

bT (t, t0; λ)= λ bA(t)bT (t, t0; λ) , (C.8)bT (t0, t0; λ)= bI , bT (t, t0; 1) =bT (t, t0) . (C.9)

Assume a solution of (C.8) to be of the formbT (t, t0; λ) = eΩ(t,t0;λ) (C.10)

with

Ω(t, t0; λ) =∞X

n=1λn∆ n(t, t0), ∆ n(t0, t0) = 0for all n . (C.11)

Now, using the identity (see, Wilcox, [24])

∂∂te

Ω(t,t0;λ)=( Z 1

0dsesΩ(t,t0;λ)∂∂tΩ(t, t0; λ)e−sΩ(t,t0;λ)

)eΩ(t,λ), (C.12)

one has Z 1

0dsesΩ(t,t0;λ)∂∂tΩ(t, t0; λ)e−sΩ(t,t0;λ)= λbA(t) . (C.13)

Substituting in (A13) the series expression for Ω(t, t0; λ) (C.11),expandingthe left hand side using the first identity in (C8),integrating and equating

29

the coefficients ofλj on both sides,we get,recursively,the equations for∆ 1(t, t0), ∆2(t, t0), . . . , etc.For j = 1

∂∂t∆ 1(t, t0) =A(t), ∆ 1(t0, t0) = 0 (C.14)

and hence∆ 1(t, t0) =

Z t

t0dt1 bA(t1) . (C.15)

For j = 2

∂∂t∆ 2(t, t0) +

12

"∆ 1(t, t0) ,

∂∂t∆ 1(t, t0)

#= 0 , ∆ 2(t0, t0) = 0 (C.16)

and hence∆ 2(t, t0) =

12Z t

t0dt2

Z t2

t0dt1

h bA(t2) ,bA(t1)i. (C.17)

Similarly,

∆ 3(t, t0) = 16Z t

t0dt1

Z t1

t0dt2

Z t2

t0dt3

nhhA(t1) ,A(t2)

i,A(t3)

i

+hhA(t3) ,A(t2)

i,A(t1)

io. (C.18)

Then,the Magnus formula in (C.6) follows from (C.9)-(C.11).Equation 31we have, in the context of z-evolution follows from the above discussion withthe identification t −→ z, t0 −→ z(1), T −→ z(2)andA(t) −→ −i¯hcH o(z).

For more details on the exponentialsolutions of linear differentialequa-tions, related operator techniques and applications to physical problems thereader is referred to Wilcox [24],Bellman and Vasudevan [25],Dattolietal. [26], and references therein.

30

Appendix-DAnalogies between light optics and

charged-particle optics:Recent DevelopmentsHistorically,variational principles have played a fundamental role in the

evolution of mathematicalmodels in classicalphysics,and many equationscan be derived by using them.Here the relevant examples are Fermat’s prin-ciple in optics and Maupertuis’ principle in mechanics.The beginning of theanalogy between geometricaloptics and mechanics is usually attributed toDescartes (1637), but actually it can traced back to Ibn Al-Haitham Alhazen(0965-1037) [27].The analogy between the trajectory of material particles inpotential fields and the path of light rays in media with continuously variablerefractive index was formalized by Hamilton in 1833.The Hamiltonian anal-ogy lead to the development of electron optics in 1920s, when Busch derivedthe focusing action and a lens-like action of the axially symmetric magneticfield using the methodology ofgeometricaloptics.Around the same timeLouis de Broglie associated his now famous wavelength to moving particles.Schr¨odinger extended the analogy by passing from geometrical optics to waveoptics through his wave equation incorporating the de Broglie wavelength.This analogy played a fundamentalrole in the early development ofquan-tum mechanics.The analogy,on the other hand,lead to the developmentof practicalelectron optics and one of the early inventions was the electronmicroscope by Ernst Ruska.A detailed account ofHamilton’s analogy isavailable in [28]-[29].

Untilvery recently,it was possible to see this analogy only between thegeometrical-optic and classicalprescriptions of electron optics.The reasonsbeing that, the quantum theories of charged-particle beam optics have beenunder development only for about a decade [4]- [12] with the very expectedfeature of wavelength-dependent effects, which have no analogue in the tradi-tional descriptions of light beam optics.With the current development of thenon-traditional prescriptions of Helmholtz optics [3] and the matrix formula-tion of Maxwell optics, accompanied with wavelength-dependent effects, it isseen that the analogy between the two systems persists.The non-traditionalprescription of Helmholtz optics is in close analogy with the quantum theory

31

ofcharged-particle beam optics based on the Klein-Gordon equation.Thematrix formulation of Maxwelloptics is in close analogy with the quantumtheory ofcharged-particle beam optics based on the Dirac equation.Thisanalogy is summarized in the table of Hamiltonians.In this short note it isdifficult to present the derivation of the various Hamiltonians from the quan-tum theory ofcharged-particle beam optics,which are allavailable in thereferences.We shall briefly consider an outline of the quantum prescriptionsand the non-traditionalprescriptions respectively.A complete coverage tothe new field of Quantum Aspects of Beam Physics (QABP), can be foundin the proceedings of the series of meetings under the same name [30].

D.1 Quantum Formalism of Charged-Particle BeamOptics

The classicaltreatment of charged-particle beam optics has been extremelysuccessfulin the designing and working ofnumerous opticaldevices,fromelectron microscopes to very large particle accelerators.It is natural, howeverto look for a prescription based on the quantum theory,since any physicalsystem is quantum mechanical at the fundamental level!Such a prescriptionis sure to explain the grand success of the classical theories and may also helpget a deeper understanding and to lead to better designing of charged-particlebeam devices.

The starting point to obtain a quantum prescription of charged particlebeam optics is to build a theory based on the basic equations (Schr¨odinger,Klein-Gordon,Dirac) ofquantum mechanics appropriate to the situationunder study.In order to analyze the evolution ofthe beam parameters ofthe various individualbeam opticalelements (quadrupoles,bending mag-nets,· · ·) along the optic axis ofthe system,the first step is to start withthe basic time-dependent equations of quantum mechanics and then obtainan equation of the form

ih ∂∂sψ (x, y; s) =cH (x, y; s) ψ (x, y; s) ,(D.1)

where (x, y; s) constitute a curvilinear coordinate system,adapted to thegeometry ofthe system.Eq.(D.1) is the basic equation in the quantumformalism, called as the beam-opticalequation; H and ψ as the beam-optical

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Hamiltonian and the beam wavefunction respectively.The second step re-quires obtaining a relationship between any relevant observable hOi(s) atthe transverse-plane at s and the observable hOi(sin) at the transverseplane at sin, where sinis some input reference point.This is achieved by theintegration of the beam-optical equation in (D.1)

ψ (x, y; s)= bU (s, sin) ψ (x, y; sin) , (D.2)which gives the required transfer maps

hOi (sin) −→ hOi (s)= hψ (x, y; s) |O| ψ (x, y; s)i ,=

Dψ (x, y; sin) bU †O bU ψ (x, y; sin)

E. (D.3)

The two-step algorithm stated above gives an over-simplified picture ofthe quantum formalism.There are severalcrucialpoints to be noted.Thefirst-step in the algorithm of obtaining the beam-opticalequation is not tobe treated as a mere transformation which eliminates t in preference to avariable s along the optic axis.A clever set of transforms are required whichnot only eliminate the variable t in preference to s but also give us the s-dependent equation which has a close physicaland mathematicalanalogywith the original t-dependent equation of standard time-dependent quantummechanics.The imposition ofthis stringent requirement on the construc-tion of the beam-optical equation ensures the execution of the second-step ofthe algorithm.The beam-optical equation is such that all the required richmachinery of quantum mechanics becomes applicable to the computation ofthe transfer maps that characterize the opticalsystem.This describes theessentialscheme ofobtaining the quantum formalism.The rest is mostlymathematicaldetailwhich is inbuilt in the powerfulalgebraic machineryofthe algorithm,accompanied with some reasonable assumptions and ap-proximations dictated by the physicalconsiderations.The nature oftheseapproximations can be best summarized in the optical terminology as a sys-tematic procedure ofexpanding the beam opticalHamiltonian in a powerseries of|bπ ⊥ /p0|,where p0 is the design (or average) momentum ofbeamparticles moving predominantly along the direction of the optic axis andbπ⊥is the smalltransverse kinetic momentum.The leading order approxima-tion along with |bπ⊥ /p0| ≪ 1, constitutes the paraxial or ideal behaviour andhigher order terms in the expansion give rise to the nonlinear or aberratingbehaviour.It is seen that the paraxial and aberrating behaviour get modified

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by the quantum contributions which are in powers of the de Broglie wave-length (λ0 = h/p0).The classical limit of the quantum formalism reproducesthe well known Lie algebraic formalism [31] of charged-particle beam optics.

D.2 Light Optics:Various PrescriptionsThe traditional scalar wave theory of optics (including aberrations to all or-ders) is based on the beam-opticalHamiltonian derived by using Fermat’sprinciple.Thisapproach ispurely geometricaland worksadequately inthe scalar regime.The other approach is based on the square-rootoftheHelmholtz operator, which is derived from the Maxwell equations [31].Thisapproach works to all orders and the resulting expansion is no different fromthe one obtained using the geometrical approach of Fermat’s principle.As forthe polarization:a systematic procedure for the passage from scalar to vectorwave optics to handle paraxial beam propagation problems, completely tak-ing into account the way in which the Maxwell equations couple the spatialvariation and polarization of light waves,has been formulated by analyzingthe basic Poincar´e invariance ofthe system,and this procedure has beensuccessfully used to clarify several issues in Maxwell optics [32, 33, 34].

The two-step algorithm used in the construction of the quantum theoriesof charged-particle beam optics is very much applicable in light optics!Butthere are some very significant conceptualdifferences to be born in mind.When going beyond Fermat’s principle the whole ofoptics is completelygoverned by the Maxwell equations, and there are no other equations, unlikein quantum mechanics,where there are separate equations for,spin-1/2,spin-1, · · ·.

Maxwell’s equations are linear (in time and space derivatives) but cou-pled in the fields.The decoupling leads to the Helmholtz equation which isquadratic in derivatives.In the specific context of beam optics, purely froma calculational point of view, the starting equations are the Helmholtz equa-tion governing scalar optics and for a more complete prescription one usesthe full set of Maxwell equations, leading to vector optics.In the context ofthe two-step algorithm, the Helmholtz equation and the Maxwellequationsin a matrix representation can be treated as the ‘basic’ equations,analogueof the basic equations of quantum mechanics.This works perfectly fine froma calculational point of view in the scheme of the algorithm we have.

Exploiting the similarity between the Helmholtz wave equation and the

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Klein-Gordon equation,the former is linearized using the Feshbach-Villarsprocedure used for the linearization of the Klein-Gordon equation.Then theFoldy-Wouthuysen iterative diagonalization technique is applied to obtaina Hamiltonian description for a system with varying refractive index.Thistechnique is an alternative to the conventional method of series expansion ofthe radical.Besides reproducing all the traditional quasiparaxial terms, thismethod leads to additionalterms,which are dependent on the wavelength,in the optical Hamiltonian.This is the non-traditional prescription of scalaroptics.

The Maxwellequations are cast into an exact matrix form taking intoaccount the spatialand temporalvariations of the permittivity and perme-ability.The derived representation using 8 × 8 matrices has a close algebraicanalogy with the Dirac equation,enabling the use ofthe rich machineryofthe Dirac electron theory.The beam opticalHamiltonian derived fromthis representation reproduces the Hamiltonians obtained in the traditionalprescription along with wavelength-dependent matrix terms,which we havenamed as the polarization terms [35].These polarization terms are alge-braically very similar to the spin terms in the Dirac electron theory andthe spin-precession terms in the beam-opticalversion of the Thomas-BMTequation[9].The matrix formulation provides a unified treatment of beamoptics and light polarization.Some wellknown results of light polarizationare obtained as a paraxial limit of the matrix formulation [32, 33, 34].Thetraditional beam optics is completely obtained from our approach in the limitof small wavelength,–λ −→ 0, which we call as the traditional limit of our for-malisms.This is analogous to the classical limit obtained by taking ¯h −→ 0,in the quantum prescriptions.

From the Hamiltonians in the Table we make the following observations:The classical/traditionalHamiltonians ofparticle/light optics are modifiedby wavelength-dependent contributions in the quantum/non-traditional pre-scriptions respectively.The algebraic forms ofthese modifications in eachrow is very similar.This should not come as a big surprise.The startingequations have one-to-one algebraic correspondence:Helmholtz ↔ Klein-Gordon;Matrix form of Maxwell↔ Dirac equation.Lastly,the de Brogliewavelength,λ0, and–λ have an analogous status, and the classical/traditionallimit is obtained by takingλ0 −→ 0 and–λ −→ 0 respectively.The parallelofthe analogies between the two systems is sure to provide us with moreinsights.

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Table A.Hamiltonians in Different Prescriptions

The following are the Hamiltonians, in the different prescriptions of light beam op-tics and charged-particle beam optics for magnetic systems.bH 0 ,pare the paraxialHamiltonians, with lowest order wavelength-dependent contributions.

Light Beam Optics Charged-Particle Beam Optics

Fermat’s Principle

H = − n2(r) − p2⊥1/2

Maupertuis’ Principle

H = − p20 − π2⊥1/2− qAz

Non-Traditional Helmholtz

bH 0 ,p=− n(r) +12n0 bp

2⊥

− i–λ16n30

hbp2⊥ , ∂

∂zn(r)i

Klein-Gordon Formalism

bH 0 ,p=− p0 − qAz + 1

2p0 bπ2⊥

+ i¯h16p40

hbπ2⊥ , ∂

∂z bπ2⊥i

Maxwell, Matrix

bH 0 ,p=− n(r) +12n0 bp

2⊥

− i–λβΣ · u+ 1

2n0–λ2w 2β

Dirac Formalism

bH 0 ,p=− p0 − qAz + 1

2p0 bπ2⊥

− ¯h2p0 µγΣ⊥ · B⊥ + (q + µ) ΣzB z

+ i¯hm 0cǫBz

NotationRefractive Index,n(r) = cp ǫ(r)µ(r)Resistance,h(r) =p µ(r)/ǫ(r)u(r) = − 1

2n(r)∇n(r)w(r) = 1

2h(r)∇h(r)Σ and β are the Dirac matrices.

bπ⊥ = bp⊥ − qA⊥µa anomalous magnetic moment.ǫa anomalous electric moment.µ = 2m0µa/¯h ,ǫ = 2m0ǫa/¯hγ = E/m0c2

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[27]D.Ambrosini,A.Ponticiello,G. Schirripa Spagnolo,R.Borghiand F.Gori, Bouncing light beams and the Hamiltonian analogy, Eur.J. Phys., 18 284-289 (1997).

[28]P. W. Hawkes and E. Kasper, Principles of Electron Optics, Vols. I andII (Academic Press, London, 1989); P. W. Hawkes and E. Kasper, Prin-ciples of Electron Optics Vol.3:Wave Optics (Academic Press,Londonand San Diego, 1994).

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[30]Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop onQuantum Aspects of Beam Physics, (04–09 January 1998, Monterey, Cal-ifornia USA),Editor: Pisin Chen,(World Scientific,Singapore,1999),http://www.slac.stanford.edu/grp/ara/qabp/qabp.html;Proceedingsof the 18th Advanced ICFA Beam Dynamics Workshopon Quantum Aspects of Beam Physics (15–20 October2000,Capri,Italy),Editor: Pisin Chen,(World Scientific,Singapore,May 2002),http://qabp2k.sa.infn.it/;Workshop Reports:ICFA Beam Dynamics Newsletter, 16, 22-25 (April1998); ibid 23 13-14 (December 2000);Joint 28th ICFA Advanced Beam Dynamics & Advanced & Novel onQuantum Aspects of Beam Physics (7–11 January 2003, Hiroshima Uni-versity, Japan), http://home.hiroshima-u.ac.jp/ogata/qabp/home.html.

[31]See,e.g.,the following and references therein:Lie Methods in Optics,Lecture notes in physics No. 250, and Lecture notes in physics No. 352,(Springer Verlag, 1986 and 1988).

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