Supersymmetry in Helmholtz optics

19 September 1994 PHYSICS LETTERS A

ELSEVIER

Physics Letters A 193 (1994) 51-53

Supersymmetry in Helmholtz optics * Sergey M. Chumakov

1, Kurt Bcrnardo Wolf

Instttuto de Investtgacwnes en Matemdtwas Aphcadas yen S~stemas, UmverstdadNactonalAutrnoma de M~xwo, Apartado Postal 139-B, 62191 Cuernavaca, Mexwo

Received 2 February 1994, accepted for pubhcatlon 18 July 1994 Communicated by A R Bishop

Abstract A planar wavegmde with a specific refractive index proffie admits supersymmetry Physically, opttcal supersymmetry describes two hght beams of different colors that form a standing periodic interference pattern along the wavegulde ax~s The supersyrnmetry ~s exact m the paraxlal approxtrnaUon and broken beyond

1. Introduction There exists a correspondence between mechanics and optics (classical or quantum, geometric or wave), m the paraxial regime [ 1 ] a n d in the global one [ 2 ]. Consequently, m a n y notions can be transported from one to the other, such as coherent a n d squeezed states (see, e.g., Refs [3,4] ). Here we describe an optical system that exhibits s u p e r s y m m e t r y in the sense o f supcrsymmetncal q u a n t u m mechanics ( S U S Y Q M ) S u p e r s y m m e t r y in q u a n t u m mechanics connects two H a m i l t o n l a n s with the same spectrum except for the ground state ( u n b r o k e n s u p e r s y m m e t r y [ 5 ], see R e f [6], a n d references therein.) S U S Y has been successfully a p p l i e d in a t o m i c [ 7 ] a n d nuclear physics [ 8 ]. T h e s u p e r s y m m e t r i c form o f the D i r a c equation in an external field [ 9-11 ] presents the Dirac equation as the square root o f the K l e i n - G o r d o n equation; the k i n d o f s u p e r s y m m e t r y we have here is analogous a n d finds the square root o f the H e l m h o l t z * Work supported by project DGAPA-UNAM IN 104293 ' On leave from the Central Bureau of Unique Device Designing, Russian Academy of Sciences, Moscow, Russian FederaUon

equation. The s u p e r s y m m e t n c structure o f Helmholtz optics describes light p r o p a g a t i o n in a waveguide a n d a d m i t s a very clear physical interpretation. We consider optical waveguldes along the z-axis, i.e., m e d i a that are lnhomogeneous only in the x-direction. F r o m the wave equation in 2 + 1 d i m e n s i o n s for solutions o f t i m e frequency u we have the Helmholtz equation [0 2 + 0 2 + v2n2(x) / c 2 ] f ( x , z ) = 0

(1)

In Section 2 we present the s u p e r s y m m e t n c structure o f ( 1 ), in Section 3 a physical reinterpretation, and some concluding r e m a r k s in Section 4. N o t e that until Section 3 we do not use the paraxial approximation.

2. Supersymmetric structure We write a system o f two first-order equations for a t w o - c o m p o n e n t wave function o f the x- and zcoordinates,

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S M Chumakov, K B Wolf~ Phystcs Letters A 193 (1994) 51-53

52

(2) where k is a constant and

to be a normahzable solution of one of the two equations v+ ~ = 0 or v_ q~2= 0 Therefore, SUSY connects solutions of Helmholtz equations of two different waveguides with index profiles

v2n2(x)/cZ=k2-T- W x - m 2

v+_- + O x + W ( x ) ,

(6)

with an arbitrary function W ( x ) The second z-derivative involves the square of this 2 × 2 matrix,

/-k2+v+v

3. Physical reinterpretation

0

o

Therefore we have two different Helmholtz equations for the components,

We now reinterpret these formulas in a second way to describe two different beams in the same waveguide We choose

( O~ +02 + k 2 - W x - W2) % = 0 ,

W(x) =wx.

(az2 + a ~ + k ~ + w x - w2) ~u~= 0 ,

(3)

where Wx-=d W / d x Now, for a given wavenumber x in the z direction that is common to both wavefunctlons, we write %.2(x, z ) = ~ l 2(x) e . . . .

(4)

Eqs. (3) then become the eigenvalue equations for the components, ( - - 0 2 - Wxdt- W 2 ) ~ ) l = ( k 2 - / ¢ 2 ) t ~ 1

( -02-

,

W~+ W 2 ) q h = ( k 2 - x 2 ) @ 2 .

(5)

Now we introduce the supersymmetnc structure by considering the component q~ as representing the "boson" and @2 the "fermlon" sectors of the supersymmetric Hamlltonlan exgenfunctlons. The supercharge operators are

o) The supersymmetrlc Hamlltonxan, defined as the antxcommutator of the two supercharges, as

+ vk_), The eigenvalue equations for H, are then V_ V+ ~J~1 = ( k 2 - x 2 ) ~ l

,

v+v_ ~2 = (k2--K'2)~2

These forms coincide with Eqs. (5) and have the same spectrum except for the ground state, which has

(7)

The eigenvalue equations (5) are then two Schrbdlnger equations for harmonic oscillators with displaced energy levels,

( -02x+W2X2)q~,,2 = (k2-7- w-x2)q~l,2.

(8)

Meanwhile, seen as Helmholtz equations, Eqs (8) correspond to

vZn2(x)/cZ=kZT- w - w Z x 2 ,

(9)

with the same n ( x ) In the absence of material disperslon, n is independent of v This is an elhptw index-profile wavegulde; it approximates the usual parabohc index-profile wavegulde in the paraxlal regime [ 12 ]. The elgenvalues of the operator on the left of (8) are well known to be

Em=w(2m+l),

m=0,1,2,

(10)

We now find conditions for a supersymmetrlc pair of Helmholtz solutions in the same waveguide. Into this wavegulde n 2(x) = n oz - n 21x 2 we inject two light beams with slightly different time frequencies

v21,2= Vo2(1+ e ) , and will call xl,z the correspondingly slightly different space frequencies along the z-axis To fulfill Eqs. (9) we substitute k = no Vo/ C, w= n l Vo/ C and find the conditions

nl w n o k - k 2'

v ~ - v 2 =2E v~

(11)

This determines the frequency shift in terms of the transverse index profile of the waveguide.

S M Chumakov, K B Wolf~PhyswsLettersA 193 (1994)51-53 From the standard quantum harmomc oscillator solution we know that the Gaussian beam w~dth in the x-direction is A x = ( 2 w ) - 1/2. Therefore, the parameter ~= [ 2 k 2 ( A x ) 2 ] - 1 ~ (wavelength/beam width) 2 should be small. The two light beams with the frequenoes gl.2 obey the Helmholtz equations with

p21,2n2(x)/C2 = k 2 ~ w - w 2 x 2 ( l-T-e) • When e--,0, these two Helmholtz equations become a supersymmetrlc pair that is slightly broken by the term ~W2X 2. It is easy to see that SUSY has the same accuracy as the paraxlal approximation, examinatmn of Eqs. (8) and (10) shows that the z-propagation wavenumbers of the two beams are x2,2(m)=k2-w[(1 +~)(2m+l)+

or

k

l _ ~ ( m + ½+ ½)_ ½E2(m+ ½+ ½)2 __

__

Acknowledgements

--

T-~2(m+½)+...

References

The term of order ~= w / k 2 exhibits exact SUSY, except for the ground state x2 ( 0 ) = k, the z-wavenumbers of the two beams coincide,

xl(m--1)=lc2(m)+O(~2),

different refractive radices (3) form a supersymmetrlc pair in the sense of SUSY QM. Supersymmetry also describes the propagation of light beams ofdtfferent colors in the same wavegulde when the transversal index profile is elliptic This profile determines the difference ofhght frequencies SUSY is exact in the paraxlal approximation, and broken beyond. In the paraxlal regime, supersymmetric hght beams of different colors have the same wavelength along the axis of the waveguide, giving rise to a stable interference pattern.

We would like to thank Dr. V.V Semenov (Institute of Spectroscopy, Russian Academy of Sciences, Troltsk) for a useful discussion, and Drs. D.D Holm (Los Alamos National Laboratory) and V.P Karassiov (P.N Lebedev Physics Institute, Moscow) for interest in this work.

1] ,

m--0, 1,

Xl,2(m)

53

re=l,2,..

,

(12)

The two terms of order ~2 respectively gave a nonlinear correction to the paraxial approximation and break supersymmetry. Therefore, SUSY IS exact m the paraxlal approximation and is broken beyond this regime. Thus, supersymmetry (12) connects hght beams of different frequencies v2,2 in the same waveguide (Eq. ( 11 ) ), but having the same wavelength 27t/x in the propagation direction z (Eq ( 4 ) ) . These two beams form a stable interference pattern along the wavegulde axis.

4. Conclusions We have considered the propagation of light in a planar waveguIde ruled by the two-dimensional Helmholtz equation. Two Helmholtz equations with

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