Modes in Helical Gas Lenses

J. A. Arnaud

Helical gas lenses incorporate four coaxial helices at temperatures + T, - T, + T, and - T, respectively.Because of the resulting change in refractive index of the gas filling the space inside the helices, opticalbeams can be guided by this system over long distances. A general expression for the modes of propaga-tion is given; it involves Hermite polynomials in two complex variables. For small temperature dif-ferences the mode fields reduce to Laguerre-Gauss functions. Calculated irradiance patterns are shownfor various mode numbers and various values of T.

1. Introduction

The focusing properties of electrostatic lenses in-corporating four coaxial helices at potentials + V,-V, +V, and -V, respectively, have been knownfor a long time in the technology of particle acceler-ators. 1 These electrostatic lenses evolved from theconcept of strong focusing according to which a periodicsequence of converging and diverging lenses with equalabsolute powers has a net focusing effect, provided theperiod does not exceed a certain critical value. Tienand others2 have discussed the application of thistechnique to the guidance of optical beams, the fourhelices being raised at alternately high and low tem-peratures. Because of the difference in temperature,gradients of refractive index are created in the gasfilling the space inside the helices. The gas thus actsas a quadrupole lens whose principal axes rotate alongthe system axis. Alternatively, the gas can be re-placed by a liquid with low optical losses. Refractiveindex gradients of the type considered can also be in-duced in electrooptic materials by dc electric fields.

Tien and others2 gave an approximate expressionfor the field of the fundamental mode of propagation,applicable when the difference in temperature betweenthe helices is small. A more general expression hasbeen recently obtained by iMiari6A on the basis of thescalar Helmholtz equation. In this paper, we shalluse a quasi-geometrical optics approach based on theconcept of complex point eikonal. 4 Explicit expres-sions for modes of arbitrary order are obtained. (Pre-liminary results were given in Ref. 5.)

In two-dimensional systems the modes of propaga-tion of scalar waves can be represented by the product

The author is with Bell Telephone Laboratories, Inc., Craw-ford Hill Taboratory, Holmdel, Now Jersey 07733.

Received 1 March 1972.

of a function of Gauss and Hermite polynomials inone real variable, as was first shown by Boyd andGordon.6 In the case of systems with rotational sym-metry, Laguerre polynomials are to be introduced(Goubau and Schwering7 ). In the more general caseof systems such as the helical gas lens that lack merid-ional planes of symmetry, it is necessary to introduceHermite polynomials in two complex variables.8' 9 Inorder to make this paper self-contained, the theoryof modes in Gaussian optics is recalled.

I. Modes in Gaussian Optics

As is well known, the eikonal equation of geometricaloptics is obtained by substituting in the wave equa-tion field components of the form

,p(r) = G(r) exp[ikS(r)], (1)

where k = 2r/X denotes the free-space propagationconstant and r a point in space. Keeping, only theterms with the highest power in k, an algebraic rela-tion between the components of VS is obtained. Inthe general case of lossy media this relation, called theeikonal equation, involves complex parameters andthe solutions S(r) are complex. Even in lossless media,complex solutions are of interest to account for diffrac-tion effects.

The component S/Iz of VS on the z axis of a x1x2zCartesian coordinate system can be considered a func-tion of bS/'xi, 6S/aX 2, x, X2, and z. We shall makethe approximation that S/az is at most of seconddegree in oS/lxi, S/Cx2, x, and x2. This assump-tion can often be made for paraxial beams providedthe eikonal equation is free of singularities in the do-main of interest.

Let us consider the case of beams propagating ina direction close to the z axis in isotropic media. Theexact eikonal equation is

2514 APPLIED OPTICS / Vol. 11, No. 11 / November 1972

d = 1, (6a) + U2 = N, (6b)

as/az

Fig. 1. This figure ilustrates the parabolic approximationused for the eikonal equation in the case of beams propagating

along the z axis in isotropic media.

(aS/az)2 + (is/1X1)2 + (S/aX 2)2 = n2(X,x2Z) (

where the refractive index n can be written approxi-mately

(XlX2,Z) - 1 + 2 iN(z)x, (3)

if we assume, for simplicity, that n(O,O,z) is unity andthat the medium is aligned. We have introduced amatrix notation in Eq. (3) where column vectors andsquare matrices are represented by lower- and upper-case boldface letters, respectively, and tildes denotetransposition. x represents a point with coordinatesx1, x2, and N (z) is a 2 X 2 symmetric matrix whichcharacterizes the focusing properties of the mediumat some plane z; positive focusing effects, for instance,correspond to negative real definite matrices N.

Equation (2) can be rewritten, within the parabolicapproximation (illustrated in Fig. 1)

V + UV = 0, (6c) W + V = 0, (6d)

where the upper dots denote differentiation with re-spect to z.

The solution of Eq. (6a) is d = z. Equation (6b)is a matrix Riccati equation that can be solved in closedform for special functions N(z) and, in particular, whenN is constant in a fixed or rotating coordinate system.

Let us rewrite Eqs. (6b)-(6d) for the case where thex1,x2 coordinate system rotates at a spatial rate r aboutthe z axis. A rotation Tz of the x1,x2 coordinate sys-tem about the z axis is conveniently expressed in matrixnotation by the transformation

X - e-Tzx (7a)

where

-[ 1 ci (7b)

To obtain the transformation of the parameters de-fining the point eikonal or the refractive index, it sufficesto specify that these scalar quantities have the samevalues in the fixed and rotated coordinate systems.The laws of transformation are therefore obtained byintroducing Eq. (7a) in Eqs. (3) and (5). We get

N eT NeTZ, (8a) U --. eTzUeTZ, (8b)

V e-TzV, (8c) W - W. (8d)

Upon substitution of these expressionsgeneralized differential equations for U,are obtained:

I - TU + UT + U2 = N,

in Eq. (6),V, and W

(9a)

oSloz = 1 + iNx - (lS/bx) (Ms/x), (4)

where b/ox denotes the gradient operator in the x, x2plane.

Let S(r;r') denote the optical length of a ray goingfrom a point r' to a point r; S is called the point eikonal.Within the approximation of Gauss this function is,at most, quadratic in x and x'. Thus, the point eikonalcan be written

S(r;r') = d + iUx + Vx' + li'Wx', (5)

where U and W are 2 X 2 symmetric matrices and Vis a 2 X 2 matrix.

d, U, V, and W obey ordinary differential equationsthat are readily obtained by substituting Eq. (5) inEq. (4). We get

V - TV + UV = 0, (9b)

(9c)W + VV = O.

Because these equations are of first order, specifica-tion of U, V, W at some plane (say z = 0) uniquelydefines these matrices everywhere.

To obtain the modes of propagation of optical beamsin lenslike media we simply have to expand the Greenfunction in power series of the source coordinates.1 0

An asymptotic expression for the field radiated by apoint source located at some point r' in the medium,valid in the limit k -- - (short wavelengths), has beenobtained by Van Vleck" in connection with quantum-mechanical problems. 1

2 This expression is4 -(r;r') = aS(r ;r')1bxjx1 exp[ikS(r;r')], (10)

November 1972 / Vol. 11, No. 11 / APPLIED OPTICS 2515

I slax

(2)

Hemim2(z) ap.O (-a~)aml(y a, 12 - AlMs 1- y-+ a +m2- Y -1#+ )!= E'= 2(a+#ol,8!(- a'- )!(Ml - or-a + !MM2 - - + )!'

where the vertical bars denote a determinant. Thephase of the square root is defined by continuity fromsome reference plane. 4,(r;r') in Eq. (10) is a scalarwavefunction such that fi*ds represents the powerflowing through a small area ds. In the case of opticalbeams propagating in a direction close to the z axisin isotropic media, 4' can be taken as n1E, where ndenotes the refractive index and E the magnitude ofthe electric field, assumed linearly polarized. WhenEq. (3) holds, n is almost unity and 4' need not be dis-tinguished from E.

In the case of lossless media and real eikonals, thephysical significance of the exponential term in Eq.(10) is straightforward because S represents in thatcase the optical pathlength from r' to r (phase shiftsat caustics and focal lines being taken into account).The determinant in front of the exponential, on theother hand, follows from power conservation require-ments. Although these interpretations are not ap-plicable to complex eikonals, the expression, Eq. (10),remains a solution of the wave equation, provided therefractive index is analytic for complex r, an assump-tion that can almost always be made.

It turns out, very fortunately, that Eq. (10) is anexact rather than asymptotic solution of the parabolicwave equation when S(r;r') is at most quadratic in xand x' for every z, z', i.e., within the approximationof Gauss, Eq. (5). 15 Upon substitution of this qua-dratic form, Eq. (5), in Eq. (10) the field radiated by apoint source is found to be

(r;r') = V1I exp[ik(z + 12Ux + Vx' + ti'Wx')I. (11)

Note that because V is complex and varies with z, thedeterminant in front of the exponential contributes tothe variation of both the phase and amplitude of thefield.

We define now the modes of propagation at someplane z as the coefficients of the expansion of 4/(r;r')in power series of icxl' and ikx 2 '. Modes can there-fore be viewed as the fields radiated by imaginarymultipoles. Explicit expressions are easily obtainedif we recall the definition of Hermite polynomials interms of their generating functions

exp(ij7 - 142ij) = ml1=O in i2. (t;o) (12)

where + denotes a 2 X 2 symmetric matrix and n, denote vectors. Explicitly we have9

where exp. means that the series terminates when oneof the exponents becomes equal to zero. It is impor-tant to notice that for any diagonal matrix D with ele-ments X1,X2 we have

Hem1m2 (D;D4D) = XlM X2M2Hem- 2(;4), (14)

a relation that readily results from the definition Eq.(12) or Eq. (13).

Let us make use of these mathematical results with

_ kx, a -us Vx, 0 us iW/k, (15)

and expand the right-hand side of Eq. (11) in powersof ikxi', ikX2'. The coefficients are, to within un-important factors,

lmim2(xz) = ViI exp[ik(z + 2ZUx)Hemm2(V;iW/k). (16)

Equation (16) gives the axial modes of propagationin Gaussian optics in their general form. We notethat these fields depend on as many as twenty-two realscalar parameters whose variation with z is definedby Eqs. (6a)-(6d). To be more specific, we may as-sume that the optical system under consideration oc-cupies the half-space z > 0 the sources being locatedat some axial point (z < 0) as shown in Fig. 2. Themedium on the left of the input plane z = 0, called amode-generating system, is chosen such that the desiredfield configurations are produced at the input plane.Note that the set of modes given in Eq. (16) is bior-thogonal to another infinite set denoted 4' mlIm, (x) (seeRef. 9). Arbitrary incident fields Vf(x) can thereforebe expanded in series of the modes 4,mm2 (x), the coeffi-cients amm, of the expansion being evaluated by in-tegrating the product of 4(x) and 4,6m, (x) at the inputplane.

Let us consider first the fundamental solution, ml= m2 0. In that case, the Hermite polynomialsin Eq. (16) are unity and the field configuration at aplane z depends only on the symmetrical matrix U(z).The curves of constant irradiance are ellipses if the

MODE GENERATINGSYSTEM OPTICAL SYSTEM

Fig. 2. The optical system under study is shown in the z > 0half-space and the mode generating system in the z < 0 half-space.

2516 APPLIED OPTICS / Vol. 11, No. 11 / November1972

I m

Fig. 3. Transformation of a Gaussian beam with circular sym-metry (I), into an astigmatic Gaussian beam (), and a beam

with general astigmatism (III).

exp(az), where a denotes a constant, because theseinvariant solutions form a simple basis for the expan-sion of arbitrary field configurations.

It is clear from Eq. (16) that the field configurationof the fundamental mode (ml = m2 = 0) is invariant ifthe matrix U does not depend on z. It is easily verifiedthat if we specify that U = 0, the solution of Eq. (9a)is,

U = tanvi(cosv - sinv)1 ]

i(cosv + sinv)JJ(18)

imaginary part of U is positive definite, an assump-tion that has to be made in order that the mode powerbe defined. The wavefronts, on the other hand, mayhave either a positive curvature (converging waves),a negative curvature (diverging waves), or be saddle-shaped, depending on whether the real part of U isnegative definite, positive definite, or indefinite. Itshould be noted that, in general, the principal axesof the constant phase curves, ellipses, or hyperbolas,do not coincide with those of the constant irradiancecurves. These rather unusual mode configurationsare observed, for instance, when ordinary Gaussianbeams with circular symmetry are launched into asystem of cylindrical lenses oriented at some angledifferent from zero, modulo r/2, with respect to oneanother, 7 as illustrated in Fig. 3. The irradiancepatterns when ml and m2 are different from zero arerather complicated. In general, the wavefronts do notexhibit simple shapes as in the separable case, and theyare different for different mode numbers.

We now apply these general results to helical gaslenses.

111. Modes in Helical Gas Lenses

Helical gas lenses incorporate fur coaxial helicesat temperatures + T, - T, + T, and - T, respectively,as shown in Fig. 4(a). Neglecting aberrations, therefractive index distribution is approximated by ahyperbolic law of the form2' 3' 5

where we have defined

sin(2p) = 7-2 (19)

Assuming that the medium is lossless, i.e., that isreal, the angle v defined in Eq. (19) is real when thestability condition

7 < 72 (20)

is satisfied. Because we are interested only in beamsthat carry finite powers, we can restrict ourselves tovalues of v included between 0 and 7r/4. Equation(18) coincides with an expression recently obtainedby Mari6' using a totally different approach and, inthe limit where i7 is very small compared with 2, withan approximate expression given in Ref. 2.

Because U is a constant we can easily integrate Eq.(9b):

n(X1,X2) = 1 - (X12 _ x 2 ) -1 + 2iNx,

-N n 2N =,

_ ° a

(17a)

(17b)

in an x,,x2 coordinate system that rotates at a spatialrate r = 27r/p about the z axis, p being the period ofthe helices; a positive helicity is assumed. is pro-portional to the temperature T of the helices; it is realif the optical losses in the gas are negligible. Thisassumption, however, need not be made now. Forsimplicity, we assume that the optical waveguide hasa straight axis and that the refractive index is unity onthat axis. Because N is constant in the rotating co-ordinate system, we shall use the differential equations,Eqs. (9a)-(9c), obeyed by U, V, and W in that system.We are particularly interested in solutions of the fieldequation that are independent of z except for a factor

f -f f o

AD Z~~~~~~~~~~~~~~~~~~~14-p/2 -e} -- p /2-..

(b)

Fig. 4. (a) Helical gas lens (H = hot, C = cold) with helicesof period p. (b) View in a meridional plane of symmetry of asequence of thin quadrupole lenses (f = 2p and -2/1p al-ternately), whose properties are analogous to those of helical

gas lenses.

November 1972 / Vol. 11, No. 11 / APPLIED OPTICS 2517

where

f- z

(a)

V = e(T-U)zV, (21)

where Vo denotes a constant matrix, to be defined later.The matrix T - U can be diagonalized using Eqs. (7b)and (18) and written

T - U = MDoM-', (22)

where

Now substituting Eq. (29), with D(z) given by Eq.(31b), in the differential equation, Eq. (9c), and usingEq. (31a), we get a linear equation for Wo:

DoWo + WoDo + MMK2 = 0, (32)

which has the solution

iWo/k = [(1/sinv - 1/cosv)'i cotv

i cotv(1/cosv + 1/sinv)-'j (33)

M -F 1 i(cosv - sinv)1Li(coss + sinv) 1 1'

Do e [i(cosv- sinv) 0 s-i(cosv + sinv)].

The expression for the V matrix given in Eq. (21) canconsequently be rewritten

V = MeDOZM1V,. (25)

The factor VjI in Eq. (16) becomes

IVI = V4ji exp[jTrace(Do)z]. (26)

The propagation constant of the fundamental mode istherefore, from Eqs. (24), (26), and (16),

= k - T sinv.

(23) Introducing these expressions, Eqs. (18), (31), (23),and (24), in the general expression of the field Eq. (16),the complete solution is finally obtained:

(24)

imm 2(xz) = exp(ikz)

X exp{irz[(ml + 1/2)(cosv - sinv) - (2 + 1/2)(cosv + sinv)]

X exp{ -2 tanv[(cosv - sin'V)x12 + (cosv + sinv)x22]}

X exp[i(tan)XXjl

X He. 1-2[xI + i2(cosv + sinv),x2 + i(cosv - sinv);iWo/k],

(34a)

where

X2- KX(27)

Let us now look for solutions of Eq. (9c) correspond-ing to invariant higher order modes. The Hermitepolynomial introduced in Eq. (16) is essentially in-variant, as we have seen [Eq. (14) ], if V and W assumethe forms

V(z) = D(z)Vo,

W(z) = D(z)WoD (z),

(28)

(29)

where D(z) denotes an arbitrary diagonal matrix andVO, Wo constant matrices. V(z) has the desired form,Eq. (28), if we choose

Vo = KM,

(34b)

X2(J

. . ...t j , . ' - X..... ......:. ...... . . ................

(a)

(30)

the factor K -- (kr)l being introduced for later con-venience. V(z) becomes, with that choice for Vo,

D(z) _ eDoz

(31a)

(31b)Fig. 5. (a) Irradiance pattern of the fundamental mode. (b)

Wavefront f the fundamental mode.

2518 APPLIED OPTICS / Vol. 11, No. 11 / November1972

where

V(z) = D(z)M,

z

(b)

and iWo/k is given by Eq. (33). The explicit ex,sion for Hermite polynomials in two variables is EinEq. (13).

The first exponential term in Eq. (34) correspondthe geometrical optics phase shift. The second'ponential term expresses the phase advance resufrom diffraction. This phase term is of particularportance when the helical gas lens is incorporatedresonator, because it determines the resonantquencies. In optical waveguides, knowledge ofphase term is also important because it allows compute the transformation of optical beams tha-not in a single mode. It should be noted that the gvelocity is for all modes, according to this expresequal to the velocity of light. This is a generalsequence of Gaussian optics.

The third exponential term in Eq. (34) is prohthe most important from a practical point ofbecause it defines the size of the beam. The Icontour, corresponding to a field amplitude redby a factor e = 2.718. . . at some transverse p.is an ellipse whose major and minor semiaxes arespectively,

w = [(kr/2) tanv(cosp - sinv)] , (xi axis)

and

W2 = [(kr/2) tanv(cosv + sinv)] l, (x2 axis).

T, an -+; En _ LA - L- B k

It iS moiuereSuing U oe Mnau une argest eam alr-width (w1) is to be found in the direction of the focusing,i.e., the x, axis, an unexpected result. The beamirradiance pattern is shown- in Fig. 5(a). Equation(35a) can be alternatively written

pres-Iiven

ds toi ex-Itingr m-in a

If we assume that the radius of the helices is p =2wo (a value large enough to ensure negligible inter-ference with the optical beam), a relation is obtainedbetween the total difference in refractive index An atadjacent helices and p/X that reads

An - 20(p/X)-2. (38)

fre- For most liquids (dn/dT - 5 X 10-4 /0 C) and p/Xthis = 100, this difference in refractive index corresponds

is to to a total temperature difference of about 4C. Int are the case of carbon dioxide at a pressure of 10 atmo-roup spheres the temperature difference required is aboutsion, 50'C. The main advantage of confining optical beamscon- by linear refractive index gradients rather than by

total reflection, as in cladded fibers, is that group dis-iably persion is eliminated to a large extent.view Let us now consider the higher order modes of prop-)eam agation. We first note that the irradiance patternsuced (Av*) are symmetrical with respect to the x and x2lane, axes. The same is not true, however, for the wave-, re- fronts. In the limit where v 0 the diagonal terms

of W0 vanish. Thus, the triple sum in Eq. (13) re-duces to a single sum over y. It is not difficult toshow that, in that case, the field of higher order modes

(35a) is represented by Laguerre-Gauss functions [with anexp(ihk) azimuthal dependence]. The irradiance pat-terns therefore exhibit a circular symmetry for smallvalues of . Irradiance patterns were calculated fromEq. (34) for various values of v, ml, and M2 . They areshown in Fig. 7 for ml = M2= 1; ml = 2, m2 = 3;

(35b) m1 = 10, m2 = 1, and ml = M2= 6. In each case threevalues of v- 5 °, 300, and 40°-have been considered.As v increases the beam at first decreases in size andlater becomes elongated along the xi (focusing) axis.

201-

k?-lwl2 = 87r(flp2) CoS2v(Cosv - sinp)-l. (36)

For comparison, let us consider a sequence of thinquadrupole lenses rotated by 900 every half-period,having the same optical thickness per section as thehelical gas lens [i.e., f = 2/77 p, see Fig. 4(b)]. Themaximum beam radius w3 in this system is easily foundto be

kInw32 = 8(qp2)4-[(1 + p'/8)/( - p'8)

10

(37)I I I I I

0 1 2 37 /2 p

4 5 6

The variations of kwil2 and kw 32 are shown inFig. 6 as functions of 77 p. This figure shows thatthe minimum beam sizes w are roughly the same inboth cases, though they occur for different periods.

Fig. 6. Comparison between the maximum beam radius in thehelical gas lens (w1 ) and in a sequence of thin quadrupole lenses(w3), in reduced coordinates, as a function of the period p. - isproportional to the temperature of the helices and k = 27r/X.

November 1972 / Vol. 11, No. 11 / APPLIED OPTICS 2519

-

Fig. 7. Mode irradiance patterns in helical gas lenses for various mode numbers and v = 50, v = 30°, v = 400. The coordinates are(kr)"2x1,2.

These irradiance patterns do not resemble the modepatterns observed in the separable case (see, for in-stance, Fig. 7 of Ref. 18). The behavior of modes withlarge mode numbers can, in principle, be described onthe basis of Keller's asymptotic theory 9 ; this asympto-tic theory, however, will not be discussed here.

The main practical difficulty in using helical gaslenses for the guidance of optical beams over long dis-tances is that the beam is eventually intercepted by thewalls of the guide if the system suffers from randombends. As is well known, the axes of optical beamspropagating in misaligned systems follow classical ray

trajectories, provided the approximation of Gauss isapplicable.2 0 The effect of known misalignments istherefore not difficult to evaluate; this evaluation,however, lies outside the scope of this paper.

The author expresses his thanks to J. B. Keller forclarifying discussions. Numerical calculation of themode irradiance patterns was made by W. Mammel.

This paper is the text of a talk given at the New YorkUniversity Seminar on Applied Mathematics, October1971.

2520 APPLIED OPTICS / Vol. 11, No. 11 / November 1972

References1. See, for instance, A. Septier, in Advances in Electronics and

Electron Physics (Academic Press, New York, 1961, pp. 85-205.

2. P. K. Tien, J. P. Gordon, and J. R. Whinnery, Proc. IEEE53, 129 (1965).

3. P. Mari6, Ann. T6lcommun. 25, 320 (1970). The quantitieso' and or' in Eq. (39) of this paper should be interchanged (P.Marie, C.N.E.T., France, private communication). Withthis correction, the results given there coincide with ours.

4. The concept of complex eikonal (and complex rays) has beenintroduced by J. B. Keller, "A Geometrical Theory of Dif-fraction," in Proc. Symp. Appl. Math (McGraw-Hill, NewYork, 1958), Vol. 8, pp. 27-52, to obtain asymptotic formsof diffracted fields.

5. J. A. Arnaud, Proc. IEEE Lett. 59, 1378 (1971).6. G. D. Boyd and J. P. Gordon, Bell Syst. Tech. J. 40, 489

(1961).7. G. Goubau and F. Schwering, IRE Trans. Antenna Prop-

agation AP9, 248 (1961).8. J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).9. J. A. Arnaud, J. Opt. Soc. Am. 61, 751 (1971). (The

notation for Hermite polynomials in two variables differsslightly from the one used in the present paper.)

10. For a general discussion of the relation between modes andGreen functions, see B. Friedman, Principles and Techniquesof Applied Mathematics (Wiley, London, 1960).

11. J. H. Van Vleck, Proc. Nat. Acad. Sci. U.S. 14, 178 (1928).

12. The formal equivalence between scalar optics and quantummechanics has often been pointed out. See, for instance,Ref. 13, pp. 224-225, and Ref. 14.

13. A. Messiah, Quantum Mechanics (North-Holland, Amster-dam, 1961).

14. J. P. Gordon, Bell Syst. Tech. J. 45, 321 (1966).15. This is an almost obvious consequence of Eq. (13) of Ref. 11.

For the case of real eikonals in two dimensions, see Ref. 16,p. 108 (English translation); and for a general discussionsee P. Choquard, Helv. Phys. Acta 28, 89 (1955).

16. I. I. Goldman and V. D. Krivehenkov, Problems in QuantumMechanics, (Pergamon Press, Oxford, 1961).

17. J. A. Arnaud and H. Kogelnik, Appl. Opt. 8, 1687 (1969).18. H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).19. J. B. Keller and S. J. Rubinow, Ann. Phys. 9, 27 (1960).20. See Ref. 16, p. 107; Ref. 13, pp. 218 and 446; or Ref. 2

for the two-dimensional case, and J. A. Arnaud, Appl. Opt.8, 1909 (1969), for the case of optical systems lacking merid-ional planes of symmetry.

The 2nd Congress of the Association Internationale de la Couleur COLOUR 73 University of York, England,2-6 July, 1973

Provisional ProgrammeThe programme will consist of general sessions in themornings in the large Central Hall of York University, andspecialist sessions in the afternoons in smaller lecturetheatres in the Physics Department and in VanbrughCollege.At the general sessions broad surveys on various aspects ofcolour will be given by invited lecturers, while in thespecialist sessions contributed papers on more detailedaspects of colour will be presented. Papers on similar topicswill be grouped together and parallel sessions will bearranged in the afternoons if required, depending on thenumber of papers submitted. The time-table will includeperiods for discussion at both general and specialistsessions. It is hoped in this way that the wishes of the AICon the organization of the Congress, as expressed inStockholm in 1969, will be met.Facilities will be provided for the display of books, charts,photographs and other illustrative material in the Concoursein the Physics Department, but it is not intended to stagean exhibition of scientific equipment.It is expected that the opening session of the Congress willtake place at 9.30 am on Monday, 2 July, while the closingsession will finish not later than 4.00 pns on Friday, 6 July.

Invited Survey LecturesThe following speakers have accepted invitations to givesurvey papers on the topics indicated:

Dr L. Gall (BASF, Ludwigshafen, German FederalRepublic): 'Colorant Formulation'.

Dr R.W.G. Hunt, (Kodak Limited, Harrow, England):'Problems in Colour Reproduction'.

Professor T. Indow (Keio University, Tokyo, Japan):'Colour Atlases and Colour Scaling'.

Dr J.N. Lythgoe (Sussex University, Brighton, England):'Colours under Water'.

Dr E.F. MacNichol (National Institutes of Health,Bethesda, USA): 'Colour Discrimination Processes inthe Retina'.

Dr M. Marre (Universitdt Augenklinik, Dresden, GermanDemocratic Republic): 'The Investigation of AcquiredColour Vision Deficiencies'.

Dr P.L. Walraven (Institute for Perception, Soesterberg,Holland): 'Theoretical Models of the Colour VisionNetwork'.

Dr G. Wysecki (National Research Council, Ottawa,Canada): 'Current Developments in Colorimetry'.

It is also hoped to include an invited lecture on 'Colour inthe Environment'.

Contributed Papers and Papers CommitteeOffers of papers, which may deal with any aspect of thescience, art and technology of colour, must be submitted tothe Secretariat with 6 copies of an extended abstract ofnot more than 1,000 words by 31 October, 1972. The abstracts,which will be published in the Proceedings, may be inEnglish, French or German and may include one diagram.They will be considered by the Papers Committee, who illbe responsible for their acceptance, their grouping togetherand the allocation of time for their presentation. Themembership of the Papers Committee is as follows:

Dr R.W.G. Hunt (Great Britain)-ChairmanMr C.J. Bartleson (USA)Mr G. Bertrand (France)Dr H. Terstiege (Germany).

Preprints and ProceedingsThe Proceedings of the Congress, consisting of the surveylectures and the extended abstracts, will be published byAdam Hilger Limited, who will also supply preprints of thelectures and abstracts in advance of the meeting. The costof the preprints and proceedings will be included in theregistration fee. Authors of contributed papers will be freeto publish the complete version of their papers in anyjournal of their choice if they wish to do so.

AccommodationThe main accommodation for participants will be inLangwith, Derwent and Alcuin Colleges on the UiversityCampus. These are modern buildings with excellentaccommodation for both single personas and marriedcouples. The cost per person including all meals and gratuitywill be about £3-50-£4-00 per night, and rooms will beavailable from Sunday, I July to Friday, 6 July in Langwithand Derwent Colleges and from Sunday, I July to Saturday,7 July in Alcuin College.A very limited number of rooms hase been reserved in theViking Hotel in the City of York. This hotel is about twomiles from the Campus. The cost of a double room in thehotel will be about £9-00 and for a single room about £6-00including gratuity but excluding all meals. Arrangementswill be made for participants who stay at the Viking Hotelto have their lunch at the University.

Social ProgrammeA full social programme will be arranged and participantsare invited to bring members of their families with them.The City of York was for a long time the second city inEngland, and has many ancient and historic buildings.York Minster is famous for its stained glass windows, andthe surrounding country is some of the most lovely inBritain, with many places of great interest and charm. Italso happens that performances of the famous medievalYork Mystery Plays will b taking place at the time of theCongress and arrangements will be made for participantsto make advance bookings if they wish to do so.

Registration FeeThe registration fee for participants is expected to be £1600,but a firm announcement will be made in the second circular.This fee will include the cost of the Preprints and theProceedings. The registration fee for accompanying personswill be £4-00.

Preliminary RegistrationPreliminary registration for the Congress may be made bycompleting and returning the attached form. This is not abinding registration, but the second circular giving furtherdetails of the Congress and including the final registrationform will only be sent to those who have submitted thepreliminary registration form. The second circular will besent ott in January 1973. It will greatly assist the OrganizingCommittee if as many preliminary registration forms aspossible are returned by June 1972.

CorrespondenceAll enquiries, correspondence, and abstracts of papers shouldbe addressed to:

Professor W.D. Wright,(AIC Colour 73),Applied Optics Section,Imperial College,LONDON sw7 2sZ,England.

The Congress is being organized under the generaldirection of the President of the AIC, Professor Yves LeGrand, and of the AIC Executive Committee, and under thedetailed direction of an Organizing Committee whosemembers are drawn from The Colour Group (GreatBritain), the Society of Dyes and Colourists, the Oil andColour Chemists Association and the British ColourCouncil. The Chairman of The Colour Group, Mr R. W.Brocklebank, is Chairman of the Organizing Committee.The Secretariat for the Congress is being set up in theApplied Optics Section, Imperial College, under ProfessorW.D. Wright.

November 1972 / Vol. 11, No. 11 / APPLIED OPTICS 2521