properties of periodic gas lenses

Properties of Periodic Gas LensesBy D. MARCUSE

(Manuscript received June 18, 1965)

Gas lenses are being considered as focusing elements of beam-waveguides.Since very many lenses are neededto form a longwaveguide, it is reasonableto consider periodic arrangements of gas lenses. Such periodic structuresmight operate with a gas stream which flows through all of the lenses insuccession. A periodic temperature distribution in the gas results which isdifferent from that of single gas lenses considered in two earlier papers.3 ,4

This paper analyses the my optics properties,'such as focal length andprincipal eurfacee of the gas lenses, of two types of alternating gradientfocusing 81Jstems. One system consists simply of a succession of hot andcold tubes, The other system results [rom. the ji1'st by insertion of heat in-sulating tubes of equal length betweenthe hot and cold tubes.

I. INTRODUCTION

The beam-waveguide described by Goubau' appears as a promisingdevice to transmit light over long distances, However, to reduce thepower loss due to absorption and reflection, which is inevitable withlenses made of solid dielectrics, gas lenses have been proposedt-! insteadof the solid lenses used by Goubau.

Two earlier papersv' discussed the properties of a particular type ofgas lens, This tubular gas lens consists of a warm tube into which acooler gas is blown. The thermal gradients in the gas lead to densitygradients which give the structure the properties of a positive lens.

Ref. 4 discusses the focal length and principal surface of this gas lensfor the case that the gas enters the lens at a constant temperature. Itwas shown that this device, when operated under optimum conditions,acted as an optically rather thin lens with moderate lens distortions.

The present paper extends the earlier analysis in several ways. Weconsider periodic lens structures. Such a structure results if hot and coldtubes are alternated to form a long, periodic structure. The gas is heatedand cooled periodically giving rise to periodically arranged positive andnegative lenses. A periodic structure of this type represents an alter-

2083

2084 'l'HE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1965

nating gradient focusing system." In general, the temperature of the gasentering the hot or cold tubes will not be constant over the cross sectionof the tube so that the earlier results are no longer applicable. The gastemperature in the periodic structure will also be periodic and will de-pend on the temperatures of the hot and cold tubes as well as on the flowvelocity.

It is our aim to compute the temperature distribution in such periodicstructures and use it to determine the properties of the equivalentlenses which describe the ray optics of the alternating gradient focusingsystems.

The equivalent lenses are rather complicated. They are neither op-tically thin nor free of distortions. Further investigations are requiredto determine the guidance properties of an alternating gradient focusingsystem with imperfect lenses of this type.

We discuss two types of periodic gas lens systems. In one case weassume that hot and cold tubes of equal length are directly adjacent toeach other. The other type is an alternating gradient beam-waveguidewhich consists of hot and cold tubes which are separated by tube sec-tions made of an ideally heat insulating material. For simplicity it isassumed that the insulating sections are as long as the hot and coldtubes. The assumption of a perfectly heat insulating material is an over-idealization since hardly any material conducts heat more poorly thangases. It is intended as an approximation to the real situation of im-perfect heat insulators.

In the insulating tube sections, the gas has a chance to equalize itstemperature. As it does so rather rapidly, we again have the case of hotand cold tubes being fed by an input gas at a constant temperature.However, the insulating sections act also as lenses in the same sense asthe hot or cold tubes by which they are preceeded. Therefore, it is notsurprising that some improvement of efficiencyresults if heat insulatingtube sections are used to separate the hot and cold tubes. But this ad-vantage is not very striking; and, since this analysis assumes ideallyinsulating tubes, it is not certain how much of a real advantage can begained by using this construction. Considerably more experience isneeded before a decision can be made.

This analysis again neglects all convection effects in the gas lenses.To be able to distinguish which of the two structures is being dis-

cussed we will call the structure using hot and cold tubes without heatinsulating tube sections the simple periodic structure while the secondcase which includes insulating sections will be called the extended periodicstructure.

II. RAY TRACING

PERIODIC GAS LENSES 2085

(1)

(2)

(3)

Before entering into a discussion of the simple and extended periodicstructures, the ray tracing technique used to determine the focal lengthand principal surface of the lenses will be explained.

The trajectory of the light ray in the gas lens is given by the rayequation"

d ( dr)- n- = gradnds ds

r = the position vector leading from an arbitrary origin to pointson the ray.

n = index of refraction.s = length coordinate measured along the ray.

We limit ourselves to rays which are very nearly parallel to the axisof the structure which is used as the z-coordinate so that we can replaces by z.*Assuming angular symmetry it is sufficient to consider the vectorcomponent in radial direction r perpendicular to the z-axis. Finally, weneglect the term

(an/az) (dr/dz)

because dr/dz « 1 for rays which are nearly parallel to the s-axis andalso because the variation of n in the z-direction will generally be smallerthan that in the r-direction.

(an/az) « (anJar).

With these assumptions we obtain from (1)

d21' 1 an

dz2 = nar •

However, since we are only interested in gases where n - 1 « 1 we cansafely write

d2r andz2 = ar •

The index of refraction depends on temperature in the following way:

n - 1 = (no - 1) ~o (4)

To is the absolute temperature at which no is measured while T is the* The error caused by this approximation is estimated in Ref. 4.

2086 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1965

absolute temperature for which we want to determine n. It followsfrom (4) that

On = _ (no _ 1) To aT.ar T2 ar

The value of the absolute temperature varies only slightly through-out the gas so that T can be replaced by a suitable average temperature.It is convenient to choose To equal to this average temperature whichshould be chosen as

To = HTh + To)

Th = temperature of hot tubesTo = temperature of cold tubes

This leads us to the ray equation

d2r no - 1 aTdz2 = --rr;- ar .

(5)

(6)

Equation (6) is our starting point for the ray optics of the gas lenses.Since aTlar is a complicated function of r and z, it is difficult to solve(6) analytically so that we content ourselves with numerical solutionsobtained by means of an electronic computer.

Rather than expressing our results as functions of z, we want to ob-tain them as functions of the on-axis gas velocity Vo normalized by asuitable constant V. (This representation was also used in Refs. 3 and4.) We define V(L) by

with

volV(L) = aluL

a = klavopcpa = tube radiusL = tube lengthk = heat conductivity of the gasp = (average) gas densityCp = specific heat at constant gas pressure.

(7)

(8)

Equation (7) shows that volV (L) is inversely proportional to the lengthof the tube. Therefore, it is convenient to introduce a variable

u(z) = aloz (9)

PERIODIC GAS LENSES

which at z = L equals

W = ui L) = V~~) .

Using

x = ria

equation (6) becomes

d2x 2 dx 1 no - 1 iJTdu2 + udu = -;4 u2To iJx .

2087

(IO)

(11)

(I2)

Equation (12) is used to obtain x and dx/du as a function of W fromwhich the focal length f and principal surface p can be computed.

As shown in Fig. 1 (a), we follow the ray from a point Z = Zl to Z =Z2 , corresponding to U = Ul and U = U2, through the tube anticipatingthe case of lenses which are not bounded by planes through the ends ofthe tube but by surfaces inside of the tube to be defined later.

The definition of-focal length and principal surface can be seen fromFig. 1 (a). The principal surface is obtained by following a ray, which isincident parallel to the axis (dx/dz = 0 at z = Zl), through the lens.If we extend the direction of the ray entering the lens and the directionof the ray leaving the lens at Z = Z2 by straight lines back into the lenswe obtain a cross-over point which defines a point on the principalsurface. The distance p of this point measured from the beginning of thetube as a function of Xl , the input position of the ray, describes theprincipal surface. The distance p ; for rays traveling in the same direc-tion as the gas flow is given" by (Fig. 1 (a»

X(Z2) - X(ZI)p.; = Zl + L - (dX) (I3)

dz Z=Z2

or, expressed in terms of u and TV rather than L

P+ _ Zl X(U2) - X(Ul) wL - L + 1 + 2 (dX) .

1(2 -du "="2

(14)

We define the focal length as the distance from the intersection of theincoming ray (extended in a straight line) with the principal surfaceto the point at which the outgoing ray crosses the axis of the structure.


-LIGHT RAY

PRINCIPALSURFACE',

" TUBE

t..-----~~~~~

-----T--T---DIREg;'ON I IGAS FLOW k--------- --L-- ---1 I

z=o z=z. I I II z=L Z=Z2

~----P+(J:I) ----+-----------+'+----------->/(a)

DIRECTIONOF

GAS FLOW-PRINCIPALSURFACE',

\\ TUBE

LIGHT RAY..-

Fig. 1 - Geometry of a single gas lens showing the definition of focal lengthand principal surfaces.

Therefore, we haveX(Zl)

(::).=.2 (15)

or, in terms of u and W,

(16)

Similarly, we obtain from Fig. 1 (b) the principal surface and focal


length for the ray traveling in opposite direction to the gas flowp_ ZI X(U2) - X(UI)L = L - lV 2 (dX) (17)

UI -du U=UI

and

f- X(U2)£ = - W 2 (dX) (18)

Ul -du U="I

The principal surfaces p; and p.: do not, in general, coincide. If theyare identical the lens is called optically thin. The separation betweenthe two principal surfaces is an indication of the optical thickness of thelens.

III. THE snifPLE PERIODIC STRUCTURE

3.1 Temperature Distribution

The simple periodic structure consists of alternating hot and coldtubes (Fig. 2). In order to compute the equivalent positive and negativelenses of this structure we first have to determine the temperaturedistribution.

The temperature distribution is given by a series expansion" which,in the hot tube, reads ..

Tl(x,u) = T h - L AnRn(x) exp (-f3n2/U) (19)n=O

1JT _ Vo00 > 1/ > - 1'(£)

with Th being the wall temperature of the hot tube and 1t = a/UZj and

I HOT I COLD I HOT I COLD IGAS I I I I IFLOW- I I I I I

I I I I Iz=o z=L z=2L z=3L z=4L

Fig. 2 - Sequence of hot and cold tubes comprising the "simple periodicstructure" .


in the cold tube

(20)

withW> u > jW

T c = wall temperature of cold tubeW = a/uL.

The functions B« and the eigenvalues Pn are discussed in the appendix.The coefficientsAn and Bnhave to be determined so that the temperatureis a periodic function in the simple periodic structure of Fig. 2.

To simplify the determination of An and B; we limit ourselves tosufficiently long tubes or slow enough flow velocities so that the firstterm of the series expansions (19) and (20) are sufficient to describethe temperature distribution at the end of each tube accurately. Thiscondition is expressed by the requirements

VoW = V(L) < 10. (21)

The periodicity condition requires that the temperature at the end ofthe cold tube (z = 2L or u = tW) equals the temperature at the be-ginning of the hot tube (z = 0 or u = 00)

T1(x,00) = T2(x,tW). (22)

In addition, we have to require that the gas temperature passes con-tinuously from the hot to the cold tube

T1(x,W) = T2(x,W). (23)

The conditions (21) to (23) allow the determination of the constantsA.n and En

and

with

A." =

Bn = - An

(24)

(25)

= {01i)onn=On ¢ O.

The eigenvalues Pn and the derivation fJRlfJP are given in the appendix.


The temperature distribution in the hot tube is shown in Fig. 3 (a)for vo/V (L) = 5 and in Fig. 3 (b) for vo/V (L) = 10 as a function ofnormalized radium x = r/a. The various curves in each figure correspondto different positions along the tube axis. At z = 0 the temperaturedistribution is identical to that at the end of the cold tube. The tem-perature is cold on the wall and warmer in the center of the tube. Aswe follow the temperature distribution deeper into the hot tube we seethat the temperature on the wall changes instantly from its value equalto the wall temperature of the cold tube to that of the hot tube. How-ever, the slope of the temperature distribution close to the tube axisremains negative for quite some length. This means that the gas in thehot tube acts like a negative lens close to the input end of the tube. Ittakes some distance to reverse the negative temperature gradient whichthe cold tube imparted to the gas. In fact, there exists a neutral surfacein the hot as well as the cold tube which is defined by the points wherethe temperature gradient aT/a:r = O. On this surface the gas acts neitheras a positive nor negative lens. The neutral surface separates the region

1.00

0.9S

0.96

0.94T

Th0.92

0.90

O.SS

0.S6

Fig. 3(a) - Temperuture distribution in the simple periodic structure as afunction of normalized radius x = ria at various cross sections z/L in the tube.The normalized flow velocity is vo/V(L) = 5 and (T h - T,)/Th = 0.155.

Fig. 3(b) - Same as Fig. 3(a) with vo/V(L) = 10.


of the positive from the negative lens. It has the same shape in the hotas well as the cold tube and the distance between corresponding pointsof these surfaces in either tube is L, the length of the hot and cold tubes.The temperature distribution TIT" in the cold tube is obtained byreflecting each point of the temperature distribution of Fig. 3(a) or3 (b) on the line parallel to the x-axis at TIT" = T" + T ./2T" . Theneutral surfaces, zlL as a function of x, for various values of flow veloc-ity are obtained by rotating the curves of Fig. 4 around the s-axis, Athigh gas velocities (voIV(L) = 10) the neutral surface extends almostto the half way point into the hot and cold tubes.

3.2 Focal Length and Principal Surface

To calculate effective lenses which describe the ray optics propertiesof the hot and cold tubes it is not permissible to trace rays through eachtube and compute focal length data from the ray trajectory since eachtube functions as a combination of positive and negative lenses. It ismore reasonable to trace rays from one neutral surface to the nextsince the gas between two neutral surfaces acts entirely in one senseeither as a positive or negative lens.

0.5,........-----;,....-----,---.-----.----.

zt

1.0

Fig. 4- Shape of the neutral surfaces for various values of voIV(L).


We inject a ray at a point Zl , x (Zl) on the neutral surface with a slope(dx/dZ)Z_ZI = 0 and follow it to the point Z2 = Zl + L, x (Z2). This pointdoes not, in general, lie on the next neutral surface since the ray movesfrom its entrance position, X(Zl) ~ X(Z2)' However,this point Z2, X(Z2)lies sufficiently close to points on the next neutral surface that this raytracing procedure seems justified.

Our present discussion explains the meaning of the points Zl and Z2(or correspondingly Ul and U2) introduced in (13) through (18) andshown in Fig. 1.

The slope and positions of rays entering at Z = Zl with dxldz = 0were computed at Z = Z2 by numerical integration of (12). The tem-perature distribution entering into (12) is given by (19) and (20).The values of the slope and the ray position were then used to calculatethe focal length and principal surface from (14) and (16). The raystraveling in the direction opposite to the gas flow were launched atZ = Z2, X(Z2) on the neutral surface with the slope (dxldz)z=:2 = 0and their slope and position at ZI = Z2 - L, X(Zl) was used to calculatep_IL and I_IL from (17) and (18).

It is apparent from (12) and (24) that all of our results depend on aparameter

(26)

However, we like to plot our results as functions of W = vo/V (L) whichis contained in a . It is therefore convenient to write

and use

D = Cr~~)Y C (~)

(L). ( )T" - t: (L)2C a = no - 1 -----ro- a

(27)

(28)

to characterize the focusing power of the lens. * Fig. 5 shows the focallength 1divided by the length L of the tubes as a function vo/V (L). Thesolid curves represent the positive, the broken curves the negative lens.The focal length of the negative lens is shown as a positive quantity.These curves were computed setting x (Zl) = 0.1. The positive andnegative lenses have almost equal focusing power for small values ofC(Lla). The negative lens has more focusing power for larger values

• In reference 4 C(L/a) was defined slightly differently. There, T h - T c wasreplaced by T h - T', iT', temperature of input gas).


10B642

l\ :ll,=O.1

\ c(~)=~ 0.05 -----~-- ------ ------

l-

f- \f- \.f-\'~ 0.3......-- ------ ------ ------

\~

~,

0.75

If-...... - ------ ----- --

f-0.80.6

o

2

tOO80

4

tof[ 8

6

20

40

60

Fig. 5 - Normalized focallengthf/L of the positive (solid lines) and negative(broken lines) lenses of the simple periodic structure as functions of the normal-ized flow velocity vo/V(L).

of this parameter. The temperature gradients are equal but opposite insign for either of these lenses so that one might expect that they shouldhave equal focusing power. The discrepancy is explained by consideringthat the rays travel through different parts of the lenses. In the positivelens the ray starting at x (Zl) = 0.1 moves closer towards the lens axiswhile the ray in the negative lens, starting at the same point, movesaway from the axis and toward the wall.

The minima of the focal length curves are explained by the fact thatwe have no lens action if the gas is stationary, va/VeL) = O. The lensbegins to function with increasing gas flow. But, if the gas finally flowsso fast that the on-axis temperature does not have time to follow, lensaction ceases again. Interpolation of curves for parameter values otherthan those shown in Fig. 5 is facilitated by noting that the focal lengthis nearly proportional to [C(L/a)]-l.

The focal length of an ideal lens does not depend on the input positionX(ZI) of the ray. Plotting f / L as a function of z should result in a straight


line parallel to the x-axis. That gas lenses are not ideal lenses is shownin Figs. 6(a) through 6(e). These figures contain three different types ofcurves. The solid curves represent the positive lens for rays traveling inthe same direction as the gas while the dotted curves represent raystraveling opposite to the gas flow. The dash-dotted curves give theresults for the negative lens and rays in the positive gas direction. Therays opposite to the gas flow in the negative lens have been ommitted.They can be visualized by the fact that the curves in the two directionscoincide at x = O. At x = 0.9 the curves for the negative lens join upwith the dotted lines of the positive lens. The lines for the negative lensdo not all extend to x = 0.9 because the ray in the negative lens movestoward the wall and may hit it before it travels its full length if the lensis toostrong and.lf.fhe.tay.started out sufficiently closetothe wall.

Figs. 6(a) through 6(e) show that there are focal length distortions forsmaller values of vo/V(L). For vo/V(L) = 6 and 8 the focal length curvesare substantially parallel to the z-axis. (We see, furthermore, that thefocal length for the two directions of propagation coincide more closelyfor smaller values of C(Lja) and x.

The principal surfaces are shown in Figs. 7(a) through 7(e). Themeaning of the solid, dotted and dash-dotted curves is the same as ex-plained above. The dash-dotted lines for the negative lens for the lowvalues of C(L/a) coincided very nearly with the solid line for the positivelens and was omitted. Also not shown are the corresponding curves forthe negative lens for rays traveling against the gas flow. The principalsurfaces are far from being plane. It is also apparent that for mostvalues of C(Lja) and x, the two principal surfaces for the two directionsof the beams don't coincide too closely. This shows not only that thelenses comprising the simple periodic structure have considerable dis-tortions but also that they are not optically thin under all conditions.

Fig. 8 shows the dependence of the point x = 0.1 of the principalsurfaces on the flow velocity vo/TT(L). The principal surfaces move toz = 0 for vanishing flow velocities and extend far into the tube for largevalues of vo/V(L).

IV. THE EXTENDED PERIODIC STRUCTURE

4.1 Temperature Disiribuiion.

The extended periodic structure is shown in Fig. 9. It consists ofalternating positive and negative lenses which are separated by piecesof insulating tubes of equal length.

The temperature distribution, '1'3, in the hot or cold tubes is well

1.00.80.80.40.2

(b)

c(la)=--

;:::;o.:~_- :::.-

>----v._0__4I-V(L) - -0.3 --::,;:: ,,-----

~, --0.75 -;.-- .........---- ---..,;:

Cd)

c(!a)=0.05 ---- :&:l'

I-v.o -8I--V(l)-

-~.~- ----- ----- -0.75 ---- ------ ---.::::

1.0 00.80.60.40.2

0(a)

0 c(~)= ~~_<:c:~~ ---

0

-~-2--- V(l)-

0.3 ~------

0.75 ----.:::;---- .=;.-

1---~-

(e)0

c (!a)= --0.05 -----=:; Ii<=""-I--- VoI---V(l)=6

0.3 ---- --- ----->-

0.75 ---- ----- ---::;;1--

4

108

6

20

2

2

4

2

1088

4

I0.80.8

80

4

6

I0.80.6

o

fL

.il

na le)-a

c(j)=0 0.05-- -----~081-- Vo61-- Vel) =10

0.3---- ----- ---- --I---- --0.75 ---- f.--_..-- -;:;;;;;;:_I-

I

2

e

2

4

4

0.6

0.8

iL

0.4o 0.2 0.4 :c 0.6 0.8 1.0

Fig. 6 - Normalized focal IengthflL as a function of the ray's position x = riafor rays ill the positive lens traveling (1.) with the gas stream (solid lines), and(2.) against the gas stream (dotted lines); and for rays in the negative lens travelingwith the gas stream (dash-dotted lines).

2O~G

PERIODIC GAS LENSES 209i

known3•7 if we can assume that the input gas is at a constant tempera-

ture.

T3(x,u );-. Rn(x) 2

Til' + 2(Tw - T i ) ~ (oR) exp (-{3n /u)n-O (3 n

n o{3 %~1fJ=fJ..

oo>u>W

(29)

(30)

Til' = either T«, temperature of hot tube, or T c, temperatureof cold tube.

T i = input temperature to the hot or cold tubes.W = vo/V(L) with L length of hot or cold tubes.

The temperature distribution, T4 , in the insulating sections is given interms of U-functions and their eigenvalues 'Y which are defined in: theappendix.

T 4(x ,z ) = A o + '&1 A ..U,,(x) exp [ -'Y/ (~ - ~)]

TV> u> !W.The expansion coefficients have to be determined from the condition

(31)

Since the exponential functions appearing in (29) and (30) decrease verympidly-with-decreasing-values-of u, it is sufficient.to.consider.only.thefirst term in the expansion at the end of each tube. This is justified if

A o =

o ~ TV < 10.

Condition (31) leads to

Til' - 8(Tw _ Ti

) Ro'(l) exp (-{3NW)

(303eo~o)=1

fJ=fJo

and for n T6 0

An = -4(TII' _ T.) 'YnRo'(l) exp (_{302/l!')• {3 (2 (3 2) (oRc 0U)o 'Yn - 0 -'-o{3o O'Yn %=1

where we have used the notation F' = dF lax,

(32)

(33a)

(33b)

1.00.8:r

0.60.40.2

(d)~-..~

<,......~...~... ...

~~, ... ...

~... ...~>,"" ...,,

~(!aF1\'...

" """ """ ""

\::0.75_

..!2.... =8 \ "" ',~.3

Vel)...

',,0.05

\0.75~~:75

0.05

~

1.000.80.6I.O.---..----.,.--...,...--""T""----,

0.4:r0.2

6

(a)

voVel) = 2

r-----_=.--.;.--

....:-- ... c(~)=--"" --~~

.:'f.75~.05

~5 0.750.05

9

(C)

vo

7~Vel) =6

--~-r---: ~~~, ----~

... ..." ... ' ....",~:... c(!a)=... ...\'\; ... ...0.75

~"''''...... ...\ ...... ......~.3

\ ~~.050.75

~.75

0.05

0.5

0.5

o.

0.4

0.4

0.1

O.

o.

O.O!

0.8

0.3

0.2o

ElO.6

£.l

0.3

1.0

(e)

,-,",c(~=

......,0.75

,

0.30L--....L.---'---.L...---,J':-~-,J

0.51-----11----+--''.--j-~''*"7'::-i

0.81-----1H~7t""--+--+---l

0.6

O.4f----,---+---;-j-----.3id-

J:

Fig. 7 - The principal surface for rays in the positive lens traveling (1.) withthe gas stream (solid lines), and (2.) against the gas stream (dotted lines); andfor rays in the negative lens traveling with the gas stream (dash-dotted lines).

2098

.£L

PERIODIC GAS LENSES

I.or----r----r-----r-----r----,

vaV(L)

2099

Fig. 8 - The dependence of the point of the principal surface at x = 0.1 onthe gas velocity. The solid lines represent the positive lens, the dash-dotted linesthe negative lens for rays traveling with the gas stream.

The temperature distribution in the hot and cold tubes can be in-ferred from curves shown in Ref. 3. The temperature distribution inthe insulating tubes is shown in Fig.lO(a)foru(L/a) = 0.15 (W = 6.67)and in Fig. lO(b) for u(L/a) = 0.05 (W = 20) for various valuesof u[(z - L)/a]. z - L is the length coordinate counting from the be-ginning of the insulating tube. The curves show the temperature as afunction of radius at different distances from the input to the insulating


COLD

INSULATION INSULATIONI I_j l _HOT

INSULATIONI___t. _HOT

INSULATIONI______.:f___ COLD

IZ=O

Fig. 9 - The hot, cold and insulating tubes comprising the "extended periodicstructure".

1.0

00

0.8

Lail =0.05

~=0.857

0.4 0.8

:x:=~

0.2

(b)

1.0 00.80.4 0.6

:t=.!:a

0.2

(a) /'aZ~L .:

~

00

~~V~0.02 /'~

a.!a = 0.15

T· f----

Tl~=0.857

~~"<~-f- .-f- f---0.86

0.90

0.94

0.84o

0.88

T

Th 0.92

1.00

0.98

0.96

Fig. lO(a) - Temperature distribution in the insulating tubes of the extendedperiodic structure following a hot tube. u[(z - L)/a] is the normalized lengthmeasured from the beginning of the insulating tube. The ratio of input tempera-ture to the preceeding hot tube To over its wall temperature Th is ToITh = 0.857.The normalized length of the tubes is uL/a = 0.15.

Fig. 10(b)- Same as Fig. 10(a). uL/a = 0.05. The dotted line is the actual tem-perature distribution at the end ofthehot tubewhile the line with 0" [ (z - L)/a) = 0is the temperature distribution at the end of the hot tube which results fromdropping all but the first term in the series expansion of (29).

PERIODIC GAS LE:\SES 2101

tube. At z - L = 0, the temperature distribution is identical to thatat the output end of the hot tube feeding the insulating tube. It isapparent that the temperature equalizes rather rapidly. For practicalpurposes we can say that the temperature has reached a constant valueat 0"[ (z - L) /a] ~ 0.1. If we consider insulating tubes of length L,equal to the length of the hot or cold tubes feeding them, we obtainconstant output temperatures of the insulating tubes for values 0 ~

TV < 10. The hot and cold tubes are fed by gas at a constant tempera-ture a."! long as these conditions are met. Therefore, we are justifiedin using (29) which has been derived for the case that the gas at theinput end of the tube is at a constant temperature.

In the periodic structure of Fig. 9, the input temperature T to thehot and cold tubes are not arbitrary. They adjust themself to satisfythe periodicity condition

(34)

With the help of (34) we can calculate the input temperature Tillof the hot or Tie of the cold tube from (29), (30) and (33).

We obtain from (30) and (33a), for the constant output tempera-ture of the insulating tube following the hot tube, (TV < 10 is assumedRO t.hn.t-un-exponential terms exp (_",.2lTV) can be ii-eglected)

T _ 8(T _ T. ) HO/O) exp (-PNTV)h Illh 1303 (oRo)

013 :<=1IJ=IJo

and also

Here we have two equations which allow us to determine the two quanti-ties T ih and Tic.

It is convenient to express them in the form


and

Tic - T c _ Th - T ihTh - T c - Th - T c •

(35b)

A plot of (35a) as a function of W = vo/V(L) is shown in Fig. 11.

4.2 Focal Length and Principal SU1jace

The following graphical representations show the focal length andprincipal surface of one hot and insulating or one cold and insulatingtube. The extended periodic structure is thus transformed into a systemof equivalent positive and negative lenses in the same way as in the caseof the simple periodic structure. Fig. 12 shows the dependence of thenormalized focallengthf/L on the normalized flow velocity vo/V(L) fora ray entering at 1'/a = 0.1. The length L is that of the hot tube and notthe total length of the combination of hot and insulating tubes whichhas the length 2L. The solid curve in Fig. 12 shows the focal length ofthe combination of hot and insulating tubes while the dotted curveshows the focal length of the hot tube alone for comparison. It is obviousthat the insulating tube adds to the focusing power of the gas lens. Weterminated the curves at vo/V(L) = 10 since we wanted to remain inthe domain of (32) where our simplifying assumptions used to calculatethe temperature distribution are valid. Fig. 13 shows the corresponding

1.2

1.0

0.6

<,<, ---r---

0,4o 2 4 6 8 10

Fig. 11 - Temperature difference between the hot tube T h and the input tem-perature to the hot tube Tih divided by the temperature difference T h - T cbetween hot and cold tube as a function of normalized gas velocity vo/V(L).


.\\ :J:.=O.f

\ C(f) =

~-_....-...... -.E:2= __ ------r---\

\r- \ \r-\ ~ 0.3 ------ -----------

'\<; 0.75 --------- ----I

4

2

20

40

tOO80

eo

iL to

8

6

o 2 4Vo

V(L)

6 8 10

~. Fig. 12 - Normalized focal length IIIJ of the positive lenses of the extendedperiodic structure (solid lines) and of the hot tubes alone (dotted lines) as func-tions of normalized flow velocity voIV(L).

curves for the cold tube resulting in a negative lens. A comparison of thetwo figures shows that the negative lens is more powerful than thepositive lens for corresponding values of C(Lla).

Figs. 14(a) and (b) show the dependence of the focal length (measuredfrom the principal surface) on the input position, x = ria, of the ray forthe hot plus insulating tubes. The solid lines represent rays traveling inthe same direction as the gas while the dotted curves show the focallength of rays traveling in opposite direction to the gas flow.

The shape of the principal surfaces for the hot plus insulating tubeare shown in Fig. 15(a) and (b). The distance p of points on the principalsurface is measured from the gas input end of hot tube. The solid anddotted lines again represent rays traveling with and against the gas flowrespectively. The principal surface is far from being a plane for largervalues of vo/V(L). The two principal surfaces do not coincide exactlywhich means that the lens has Rome optical thickness for larger valuesof C(Lla).

The corresponding negative lens shows very similar distortions andhas therefore been omitted.

to86Va

Vel)

42

II

1\\ :1:.=0.1

\c(~)=

~~~~~~~__~o..5__ ------------\

- \- \\- \ ~ ~ ... ~

0.3 ~~~

~~- ----- ------\ ----~-,

~~

- ~- ------ ~------ ~5--r---

I

0.8

0.6

0.4o

80

60

2

4

10

i. 8l 6

20

40

Fig. 13- Normalized focal length ilL of the negative lenses of the extendedperiodic structure (solid lines) and of the cold tubes alone (dotted lines) as func-tions of the normalized flow velocity voIV(L).

tOO80

60

40

20

.i tol 8

6

4

2

(a)Va

C(~)=V(l)= -::;:;:-l-3-0.05 -~

~0.05

-~

L.3- /0.3

~~~6'4..5~.;::;---- ---

(b)

c(~)=~

~o.:.o..s-:::~

Va =toVel)

0.3 ---- ~-

---- ----- L..-

Io 0.2 0.4 0.6

:I:0.8 1.0 0 0.2 0.4 0.6

:I:0.8 1.0

Fig. 14- Dependence of focal length on the ray's input position for thepositive lenses of the extended periodic structure for rays traveling with the gasflow (solid lines) and against the gas flow (dotted lines).

2104


1.00.60.60.40.2

-----...(b)

~ "~'~', '0o ,;s'

.....~'c(~) =0.3 \ ~\,

~' ..!2.- =10\ ' Vll),,"~'~-, ,\ ','.\' ~, ,\',

1.000.60.60.40.2

I

(a)

9C(Ja)=

1=;-0.;.3

~~.. vo =6.450.05

~bu- ...... Vel)

~,~"~~'~, ,\ ,',~

10...51:0 5 vo ~--=2 "',

"-- ~ V(l)0.7!>

~~I---

I.

t.o

0.2o

0.

0.8

0.5

0.3

0.4

0.7PI

0.6

Fig. 15 - The principal surface of the positive lenses of the extended periodicstructure for rnys traveling with the gas flow (solid lines) and against the gasflow dotted lines.

V. COMPARISON OF THE TWO PERIODIC STRUCTURES

In order to compare the two periodic structures let us assume thattheir equivalent lenses are spaced at the same distance D. For the simpleperiodic structure D, the distance between a positive and the next nega-tive lens is equal to the length L of the individual tubes. In the extendedperiodic structure D = 2L. I t seems fair to compare both structuresunder the condition that the actual gas velocities in either one of themare identical. However, this assumption requires some rescaling of thedata of the extended structure. If we operate the simple structure ata certain value of vo/V(D) the corresponding value for the extendedstructure will be different since Vo is the same but in the extended struc-ture D = 2L. It is apparent that

(V~U~..nded ~ V~) ~ 2 V~j· (36)


A similar transformation has to be done on C. A certain value of C (D / a)of the simple periodic structure corresponds to a value of

(L) (D) 1 (D)C - = C - = -C -a extended 2a 4 a'

(37)

Since IlL is nearly proportional to C-1 for small values of C it isconvenient to compare the values of

(D) I [(L) IJC - '- = 2 C - -a D a L extended'

This comparison is shown in Fig. 16. For the same values of Th - T e

and Vo, the extended structure has the longer focal length because itsfictive hot (or cold) tube is only half as long as that of the simplestructure. The curve of Fig. 16 for the extended structure is not very

2.0

I.S

1.6

1.4

1.2

1.0

o.e

l\\' EXTENDED

I'\.TRUCTURE

r-."'-r-;r---

~'"'"STRUCTURE

~

0.6o 2 4

VoV(D'

6 to

Fig. 16- Comparison between the focal lengths of positive lenses of thesimple and extended periodic structures as functions of the same normalized gasvelocity vo/V(D).


accurate for values of vo/V CD) > 5 because the value of vo/V (D) = 10corresponds to vo/V(L) = 20 for the extended structure. For such alarge value of the normalized flow velocity, our assumption of a constantinput temperature to the hot (or cold) tube is incorrect.

To be able to compare the efficiencies of the two systems we need toknow the power consumption of one positive lens of either of themassuming that it requires no additional power expenditure to keep thecold tubes at the temperature T c »

This power consumption is given by

P = 21rpcpa2{ ([T(x,u)]z=L - [T(x,u»)._o}xv(x) dx (38)

v(x) is the gas velocity as a function of x. For viscous, laminar flow

(39)

2P.

Using (19) and (24), we obtain for the power consumption per hot tubeof the simple periodic structure

{

. 16R'(1) }

= V~~) 1 - fJo3eO~o)Z=1 (l + exp (fJo2/W» (40)

and with the help of (29) for the corresponding power consumptionin the structure composed of extended tubes (assuming that the firstterm in the series of (29) describes the temperature distribution atz = L sufficiently well), we obtain

. 2Pcxt _ ~_ TIl - Tih{l _ 8Ro'(1) 2}1rkD(TIl - Tc ) - V (D) 1'1, - T. fJo3 (ORO) exp t-s, /W) . (41). ofJ z=1

The expression (1'1' - T iI, ) / (Til - 1'.) has to be substituted from (35a).-The quantities represented by (40) and (41) are plotted in Fig. 17 asJunctions of vo/V (D). The positive lenses of the extended structureconsume less power than those of the simple structure for equal amounts'of gas flowing through them. This is not surprising considering thatthe hot tubes of the extended structure are only half as long as thoseof the simple structure.

To compare the two structures we require that their lenses have equalfocal length which we achieve by adjusting the temperature difference be-tween the hot and cold tubes. We then plot the ratio of the resultingpower consumptions. This plot is shown in Fig. 18. The extended struc-ture requires less power than the simple one and this ratio improves as


5r----....----,-----,-------,r----,

fOe6~V(D)

42

41----+-----i---+-:7"'''--j-------t

~31_--_+--~:j_---+---+_--___jI0.'=N '-'o

.:£to 21-----4I'----:::iI"'''F'-----+---+-----i

Fig. 17- Comparison between the power consumptions of the positive lensesof the simple and extended periodic structures as functions of the same normal-ized gas velocities vo/V(D).

1.2

1.0

PeXT 08P. .s

0.6

--~r-,<,

'--0.4

o 2 4 6 8 to

Fig. 18- The ratio of the power consumption of the positive lenses of theextended periodic structure over the power consumption of the positive lensesof the simple periodic structure for equal focal length as a function of normalizedgas velocity vo/V(D).

PERIODIC GAS LEXSES 2109

the flow velocity.increases, It should be remembered, however, that thefocal length of the extended structure (Fig. 16) is inaccurately repre-sented for values vo/V (D) > 5. This inaccuracy carries over to Fig. 18.Nevertheless, it is clear that the efficiency of the extended structure ismore than 20 per cent higher than that of the simple structure forvo/V(D) > 6. The additional focusing which the insulating tube sectionsprovide pays off to some extent to make the extended structure moreefficient. The improvement is not as large, however, as one might havehoped and it may be considerably poorer if the insulating tubes have thefinite heat conductivity of a real material.

VI. ACKNOWLEDGMENT

The author gratefully acknowledges the help of Mrs. C. L. Beattiewho wrote the machine programs for the calculation of the R- and U-functions and for part of the ray tracing.

APPENDIX

The functions to be discussed in the appendix are solutions of thedifferential equation

d2F +.! dF + k2(1 _ x2 )F = O. (42)

dx2 x dx

The solutions of (42) are related to Whittaker's functions W.,I' by

F = ~ W(k/4),o(kx2) .

x

The differential equation (42) stems from an approximate formulationof a heat transfer problem.' Assume that a gas at a different tempera-ture is blown into a tube with a circular cross section. The gas is sup-posed to flow as a viscous fluid in laminar flow with a velocity profile

v(1') = Vo (1 -~)

r = distance from tube axisa = tube radius.

(43)

The stationary state of the temperature distribution T is obtained from

(44)

with a being a constant which contains the heat conductivity, density


and specific heat at constant pressure - all of which are assumed tobe temperature independent which, strictly speaking, is not true.

The velocity has only a longitudinal component Vz given by (43).In polar coordinates r, <p, z taking a!iJ<p = 0 (44) can be written

a {a2T +! aT + a

2T}

= Vo(1 _~) r: (45)ar2 r ar az2 a2 az

It is often permissible to neglect a2T /ai compared to the term on theright hand side of (45). Taking

r = ax

and

T(r,z) = F(x) exp (-AZ)

we get from (45)

d2F +! dF + a

2v OA (l _ x 2 )F = O.dx2 xdx a

Finally, taking

(46)

(47)

(48)

(49)

we recognize that (48) is identical to (42).We are interested in two types of heat transfer problems:(1.) The tube through which the gas flows may be kept at a constant

temperature T", . We then have the boundary condition T(a,z) = T1D •

It is more convenient to introduce a new variable

8(r,z) = T", - T(r,z). (50)

We can replace T by 8 without changing any of equations (44) through(47). However, the boundary condition now becomes simply

8(a,z) = O.

The functions satisfying this boundary condition are designated by

F(x) = R(x)

and (51) becomes

R(l) = o.

(51)

(52)

(53)

The R functions have been studied in some detai1.7•8

(2.) The second type of problem involves a tube which is a perfect


heat insulator. That means that no heat flows into or out of the walls.This assumption requires that

(aT) = 0ar r=a •(54)

A second class of functions is obtained by setting F (x) U (x). TheU-functions satisfy the same differential equation but are defined bythe boundary condition

(dU) = O.dx z=1

For convenience, both functions are normalized so that

R(O) = U(O) = 1.

(55)

(56)

The U-functions have not been studied to the knowledge of the author.For series expansions of arbitrary heat distributions in terms of either

the R or the U functions we need their orthogonality relations and anynumerical evaluation of heat transfer problems requires the knowledgeof the eigenvalues and numerical values of these functions.

The eigenvalues belonging to the R functions R" (x) will be designatedas

and those of U; (x) will be designated by

k" = '1".

(57)

(58)

Orthogonality Relations

Let F; with eigenvalue len designate either an R; or a U; function.We proceed to show that

f x(l - x2)Fn(x)Fm(x) dx = 0 for n ~ m. (59)

We have

(60)

and

(61)


We multiply (60) by xFm and (61) by xFn , subtract and integrate:

(kn2 - km2) f x(l - x2)Fn(x)Fm(x) dx

We perform partial integrations and obtain for the right hand side ofthis equation

-[xFmFn' - xFnF".']~ + II {FmFn' - r»: + x(Fm'Fn' - Fn'Fm')

- (FmFn' - FnFm')I dx = (FmFn' - FnFm')Z=1 = O.

The last part of the equation follows from the boundary condition (53)or (55), depending whether F stands for an R or a U function. Thiscalculation proves (59).

Next, we calculate the value of (59) in the case n = m.

11 x(l - x2)[Fn(x)]2 dx = lim (Fm~n~ =:n;m')o km ....kn n m z=l

= (aFn F' - F !.... F')ak n n ak n

k ",=1.2 n k=kn

For F; = R; we get

(62)

and for F; = U«

Finally, we need to determine the value of the integral over the prod-ucts RnUm :

(64)

(65)


«(3n2- "'1m2

) { x(l - x2)R,,(x) Um(x)dx

= { {x(R"Um" - UmRn") + u.o; - UmRn'}dx

or using (53) and (55),

{ x(l - x2)Rn(x)Um(x)dx = 2 1 2 [UmRn'],,=l . (66)o "'1m - {3n

Calculation of the Rand U Functions and Their Eigenvalues

We make the series expansion00

F(x) = L C2vX2V

•v=o

(67)

For the problem of interest to us F (x) has to be an even function of x,for that reason only even powers of x appear in (67). The normalization

F(O) = 1

requires that

Co = 1.

The substitution of (67) into (42), using (68), leads to

C2 = -1 k2.

and

(68)

k2

C2v = (2U)2 {C2v-4 - C2v- 2 } foru ~ 2. (69)

The parameter k has to be chosen so that either F (1) = 0 or F' (1) = 0results, depending whether k and F shall represent {3 and R 01'"'1 and Urespectively.

The fact that k2 enters all coefficients C2v makes the determinationof {3n and "'In very tedious.

A further difficulty results from the fact that the coefficients C2v

grow to very large values particularly for the larger values of {3n and'Yn before they decrease again. The series (67) does not converge readilyfor values of x close to 1. In fact, it proved impossible to compute morethan the first eight Rand U functions from (67) on the IBM 7094 com-


puter even using double precision since the absolute value of Rand Uremains between zero and one but the coefficients C2• grow to valuesabove 1020• The series (67) can be used to compute R« and U« for xin the range 0 ;;;i; x ;;;i; 0.5 since the powers of x decrease rapidly enoughto keep the value of the product C2• x2

' within manageable proportions.However, in order to cover the whole range 0 ;;;i; x,;;;i; 1 it proved

necessary to use the following series expansion:co

F(x) = ~ D.y" with y = 1 - z.=0

(70)

(71)

to calculate Rand U in the range 0.5 ~ x ~ 1.Equation (42) expressed in terms of y reads

( ) d2F dF 2( 2 3)1 - Y -d2 - d- + k 2y - 3y + y F = O.y Y

The coefficients Do and D1 have to be properly chosen to satisfy theboundary conditions at x 1. For F = R we require R (1) = 0 sothat

Do = 0results. For F = U we require U' (1) = 0 so that

D1 = 0

results.The substitution of (70) into (71) yields

(72)D.. = v(v ~ 1) {(v - 1)2 D._1 - k2(2D.._3 - 3D..-4 + D..- o) I·

The eigenvalue k = (3 or k = " and the coefficient D1 or Do must be'chosen so that F as well as F' are continuous at x = 0.5 where both seriesexpansions should coincide.

By breaking the range of x into two parts and using different seriesexpansions to cover both parts of the range it was possible to computethe function and their eigenvalues. Table I shows the eigenvalues(3n and v; as well as the values of iJRnliJ(3, -D1 = Rn' . (iJjiJ,,)Un ' andDo = U« all taken at x = 1. These values are needed to evaluate theintegrals (62), (63) and (66).

The values of iJRnjiJ(3 and so: jiJ" were obtained from differentiationof the series (67) and evaluating it at x = 1. The terms of the differ-


entiated series grow very large so that only the first eight values ofaRja{3 and the first six values of au'ja'Y could be obtained. The remain-ing values of aRja{3 were calculated from the approximation"

(aR) _ (-1)" 11" (73)a{3 =1 - 62/3 I' m{3"1/3fJ-fJn

which is in good agreement with the values obtained by machine calcu-lation for larger values of n. An approximation of auj a-y can be obtainedby using approximations similar to the ones used for the R functionsin Ref. 8. One gets

(au')a'Y =1

(74)

However, this approximation is not very good for n ~ 15 so that weused the equation

(au')a'Y =1

1 6 A( 1)" 11"'Y '" p= - - 6!r (.I.) + /-J -; .

3 p=1 'Y(75)

The coefficients Ap were determined from the first six values of au'ja'Ywhich were obtained from a machine calculation. Their values are givenat the bottom of Table 1.

TABLE I

II ~ R,,' (I) CR(l») Un(l) ("u'(I»)~ P-Pn 'Yn iJ-y 'Y-'Yn

0 2.70436 -1.01430 -0.50090 0 11 6.67903 1.34924 0.37146 5.06750 -0.492517 0.978162 10.67338 -1.57232 -0.31826 9.15760 0.395509 -1.247203 14.6711 1. 74600 0.28648 13.1972 -0.345874 1.435224 18.6699 -1.89090 -0.26449 17.2202 0.314047 -1.584865 22.6691 2.01647 0.24799 21.2355 -0.291252 1.711276 26.6686 -2.12814 -0.23491 25.2465 0.273806 -1.821647 30.6682 2.22038 0.22485 29.2549 -0.259853 1.920428 34.6679 -2.32214 -0.21548 33.2615 0.248332 -2.010379 38.6676 2.40274 0.20779 37.2669 -0.238591 2.09330

10 42.6667 -2.48992 -0.20108 41.2714 0.230199 -2.1704511 46.6667 2.56223 0.19516 45.2752 -0.222863 2.2427512 50.6667 -2.64962 -0.18988 49.2785 0.216371 -2.3108813 54.6667 2.70216 0.18513 53.2813 -0.210569 2.3753914 58.6667 -2.76421 -0.18083 57.2837 0.205216 -2.43671

.ttl = -4.881355A. = 1.536461 102

.11 3 = -2.838383 103

.11 4 = 2.838701 104

.1 5 = -1.420240105

.11 6 = 2.728875105


An approximate formula for f3 iss

The f3 values of Table I for n E;; 11 have been computed from (76).The lowest order U function is a constant

Us = 1 with 'Yo = O.

The 'Y values should converge to

'Yn = 4n +!.

(76)

(77)

(78)

This expression can be derived by methods analogous to those used inRef. 8. For unknown reasons this approximation is much poorer thanthat for f3n • However, it appears from the values of Table I that 'Y..will converge to (78) for very high values of n.

REFERENCES

1. Goubau, G., and Schwering, F., On the Guided Propagation of Electromag-netic Wave Beams, IRE Trans. AP-9, May, 1961, pp. 248-256.

2. Berreman, D. W., A Lens or Light Guide Using Convectively Distorted Ther-mal Gradients in Gases, B.S.T.J., 48, July, 1964, pp. 1469-1475.

3. Marcuse, D., and Miller, S. E., Analysis of a Tubular Gas Lens, B.S.T.J. 48,July, 1964, PP. 1759-1782.

4. Mareuse, D., Theory of a Thermal Gradient Gas Lens. Paper delivered at the1965 IEEE-G-MTT Symposium, to be published in the IEEE Trans-G-MTT,Nov., 1965.

5. Miller, S. E., Alternating-Gradient Focusing and Related Properties of Con-vergent Lens Focusing, B.S.T.J., 48, July, 1964, pp. 1741-1758.

6. Born, M., and Wolff, E., Principles of Optics, Pergamon Press, New York,1959, p. 121 (Equation (2».

7. Jakob, M., Heat Transfer, 1, John Wiley, New York, 1949, pp. 451-464.8. Sellers, J. R., Tribus, M., and Klein, J. S., Heat Transfer to Laminar Flow

in a Round Tube or Flat Conduit - The Graetz Problem Extended, Trans.Am. Soc. Mec. Eng., 78, 1956, pp. 441-448.

properties of periodic gas lenses

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