Closed formulae for the wave number of Sommerfeld's wire line with a large radius

Accepted Manuscript Closed formulae for the wave number of Sommerfeld's wire line with a large radius Denis Jaisson PII: DOI: Reference:

S1350-4495(16)30667-3 http://dx.doi.org/10.1016/j.infrared.2017.02.013 INFPHY 2244

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Infrared Physics & Technology

Received Date: Revised Date: Accepted Date:

24 November 2016 23 February 2017 23 February 2017

Please cite this article as: D. Jaisson, Closed formulae for the wave number of Sommerfeld's wire line with a large radius, Infrared Physics & Technology (2017), doi: http://dx.doi.org/10.1016/j.infrared.2017.02.013

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Closed formulae for the wave number of Sommerfeld's wire line with a large radius Dr. Denis Jaisson independent consultant [email protected]

Abstract Simple closed formulae are derived for the number of a harmonic wave traveling along a straight conducting wire, whose diameter may be larger than that the wavelength in vacuum. Those formulae are validated against a numerical method which solves the propagation equation exactly. Two examples show that the frequency-range of application of the proposed formulae extends into the infrared.

Keywords surface waves, transmission lines, Sommerfeld wire, Goubau line, continued fractions, Bessel functions, THz, infrared, Drude's law

Fig. 1. Sommerfeld's wire line.

1 Introduction The transmission line of Sommerfeld [1], also known as Goubau line [2], is sketched in Fig.1. It is a straight conducting round wire with radius a, which stretches through vacuum along the z-axis. Owing to its low loss and low dispersion, it is a favourite waveguide for operation at terahertz (THz) frequencies [3]. It spares one from using some of the bulky optical parts that are customary in THz systems, while lending itself to the propagation of subpicosecond pulses [4]. Let k 0 and γ z be the respective wave numbers of vacuum and of that wire's fundamental TM 0 mode of propagation, at frequency f. Until recently one has used an algorithm [5,6] to solve the equation for γ z , except when k0 a  1 [7]. That equation involves Bessel functions of a complex variable, whose computation is difficult and time-consuming. The closed formulae for γ z , that Jaisson published last year [8] are for the case where k 0 a equals at most one wavelength. That ballpark limit was set by the approximations that Jaisson applied to the Bessel functions. In many cases, it is not restrictive, from the user's point of view. But when f  1 THz , going into the infrared range [9], the requirement for mechanical strength requires that the wire be electrically thicker, that is k0 a  π . For lack of more general approximations for the said Bessel functions, one must treat the case of the so-called thick wire separately, as is done in this paper. In Section 2, one approximates the propagation equation of the TM0 mode, to a cubic equation. The latter is easily solved, for it involves no Bessel function. One determines the frequency-range where the formulae thus obtained for γ z are valid, in Section 3. In Section 4, one compares the values obtained for γ z by means of those formulae, to the solution of the exact propagation equation, in two examples.

2 Approximate propagation equation In this Section one recalls the propagation equation of the TM 0 mode of Sommerfeld's wire line. Then one approximates the Bessel functions involved, whereby one obtains an equation which is solvable by hand. Let   2 π f , and let factor exp( j t ) express the harmonic fields' dependence on time t, as a convention ( j  j  1) . One considers a wire made out of a non-magnetic metal with conductivity  and relative permittivity [9,10]

 r  1   / j  0

(1)

where  0 is the permittivity of vacuum and where |  |    0

(2)

for f  100 THz , and even beyond depending on the wire's metal. Conductivity  varies with  according to Drude's law [9,10]:

   0 / (1 j )

(3)

where  is the mean time between two collisions of conducting electrons. Maier and Ashcroft have a minus sign in front of term j   in their formula for Drude's law, by the way, because they used factor exp(  j t ) in their convention for timedependence. Define radial number γ ρ inside metal and vacuum respectively: 2 2 1/2   γ m  (  r k0  γ z ) ,   a γρ   2 2 1/2   γ v  ( k0  γ z ) ,   a

(4)

with Im(γ v )  0 . k 02   2  0  0 , where  0 is the permeability of vacuum. Define u  γ m a and v  j γ v a with Re(v)  0 . The propagation equation of the fundamental TM0 mode is [1,2]

v 

u J 0 (u ) K1 (v)  r J1 (u ) K 0 (v)

(5)

where K n is the modified Bessel function of the second kind and order n, and where J n is the Bessel function of the first kind and order n [11]. According to (1) and (4), γ 2m  γ v2  γ a2 with γ a2   k 02 (1   r )

(6)

which defines γ a2 . Equation (6) makes it clear that computing γ z amounts to solving (5) for v and getting γ z from (4). Solving (5) numerically reveals that

| v |  | γ a a | , k 0 a

(7)

as long as (2) holds. Therefore, according to (2), (6) and (7), one has γ m  γ a   j k 0 (1   r )

1/2

.

(8)

As far as equation (5) is concerned, which root of γ a2 is used in (8) matters not, since factor u  J 0 / J1 is an even function of u . According to (1), (2) and (8), | γ m a |  1 when k 0 a  π . Therefore J 0 / J1  j , if one picks the root of γ a2 that lies in the fourth quadrant. Equation (5) thus simplifies to

v  K 0 (v) / K1 (v)   j γ a a /  r .

(9)

where one has accounted for γ m  γ a (8). While striving to simplify ratio K 0 /K1 in (9) the author looked at the published approximations [11] that Cao and Jahns used to solve exact (5) numerically. Those approximations involve polynoms whose degree, either is too large to allow for (9) to be solved by hand, or imposes too large a lower limit on | v | and therefore on

k 0 a . Below, one derives a wide-range approximation for K 0 /K1 , from limit values [11]

  v  ln(v) , | v |  1 K 0 / K1   .  1 , | v |  1

(10)

First, one bridges between those two limits, by writing

K 0 / K1  v  ln(1  1 / v ) ;

(11)

equation (11) is exact for | v |  0  and  respectively. As one will see in the examples, | v | is large in part of the spectrum of interest, by the way. That prevents one from solving (9) as modified by (11), by using Lambert's function [12] like Jenn [7] and Jaisson [8] did for thinner wires. Continued fraction [11] 1/ v 1/ v 1/ v  1 ln1     v 1 2  3   

(12)

provides a means to approximate the logarithm in (11), that imposes no upper limit on | v | , while the lower limit is small enough for the present purpose, as one will see. In equation (9) one uses combined (11) and (12) while truncating the continued fraction to order 3. (9) thus becomes a much simpler cubic equation:

v'  (Z  1 / 4) v'  3 Z v' / 4  Z / 24  0 3

2

(13)

where unknown v' differ from exact v by v  v' v , and where one defines Z  k 0 a (1   r )

1/2

/r .

(14)

One will show in the examples the error that is incurred by χ (v)  v  K 0 (v) / K1 (v) when one approximates  by means of (11,12). Note, while reflecting on previous work, that Jaisson could not have used combined (11) and (12) to treat the case of the thinner wires [8], because the validity limit of the variable of the truncated fraction from (12) is finite. On the other hand, the upper limit of the validity range of (11,12) is infinite, contrary to that of Jaisson's previous formulae. Note also that there existed a continued fraction for K1 / K 0 [13] before one used the one from (12) to approximate K 0 /K1 . Yet the former fraction leads to less accurate results, as one must truncate it to order 2, not 3 as in (12), to obtain a cubic equation like (13). Solution v' of equation (13) has 3 branches v'  v1 and v'  v given by [14]

v1  s  s  ( Z  1 / 4 ) / 3

(15)

and

v    ( s   s  ) / 2  ( Z  1 / 4) / 3

(16)

 j 3 (s  s ) / 2 where one defines

s  ( r  ( q 3  r 2 )

1/2 1/3

)

r  Z ( Z  1 / 4) / 8  Z / 48  ( Z  1 / 4) 3 / 27 .

(17)

q  Z / 4  ( Z  1 / 4) / 9 2

One finds out which branch satisfies condition Re(v)  0 , by looking at v1 and v  when f goes to zero. That is valid mathematically, although (13) was derived with a non-zero limit for f in mind, by the way. According to (1), (2), (3) and (14),

Z  k 0 a (  0 /  0 )

v   ( 0 a / 6)

1/2

(  0 /  0 ) 3

1/2

1/4

e

e

j3π / 4

j7 π / 8

when

f  0 .

Equations

(15)

and

(16)

then

become

  v1 and v   1 / 4 , respectively. The only branch that satisfies Re(v)  0 is v   v1 .

Therefore γ z is given in general by (4), as

γ z  (  v 2 / a 2  k 02 )

1/2

(18)

with v  v1 . One picks the root of γ 2z for which   Re( γ z )  0 and   Im( γ z )  0 , if the wave is travelling in the direction of increasing z. Loss factor  is the most critical characteristic of Sommerfeld's wire line. γ z and  incur respective errors γ z and

  Re(γ z ) , when one approximates equation (5) to (13). Let e  | Re(γ z ) /  | be the relative error of  , and let espec be its specified maximum. The frequency-range f min  f  f max where e  espec is determined in the next section.

3 Frequency-range of validity One expresses γ z in terms of v  v' v , by taking differential γ z / v at v  v  in (18), and by approximating γ z / v to γ z / v :

γ z  v'  v / k 0 a 2

(19)

where one has approximated γ z to k 0 according to (7) and (18). Determining

f min and f max is about deriving simple

expressions for v' , v and  , which facilitate the reverse calculation of f when e  espec . Simplicity will stem from those expressions being true only when f is close to respective f min and f max . The basis for the derivation of f min will be equation (9), not (5), because (9) is simpler, and because approximations (8) and J 0 / J1  j contribute to error e when f  f min , a lot less than (11) and (12) do. Recall that χ (v)  v  K 0 (v) / K1 (v) , and define χ' (v' )  v'  K 0 (v' ) / K1 (v' ) as the approximate value that  takes when one uses (11) and the truncated continued fraction from (12), in equation (9). The latter then becomes

χ' (v' )   j γ m a /  r ;

(20)

equations (13) and (20) are equivalent. In order to express v in terms of χ  χ' χ , one extrapolates χ'  χ  χ from , according to χ' (v' )  χ' (v' )  χ( v  v)  χ( v' ) . (21)  χ' (v' )  χ( v)  v   χ /  v Using (9) and (21) in (20), one obtains v 

χ (  χ /  v ) v'

(22)

where evaluating [11]

χ     v'   v  v'

 K0   K1

2

 K   2  0  v'  K1

   v'  v'

(23)

at v' instead of v brings about a negligible error of the order of (v) 2 . Equations (19) and (22) yield

 γ e  Re  z   

 v'  / k  0   Re    (  χ /  v )   v' 

   . 

(24)

One assumes that | v' |  1 when f  f min , and that | v' |  1 by the time f reaches f max ; that assumption will be justified by the examples. Define min  2 π f min and  0  (  0 a ) 2/3 (  0 /  0 )1/3 . According to equations (1), (3), (14), (15) and (16), solution v'  v1 (15) of (13) simplifies to

v'  v1 e

jπ /8

( 1  j min  ) ( min  0 ) 3/4 / 6

(25)

when    min . Equations (1), (3), (7), (14), (18) and (25) then yield

  Re ( j k 0  j v' 2 / 2 k 0 a 2 )  ( min  0 /  0 )1/2 ( 1  ( min  ) 2 )1/4 cos(  ) / 12 a

(26)

Fig. 2. e (  ) and  (x) () ,  m ( x) (  ) for e  10% where one defines   π / 4  tan 1 (  ) / 2 . letting espec  10 % for engineering purposes, one computes   min  0 in terms of   min  numerically, using χ  χ'  v'  K 0 / K1 from (11) and (12), (  χ /  v ) v' from (23), v' from (25), and  from (26), in e  espec (24). One thus obtains function  ( ) shown in Fig.2 over interval 0    20 . That interval was deemed wide enough for all practical purposes. Contrary to Ω and  , the relationship  ( ) is independent of wire's parameters a ,  0 and  . In order to evaluate  min by hand, one devises a simple mathematical model for  ( ) :

 m ( )   0 / (1   /  0 )1/2 ;

(27)

the curve of  m ( ) fits that of  ( ) for  0  0.165 and  0  2 .25 . It is shown in Fig.2 where one can see that

e  | 1   m /  | , the relative error of  m , is less than 6 %. Letting min  0   m (min  ) and raising both sides of (27) to the square, one obtains a cubic equation for  min :

min   0 min /    0  0 /  0   0 . 3

2

2

2

(28)

Define 3   0 / 3 , 03  33   0  0 / 2 02 ,  6  06  36 , 3  cos 1 ( 03 / 33 ) and   ( 03  3 )1 / 3 . The three 2

solutions of (28) are all real numbers when  6  0 . Only one of them is real when  6  0 [14]. The general formula for the real solutions of (28) is

 min  2 π f min

   3  n π   6    1 ,    0   3  2  cos    3       6         3 ,    0

(29)

with n  0 or  1 . One lets n  0 when 06  36 , upon noticing that min   0 /  0 in the limit   0  , according to (28). As far as lone combined (11) and (12) are concerned, v has no upper limit of validity. Yet f max is finite, for there comes a point, while f increases far away from f ´min , where approximation γ m  γ a (8) is too coarse, as equation (3) does suggest; bear in mind that | u , v |  1 then. On the one hand, equation (9) looses accuracy, as it is based on (8). On the other hand, (9) and (13) become equivalent, as (10) and combined (11) and (12) then say the same thing, that is K 0 / K1  1 , even if the continued fraction from (12) is truncated. In other words,

v'   k 0 a ( 1   r )1/2 /  r

(30)

( Re(v' )  0 ) is the solution of (13) as much as of (9) when f  f max . To sum things up, e reaches specified maximum espec while the discrepancy between v' from (30) and solution v of exact equation (5) keeps growing.

In order to compute v at f max , one first rewrites (5) for a general f : 2

v  2



γa a

2

1   r  J 0 (u )  K1 (v) / J1 (u )  K 0 (v )



.

(31)

2

Then one accounts for (6) and for J 0 / J1  j and K 0 / K1  1 , to simplify (31) to

v  k 0 a / (  1   r )1/2

(32)

when f  f max ; let condition Re(v)  0 and equations (1) and (3) determine the sign of v. Incidentally, equations (4) and (32) yield

γ z   j k 0 / (1  1 /  r )1/2

(33)

which indicates that γ z is independent of wire's radius a when f  f max . γ z then equals the wave number of a transversemagnetic wave guided by the interface of air and of a semi-infinite plane conductor with same relative permittivity  r as the wire's metal [10]. Using γ z from (19),   Re( γ z ) from (18) and (32), v' from (30), and using v  v  v from (30) and (32), in e  | Re(γ z ) /  | , one obtains:

Im (1  1 /  r )1/2 /  r    r (1   r )  - 1/2    e  . / Im (1  1 /  r ) - 1/2   

(34)

For a given metal, one computes f max numerically, once and for all, by letting e  espec in (34) where  r is given by (1) and (3).

4 Examples In this section one compares the values obtained for γ z by means, respectively, of approximate (15) and (18), and of exact propagation equation (5), in two examples. Orfanidis solved (5) in the lower THz-range, by running an algorithm which kept replacing v in the right-hand side of (5), with the value from the previous iteration [5]. That algorithm fails to converge when the real part of  r is not much smaller than the imaginary part, as is the case in the infrared. The author of the present paper attained convergence in that range too, by letting Orfanidis' algorithm run (31) instead of (5), while making sure that condition Re(v)  0 were satisfied. The reason why (31) works better than (5) has been explained by Kekatpure et al [15]. It falls outside the scope of the present paper, since the latter is about avoiding using an algorithm to compute γ z .

4.1 Silver wire Consider a wire made out of silver (Ag) with 2 a  200 μm , at a 77°K temperature where  0  330 10 6 S/m and   0.2 ps [9] and where (34) predicts that f max  884 THz . Fig.3 shows the exact and approximate values of  , derived from respective exact v (31) and approximate v'  v1 (15). It shows also the value that  would take if one used Jaisson's formulae for thin wires. Fig.4 shows that the curves of e drawn on the basis of, respectively, (15) and Jaisson's earlier formulae, conveniently cross at e  espec  10 % . It also shows how e as derived from v'  v1 (15) correlates with error

e  | χ' / χ | of χ' . Equation (29) with  6  0 and n  0 predicts that f min  7.86 THz and   min   9.9 . In fact, e  4 % at f  f min , whereas e  10 % at f  4.99 THz where k 0 a  10 .5 . f min as predicted by (29) is conservative because, at that frequency, | v' |  0.17 , that is not as small a value as one assumed when (29) was derived. On the other hand, one confirms that e  10 % at f  f max as predicted by (34). Fig.5 shows frequency-dispersion    / k 0  1 of

  Im( γ z ) , as computed using (15), (30) and Jaisson's formulae, respectively.

Fig. 3.  from (5) (- - -), (15) (___) and Jaisson [8] (. . .).

Fig. 4. e  (- - -) , e from (15) (___) and from Jaisson [8] (. . .).

Fig. 5. Dispersion of  from (5) (- - -), (15) (___) and Jaisson [8] (. . .).

Fig. 6. k 0 a (. - .) , |  /   0 | (. . .) and | u | (- - -) , | v | (___) from (5).

In the course of the derivations that were performed in the previous sections, several assumptions were made about  , k 0 a , u  γ m a and v  j γ v a . For the sake of academic interest if nothing else, Fig.6 shows how well those assumptions hold, depending on the position of f with respect to f min and f max .

4.2 Beryllium wire Consider a wire made out of beryllium (Be) with 2 a  200 μm , at a 273°K temperature where  0  36 10 6 S/m and   5.1 fs [9], and where (34) predicts that f max  1820 THz . Note that  0 and  are, respectively, 10 and 40 times less than in silver at 77°K. By the look of Fig.7 to 10, the analysis of the simulation results is qualitatively the same as in the first example. Equation (29) with  6  0 predicts that f min  8.24 THz and   min   0.26 ; | v' |  0.17 at f  f min , again. In fact e  5 % at that frequency, whereas e  10 % at f  5.12 THz where k 0 a  10 .7 ; (29) is conservative in this example too. On the other hand, one confirms that e  10 % at f  f max ( | v' |  1900 ) . It makes sense to switch from v' (30) to v (32) to compute γ z , when f goes through f max , or so it seems. Yet  is large, then, as is illustrated by the above examples. Indeed, using either one of those wire lines over some distance d that requires the wire to be thick for the sake of mechanical strength, that is d  20 a  2 mm roughly, is pointless when f approaches f max . That is why the author, having in mind the applications for Sommerfeld's wire, listed by Wang and Mittleman [3,4], contented himself with Drude's law (3) although the latter brakes down when one goes from the range of the infrared to that of the visible light waves [10].

Fig. 7.  from (5) (- - -), (15) (___) and Jaisson [8] (. . .).

Fig. 8. e  (- - -) , e from (15) (___) and from Jaisson [8] (. . .).

Fig. 9. Dispersion of  from (5) (- - -), (15) (___) and Jaisson [8] (. . .).

Fig. 10. k 0 a (. - .) , |  /   0 | (. . .) and | u | (- - -) , | v | (___) from (5).

5 Conclusion Simple formulae have been derived for computing wave number γ z of Sommerfeld's wire line when the latter is electrically thick. As they involve none of the Bessel functions from the exact propagation equation, and they are not algothmic, one can write them in the equation block of a simulator of high-frequency or infrared systems, without linking to a mathematical solver such as MatLab. Their frequency-range of validity starts where formulae derived previously for thin wires loose validity. It stops where propagation is not practical because the attenuation is too strong. That is why f max has been left out of Table 1 which summarises the proposed formulae in the order of coding in a computer. Evaluating f min was as demanding as computing γ z , the main number of interest. It was worth the effort, though, as the formulae for f min are simple and reliable. TABLE 1 FORMULAE FOR COMPUTING γ z

 0  (  0 a ) 2/3 (  0 /  0 )1/3 ,  0  0 .165 ,  0  2 .25 3   0 / 3 , 03  33   0  0 / 2 02 ,  6  06  36 , 3  cos 1 ( 03 / 33 ) ,   ( 03  3 )1 / 3 2

6 ! 1   3 ( 2  cos(3 / 3 )  1) ,   0 f  f min   6 2π       3 ,   0 

  2 π f ,  r  1   0 / j  0 ( 1  j   ) , Z  k 0 a (1   r )

1/2

3 2 /  r , r  Z (Z  1 / 4) / 8  Z / 48  (Z  1 / 4) / 27 , q  Z / 4  (Z  1 / 4) / 9 ,

s  ( r  ( q 3  r 2 )

) , v  s  s  ( Z  1 / 4 ) / 3

1/2 1/3

γ z  (  v 2 / a 2  k 02 )

1/2

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