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A comparative study of rectangular and circular dielectric-coated metallic waveguides for CO2 laser light: Theory
Volume 68. number 1
OPTICS COMMUNICATIONS
1 September 1988
A COMPARATIVE STUDY OF RECTANGULAR AND CIRCULAR DIELECTRIC-COATED METALLIC WAVEGUIDES FOR CO2 LASER LIGHT: THEORY Mitsunobu MIYAGI and Satoru KARASAWA Department of Electrical Communications, Faculty qf Engineering. Tohoku University. Sendai 980, Japan Received 16 February 1988; revised manuscript received 16 May
A theoretical study has been conductedto explore the possibility of realizing flexible rectangular waveguideswith low losses. It is shown that rectangular waveguides with low losses whose width and height are nearly the same can be realized by coating dielectric layers on the inside walls.
1. Introduction There are many industrial and medical applications for high-powered CO2 lasers [ 1,2]. Although much research has been conducted to develop flexible low-loss transmission media, only a few kinds of solid fibers or hollow-core waveguides are presently available for practical use. Hollow waveguides seem promising for high power applications [3-5 ]. A rectangular hollow metallic waveguide proposed by Garmire et al. [ 1 ] has a low transmission loss and low bending loss when the guide is bent so that the electric field of the guided mode is perpendicular to the plane of curvature. However, due to the flat shape of the cross section of the waveguide, it cannot be bent arbitrarily, which makes the waveguide inconvenient for practical use. A dielectric-coated metallic waveguide [ 6] seems to overcome the above inconvenience while preserving low transmission losses. In order to reduce bending losses, the waveguide should be designed so that the straight waveguide loss becomes small [ 7 ]. This means that the dielectric coating as well as the metallic material should be chosen properly, or multiple layers of dielectrics should be used. Therefore, further extensive work is required for the fabrication process compared with that conducted in fabricating waveguides with an inner dielectric layer [7]. In this communication, we propose a simple dielectric-coated rectangular waveguide whose height and width are almost identical in size. The wave18
guide is composed of two pairs of metal strips, one of which is coated with a thin dielectric layer. Similar structures have already been described in the literature [8,9 ]. However, in the present proposal, only one pair of strips is coated compared with the waveguide in ref. [8], and the height and width of the waveguide are almost identical in contrast to ref. [ 9 ]. The losses of circular and rectangular metallic waveguides with dielectric layers are compared.
2. Attenuation in straight waveguide We first compare the straight waveguide loss of the HE~ mode in the circular waveguide with radius T and that of the E]'I mode in the rectangular waveguide with T * × T* (see fig. 1 ) coated by a dielectric layer so as to satisfy the minimum loss conditions [ 10]. Let the refractive indices of the hollow core and the dielectric coating be no ( --- 1 ) and a~no, and the complex refractive index of the metal be n o ( n - i x ) . When the thickness d of the dielectric layer is designed so as to satisfy the minimum loss condition [6 ] (a~-l)l/2nokod=tan-J[al/(a~-l)l/4
],
(1)
the attenuation constant c~c of the HE~ mode is expressed by [6 ] U2 oi~=lnoko ( n o k o T )
( ( a 2 ~ i ) 1 / 2 a 2 )2 14 nn2-~_ x 2~
(2)
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
Volume 68, number 1
OPTICS COMMUNICATIONS
1 September 1988
I
2T~_/-~T
f
o,s - ~ - ~ ~ T % T
Dielectric layer
Dielectric layer
(a)
o
(b)
i
i
2
3 1211
Fig. 1. Dielectric-coated metallic waveguide with circular (a) or rectangular (b) cross section.
where/Co is the wavenumber in v a c u u m and Uc is the first zero of Jo(x) = 0 , i.e., 2.405. On the other hand, when the thickness d of the dielectric layer is designed so as to satisfy the m i n i m u m loss condition [10]
( a~ - 1 ) ~/2nokod= n / 2 ,
(3)
the attenuation constant ogr of the E y, mode is expressed by summing the attenuation constants in the corresponding slab waveguides [ 1,10] as follows u~
(
a4 ~
n
oq=noko (nokoT.) 3 1 + a2-7-1~_1j n2+K 2 ,
(4)
where ur is equal to =/2. The transmission loss becomes 0.063 d B / m for a germanium-coated nickel waveguide with T * = 0 . 5 m m at 2 = 10.6 ~tm, where we have used a ~ = 4 . 0 and n - i x = 9 . 1 - i 3 4 . 4 . Let's compare the losses o f the two waveguides. From eqs. (2) and (4), we have 2
3
4
9
a~ (ur~ ( T'] l+a,/(ay-1) ~cc=\-~c] \--~.] [1.t..a2/(a~_l)l/212.
(5)
Fig. 2 shows a J a c as functions ofa~ for T*=Tand 2T*=x/~ T corresponding to the same cross sectional area of both waveguides. It is seen that ar is always smaller than ac.
Fig. 2. O~r/OQ a s functions ofa~. edge guidance condition is satisfied i.e., the power o f the guided mode is confined near the outer edge of the waveguide and the bending loss is proportional to R -~ [7]. By noticing that the loss in bent rectangular waveguide is obtained by summing the losses in the corresponding slab waveguides, we can express the attenuation constants of the E[~ mode, when the bending plane is parallel to the (x, z) or (y, z) plane, as follows
c¢r= [n/(n2+x2)] ( l / R ) 1
(x, z) plane,
× aa/(a~-l)
(6)
(y,z) plane.
On the other hand, for the circular dielectric-metallic waveguide, one can express ac as follows Olc - n- 2 -+ x 2 R
1 + ( a ~ - l) w2 l + c
[l + [ a l / ( a ~ - l )l/2]c, ×(c+a~/(a21-1) ~/2 ,
(x,z) plane, (y,z) plane,
(7)
where c is measure of the coupling effect between transverse and perpendicular field components to the metallic wall and is given by
c=l/(2nokoT~/R-1)-l/(n,fX-1),
(8)
and X is defined as 3. Attenuation in sharply bent waveguide
We consider the case where the waveguide is bent sharply with uniform curvature R so that the so-called
X = ( 2 / n ) 2 (nokoT)3/nokoR.
(9)
Detailed derivations of eqs. (7) and (8) will be reported elsewhere [ 11 ]. Fig. 3 shows oq/ac as functions o f X. One can see 19
Volume 68, number 1
OPTICS COMMUNICATIONS
R (m) 5 2 ,,,
,
I ,
0.5 ,
,
,
,
0.3
,
1 .8
- ~
)
2.4
~ ( y , z)ptone
1.28
/~
1.28
.2
(x,z)plane L
0
i
2'0
i
4 i
40
6~3 X Fig. 3. ~,/~c of bent waveguide as functions of X. The upper scale corresponds to the bending radius in the case of 2= 10.6 lam and T= 500 lam. t h a t the b e n d i n g loss o f a r e c t a n g u l a r w a v e g u i d e can be r e d u c e d by c o a t i n g the i n n e r m e t a l l i c surface.
4. C o n c l u s i o n
Circular and rectangular metallic waveguides with
20
an i n n e r dielectric layer h a v e b e e n c o m p a r e d . It is s h o w n that low-loss r e c t a n g u l a r w a v e g u i d e s can be a c h i e v e d w h o s e w i d t h a n d h e i g h t are a l m o s t o f the s a m e size. T h e r e f o r e , the r e c t a n g u l a r w a v e g u i d e s can be b e n t in a r b i t r a r y d i r e c t i o n s while p r e s e r v i n g low t r a n s m i s s i o n losses.
References
.z,
0
1 September 1988
[ 1] E. Garmire, T. McMahon and M. Bass, IEEE J. Quantum Electron. QE-16 (1980) 23. [2 ] D.A. Pinnow, A.L. Gentile, A.G. Standlee, A.J. Timper and L.M. Hobrock, Appl. Phys. Lett. 33 (1978) 28. [3] H. Ishiwatari, M. Ikedo and F. Tateishi, IEEE J. Lightwave Technol. LT-4(1986) 1273. [4] U. Kubo, Y. Hashishin and M. Nakatsuka, 4th Congress of International Society for Laser Surgery, 1981, Tokyo, Japan. [51 D. Gal and A. Katzir, IEEE J. Quantum Electron. QE-23 (1987) 1827. [6] M. Miyagi and S. Kawakami, IEEE J. Lightwave Technol. LT-2 (1984) 116. [ 7 ] M. Miyagi, K. Harada, Y. Aizawa and S. Kawakami, SPIE 484 (1984) 117. [ 8 ] M.B. Levy and K.D. Laakmann, SPIE 605 (1986) 57. [9] J. Gombert and M. Gazard, Optics Comm. 58 ( 1986 ) 307. [ 10] M. Miyagi, A. Hongo and S. Kawakami, IEEE J. Quantum Electron. QE-19 (1983) 136. [ 11 ] S. Karasawa and M. Miyagi, unpublished.
OPTICS COMMUNICATIONS
1 September 1988
A COMPARATIVE STUDY OF RECTANGULAR AND CIRCULAR DIELECTRIC-COATED METALLIC WAVEGUIDES FOR CO2 LASER LIGHT: THEORY Mitsunobu MIYAGI and Satoru KARASAWA Department of Electrical Communications, Faculty qf Engineering. Tohoku University. Sendai 980, Japan Received 16 February 1988; revised manuscript received 16 May
A theoretical study has been conductedto explore the possibility of realizing flexible rectangular waveguideswith low losses. It is shown that rectangular waveguides with low losses whose width and height are nearly the same can be realized by coating dielectric layers on the inside walls.
1. Introduction There are many industrial and medical applications for high-powered CO2 lasers [ 1,2]. Although much research has been conducted to develop flexible low-loss transmission media, only a few kinds of solid fibers or hollow-core waveguides are presently available for practical use. Hollow waveguides seem promising for high power applications [3-5 ]. A rectangular hollow metallic waveguide proposed by Garmire et al. [ 1 ] has a low transmission loss and low bending loss when the guide is bent so that the electric field of the guided mode is perpendicular to the plane of curvature. However, due to the flat shape of the cross section of the waveguide, it cannot be bent arbitrarily, which makes the waveguide inconvenient for practical use. A dielectric-coated metallic waveguide [ 6] seems to overcome the above inconvenience while preserving low transmission losses. In order to reduce bending losses, the waveguide should be designed so that the straight waveguide loss becomes small [ 7 ]. This means that the dielectric coating as well as the metallic material should be chosen properly, or multiple layers of dielectrics should be used. Therefore, further extensive work is required for the fabrication process compared with that conducted in fabricating waveguides with an inner dielectric layer [7]. In this communication, we propose a simple dielectric-coated rectangular waveguide whose height and width are almost identical in size. The wave18
guide is composed of two pairs of metal strips, one of which is coated with a thin dielectric layer. Similar structures have already been described in the literature [8,9 ]. However, in the present proposal, only one pair of strips is coated compared with the waveguide in ref. [8], and the height and width of the waveguide are almost identical in contrast to ref. [ 9 ]. The losses of circular and rectangular metallic waveguides with dielectric layers are compared.
2. Attenuation in straight waveguide We first compare the straight waveguide loss of the HE~ mode in the circular waveguide with radius T and that of the E]'I mode in the rectangular waveguide with T * × T* (see fig. 1 ) coated by a dielectric layer so as to satisfy the minimum loss conditions [ 10]. Let the refractive indices of the hollow core and the dielectric coating be no ( --- 1 ) and a~no, and the complex refractive index of the metal be n o ( n - i x ) . When the thickness d of the dielectric layer is designed so as to satisfy the minimum loss condition [6 ] (a~-l)l/2nokod=tan-J[al/(a~-l)l/4
],
(1)
the attenuation constant c~c of the HE~ mode is expressed by [6 ] U2 oi~=lnoko ( n o k o T )
( ( a 2 ~ i ) 1 / 2 a 2 )2 14 nn2-~_ x 2~
(2)
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
Volume 68, number 1
OPTICS COMMUNICATIONS
1 September 1988
I
2T~_/-~T
f
o,s - ~ - ~ ~ T % T
Dielectric layer
Dielectric layer
(a)
o
(b)
i
i
2
3 1211
Fig. 1. Dielectric-coated metallic waveguide with circular (a) or rectangular (b) cross section.
where/Co is the wavenumber in v a c u u m and Uc is the first zero of Jo(x) = 0 , i.e., 2.405. On the other hand, when the thickness d of the dielectric layer is designed so as to satisfy the m i n i m u m loss condition [10]
( a~ - 1 ) ~/2nokod= n / 2 ,
(3)
the attenuation constant ogr of the E y, mode is expressed by summing the attenuation constants in the corresponding slab waveguides [ 1,10] as follows u~
(
a4 ~
n
oq=noko (nokoT.) 3 1 + a2-7-1~_1j n2+K 2 ,
(4)
where ur is equal to =/2. The transmission loss becomes 0.063 d B / m for a germanium-coated nickel waveguide with T * = 0 . 5 m m at 2 = 10.6 ~tm, where we have used a ~ = 4 . 0 and n - i x = 9 . 1 - i 3 4 . 4 . Let's compare the losses o f the two waveguides. From eqs. (2) and (4), we have 2
3
4
9
a~ (ur~ ( T'] l+a,/(ay-1) ~cc=\-~c] \--~.] [1.t..a2/(a~_l)l/212.
(5)
Fig. 2 shows a J a c as functions ofa~ for T*=Tand 2T*=x/~ T corresponding to the same cross sectional area of both waveguides. It is seen that ar is always smaller than ac.
Fig. 2. O~r/OQ a s functions ofa~. edge guidance condition is satisfied i.e., the power o f the guided mode is confined near the outer edge of the waveguide and the bending loss is proportional to R -~ [7]. By noticing that the loss in bent rectangular waveguide is obtained by summing the losses in the corresponding slab waveguides, we can express the attenuation constants of the E[~ mode, when the bending plane is parallel to the (x, z) or (y, z) plane, as follows
c¢r= [n/(n2+x2)] ( l / R ) 1
(x, z) plane,
× aa/(a~-l)
(6)
(y,z) plane.
On the other hand, for the circular dielectric-metallic waveguide, one can express ac as follows Olc - n- 2 -+ x 2 R
1 + ( a ~ - l) w2 l + c
[l + [ a l / ( a ~ - l )l/2]c, ×(c+a~/(a21-1) ~/2 ,
(x,z) plane, (y,z) plane,
(7)
where c is measure of the coupling effect between transverse and perpendicular field components to the metallic wall and is given by
c=l/(2nokoT~/R-1)-l/(n,fX-1),
(8)
and X is defined as 3. Attenuation in sharply bent waveguide
We consider the case where the waveguide is bent sharply with uniform curvature R so that the so-called
X = ( 2 / n ) 2 (nokoT)3/nokoR.
(9)
Detailed derivations of eqs. (7) and (8) will be reported elsewhere [ 11 ]. Fig. 3 shows oq/ac as functions o f X. One can see 19
Volume 68, number 1
OPTICS COMMUNICATIONS
R (m) 5 2 ,,,
,
I ,
0.5 ,
,
,
,
0.3
,
1 .8
- ~
)
2.4
~ ( y , z)ptone
1.28
/~
1.28
.2
(x,z)plane L
0
i
2'0
i
4 i
40
6~3 X Fig. 3. ~,/~c of bent waveguide as functions of X. The upper scale corresponds to the bending radius in the case of 2= 10.6 lam and T= 500 lam. t h a t the b e n d i n g loss o f a r e c t a n g u l a r w a v e g u i d e can be r e d u c e d by c o a t i n g the i n n e r m e t a l l i c surface.
4. C o n c l u s i o n
Circular and rectangular metallic waveguides with
20
an i n n e r dielectric layer h a v e b e e n c o m p a r e d . It is s h o w n that low-loss r e c t a n g u l a r w a v e g u i d e s can be a c h i e v e d w h o s e w i d t h a n d h e i g h t are a l m o s t o f the s a m e size. T h e r e f o r e , the r e c t a n g u l a r w a v e g u i d e s can be b e n t in a r b i t r a r y d i r e c t i o n s while p r e s e r v i n g low t r a n s m i s s i o n losses.
References
.z,
0
1 September 1988
[ 1] E. Garmire, T. McMahon and M. Bass, IEEE J. Quantum Electron. QE-16 (1980) 23. [2 ] D.A. Pinnow, A.L. Gentile, A.G. Standlee, A.J. Timper and L.M. Hobrock, Appl. Phys. Lett. 33 (1978) 28. [3] H. Ishiwatari, M. Ikedo and F. Tateishi, IEEE J. Lightwave Technol. LT-4(1986) 1273. [4] U. Kubo, Y. Hashishin and M. Nakatsuka, 4th Congress of International Society for Laser Surgery, 1981, Tokyo, Japan. [51 D. Gal and A. Katzir, IEEE J. Quantum Electron. QE-23 (1987) 1827. [6] M. Miyagi and S. Kawakami, IEEE J. Lightwave Technol. LT-2 (1984) 116. [ 7 ] M. Miyagi, K. Harada, Y. Aizawa and S. Kawakami, SPIE 484 (1984) 117. [ 8 ] M.B. Levy and K.D. Laakmann, SPIE 605 (1986) 57. [9] J. Gombert and M. Gazard, Optics Comm. 58 ( 1986 ) 307. [ 10] M. Miyagi, A. Hongo and S. Kawakami, IEEE J. Quantum Electron. QE-19 (1983) 136. [ 11 ] S. Karasawa and M. Miyagi, unpublished.