-1000 -500 0 500 1000 squared driving voltage, V2

Fig. 4 Micromirror displacement against driving voltage

Samples: 0 1 , 0 2 , A3, A 4

Dcrice clzuracteristics The micromirror displacement against driv- ing voltage IS shown in Fig 4. It is linear against the square of the driving voltage The signal displacement indicates the direction of the niirror (-. forward, t. backward relative to the LD facet), and total deflection of - l o p as the sum of forward and backward motion was obtained at <30V driving voltage.

squared driving voltage,V2

Fig. 5 Wavelength tunnbilrty againsr driving voltage

I,, = 20mA

Fig. 5 shows the wavelength tunability against the driving volt- age of the micromirror. The LD was operated at 20mA CW cur- rent, which was twice the threshold current. The L D lased with mutimode spectra and the lasing peak wavelength decreased with increasing external cavity length. A total tunable range of 20 nm was obtained from the total forward and backward motion of the micromirror. For any lasing spectrum, the wavelength increased continuously for -1 nm with increasing cavity length, and then jumped to the next spectrum. In general, the lasing wavelength h is given by h = 2nUm (m: integer), where n is the refractive index and L is the cavity length. From this equation, we estimated that the mirror can be controlled within 2nm precision, because the wavelength could be varied in 0.Olnm steps by changing the mir- ror displacement.

Conclusion; A tunable laser diode with an Ni micromirror was fabricated by Ni surface micromachining. The mirror acts as an external cavity of the LD, and 20nm wavelength tuning was attained. The mirror displacement precision was estimated to be within 2 nm based on the Wavelength variation.

0 IEE 1996 Electronics Letters Online No: 19960801 Y. Uenishi, K. Honma and S. Nagaoka (NTT ZnferdiscQlinary Research Laboratories, 3-9-11 Midori-cho, Musashino, Tokyo, 180 Japan)

24 April 1996

References

1 BROOKS. R E : ‘Micromechanical light modulators on silicon’, Opt. Eng., 1985. 24, (l), pp. 101-106

WOLFFENBUTTEL. R F., and MIDDELHOEK, s.: ‘Surface micromachined tunable interferometre array’, Sens. Actuators A, 1994. 43, pp. 17-23 LI\. L.Y. LEE. s.s, PETER, K S J , and wu. M.c.: ‘Three dimensional micro-Fresnel optical elements fabricated by micromachining technique‘. Electron. Lett., 1994, 30, ( 9 , pp. 448-449 ~ i h . L Y.. LEE. s s . PISTER. K.S J , and wu. M c : ‘Micromachined three dimensional micro-optics for integrated free-space optical system’, IEEE Photonics Technol. Lett., 1994, PTL-6, pp. 1445-1447

5 UENISHI. Y TSLGAI. M , and MEHREGANY, M.: ‘Hybrid-integrated micromirror of laser-diode fabricated by (1 10) silicon micromachining’. Electron. Lett., 1995, 31, (12), pp. 965-966

mechanical devices fabricated by anisotropic etching of (1 10) silicon’, J. Micromech. Microeng., 1995, 5 , pp. 305-3 12

2 ARATANI. K FRENCH, P J., SARRO, P.M., POENAR, D.,

3

4

6 LENISHI, Y., TSUGAI, M., and MEHREGANY, M.: ‘Micro-opt0

Analytical formula for calculating the coupling characteristics between parallel coplanar lines

K.-K.M. Cheng

Indexing terms: Coplarzar waveguide.s, Waveguide theory

For the first time, a CAD-oriented analytical formula for the evaluation of the quasi-static coupling characteristics between parallel coplanar lines is presented. The analysis is based on a sequence of conformal transformations, and the derived expressions show excellent accuracy compared to the results generated by a spectral domain approach.

Introduction: Owing to the increasing popularity of coplanar waveguides for the design of hybrid and monolithic microwave integrated circuits, the need for accurate characterisation of the structures has increased. Quasistatic solutions to the coplanar transmission lines have been reported using various approaches, such as the finite difference method [l] and the spectral domain technique [2]. The conformal mapping method (31 has also been used to obtain the characteristic parameters of single coplanar lines. In this Letter, a new and accurate closed-form formula for evaluating the coupling characteristics between parallel coplanar lines is proposed.

: d j : c . :

, .

D *----*

t+’ E r

(93711/

Fig. 1 Parallel coplanar line structure

Analysis: The structure to be analysed is shown in Fig. 1, where the two coplanar lines are separated by a ground plane of width 20. All conductors are assumed to be infinitely thin and perfectly conducting. It is assumed that the air-dielectric interfaces, where all the conductors are located, can be dealt with as though perfect magnetic walls are present in theni. The even- and odd-mode capacitances per unit length of the structure can thus be consid- ered as the sum of the capacitances in the upper region (air) and lower region (air and dielectric layer). The lower region capaci- tance is then evaluated by the approximate technique suggested in [3] as the sum of the free-space capacitance in the absence of the

1208 ELECTRONICS LETTERS 20th June 1996 Vol. 32 No. 13

dielectric (Fig. 2a and b) and the capacitance of the dielectric layer (Fig. 2c and 4, assumed to have permittivity (E?-1). Note that the structure in the t-domain shown in Fig. 2a is simply an asymmetri- cal coplanar line, and its electrical characteristics has been

0 a b c d 0 a b c d r---: z-plane ; I--

C d - 37 #

Fig. 2 Odd and even mode of coplanar lines

r 1 - I I I I

C I 1 1 I I I I I

P3 = (P1 + h ) / 2 (10) where E, is the dielectric constant of the lower-half region in a. Simple and accurate formulas are available [SI for the solution of eqns. 1 and 3-9.

Table 1: Comparison of even- and odd-mode characteristic imped- ance values for parallel coplanar lines, calculated by pro- posed method and a spectral domain approach (d - c) = (c - b) = (b - a ) = h, E, = 10

This Letter This Letter 75.4 74.0 65.2 64.3

0.2 72.3 71.2 67.3 66.2 0.5 70.1 69.0 69.2 68.1

Discussions: Table 1 shows the even- and odd-mode characteristic impedance values of parallel coplanar lines, evaluated by both the proposed method and a spectral domain approach [6]. It is observed that the discrepancies between the two sets of values are practically small (both <2%) for different a/d ratios. Hence, cou- pling coefficients [3] calculated from the two sets of impedance values are found to differ by <0.2dB.

0 IEE 1996 Electronics Letters Online No: 19960809

K.-K.M. Cheng (Departnzent of Electronic Engineering, The Chinese University of Hong Kong, Shutin, Hong Kong)

3 April 1996

described elsewhere [4]. For the configurations in the t-domain depicted in Fig. 2h and d, further transformations (Fig. 3) are required in order to obtain their line capacitance values. Referring to Fig. 3, the structure in a is first converted into the one shown in b by the following expressions:

(5)

Fig. 3 Capacilante evaluation h j conformal tran forination

References

where F($, k ) is the incomplete elliptic integral of the first kind, written in Jdcobi notation. A magnetic wall is then assumed to be present at the centre of the slot, and hence the structure in b can be considered as the sum of two capacitances, as depicted in e. As a result, the capacitance per unit length of the configuration shown in a is given by

ELECTRONICS LETTERS 20th June 7996 Vol. 32

1 HATSUDA, T.: ‘Computation of coplanar-type strip-linc characteristics by relaxation method and its application to microwave circuits’, ZEEE Trans. Microw. Theory Tech., 1975, MTT-23, (lo), pp. 795-802

2 BOIX, R.R., and HORNO, M.: ‘Modal quasistatic parameters for coplanar multiconductor structures in multilayered substrates with arbitrary transverse dielectric anisotropy’, IEE Proc. H , Microw. Antennas Propag., 1989, 136; (l), pp. 76-79

3 GHIONE, G., and NALDI. c.: ‘Coplanar waveguides for MMIC applications: effect of upper shielding, conductor backing, finite- extent ground planes, and line-to-line coupling’, IEEE Trans. Microw. Theory Tech., 1987, MTT-35, ( 3 ) , pp. 260-267 CHENG, K.K M., and ROBERTSON, T D.: ‘Simple and explicit formulas for the design and analysis of asymmetrical V-shaped microshield line’, IEEE Trans. Microw. Theory Tech., 1995, MTT-43, (IO), pp. 2501-2504

4

5 CHENG, K.K.M., and ROBERTSON. I.D.: ‘Quasi-TEM Study of microshield lines with practical cavity sidewall profiles’, IEEE Trans. Microw. Theory Tech., 1995, MTT-43, (12), pp. 2689-2694 CHENG. K.K.M., and ROBERTSON, I.D.: ‘Numerically efficient spectral domain approach to the quasi-TEM analysis of supported coplanar waveguide structures’, IEEE Trans. Micro~w. Theory

6

Tech., 1994, MTT-42, (lo), pp. 1958-1965

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