Qt �1

2Q �

�l

�g��g

� �2�1

2B� 2 � 1� . (51)

For high Q resonance cavity, there is B� � 1 in general, whichyields

Qt ��l

�g��g

� �2

� �1

2B� 2� . (52)

5. CONCLUSION

Extending the Foster theorem to apply to any lossy system is ameaningful subject. The generalized Foster theorem presented inthis paper represents the current research in this field. The systemadopted the network parameters conveniently in calculation andclearly in physical conception. The example given in this paperindicates the slope of active function may be negative in a lossysituation.

The development of the concept of Network-Q will be veryuseful, both in principal and in practice. The Network-Q will beintroduced into the study of open resonance systems and furtherstudy will be reported in other papers.

REFERENCES

1. R.E. Collin, Foundations of microwave engineering, McGraw-Hill,New York, 1992.

2. Q. Chu and C. Liang, General forster theorem, Xidian University, Vol.22, 1995, pp. 435–437.

3. M. Wind and H. Rapaport, Handbook of microwave measurements,Polytechnic Institute, Brooklyn, NY, 1963.

© 2003 Wiley Periodicals, Inc.

DESIGN OF LONG-PERIOD GRATINGS:NECESSITY OF A THREE-LAYER FIBERGEOMETRY FOR CLADDING MODECHARACTERISTICS

Rashmi Singh, Harish Kumar, and Enakshi Khular SharmaDepartment of Electronic ScienceUniversity of Delhi South CampusNew Delhi-110021, India

Received 19 September 2002

ABSTRACT: In this paper, we show that a complete three-layer analy-sis is necessary to characterise the cladding modes in order to obtaingrating period, resonances, and coupling length in the design of long-period gratings (LPGs), and that the simplified two-layer fiber geometryused by many authors leads to incorrect designs. This is illustrated bydesign calculations corresponding to actual applications of long-periodgratings as sensors and gain equalisation filters for EDFA. © 2003Wiley Periodicals, Inc. Microwave Opt Technol Lett 37: 45–49, 2003;Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/mop.10820

Key words: long-period fiber grating; resonances; three-layer fiber ge-ometry; two-layer approximation; erbium-doped fiber amplifiers

1. INTRODUCTION

A long-period grating (LPG) couples light from the fundamentalguided-core mode to forward-propagating cladding modes. Thelight coupled to the cladding decays quickly due to scatteringlosses, thus leaving lossy bands in the guided-core mode observed

at the output. The transmission spectrum of a long-period grating,therefore, consists of distinct resonant peaks, which are veryattractive for both telecommunications and sensing applications.Various devices based on LPG have been demonstrated, such asband-rejection filters [1], EDFA gain equalizers [2], and fibersensors [3–5]. A long-period fiber grating essentially consists of afiber in which the core has periodic refractive index changes thatcouple light from the bound core mode to lossy cladding modes.The typical geometry of the fiber without any acrylic jacket hasthree distinct regions: a core of diameter 5 �m, cladding ofdiameter 125 �m, and ambient (air). For simplifying the analysis,many authors use a simple two-layer geometry to calculate prop-agation characteristics of the cladding modes for the design ofthese gratings. For example, Vengsarkar, et al. [1] calculatedcladding mode propagation constants using the eigenvalue equa-tions for a simple multimode step-index structure with a corediameter of 125 �m and ambient as cladding to ignore the effectof the core. In [4], Lee, et al. used a geometric-optics approxima-tion [6] to calculate the mode indices of the core and the claddingmodes. This approximation essentially takes into account the phaseshift, which is due to the total internal reflection at the interfacebetween the cladding and the ambient for cladding modes, whilethe core-mode effective index is calculated by taking into accountthe total internal reflection at the interface between the core and thecladding.

In this paper, we explicitly show that there is a significantdifference in the propagation characteristics of the cladding modesobtained by the two-layer approximation and the complete three-layer geometry, which leads to erroneous designs for the applica-tion of LPG in sensors and as gain-flattening devices. We use thescalar-mode analysis, which is adequate for cladding modes of alarge diameter fiber, as also pointed out by Fielding [7]. It may bementioned that some authors [8, 9] have previously used the vectormodes analysis in the three-layer geometry for design of LPGs.

2. THE THREE-LAYER FIBER GEOMETRY

In the two-layer approximation used by various authors [1, 3–6],the refractive index profile used to evaluate the propagation char-acteristics of cladding modes is given by

n � n2 r � acl

� na�1 for air� r � acl, (1)

whereas for the core-guided mode it is given by

n � n1 r � aco

� n2 r � aco, (2)

where aco and acl are the core and cladding radii, respectively, andn1, n2, and na are the respective refractive indices of the core,cladding, and ambient.

The complete refractive index profile for evaluating the prop-agation characteristics of both core and cladding modes is given by

n � n1 r � aco

� n2 aco � r � acl

� na r � acl. (3)

In case of the complete three-layer geometry, both the cladding-ambient and core-cladding interfaces are considered for calcula-

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 37, No. 1, April 5 2003 45

tion of the effective indices and modal fields of the modes. We usethe scalar LP-mode analysis, which is adequate for cladding modesof a large diameter fiber [7].

Table 1 is a comparison of the effective indices of some LPmodes obtained by the two-layer approximation with those ob-tained by the three-layer analysis at � � 1.45 �m. The effectiveindices of different cladding modes differ significantly for bothgeometries, whereas the effective index for the core-guided mode(now referred to as the LP01 mode) are identical in both geome-tries. The difference in the effective indices of the correspondingmodes obtained by the two geometries leads to a difference in thegrating periods � � 2�/(core � clad) (that is, the periodicity ofthe induced refractive index changes) required for coupling. Thegrating periods for coupling from LP01 to a few LP0n modes at � �1.45 �m are also tabulated in Table 1. The grating periods forlower-order mode coupling show larger differences, whereas thedifference in grating period for higher-order mode coupling isrelatively lower. The fields and the corresponding eigenvalueequation for the cladding modes in the three layer-geometry aresummarised in the Appendix.

The normalised fields corresponding to a few cladding modeshave been plotted in Figure 1. The differences in the modal fieldsare very large in the core region where the photo-induced refrac-tive index change �n occurs for the grating, and it is these fieldsthat contribute to the coupling coefficient between the core andthe cladding modes.

3. EFFECT ON THE DESIGN PARAMETERS OFLONG-PERIOD GRATINGS

As seen in the previous section, the difference in grating periodsobtained by the three-layer geometry and the two-layer approxi-mation can be as large as 30 �m. In this section, we explicitlyshow that the use of the two-layer geometry to characterise clad-ding modes leads to incorrect design parameters for long-periodgratings. We consider both lower-order mode coupling (couplingbetween the fundamental core mode and a lower-order claddingmode) and higher-order mode coupling (coupling between thefundamental core mode and a higher-order cladding mode), both ofwhich find applications in sensors and in gain flattening of erbium-doped fiber amplifiers.

Figure 2 is a plot of the grating period versus wavelength forcoupling to different modes at different wavelengths, obtained bythe three-layer geometry and the two-layer approximation. Incoupling to lower-order modes, only one wavelength is coupled fora given grating period, while in coupling to higher-order modes,two different wavelengths can be coupled for a given gratingperiod. For coupling to lower-order modes (LP02, LP04, and LP06),the actual and the approximate curves in Figure 2 show significantdifferences. For example, for a grating fabricated with a period of360 �m coupling resonances will be obtained at 1210 nm, 1250nm, and 1361 nm, while the two-layer approximation would pre-dict resonances at 1220 nm, 1275 nm, and 1412 nm. The gratingperiod curves using the three-layer geometry are always higherthan those using the two layer approximation.

For higher-order mode coupling, the grating period first in-creases and then decreases with wavelength. In this case also, the

TABLE 1 Effective Indices and Corresponding Grating Periods of Different Modes as Obtained by the 3-Layer Geometry and the2-Layer Approximation for a Typical Set of Fiber Parameters: aco � 2.5 �m, acl � 62.5 �m, na � 1.0, n1 � 1.458, n2 � 1.45, and � �

1.45 �m

Mode

Effective Index ne � /k0 Grating Period � (�m)

3-Layer Geometry 2-Layer Geometry 3-Layer Geometry 2-Layer Geometry

LP01 1.452332 1.452332LP02 1.449954 1.449858 610 586LP03 1.449801 1.449650 573 541LP04 1.449544 1.449351 520 486LP05 1.449184 1.448959 460 430LP0,12 1.443913 1.443624 172 166LP0,13 1.442776 1.442489 152 147LP0,14 1.441543 1.441262 134 131LP0,15 1.440216 1.439940 119 117

Figure 1 Comparison of normalised modal fields of a few LP0m modes(m marked on curves) in the 2-layer approximation (with circles) and the3-layer analysis (continuous)

Figure 2 Grating period � versus wavelength for coupling from LP01 toLP0m modes (m marked on curves) as obtained by the 2-layer approxima-tion (with circles) and the 3-layer analysis (continuous)

46 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 37, No. 1, April 5 2003

grating period curves using the three-layer geometry differ signif-icantly from those obtained using the two-layer approximation.This results in incorrect prediction of resonances and their sepa-ration; for example, a grating fabricated with a period of 118 �mhas two resonances at 1110 nm and 1535 nm, that is, a separationof 425 nm, while the two-layer approximation would predictresonances at 1250 nm and 1375 nm, a separation of only 125 nm.

Table 2 shows the position of resonance peaks, in coupling ofpower from core LP01 mode to LP08 cladding mode, obtained bythe three-layer geometry and the two-layer approximation for atypical LPG of � � 290 �m at different ambient indices. Theresonance peaks shift towards lower wavelengths as the ambientindex increases. This response of LPG to ambient refractive indexchanges is used for application as a refractive-index sensor. Thepeaks predicted by the two-layer approximation always occur athigher wavelengths than those using the three-layer geometry.Table 2 also depicts the position and separation of resonance peaksin coupling of power from core LP01 mode to LP0,15 claddingmode obtained by the three-layer geometry and the two-layerapproximation for a typical LPG of � � 118 �m at differentambient indices. The resonance peaks shift in the opposite direc-tion as the ambient index increases; their separation is a strongfunction of ambient index. This phenomenon can be used tocalibrate LPG as refractive index sensor with enhanced sensitivity,based on measurement of separation of the dual peaks [10]. Boththe position of the peaks and their separation are incorrectlypredicted by the two-layer approximation.

4. EFFECT OF THE DESIGN INACCURACIES ONACTUAL APPLICATIONS

As mentioned earlier, LPG finds interesting applications as refrac-tive index sensors and as gain flattening devices in erbium-dopedfiber amplifiers, each of which require an accurate design and apredictable response. In this section, we show the effect of designinaccuracy on these actual practical applications.

4.1. Refractive-Index SensorsThe coupling or transmission characteristics of the long-periodgrating are a function of the ambient refractive index for lower-order mode coupling, as well as higher-order mode coupling.Hence, these gratings can act as refractive-index sensors for boththese types of mode coupling.

4.1.1. Lower-Order Mode Coupling Sensors. In coupling to lower-order cladding modes, as the ambient refractive index increases,the resonance peaks of a long-period grating shift towards shorterwavelengths. This phenomenon can be utilised to calibrate theLPG as a refractive index sensor. If an LPG of � � 290 �m isdesigned by two-layer approximation for sensor application, theresonance peaks are expected to vary with ambient refractive indexas �(na � 1.0) � 1484 nm, �(na � 1.33) � 1478 nm, and

�(na � 1.4) � 1472 nm, as shown in Figure 3 (dotted curves).Assuming the maximum index change of �n � 10�4, the ex-pected grating length would be 4.4 cm. However, the actualresponse of the LPG will be obtained as depicted in the same figureby continuous curves (with circles). In this case, weak resonancepeaks occur at �(na � 1.0) � 1402 nm, �(na � 1.33) � 1398nm, and �(na � 1.4) � 1394 nm. For the same index change, alonger grating of length 7.7 cm is required for complete reso-nances, as shown in the figure by continuous curves.

4.1.2. Higher Order Mode Coupling Sensors. As discussed insection 3, coupling to higher-order cladding modes leads to dualresonant peaks. These peaks shift in opposite directions as theambient refractive index increases, and their separation can becalibrated for use as refractive-index sensors. If an LPG of � �118 �m (coupling: LP01 to LP0,15) is designed by two-layerapproximation for the sensor application, the resonance peaks areexpected to vary with ambient index as �(na � 1.0) � 1250 nmand �(na � 1.0) � 1375 nm, �(na � 1.33) � 1215 nm and�(na � 1.33) � 1425 nm, and �(na � 1.40) � 1190 nm and�(na � 1.40) � 1470 nm, as shown in Figure 4 (dotted curves).Assuming the maximum index change of �n � 10�4, the ex-pected grating length would be 3.5 cm. However, the actualresponse of the LPG sensor is also depicted in the same figure bycontinuous curves (with circles). In this case, weak resonancepeaks occur at �(na � 1.0) � 1110 nm and �(na � 1.0) �1535 nm, �(na � 1.33) � 1100 nm and �(na � 1.33) � 1560nm, and �(na � 1.40) � 1090 nm and �(na � 1.40) � 1590nm. For the same index change, a longer grating of length 6.5 cmis required for complete resonances. In fact, the actual change in

TABLE 2 Resonance Peaks by the 3-Layer Geometry and the 2-Layer Approximation for LP01 to LP08 and LP01 to LP0,15 ModeCoupling for Typical LPGs of � � 290 �m and � � 118 �m, Respectively, at Different Ambient Indices na

na

LP01–LP08 (� � 290 �m) LP01–LP0,15 (� � 118 �m)

3-Layer 2-Layer 3-Layer 2-Layer� (nm) � (nm) �1 (nm) �2 (nm) �� (nm) �1 (nm) �2 (nm) �� (nm)

1.0 1402 1484 1110 1535 425 1250 1375 1251.2 1400 1482 1100 1545 445 1235 1395 1601.33 1398 1478 1100 1560 460 1215 1425 2151.38 1396 1475 1095 1575 480 1200 1450 2501.4 1394 1472 1090 1590 500 1190 1470 280

Figure 3 Transmission spectra of a LPG with � � 290 �m and length4.4 cm at ambient refractive index na � 1.0, 1.33, and 1.4 as obtained bythe 2-layer approximation (dotted) and 3-layer analysis (with circles); thecontinuous line corresponds to the result of a 3-layer analysis for a gratinglength of 7.7 cm.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 37, No. 1, April 5 2003 47

the separation of peaks is much smaller than that predicted by thetwo-layer approximation.

4.2. LPG as a Gain Flattening FilterA long-period fiber grating can be used as a gain flattening filterfor an erbium-doped fiber amplifier. The typical gain spectrum ofan erbium-doped fiber is shown in Figure 5(a), corresponding tothe following fiber parameters [11, 12]: step index fiber withconstant erbium doping in the core, aco � 1.5 �m, acl � 62.5�m, n1 � 1.46, NA � 0.24, � (erbium ion concentration) �1.6 � 1025 m�3, �p (pump wavelength) � 980 nm, and aninput-pump power level of 30 mW. The emission cross section e(�s) and absorption cross section a(�s) profiles correspond toFigure 4.22 of [13], with peak values a (�s � 1.53 �m) � 7 �10�25 m2, e (�s � 1.53 �m) � 0.92 a(�s) and a(�p) � 2 �10�25 m2, where �s is the signal wavelength.

An analysis using the two-layer approximation predicts that thepeak of EDFA at 1.53 �m can be flattened by use of a long-periodgrating of � � 476 �m, and the expected grating length would be3.2 cm, assuming maximum index change of �n � 10�4 (modecoupling: LP01 to LP05), as shown in Figure 5(a) (curve with filledcircles). However, the three-layer analysis shows that for this LPG,only a weak resonance occurs at 1.47 �m as depicted in Figure5(a) (curve with unfilled circles), and hence this grating cannot beused to flatten the gain of the EDFA. However, a correct design ofthe LPG by the three-layer analysis shows that a grating with aperiodicity of 372 �m (mode coupling: LP01 to LP07) can be usedto accurately flatten the gain spectrum of EDFA, as shown inFigure 5(b).

ACKNOWLEDGMENT

One of the authors, Rashmi Singh, acknowledges financial supportfrom the Council of Scientific and Industrial Research (CSIR) anduseful discussions with her colleague Sunanda.

APPENDIX

The fields of the cladding modes (n2 � /k0 � na) correspondingto the three-layer geometry given by Eq. (3) can be expressed as[14]:

� � AJ0�Ur/aco� r � aco

� BJ0�U� r/aco� � CY0�U� r/aco� aco � r � acl

� DK0�Wr/aco� r � acl, (4)

where U � aco(k02n1

2 � 2)1/ 2, U� � aco(k02n2

2 � 2)1/ 2, andW � aco(2 � k0

2na2)1/ 2. J0, Y0, and K0 are the Bessel and

modified Bessel functions and B, C, and D can be expressed interms of A as

B ��

2AUJ1�U�Y0�U� � � U� Y1�U� �J0�U�, (5)

C ��

2AU� J1�U� �J0�U� � UJ1�U�J0�U� �, (6)

D �BJ0�U� c� � CY0�U� c�

K0�Wc�. (7)

A is obtained by the normalization condition

neff �0

�

�2rdr � 1, (8)

Figure 4 Transmission spectra of a LPG with � � 118 �m and length3.5 cm at ambient refractive index na � 1.0, 1.33, and 1.4 as obtained bythe 2-layer approximation (dotted) and the 3-layer analysis (with circles);the continuous line corresponds to the result of a 3-layer analysis for agrating length of 6.5 cm

Figure 5 (a) Gain spectra of EDFA with peak at 1.53 �m (continuouscurve): grating loss characteristics and consequent gain flattening of theLPG with � � 476 �m and length � 3.2 cm as predicted in the 2-layerapproximation (curves with filled circles); actual loss characteristics of thegrating by the 3-layer analysis (curve with unfilled circles). (b) Gainspectra of EDFA with peak at 1.53 �m (continuous curve): grating losscharacteristics and consequent gain flattening of the LPG with � � 372�m and length � 3.45 cm designed by the 3-layer analysis (curves withunfilled circles)

48 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 37, No. 1, April 5 2003

where all the integrals involved in the above expression are ana-lytical. The eigenvalue condition is given by

J� 0�U� � J� 0�U� K� 0�Wc� � Y� 0�U� c�

J� 0�U� � Y� 0�U� �K� 0�Wc� � J� 0�U� c��

J1�U� c�Y1�U� �

J1�U� �Y1�U� c�, (9)

where c � (acl/aco) and Z� m( x) � (Zm( x))/( xZm�1( x)) (Zrepresents the Bessel functions J, Y, or K).

The analytical expression for is given as

�aco

U1 � Un

k0

4�n2A1AnU1J1�U1�J0�Un� � UnJ0�U1�J1�Un�,

(10)

where Un is the value of U corresponding to the LP0n mode, andA1 and An are the corresponding normalisation constants.

REFERENCES

1. A.M. Vengsarkar, P.J. Lemaire, J.B. Judkins, V. Bhatia, T. Erdogan,and J.E. Sipe, Long period fiber gratings as band-rejection filters, JLightwave Technol 14 (1996), 58–64.

2. P.D. Greene and H.N. Rourke, Tailoring long period optical fibergratings fot flattening EDFA gain spectra, Electron Lett 35 (1999),1373–1374.

3. V. Bhatia and A.M. Vengsarkar, Optical fiber long period gratingsensors, Opt Lett 21 (1996), 692–694.

4. B.H. Lee, Y. Liu, S.B. Lee, S.S. Choi, and J.N. Jang, Displacements ofthe resonant peaks of a long period fiber grating induced by a changeof ambient refractive index, Opt Lett 22 (1997), 1769–1771.

5. H.J. Patrick, A.D. Kersey, and F. Bucholtz, Analysis of the response oflong period fiber gratings to external index of refraction, J LightwaveTechnol 16 (1998), 1606–1612.

6. M.J. Adams, An Introduction to Optical Waveguides, Wiley Inter-science, New York, 1981, pp. 223–233.

7. A.J. Fielding, K. Edinger, and C.C. Davis, Experimental observationof mode evolution in single-mode tapered optical fibers, J LightwaveTechnol 17 (1999), 1649–1656.

8. T. Erdogan, Cladding-mode resonances in short- and long-period fibergrating filters, J Opt Soc Am A14 (1997), 1760–1773.

9. R. Kashyap, Theory of Fiber Bragg Gratings, Academic Press, Lon-don, 1999, pp. 171–174.

10. X. Shu, X. Zhu, S. Jiang, W. Shi, and D. Huang, High sensitivity ofdual resonant peaks of long period fiber grating to surrounding refrac-tive index changes, Electron Lett 35 (1999), 1580–1581.

11. Sunanda and E.K. Sharma, Field variational analysis for modal gain inerbium-doped fiber amplifiers, J Opt Soc Am B16 (1999), 1344–1347.

12. R. Singh, Sunanda, and E.K. Sharma, Propagation characteristics ofsingle mode optical fibers with arbitrary complex index profiles: Adirect numerical approach, J Quantum Electron 37 (2001), 635–639.

13. E. Desurvire, Erbium Doped Fiber Amplifiers, Wiley, New York,1994, p. 252.

14. M. Monerie, Propagation in doubly clad single mode fibers, IEEETrans Microwave Theory Tech 30 (1982), 381–388.

© 2003 Wiley Periodicals, Inc.

HIGH PERFORMANCE RF INTEGRATEDPASSIVE DEVICES ON THICK OXIDESUBSTRATE USING Cu-BCB PROCESS

Inho Jeong,1 Ki-Joong Kim,1 Tong-Ook Kong,1

Jun-Seok Kim,1 Hyung-Kyu Choi,1 Choong-Mo Nam,1

Dong-Wook Kim,1 and Young-Se Kwon2

1 Telephus Inc.25-11 Jang-dong, Yuseong-guTaejeon, 305-343, Korea2 Dept. of EE, KAIST373-1 Kusong-dong, Yuseong-gu305-701, Taejeon, Korea

Received 18 September 2002

ABSTRACT: In this paper, we develop a low-cost manufacturingtechnology for RF substrate and a high-performance process technol-ogy for integrated passive devices by electrochemically forming thickoxide on Si wafer and processing Cu thick metal and BCB. Severalintegrated passive devices such as LPF, BPF, and balun are fabri-cated using this technology and they show good RF performance inspite of their small chip size. © 2003 Wiley Periodicals, Inc.Microwave Opt Technol Lett 37: 49 –52, 2003; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10821

Key words: IPD (integrated passive device); LPF; balun; Cu process;BCB; multi-layer RF passive devices

1. INTRODUCTION

Nowadays, an RF system, which has small volume or size, lowcost and multi-functional integration, is one of the most populartopics. But the RF front-end of wireless devices still contain a largeamount of discrete components, which is an important obstacle forfurther size reduction. An RF integrated passive device (RFIPD)whose components are integrated on a substrate has been anattractive option for building wireless modules. In addition, theRFIPD in this work not only reduces the cost and size by elimi-nating the need for discrete components, but also has been dem-onstrated to exhibit superior performance for on-chip components.

Of all the passive components in an RF system, the inductorsare typically the most difficult to integrate with sufficiently highquality. Inductors are key components required for impedancematching and resonator elements in RF circuits. Obviously, inte-grated capacitors, resistors, and low-loss interconnects are requiredto be able to realize small size and high performance in RF sys-tems [1].

Silicon substrate is known to have many advantages, such asinexpensive material, good thermal conductivity, and a stable andmature process technology. But its utilization has been limited inthe fast-growing wireless market by large signal loss and signalleakage through parasitic substrate capacitance. So, the insulatingproperty of an SiO2 layer is used for isolation, and in order toeffectively operate this SiO2 layer in RF applications, it needs to bethick, in order to capacitively isolate it from the underlying con-ducting Si substrate. To obtain thick oxide on silicon substrate, aporous silicon technique is considered as a potential solution. Inporous material, the lattice has a large number of its silicon atomsremoved by an electrochemical reaction, thereby producing hon-eycomb-like structure. Initial work on using porous silicon for RFapplications took advantage of the oxidation process of the poroussilicon layer [2].

In this paper, we will introduce improved Si substrate technol-ogy, called Si smart substrate or thick oxide (TO) Si substrate, and

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 37, No. 1, April 5 2003 49