length lasing stable at room temperature. Each lasing mode has aspecific polarization and thus PCs can be used to control the lasingmodes (at different wavelengths) or switch (in a reconfigurableway) to some desirable lasing wavelengths according to the re-quirement. Due to the characteristics of overlapping cavities, thelasing intensities at desirable wavelengths can be equalized byoptimizing, e.g., the lengths of EDF1 and EDF2 sections.

3. LASING RESULTS

Lasing starts at the wavelengths of �1 and �3 when the power ofLD1 increases to 25 mW, and dual-wavelength lasing can beachieved by carefully adjusting PC1. If the power of LD1 increasesfurther, the output power of the dual-wavelength laser can in-crease. Note that lasing at wavelengths �2 and �4 will not occurdue to the large loss of the outer (longer) cavity. Fixing the powerof LD1 at 89 mW, lasing also occurs at wavelengths �2 and �4

(besides �1 and �3) when the power of LD2 is larger than 37 mW.Figure 3 shows the spectrum of the laser output when four-wavelength lasing occurs. The inset of Figure 3 shows the repeatedscanning of the spectrum of the laser output, from which one seesthat the multiwavelength lasing is stable, and the variation of eachwavelength is less than 1 dBm during a scanning time of 15 min.

The output wavelengths of the present laser can be furtherreconfigured by controlling the polarization of light (besides thepowers of the pumping sources) inside the resonant cavities. In theexperiments, we have observed all the four possible configurationsof three-wavelength lasing and all the six possible configurationsof dual-wavelength lasing by carefully adjusting the states of PCs.Figure 4 shows some three-wavelength lasing (the left column)and some dual-wavelength lasing (the right column).

4. DISCUSSION AND CONCLUSION

From the above experimental lasing results, we see that a stablemultiwavelength EDF laser can be realized by using the polariza-tion hole-burning effect and overlapping cavities. In the presentmultiwavelength laser, the two MFBGs are the most critical com-ponents. The MFBGs are utilized not only to select lasing wave-lengths, but also determine the state of polarization. The combi-nation of MFBGs and overlapping cavities can give more lasingwavelengths within the limited gain region of the EDF.

In our experiment, two LDs were used as the pumping sourcesdue to the output power limitation of the LDs we have. By usingan LD of larger output power and optimizing the lengths of EDFsections, a single LD pump is also possible to excite all the lasingmodes. If more than two modes can be excited in the MFBGs, thepresent fiber laser can give lasing at more than four wavelengths.

In conclusion, we have presented a reconfigurable multiwave-length fiber laser using two MFBGs. The experimental results haveshown that the multiwavelength lasing is reconfigurable by appro-priately controlling the PCs and is stable at room temperature.

ACKNOWLEDGMENT

This work is partially supported by the Science and TechnologyDepartment of Zhejiang Province (grant No. 2004C31095) andScience and Technology Bureau of Hangzhou municipal govern-ment (grant No. 20051321B14).

REFERENCES

1. J.N. Maran, R. Slavik, S. Larochelle, and M. Karasek, Chromaticdispersion measurement using a multiwavelength frequency-shiftedfeedback fiber laser, IEEE Trans Instrum Meas 53 (2004), 67–71.

2. L.R. Chen and V. Page, Tunable photonic microwave filter usingsemiconductor fibre laser, Elect Lett 41 (2005), 1183–1185.

3. X.P. Dong, S. Li, K.S. Chiang, M.N. Ng, and B.C.B. Chu, Multiwave-length erbium-doped fibre laser based on a high-birefringence fibreloop mirror, Elect Lett 36 (2000), 1609–1610.

4. A. Bellemare, M. Karasek, M. Rochette, S. LaRochelle, and M. Tetu,Room temperature multifrequency erbium-doped fiber lasers anchoredon the ITU frequency grid, IEEE Photon Technol Lett 18 (2000),825–827.

5. K. Zhou, D. Zhou, F. Dong, and N.Q. Ngo, Room-temperature mul-tiwavelength erbium-doped fiber ring laser employing sinusoidalphase-modulation feedback, Opt Lett 28 (2003), 893–895.

6. D.R. Chen, Z.W. Yu, S. Qin, and S. He, Switchable dual-wavelengthRaman erbium-doped fibre laser, Elect Lett 42 (2006), 202–204.

7. J. Sun, J. Qiu, and D. Huang, Multiwavelength erbium-doped fiberlasers exploiting polarization hole burning, Opt Commun 182 (2000),193–197.

8. C. Zhao, X. Yang, J.H. Ng, X. Dong, X. Guo, X. Wang, X. Zhou, andC. Lu, Switchable dual-wavelength erbium-doped fiber-ring lasersusing a fiber Bragg grating in high-birefringence fiber, Microwave OptTechnol Lett 41 (2004), 73–75.

9. Q. Mao and J.W.Y. Lit, Multiwavelength erbium-doped fiber laserswith active overlapping linear cavities, J Lightwave Technol 21(2003), 160–168.

10. T. Mizunami, T.V. Djambova, T. Niiho, and S. Gupta, Bragg gratingsin multimode and few-mode optical fibers, J Lightwave Technol 18(2000), 230–235.

© 2007 Wiley Periodicals, Inc.

MEASUREMENT OF A RECIPROCALFOUR-PORT TRANSMISSION LINESTRUCTURE USING THE 16-TERMERROR MODEL

Yun Zhang,1 Kimmo Silvonen,2 and Ning H. Zhu1

1 State Key Laboratory on Integrated Optoelectronics, Institute ofSemiconductors, CAS, Beijing, 100083 China2 Department of Electrical and Communications Engineering, CircuitTheory Laboratory, Helsinki University of Technology, FI-02015 TKK,Finland

Received 22 December 2006

ABSTRACT: A new method to measure reciprocal four-port structures,using a 16-term error model, is presented. The measurement is based on5 two-port calibration standards connected to two of the ports, while thenetwork analyzer is connected to the two remaining ports. Least-squares-fit data reduction techniques are used to lower error sensi-tivity. The effect of connectors is deembedded using closed-formequations. © 2007 Wiley Periodicals, Inc. Microwave Opt TechnolLett 49: 1511–1515, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22498

Key words: 16-term error model, four-port, transmission lines, calibra-tion, error sensitivity, deembedding, scattering parameters, test fixture

1. INTRODUCTION

A common method to determine the S-parameters of a multiport isto connect two of the ports to a network analyzer, and terminate theother ports with matched loads. The accuracy is dependent on theprecision of the standards. Several approaches to correct theseinaccuracies have been reported [1–6]. To determine all the 16S-parameters of a four-port, several reconnections of the loads andS-parameters are needed. Multiport network analyzers [7, 8] willcertainly become more common in the future, but even with thiskind of equipment, the four-port calibration is rather complicated.

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 7, July 2007 1511

In this paper, the 16-term error [9–14] model and a newdeembedding procedure are applied to a measurement of a micro-strip four-port with coaxial connectors. The measurement is basedon 5 two-port calibration standards, including a Through line anddifferent pairs of Match, Short, and Open (T, M, S, O). Thestandards are always connected to the same two ports, while atwo-port network analyzer is connected to the other two ports. Thiswork-flow reduces the number of repeated connections needed. Itmay also have advantages if two of the ports are situated in adifferent medium than the others, like microstrip versus coaxial. Asimilar method has previously been used in the measurement ofbalun transformers [15]. We use the least-squares-fit (LSF) algo-rithm [16, 17] to find the S-parameters from an over-determinedset of equations. The effect of connectors is deembedded usingnew closed-form equations.

2. 16-TERM ERROR MODEL

The measurement configuration used with the 16-term error modelis shown in Figure 1, where the port numbering and other con-ventions are according to Ref. 11. The error network is considereda four-port. An ideal vector network analyzer (VNA) is connectedto ports 0 and 3, while the calibration standards Sa are alwaysconnected to ports 1 and 2. Elements of matrix Sm denote the

S-parameters measured by the network analyzer. In our case, theerror network is used as a model of the unknown four-port.

The incident, reflected, or transmitted voltage waves at theinput and output terminals are shown as ai and bi. The S-param-eters of the calibration standards are related to these waves by

�a1

a2� � Sa�b1

b2� � �Sa11 Sa12

Sa21 Sa22��b1

b2�. (1)

Similarly, the measured (uncorrected) S-parameters are equatedto the voltage waves as follows:

�b0

b3� � Sm�a0

a3� � �Sm11 Sm12

Sm21 Sm22��a0

a3�. (2)

S-parameters E and transfer parameters T (chain scatteringmatrix) of the error network are defined as follows [14]:

�b0

b3

b1

b2

� � E�a0

a3

a1

a2

� � �E1 E2

E3 E4��

a0

a3

a1

a2

�where E � �

e00 e03 e01 e02

e30 e33 e31 e32

e10 e13 e11 e12

e20 e23 e21 e22

�. (3)

�b0

b3

a0

a3

� � T�a1

a2

b1

b2

� � �T1 T2

T3 T4��

a1

a2

b1

b2

� where T � �t0 t1 t4 t5

t2 t3 t6 t7

t8 t9 t12 t13

t10 t11 t14 t15

�.

(4)

Linear equations for the error terms can be achieved using acombination of S- and T-parameters using the Speciale equation[9]

T1Sa � T2 � Sm�T3Sa � T4�. (5)

The T-parameters of the four-port are first determined by amatrix Eq. (5), which produces a set of four linear equations interms of the 16 error-parameters tij [12], as shown in Eq. (6).

�Sa11 Sa21 1 � Sm11 Sa11 � Sm11Sa21 � Sm12Sa11 � Sm12Sa21, � Sm11 � Sm12

Sa12 Sa22 1 � Sm11 Sa12 � Sm11Sa22 � Sm12Sa12 � Sm12Sa22, � Sm11 � Sm12

� Sa11 Sa21 1 � Sm21Sa11 � Sm21Sa21 � Sm22Sa11 � Sm22Sa21, � Sm21 � Sm22

Sa12 Sa22 1 � Sm21Sa12 � Sm21Sa22 � Sm22Sa12 � Sm22Sa22, � Sm21 � Sm22

� �t0

t1

· · ·t15

� � 0. (6)

In total, 15 independent linear equations are picked from fiveindependent two-port calibration measurements.

3. SOLVING AND IDENTIFYING THE S-PARAMETERS

Equation (6) is composed of four homogeneous equations that arelinear as a function of the entries of the partitioned T-matrix.Theoretically, not less than 5 different two-port standards willgenerate enough equations to solve the 16 error terms [10, 12]. Atotal of 36 alternative sets of typical calibration standards can beused [14].

The standards are connected to ports 1 and 2 as shown in Figure1. An independent set of equations is obtained, for example, byusing the following standards: T, MM, SS, OO, SO. The problemlies in solving a system of homogeneous equations C�X � 0. Thispaper uses a general and accurate method based on redundant datafor solving the error terms.

3.1. LSF MethodBecause there are redundant calibration equations, data reductiontechniques can be used to improve the accuracy and lower errorsensitivity. We use here the LSF algorithm, as defined in Refs. 16

Figure 1 The measurement configuration. Sm denotes the measuredS-matrix, while the parameters of Sa are the scattering parameters of thestandards, which are measured through the unknown four-port, consideredas an S-parameter error network E

1512 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 7, July 2007 DOI 10.1002/mop

and 17. Five standards produce 20 equations, of which 16 or moreare chosen. Suppose we choose 18 equations, e.g. all the fourequations of T, MM, OO, SS, and just the second and thirdequations of SO. We did not find any advantage by using morethan 18 equations. The equations are normalized by one of theunknown coefficients, yielding to an equation of the form W�TC �U, in which W is an 18 � 15 matrix, while TC is a 15-rowsingle-column matrix.

Provided that the matrix product W*W is not singular, asolution is found by

Tc � (W*W)�1W* � U, (7)

where superscript * denotes the complex conjugate of the trans-posed matrix.

In this paper, t12 is set equal to 1, and matrix U is generated bymoving the t12-terms to the right hand side. To increase the verysmall numerical values in some equations, each equation might bescaled, for example, by S21. This may improve the numericalaccuracy.

3.2. ReciprocityThe T-parameters are converted to S-parameters using the follow-ing matrix equation [14]:

E � �E1 E2

E3 E4� � �T2T4

�1 T1�T2T4�1T3

T4�1 �T4

�1T3� . (8)

For a reciprocal four-port, E2 � E3T. The reciprocity condition

is thus more complicated in terms of T-parameters. Because of therandomly chosen scaling factor t12, the initial results do not fulfillreciprocity. In network analyzer error models, the correct value oft12 is not needed, but here it has to be determined. Although thereis not an easy way to find out the correct scaling value in advance,it can be corrected afterwards [15].

Suppose each T-parameter should be scaled by k instead of t12

� 1. This means that E2 is scaled by the same k, but E3 is scaledby k�1, while E1 and E4 remain unscaled. Thus kE2 � k�1E3

T.This gives us the correct scaling factor k2I � E3

TE2�1, where I

denotes the unit matrix. Ideally both diagonal entries of E3TE2

�1

should be equal, while the nondiagonal ones will vanish. The valueand sign of k will vary as a function of frequency. The correct signof k has to be identified by comparing the calculated phase of oneof the S-parameters to its estimated value, based for example, onthe mechanical length of the structure. The procedure can thus beused only if the correct root can be estimated accurately enough.However, the two solutions are situated 180° apart from eachother, which makes the decision easier.

4. MEASUREMENTS

4.1. Measurement ConfigurationThe measurement network shown in Figure 2 was used to test thenew procedure in practice. The DUT we used was a coupled line

structure on a microstrip substrate. The structure was connected toa VNA and the standards by an SMA-type coaxial-to-microstripconnectors. Network analyzer HP8720D was connected to ports 0and 3, while the standards were connected to ports 1 and 2. A401-point calibration was performed from 0.05 to 6.05 GHz usingthe coaxial standards. The standards T, MM, SS, OO, SO werethen measured, with the coupled-line structure in between.

The transmission from port 1 to 0 matches well with thetransmission from port 2 to 3, as can be seen in Figure 3. Theexisting small differences are mostly due to the inaccuracies in themeasurements, connectors, and calibration standards.

4.2. Comparison With the Direct MeasurementA standard method to measure the S-parameters of a multiport isto connect two of the ports to a calibrated VNA, and terminate theother ports with matched loads [1]. The measurement of a four-port would include six repeated measurements.

A comparison between the new method and the direct mea-surement can be seen from Figure 4. The results match very well.

5. DEEMBEDDING

Reasons for making the measurements in a coaxial environmentare the availability of the calibration standards as well as theconnector type of the network analyzer. The ultimate task is often,however, to find out the electrical parameters of the microstripstructure alone. In two-port systems, it is a well-known approachto deembed the effects of the connectors and connecting lines withcascaded four-ports. In a four-port system, as well, similar ap-proaches are possible.

5.1. Effect of the ConnectorsThe first step of deembedding is to use an existing calibrationprocedure to identify the S-parameters of the connectors. Here, aThrough-Delay (TD) method [18] is used. In this case, eachconnector is thus assumed similar. The differences in soldering theconnector to the substrate may reduce the accuracy of this ap-proach.

Figure 2 The practical measurement system

Figure 3 Comprison of the phase and magnitude between S01 and S32 ofthe four-port structure

Figure 4 Comparison of the phase and magnitude between the calcu-lated (the new method) and directly measured S03

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 7, July 2007 1513

Two lines of different length are shown on the left in Figure 5.They are both measured with a calibrated network analyzer. Thecharacteristic impedance of the lines is designed to be exactly 50ohms at 2 GHz. The shorter length corresponds to the sum of thetwo connection lines we used, while the second one is 12.69 mmlonger. This small length difference was chosen to avoid phasedifference around 180°, which would cause a singularity. Theblock diagram for the calibration of the fixture is shown on theright in Figure 5. The shorter line is considered a Thru standard T,while the longer one is an unknown Delay D. Both lines are placedbetween the SMA connectors. The left and right error networks Land R are assumed to be identical (at least L21 � R21). To facilitatethe symmetry assumption, two-port R is cascaded here in theopposite direction to the other two-ports. All the terms in matricesL, R, MT, and MD are scattering parameters.

The unknown transmission of the matched Delay line is foundby Eq. (9).

ST21

SD21�

SD12

ST12�

MT11 � MD22 � MD11 � MT22 � MT11 � MT22

� MT12 � MT21 � MD11 � MD22 � MD12 � MD21

MT12MD21.

(9)

By setting ST21 � ST12 equal to 1, we get a second-orderequation for SD21 � SD12. The root choice is based on comparisonof the calculated phase to the theoretical phase of a line of the samelength as the length difference of T and D.

The S-parameters of the adapter are found by Eq. (10), where�L � L11L22 � L12L21, �R � R11R22 � R12R21, and k � R21/L21

� 1 [18]. The equation gives the S-matrix of each adapter, that is,one-half of the shorter line with the connector at one end. Theresults are shown in Figure 6.

�1 ST11MT11 � ST11 0 ST21MT12 00 ST12MT11 � ST12 0 ST22MT12 00 ST11MT21 0 0 ST21MT22 � ST21

0 ST12MT21 0 1 ST22MT22 � ST22

1 SD11MD11 � SD11 0 SD21MD12 00 SD12MD21 0 1 SD22MD22 � SD22

��L11

L22

�LkR11

kR22

k�R� � �

MT11

kMT12

MT21

kMT22

MD11

kMD22

�.

(10)

5.2. Four-Port DeembeddingThe four-port is embedded between fixture halves A and B. Herewe describe a new deembedding procedure for four-port circuits.The definition of the voltage waves and port numbering are shownin Figure 7.

Deembedding of the connectors is processed using the mea-sured T-parameters of the system, as defined in Eq. (4). By usingsimilar partitioning for the elements of matrices B and A, we getEqs. (11)–(14):

�b5

b6� � B3�a1

a2� � B4�a5

a6�3 �a1

a2� � B3

�1�b5

b6� � B3

�1B4�a5

a6�,

(11)

�b1

b2� � B1�a1

a2� � B2�a5

a6� � B1B3

�1�b5

b6� � �B2 � B1B3

�1B4��a5

a6�.

(12)

�b0

b3� � A1�a0

a3� � A2�a4

a7�3 �a4

a7� � A2

�1�b0

b3� � A2

�1A1�a0

a3�,

(13)

�b4

b7� � A3�a0

a3� � A4�a4

a7� � �A3 � A4A2

�1A1��a0

a3� � A4A2

�1�b0

b3�.

(14)

When using coaxial connectors, there is practically no leakagebetween the upper and lower ports in A and B. The quadrants ofthe embedding four-ports are thus diagonal matrices.

A1 � �L11 00 L11

� A2 � �L12 00 L12

�A3 � �L21 0

0 L21� A4 � �L22 0

0 L22�. (15)

B1 � �R11 00 R11

� B2 � �R12 00 R12

�B3 � �R21 0

0 R21� B4 � �R22 0

0 R22�. (16)

However, the procedure and equations shown are generally notrestricted to nonleaky fixtures, except that there must not be anyleakage between A and B. A similar restriction is used in “partiallyleaky multiport network analyzers ” [19]. Substituting Eq. (4) toEqs. (13) and (14) gives

�a4

a7� � A2

�1�T1�a1

a2� � T2�b1

b2�� � A2

�1A1�T3�a1

a2� � T4�b1

b2��.

(17)

Figure 5 Lines and connectors for the TD-measurement (on the left).Block diagram for the calibration of the test fixture (on the right)

Figure 6 The magnitude and phase of scattering parameters L22 and R22

assuming k � 1

Figure 7 A four-port DUT Q embedded between 2 four-port fixturehalves

1514 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 7, July 2007 DOI 10.1002/mop

�b4

b7� � �A3 � A4A2

�1A1��T3�a1

a2� � T4�b1

b2��

� A4A2�1�T1�a1

a2� � T2�b1

b2��. (18)

The unknown four-port DUT, here defined by T-parameters Q,is embedded between four-port fixture halves A and B:

�a4

a7� � Q1�b5

b6� � Q2�a5

a6�. (19)

�b4

b7� � Q3�b5

b6� � Q4�a5

a6�. (20)

After including Eqs. (11) and (12), we can identify the T-parameters of the inner four-port using closed-form equations asfollows:

Q1 � A2�1[(T1 � A1T3) � (T2 � A1T4)B1]B3

�1, (21)

Q2 � A2�1[(T2 � A1T4)(B2 � B1B3

�1B4)

� (T1 � A1T3)B3�1B4], (22)

Q3 � [A4A2�1(T1 � T2B1)

� (A3�A4A2�1A1)(T3 � T4B1)]B3

�1, (23)

Q4 � [A3T4 � A4A2�1(T2 � A1T4)](B2 � B1B3

�1B4)

� [A3T3 � A4A2�1(T1 � A1T3)]B3

�1B4. (24)

These parameters are finally converted back to S-parameters.The deembedded results are shown in Figure 8.

6. CONCLUSION

A new method for measurement of passive four-ports is presented.The method is based on modeling the four-port as a 16-term errornetwork, similar to the one used in network analyzer calibration.Comparison between the commonly used direct-measurementmethod and the new method is done. The method of least squaresis used to improve the accuracy and lower the error sensitivity. Thedeembedding of the fixture is performed by direct analytical equa-tions.

ACKNOWLEDGMENTS

The work is supported in part by the National Natural ScienceFoundation of China under 60510173, 60536010, 60536006, and60606019, and in part by the National Basic Research Program ofChina under 2006CB604902 and 2006CB302806.

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© 2007 Wiley Periodicals, Inc.

Figure 8 Deembedded results

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 7, July 2007 1515