REFERENCES The way to obtain a good design must follow several steps: 1

2.

3.

4.

5.

6.

7.

8.

S. A. Johnson, T. H. Yoon, and J. W. Ra, “Inverse Scattering Solutions of the Scalar Helmholtz Wave Equation by a Multiple Source Moment Method,” Electron. Lett., Vol. 19, 1983, pp.

S. A. Johnson and M. L. Tracy, “Inverse Scattering Solutions by a Sinc Basis, Multiple Source, Moment Method, Part I: Theory,” Ultrasonic Imaging, Vol. 5 , 1983, pp. 361-375. M. M. Ney, A. M. Smith, and S. S. Stuchly, “A Solution of Electromagnetic Imaging Using Pseudoinverse Transformation,” IEEE Trans. Medical Imaging, Vol. MI-3, 1984, pp. 155-162. D. K. Ghodgaonkar, 0. P. Gandhi, and M. J. Hagmann, “Estima- tion of Complex Permittivities of 3-Dimensional Biological Bodies,” IEEE Trans. Microwave Theory Tech., Vol. MTT-31, 1983, pp.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Wiley, New York, 1977. M. Bertero, C. De Mol, and G. A. Viano, “The Stability of Inverse Problems,” in Inverse Scattering Problems in Optics, H. P. Baltes (ed.), Springer, New York, 1980, Chap. 5. J. M. Lee, S. Y. Kim, and J. W. Ra, “A Spectral Inverse Technique for Reconstruction of Complex Permittivity Profiles,” Electron. Lett., Vol. 24, 1988, pp. 556-558. H. G. Schmidt-Weinmar, “Spatial Resolution of the Subwave- length Sources from Optical Far-Zone Data,” in Inverse Source Problems in Optics, H. P. Baltes (ed.), Springer, New York, 1978, Chap. 4.

130-132.

442-446.

Selection of the devices to be used. Study of stability conditions and design of feedback net-

works or padding if it is necessary. Design of the general amplifier structure: gain level for

each stage, mismatch supported between stages to com- pensate for gain variation with frequency, etc.

Design of matching networks. Optimization.

CAD techniques for some of these steps have been developed, mainly on the feedback design [l, 21 and in the final network design and optimization processes [3, 41, but few papers have been published on the first steps and they are usually left to designer experience and cut and try techniques IS].

This paper describes the gain, match, and noise limitations for a general linear four-pole, that make possible a fast selection of the device and feedback networks, based on a low number of parameters.

When the device limits are known it is easy to optimize a first amplifier structure where gain and noise due to each stage and mismatch supported by each interstage network can be selected without impedance computation.

2. GAIN AND POWER MISMATCH LIMITS Received 8-30-88; revised 10-19-88

Microwave and Optical Technology Letters, 2/1, 26-29 0 1989 John Wiley & Sons, Inc. CCC 0895-2477/89/$4.00

~~

MATCHING, GAIN, AND NOISE LIMITS ON LINEAR AMPLIFIER FOUR-POLES Manuel Sierra Universidad Poiitecnica de Madrid E T S I Telecominicacion Ciudad Universitaria 28040 Madrid Span

KEY TERMS Linear four-pole theory, gain and match limits

ABSTRACT Equations showing the limits of gain, input and output matching, and noise hehaviour for a linear four-pole have been developed. Easier device evaluation and amplrjer design can be done with these equations.

1. INTRODUCTION

When a low noise microwave amplifier design starts, noise temperature, gain, stability and input-output return losses are usually the most important parameters that the designer has to take into account. These design variables are related to the device characteristics and the networks used to match or to link those devices, but the complexity of the relations and the large number of degrees of freedom involved in the amplifier design, make it difficult to try direct optimization processes with success.

2A. Preuiow Equations. It is well known that some limita- tions appear on linear amplifier gain and match at the input and output ports. Two of the most common limits are the gain on stable four-poles and the impossible simultaneous match when working with some unstable devices.

A general formulation for the limits that relate the avail- able gain (C,), power gain (C,), and input and output power reflection coefficients (Rl, R 2 ) , are developed as a function of two parameters: Rollet’s stability constant ( K ) and the maxi- mum stable gain (Go).

Let us start with a linear four-pole described by the scatter- ing parameters [ S ] where Z, and Z2 are the input and output reference impedances (Re[Z,] > 0 and Re[Z2] > 0). When this four-pole is loaded by source and load impedances Z, and Z,, as shown in Figure 1, the available and power gains take the values [6]

where

A = s 1 2 s 2 2 - S12S2, ( 4)

On the other hand, the power reflection coefficients can be defined in the input and output as a function of the trans- ferred and available power on each port and computed from the set of impedances as

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 2, No. 1 , January 1989 29

FOUR-POLE [ S l dZL

Figure 1 General linear four-pole circuit

These four parameters (G,, G,, R,, and R,) depend on the device and the external loads and are very important to the amplifier design.

There exists an easy equation that relates these quantities. The transducer gain, defined as the relation between the power delivered to the output load and the power available from the generator, can be obtained from these two equations:

Before searching for other relations, let us introduce two well known parameters that are very important in the device characterization.

The stability constant ( K ) can be defined as

In linear amplifier four-poles it is assumed, with only a few restrictions, that the necessary and sufficient conditions for stability are K > 1, and lAl < 1 [7]. In general, the first one means that the device can be adapted simultaneously at the input and output ports.

The other parameter is the maximum stable gain (MSG) that can be used as a normalization gain and it is defined as

Go = lS21 /S l2 l ( 9)

These two values are invariant for any lossless network added at the input or at the output of the four-pole and do not depend on the reference impedances used to define S parameters.

2B. Demonstration. Starting from the four-pole defined in Figure 1, we can select the input reference impedance as the source ( Z , = Zs) , assuming that it lies on the stable region of the device (Re[ Z,,,] > 0), and the output reference impedance equal to the conjugate of the output (Z2 = Z,T,,). Then the following relations can be written from (1)-(6):

r7 = o r,,, = s12 = o (10)

G, = IS2,l2 (11)

R ; = Irln12 = IS,, + &&r, I ? (14 R: = Ir, 12 (13)

If the output power reflection ( R , ) is maintained constant and we change only the phase of rl~ (Z, moves along a circle), the available gain is constant ( Z , does not change), but the power reflection coefficient in the input takes values in between two limits that can be obtained from (12) as

From Equations (8)-(12) it is possible to write

Finally the following limits for R, are obtained:

Any parameter in Equation (17) does not depend on the reference impedances and then it is, in general, true.

Another problem is the conditions we demand from the real part of the reference impedances in order to limit this demonstration to positive power flow, but using generalized scattering parameters and exchangeable gain, this equation can be extended to g, or g, < 0 and R, or R , > 1.

From (7) and (17) the design limits for any of those four parameters can be computed from two of the others. The dual of Equation (17) is especially easy:

2C. Graphic Models. Equations (17) or (18) can be easily plotted in the R,-R, plane as a group of straight lines using the normalized gain as a parameter. Figure 2 shows the plot of one situation for g, equal constant, where allowed and forbid- den areas have been specified.

Figure 3 shows the rate of available and power gains in the same plane, as it was obtained from (7). Once g , and a point in the plane have been chosen, all parameters are defined.

Figures 4 and 5 show plots for several values of g, and two values of K ( K = 0.8 and l.l), where strong limits of g and R appear clearly.

Once the gain and reflection coefficients have been se- lected, we keep one degree of freedom to choose the source and load impedances for the four-pole. For instance, the source impedance must be a point on the constant g , circle, but it can be any of them. When Z, has been chosen, the load impedance is completely defined through the gain and reflec- tion conditions because Z,,, is now known.

2D. Some Examples. In the case of stable amplifiers ( K > 1 and \ A [ < l), the maximum normalized gain (MAG) can be

30 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 2, No. 1 , January 1989

"I- ALLOWED AREA

- - FORBIDDEN AREA __

I .. : . . . . . . . .

R2 )

Figure 2 Allowed area on R,-R, plane

,' ,/' 0 % / i ,/' / ,,'

n 4

I / / /' ,,,' ,/

/ /

/ - Gp/Go=-lOdE Gp/Ga=-4dB Gp/Go=-ZdB Gp/Go=-ldB Gp/Go=O. dB Gp/Go=l dB Gp/Go=2dB Gp/Go=4dB

-- /

__.-

_ - Gp/Go-lOdB

i (R2) d

OUTPUT REFLECTION COEFFICIENT

Figure 3 Power gain over available gain on R,-R, plane

written as

In the R,-R, plane, the line limit.ing the available area is straight and contains the origin ( R , = R , . g,). This point means the simultaneous input and output power matching.

When K is less than unity, gain can take any value from minus to plus infinity (negative values are related to the unstable area), but simultaneous perfect matching is not possi- ble, being impossible to reach any point inside an area sur- rounding the origin in the R,-R, plane, as shown in Figure 5.

Some of the points closest to the optimum match can be the following:

R , = OR, = 41 - K~ g, = I / K g p = K (20a)

R , = i/l-Kz R, = 0 g, = K gp = 1 /K (20b)

The two first points show a perfect match at one port. These solutions are only possible when K > 0, if g have to be positive and reflexion in the other port is less than 1.

Only when K < 0 is it possible to obtain stable solutions keeping some mismatch at both ports like in point (20c).

If K < - 1, none of the previous points can assure stability.

3. NOISE AND GAIN LIMITS

3A. Previous Equations. The four-pole noise temperature de- pends, like the available gain, on the source impedance and, of course, on the four-pole noise parameters.

The most usual way to express the equivalent noise temper- ature of the four-pole as a function of source impedance is as follows [8]:

R, = R , = (1 - K ) / 2 g, = g,, = 1 (20c) where T,,, is minimum noise temperature, Z,,, is optimum

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 2, No. 1, January 1989 31

0.0 0.5 1.0 1.5

OUTPUT REFLECTION COEFFICIENT

Figure 4 Gain-reflection limits for K = 1.1

OUTPUT REFLECTION COEFFICIENT

d

Figure 5 Gain-reflection limits for K = 0.7

32 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 2, No. 1, January 1989

\

10.

0 . 5

/ d i v

/ d i v

RELATIVE G A I N G / M S G

Figure 6 Gain-noise limits for K = 1.1, G,,/G, = - 5 dB

impedance, G,, is equivalent noise conductance, and T, is reference temperature (290 K). These four parameters (Z,,, is complex) fix the noise behaviour of the device at each fre- quency. constant gain as

The same equation can be expressed using reflection coef-

Now, if we move Z, along a constant available gain circle, it is possible to obtain from (26) the minimum and maximum of Ir,l, and from (25) the minimum and maximum noise for

(27) ficients. From Equations (3a) and (21) it is possible to write [9]

(22)

1 + guguo - K(gu + goo) * \/AA, 2g,guo(K2 - 1)

Nlmin,max =

where - I& - LPJ2 - T - Tmin

4 ~ ~ ~ 1 11 - ropt12(1 - 1rsl2) where R, = Re[ Z,] (reference impedance at the input port). A = 1 - 2Kgu + gz A , = 1 - 2Kgu0 + gio

3B. Demonstration. In a way similar to Section 2B, we will choose as reference impedance for S parameters the values

Like in the previous demonstration, the parameters used in (27) do not depend on the reference impedances and hence, these limits are in general true.

3C. Graphic Model. Figure 6 shows the minimum tempera- ture associated with each available gain for some device. The upper limit is unimportant and in fact it is negative when

Contrary to what happened with the matching limits, here, once the point in the g, - N plane has been chosen, the source impedance becomes completely defined, or no more than two values are possible when N is not minimum.

Zl = Z,,, ' 2 = Z&t I when 2, = Zopc

In this situation it can be demonstrated that

r,,, = o s,, = o K < 1.

1 - IC7I2

11 - S J . . I 2 - l & * ~ * I ~ S l 2 (23) G, = IS,lI*

Go, = G,Izs=z,,, = I%I2 (24)

(25) 4. CONCLUSION The limits presented here aid a better understanding of device behaviour, matching problems, and device selection in ampli- fier design.

Evaluation of feedback techniques or padding to get a more stable, low noise or low reflexion four-pole, can be easily done using previous equations.

When designing a multistage amplifier, it is important to know how interstage mismatch affects the overall gain and the

T - Tmin Ir,12 47FnRopt 1 - IW

-- N = -

where Go, is the available gain when Zs = Zopt.

be written from (23) and (24): Changing to normalized gain, the following equation can

(26) 1 - 1YTI2

11 - sllrs12 - giolr,12 gu = goo

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 2, No. 1, January 1989 33

final input or output mismatch. This is not a hard problem with these @quations, especially if lossless networks are used to match or link stages.

Development of these techniques and examples for design will appear in another paper.

REFERENCES 1. D. J. H. Maclean. “Improved Computer-Aided Design of Ampli-

fiers using Feedback Concepts,” IEEE Proceedings, Vol. 133, Pt. H, No. 5. Oct. 1986.

2. K. B. Niclas, W. T. Wilser, R. B. Gold, and W. R. Hitchens, “The Matched Feedback Amplifier: Ultrawide-Band Microwave Ampli- fication with GaAs MESFET’s,” IEEE Trans. M T T , Vol. 28, No. 4, April 1980, pp. 285-294.

3. B. S. Yarman, “Modern Approaches to Broadband Matching Problems,’’ IEEE Proceedings, Vol. 132, Pt. H, No. 2, April 1985.

4. H. J. Carlin and P. Amstutz, “On Optimum Broad-Band Match- ing,” I E E E Trans. C A S , Vol. 28, No. 5, May 1981.

5. H. Fukui, ed., Low-Noise Microwave Transistors and Amplifiers, IEEE Press, 1981.

6. G. E. Bodway, “Two Port Power Flow Analysis of Linear Active Circuits using the Generalized Scattering Parameters,” Microwave Journal, Vol. 10, May 1967, pp. 61-69.

7. D. Woods, “Reappraisal of the Unconditional Stability Criteria for Active 2-Port Networks in Terms of S Parameters,” I E E E Trans. C A S , Vol. 23, No. 2, Feb. 1976.

8. IRE Subcommittee 7.9 on noise, “Representation on Noise in Linear Twoports,” Proc. I R E , Vol. 48, Jan. 1960, pp. 69-74.

9. P. Penfield. Jr., “Wave Representation of Amplifier Noise,” IRA Trans. Circuit Theory, Vol. 9, Mar. 1962, pp. 84-86.

Receicied 10-1 9-88

Microwave and Optical TethnoloD Letters, 2/1. 29-34 0 1989 John Wiley & Sons, Inc CCC 0895-2477/89/$4 00

THEORETICAL ANALYSIS OF A

FIBER LASER FOUR-LEVEL SINGLE-MODE

Qing-Hai Wang and Han-Xiong Lian Optical Fiber Communication Laboratory 6eiJing University of Posts and Telecommunication Beijing China

KEY TERMS Optical fiber laser, optical fiber uniplifer

ABSTRACT The relatron between rhe output power and input pump power and then the threshold punip powser and slope efficiency of a four-leiiel single mode fiber laser IS atiulvsed in terms of the four-level rate eyuutions and the numerical results are in good agreement with the available experimental results h v M. Shimizu (1 987). Meanwhile, the theoretical model of this puper is sinpler and more reasonable than that of Shiniizu and is applied to desigti a single mode fiber laser us well as a single mode fiber uniplifier with four-leoel materiul.

1. INTRODUCTION

In recent years, laser oscillators and amplifiers in a rare- earth-doped fiber form have received increasing attention since their round geometry is adapted to fiber system application.

To develop the rare-earth-doped fiber devices such as optical fiber laser, optical fiber amplifier, optical fiber filter, optical fiber sensor, and so on, the rare-earth-doped fibers such as Nd-, Er-, Dy-, Tb-, Ce-, Eu-, Tm-, Yb-, and Pr-doped fibers have been developed and a lot of associated experiments have been published [l-51. However, only a few theoretical analy- ses were presented [6]. A mode overlap theoretical analysis has been presented by Digonnet et al. [6] and it has been used by Shimizu et al. [7] to calculate the threshold pump power and slope efficiency of a Nd-doped single mode fiber laser. How- ever, their numerical results, in which the optimum fiber length for maximum energy conversion efficiency is less than 0.1 m (which cannot provide enough space to make enough population inversion in the fiber) are not so reasonable.

In this paper, a new approximation analysis based on the four-level rate equations is presented to calculate the four-level single mode fiber laser using the same parameters as Shimizu et al.; the numerical results are in good agreement with the available experiment results. The numerical results of this paper expect that both the optimum fiber length for energy conversion efficiency and the optimum fiber length for lowest threshold pump power are at 10 m according to the parame- ters of Shimizu, which is more reasonable than that of the analysis by Shimizu.

More in detail, in this paper, the laser equations are obtained based on the fact that the change rate of the total numbers of population and the change rate of the total numbers of laser photons should be balanced. Solving these equations in optical fiber lasers, the threshold pump power PI,, and the slope efficiency S and then the relation between the output power Po,, and the input pump power P,, are obtained in the form of

This theoretical model is simple and is applicable to design the optical fiber laser as well as the optical fiber amplifier.

2. THEORY

In Figure 1 the scheme of the four-level laser system is given, where N,, Nl , N2, and N3 are the numbers of the population (atoms or molecules) per volume on level 0, 1, 2, and 3, respectively. For the sake of simplicity, it is assumed that there is only one pump band (band 3 of Figure 1) and the decays from level 3 to level 2 and from level 1 to level 0 are very fast since the lifetime of level 3 and level 1 is very short compared with the lifetime of level 2. In this case, N3 = Nk = 0, and the rate equations of’the four-level scheme in Figure 1 are

N, + N2 = N, dN2/dt = Wp N, - BpNz - N2/7

( 2 4

(2b)

( 2c) dq/dt = J B ~ N , dV - q/T. U

where it is assumed that the laser oscillates on only one fiber cavity mode or the amplifier works on only one fiber mode. Also, q is the total number of photons of stimulated emission in the fiber cavity, p is the density of q, B is the Einstein B coefficient, Wp is the pump rate, T is the lifetime of level 2, and N is the total number of population (atoms or molecules) per volume.

34 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 2, No. 1 , January 1989