measured for a 180” phase shift of the light. In this latter case, the results give n2 arising from the quantum confined Stark effect alone, while the triangular points obtained using eqn. 4 show results with an additional contribution from other effects such as band-filling and saturation of the excitons,4’ which would account for the nonlinearity in undoped MQW structures. We note that the self-photo-induced Stark effect produces a significant contribution to the overall nonlinear refractive index.

\ ” \ \ \ \ \ \ ”\ “

A\ A . -- .--.-- m--m---.

8660 8660 8680 8700 /753111 wavelength, A

Fig. 1 Nonlinear rejractive index n2 as Junction of wavelength for T E (open symbols) and T M (solid symbols) polarisations obtained by direct measurement (triangles) and by use of eqn. 3 (squares)

Fig. 2 shows the nonlinear refractive index for the TM mode derived from eqn. 3 as a function of load resistance at different wavelengths, when the diode is connected to a resistive bias circuit with a bias voltage of 2 V. As expected, it is observed that the nonlinearity increases with increasing resistance, and the effect becomes more pronounced as the

1 6 c

absorption edge is approached. By comparing with the results for TM polarisation in Fig. 1, we see that in this case, the enhancement due to the self-electro-optic effect’ overshadows any contribution from the intrinsic nonlinearity, which would be present in an undoped MQW structure.

It must be borne in mind that as the load resistance is increased, the time response of the waveguide becomes longer. It is still possible, however, to increase n2 without a noticeable increase in time response. If, for example, a series resistor of 5kR is chosen, the RC constant of the diode is -Ions assuming that the capacitance of the pin diode is around 2pF. This time constant is comparable to the measured electron- hole lifetime, which determines the response time in the case of exciton bleaching.

Conclusion: We have seen that the Stark shift of the exciton peaks in an open circuit waveguide with a built-in field of 30kV/cm gives a strong contribution to the nonlinear refractive index, when the operating wavelength is well below the absorption edge. With the waveguide connected to a resistive bias circuit, an increase in n2 is obtained without significant speed penalty.

By fabricating a waveguide, where the guiding region com- prises an nipi configuration, we expected to see an n2 consider- ably enhanced over those in Fig. 1.6 Results of such measurements will be reported in a subsequent paper.

C. THIRSTRUP’ P. LI KAM WA

2Ist August 1989

Department of Electronic & Electrical Engineering University oJSheflield Sheflield S I 350, United Kingdom

* On leave from: Laboratory for Semiconductor Technology, Techni- cal University of Denmark, 2800 Lyngby, Denmark

References

ZUCKER, J. E., HENDRICKSON, T. L., and BURRUS, C. A.: ‘Low-voltage phase modulation in GaAs/AlGaAs quantum well optical wave- guides’, Electron. Lett., 1988,24, pp. 112-113

MAW, w., WOOD, T. H., and BURRUS, C. A. : ‘Novel hybrid optically bistable switch: the quantum well self electro-optic effect device’, Appl. Phys. Lett., 1984,45, pp. 13-15 LI KAM WA, P., and ROBSON, P. N.: ‘Optical nonlinearities in a passive GaAs/AIGaAs multiple quantum well strain-induced waveguide’, IEEE J . Quantum Electron., 1987, QE-23, pp. 1962- 1967 CHEMLA, D. s., MILLER, D. A. B., SMITH, P. w., GOSSARD, A. c., and WIEGMANN, w.: ‘Room temperature excitonic nonlinear absorption and refraction in GaAsJAIGaAs multiple quantum well structures’, ibid., 1984, QE-24 pp. 26S275 CHEMLA, D. s., and MILLER, D. A. B. : ‘Room-temperature excitonic nonlinear-optical effects in semiconductor quantum-well struc- tures’, J . 0p t .Soc . Am.& 1985,2,pp. 1155-1173 KOST, A., GARMIRE, E., DANNER, A., and DAPKUS, P. D. : ‘Large optical nonlinearities in a GaAs/AIGaAs hetero n-i-p-i structure’, Appl. Phys. Lett., 1988, 52, pp. 637439

MILLER, 0. A. B., CHEMLA, D. A., DAMEN, T. C., OOSSARD, A. C., WIEG-

CALCULATION OF NUMBER OF RELAY HOPS REQUIRED IN RANDOMLY LOCATED RADIO NETWORK

Indexina terms: Telecommunications, Networks, Radio links ~

The probability of establishing a connection of a given length through a network of randomly distributed packet radio sta- tions with a given number of hops is calculated. The required density of stations to ensure that this probability is high is found, as is the expected number of hops for a given path length.

Fig. 2 Nonlinear refractive index n2 for T M polarisation as Junction of load resistance at different wavelengths

Bias voltage = 2 V Introduction: As a first step, this letter describes a means of calculating the expected number of hops required for a path of

ELECTRONICS LETTERS 23rd November 1989 Vol 25 No 24 1669

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given length, for various densities of stations. Although it is possible, with N hops, to achieve a range of N times that for a single hop, this assumes optimal siting of the stations acting as repeaters, which is evidently not going to be so.

Calculation of connection probability as function of distance and repeater station density: The network is modelled as stations uniformly distributed over a flat earth. Antennas are taken as omnidirectional, and the power and sensitivity are such that contact may be made between stations separated by a distance of up to 1 km, but not outside that radius. The density of stations is Qlrra' stations/km2, thus making each station be on average within range of Q others.

The probability of being able to make a connection with another station at distance x using n or less hops is denoted by p,(x). Obviously

1 i f x i a otherwise

[ l - p2(x)] can be found as it is the probability that there is no station within the shaded area in Fig. 1. This area is given by

A = 2a2(0 - sin 0 cos 0) (2)

where 0 = cos-' xJ2a. The probability of there beinn no station within A is exp

( - AQ/na2); therefore

1 - e-"Q"' if x > a i f x i a

as connection is possible with a single hop zero for x z 2a.

(3)

A, and hence p2 , is

a >rl I

Fig. 1

For higher values of n, p . may be evaluated iteratively as follows. The probability of reaching a station at distance r from the originating station S, with (n - 1) hops but not

Let us denote the probability of not being able to make a connection through a station within area A by q(r). If this area is increased by d A by increasing r to r + dr (Fig. 2), the prob- ability of being able to make contact through a station within d A is

( n - 2), is P.- 1(r) - Pn-Z(r).

x2 + r2 - a2

2xr 8 =COS- ' ( 5 )

Fig. 2

Hence the probability of not making a connection through any station within A + dA is

h l ( r ) - pn-h912r8 dr} (6)

Therefore

r+o

x - 0

x' + r' - a' d r } (8) x r c 0 s - l

2xr

The probability of not making a link with less hops is 1 - P " - ~ ( X ) ; therefore the probability of not making a con- nection with n hops or less is

1 - P&) = c 1 - P.- 1(x)l

C P p , _ l ( r ) - p P n - z ( r ) ] r ~ ~ ~ ~ l 2xr I-Y

Results obtained by numerical integration are shown in Fig. 3.

number of hoDs

1

5 0 - $jO

g o ' i o

n

0 2 4 6 8 distance

Fig. 3

Calculation of average number of hops: These results may be used to evaluate the average number of hops required for a given connection length, ignoring connections which cannot be made with any number of hops:

N-1

where N is the maximum number of hops allowed and p,(x) = 0. These curves are plotted in Fig. 4 with a maximum of N = 8 hops. These curves become roughly straight lines

1670 ELECTRONICS LETTERS 23rd November 1989 Vol. 25 No 24

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after x = 2, apart from the ripples, until they approach the asymptote of N . Evidently, if there were no restriction on the number of hops, they would continue to rise linearly.

"I I 01 I

2 4 6 8 1846161 distance

Fig. 4

Examination of these curves shows that they may be rea- sonably approximated by

x i a E(n) 2

where m tends to 2 for small Q and to 1 as Q becomes large, having a value of 1.5 at Q = 5. The density of stations then does not greatly affect the number of hops required between two nodes, but has a much greater effect on whether a connec- tion can be made at all.

Conclusions: It can be seen that as the possible number of hops increases, so does the probability of being able to estab- lish a connection with any particular station. However, this probability approaches a limit which depends on the density of stations. This reflects the fact that there is a finite probabil- ity of there being no other stations within range of a particu- lar station. However, this probability is small so long as the average number of stations within range exceeds around five. This is therefore a reasonable estimate of the minimum density of stations to provide a reasonable blocking per- formance. At much lower densities, the probability of being able to establish a link is so low that the network is of little use. However, at that density, relaying can usefully extend the range to many times the single hop range, even though this is less than the number of hops times the single hop range.

Acknowledgment: Much of the work described was performed while I was working on a short-term fellowship at British Telecom Research Laboratories. Martlesham Heath.

S. A. G . CHANDLER Department of Engineering University of Warwick Coventry CY4 7AL, United Kingdom

6th September 1989

WORST MONTH PREDICTION MODEL FOR TRANSHORIZON MICROWAVE INTERFERENCE

Indexing t e r n : Radiowave propagation, Interference, Micro- wave links, Modelling

A model is developed to calculate the (average) worst month clear-air microwave interference statistics for land, sea and mixed land/sea paths. The model predicts the parameter Q, which relates the worst month statistics to the average annual statistics, derived either by measurements or by pre- diction. By providing a reliable means of computing worst month statistics, the model addresses a major deficiency in the existing CCIR interference prediction procedures. The validity of the model is established by comparison with mea- sured data from 1.3GHz transhorizon propagation experi- ments in northwest Europe.

Introduction: The concept of 'worst month' is of particular importance to radio system planners, since systems usually have to meet performance criteria that are based on a percent- age of 'any month'. In the interference context, the definition of worst month is given in terms of both interfering signal level and calendar months, e.g. 'the worst month is that month, in a period of twelve consecutive calendar months, during which a preselected threshold of signal level is exceeded for the longest time'.3 As such the worst month is not necessarily the same month for all signal thresholds.

Currently, although the worst month concept is widely used when performance objectives are specified, only very limited guidance as to how such statistics may be obtained is avail- able within the CCIR texts. For interference assessment, the current CCIR Report dealing with worst month" provides only examples of typical annual-to-worst month relationships, and no definitive method of directly predicting worst month statistics is given. However, several procedures for predicting cumulative distribution curves of average annual interference levels are available in other CCIR Reports,' and this letter therefore outlines a simple method of deriving worst month clear-air interference statistics from cumulative distributions of annual transhorizon interference levels. The proposed model is derived empirically using a total database of 32 link years of measurements.

Prediction method: The prediction method is derived using data from seven transhorizon microwave radio links. Each

link originates from a transmitter located at the British Telecom Research Laboratories, Martlesham Heath, UK and operates on a frequency of 1.297GHz. Details of the receivers are given in Table 1. These experiments cover a large area of NW Europe. The particular frequency is unimportant, as annual-worst month relationships are primarily a function of the meteorology of the area, and as such are generally inde- pendent of frequency, at least across the commonly used tele- communications frequency bands.

To develop the worst month model, the yearly cumulative distributions and the worst month data from each of the seven paths have been used to derive values for the parameter known as Q, which is defined as the ratio of the worst month time percentage to the annual time percentage for which the interfering signal level is above a particular threshold.

The Q values for the set of thresholds (-45, -35, -25, -15, - 10, -5, 0 and +5dB relative to free space path loss) were calculated from data derived from each of the seven experimental links. Since microwave paths over land and sea exhibit different propagation characteristics, the links were classified into land, sea and mixed land and sea paths (as indicated in Table 1). A linear least-squares fit was then plotted through each of the three sets of Q values, and these were taken as typical for that type of propagation path. The relationships for each of these classes are given by

Land: log,,(Q) = -0.184 x log,,(P) + 0.515 ( 1 4

Mixed: log,,(Q) = -0.126 x log,,(P) + 0.598 ( Ib)

Sea: loglo(Q) = -0.106 x log,,(P) + 0.701 (IC)

where p is the percentage of the average year.

Table I

Receiver

Sparsholt Firs London Leidschendam Blavand Eindhoven Breda Nera

Country

UK UK Netheriands Denmark Netherlands Netherlands Netherlands

Path Path length type

km

199.4 Land 114.3 Land 212.6 Sea 593.5 Sea 298.0 Mixed 245.9 Mixed 259.7 Mixed

Years of

data ~

3 4 9 4 7 3 2

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~ ~ _ _ _