JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 12, NO. 11, NOVEMBER 1994 1901

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A Model for the a-Profile Multimode Optical Fiber Channel: A Linear Systems Approach

Ernest L. Walker. Senior Member, IEEE

Abstract-In this paper, a linear systems model for the a-profile class of multimode optical fibers is proposed and investigated. The modeling approach is rooted in results obtained by Gloge, et al. and is augmented by the application of basic principles from linear systems theory. Propagation losses and modal mixing were not considered in the model, owing to the superior quality of current optical fiber technology. Intermodal dispersion was the major source of dispersion considered in the model; effects due to waveguide and material dispersion were minimized by imposing a narrowband transmission assumption. Both the field transfer function and the corresponding field impulse response were calculated in a small neighborhood around the optimum value of the core parameter a. Results from this investigation, to the extent possible, were compared to those found by Gloge et al. and were found to be in agreement.

I. INTRODUCTION HIS work was motivated by the need to conduct system- T level analyses for computer communication networks

and other communication systems which employ multimode optical fibers as the transmission medium; while at the same time, taking appropriate account of the effects of impairments present at the physical layer of the system. The linear systems approach was chosen for this modeling effort because the wide range of tools already in common use for linear systems analysis could be employed to characterize transmission via the multimode optical fiber channel.

Phenomena relating to transmission via the multimode op- tical fiber are well documented ([l], [3]) and will not be discussed here. The focus of this effort was to assimilate these phenomena and the aforementioned motivational factors in such a way as to facilitate a realistic characterization of the a- profile multimode fiber at the system level. At the same time, we desired that this characterization provide sufficient detail and flexibility to be useful over the wide range of applications in which multimode fibers are employed.

The approach we have chosen is one the author believes to be novel; specifically, this approach facilitates the determina- tion of field frequency response directly from a formulation of how group delay varies with propagating mode order. While other researchers [ 11, at least in part, have found similar results, the present work reveals a more illuminating perspective on the sensitivity of transmission bandwidth and time-spread to extremely small changes in index of refraction profile.

Manuscript received April 12, 1993; revised June 8, 1994. This work was supported in part by NASA under contract 26093A.

The author was with the IBM Corporation, Research, Triangle Park, NC; he is now with the Department of Electrical and Computer Engineering, University of Southwestern Louisiana, Lafayette, LA 70504-3890 USA.

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IEEE Log Number 9404880.

We will start with a discussion of definitions and assump- tions which are presented in Section 11. Next, in Section 111, we perform the field transfer function calculation. In Section IV, we will calculate the field intensity transfer function. The next section, Section V, will be devoted to a discussion on the results of these calculations and a comparison of results to those from prior works. Finally, in Section VI, we will discuss our conclusions.

11. DEFINITIONS AND ASSUMFTIONS

In this model, we will consider the class of a-profile multi- mode fibers with no coupling between propagating modes and the propagation of cladding modes is assumed negligible. This restriction is justified due to the ever increasing quality of op- tical fiber manufacturing techniques and due to the way fibers are configured in most networks, i.e., bulk scattering is small and minimum bending radii are rarely challenged. In addition to modal mixing, we will consider the major sources of disper- sion. Material dispersion is caused by variations in the index of refraction of the fiber material with source wavelength. The waveguide effect is caused by a second-order variation in the index of refraction of the fiber material with wavelength. Intermodal dispersion is caused by differences in the velocities of propagation between the excited modes in the core region.

~ o ( f ) = IH(f)121i(f) where I o ( f ) is the intensity spectrum of the output of the system when excited by the input intensity spectrum I , (f), and H ( f ) is the transfer function for the input to output field amplitude.

The output field due to the mth propagating mode for a uniform straight lossless optical waveguide can be expressed as

For a linear intensity modulated system, we can write

+ o ( m r t ) = ,$i(m) ~ X P ( j ( 2 ~ f t - P m z ) )

when excited by a monochromatic optical field, where &(m) is the amplitude of the input field, pm is the propagation constant for the mth mode, and z is the length of the waveguide. We assume in our model that there is no mode dependent or bulk attenuation of the input field amplitude as it travels down the fiber, i.e., Pm is a real constant. Again, this is not a severe restriction, particularly for the local area network environment, due to the vastly improved quality of optical fibers over the years (losses are very low). We can therefore write the output field as

$ J ~ ( ~ L , t ) = &(m) exp { j a ~ f ( t - r(m/M))l

0733-8724/94$04.00 0 1994 IEEE

1902 IOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 12, NO. 11, NOVEMBER 1994

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where T(m/M) = Pmz/27rf is the propagation delay for the mth-order mode. We also assume in our model that the principal source of dispersion is due to differences in arrival times of the propagating modes at the end of the waveguide, i.e., intermodal dispersion. If all modes are uniformly excited by the same field amplitude Ao/M, the total response at the output will be

M

$ o ( t ) = (AO/W exp W.rrf(t - 7(m/M))} (1) m=O

where A, is a constant and A4 is the highest mode-order supported by the waveguide at the wavelength of operation. This number is large for most multimode fibers, on the order of several thousand. For the a-profile fiber, the index of refraction profile is given by [ l ]

n,[l - 2A(r/u)“I1/’ n,(l - 2A)lI2

r < a r > U. n( r ) =

Here a is the radius of the core region, no is the maximum value of the index of refraction in the core region, n,l is the index of refraction of the cladding region, and A is given by (see Fig. 1).

n: - n:l A=-----. 2 4

For this family of profiles, M is given from [ l ] as

M = (a / ( ( . + 2) )a2~’n?A

where K is the wave number and is given by

K = 2T/X.

w = 27rc/X

The wavelength X is related to the radian frequency by

where c is the speed of light in free space. With M large, (1) can be approximated by

Ilo(t) = A, exp (j27rft)J exp [ - j 2 ~ f ~ ( m ’ ) ] dm’

(as is shown in the Appendix) where m’ = m / M [9]. Now, using linear system theory [7], we can write

(2)

1

0

Ilo(t) = A, exp ( j2n f t )H( f ) . Since our interest is focused on H ( f ) , we write

H ( f ) = J’ exp (-j27rf~m’) dm’. (3) 0

We will explore the solution to (3) in the next section.

111. FIELD TRANSFER FUNCTION

In order to find a solution to (3), we need the functional form of ~ ( m ’ ) . For this, we turn to work by Gloge, et al. in [l]. In [l], a weakly-guiding approximation (A << 1) was used to solve for T ( S ) , where 6 is related to m’ by

m‘ = (S/A)a 0 < S < A (4)

p = (2 + a)/(Y.

~ ( 6 ) = A + BS + C(S’//a) Gloge’s approximation of ~ ( 6 ) is given by

( 5 )

Core

Fiber Cladding no\rc, 9

%“ T 1 u r

Fig. 1. Multimode optical fiber cross-section and index of refraction profile.

A = zn,/c

B = (zn,/c)(a - 2 /a + 2)

C = (zn0/2c)(3a - 2 / a + 2).

Now, using (4), we make (3) an integration over 6, which yields

H ( f ) = ( I / / ? ) ( ~ / L I ) ~ / ~ S ~ - ’ 0 exp [-j27rf~(S)] dS.

Substituting into (6) and completing the square on the argu- ment of the exponential, we get

H ( f ) = rJASD-’ exp [ - j 2 ~ f C ( S + B/2C)’] dS (6)

where we have defined r as I? = exp( - j2~fC[ (A/C) - (B/2C)’]}(l/p)(l/A)’.

For this investigation, we are interested in the transfer function in the region where alpha is near its optimum value, which is given in [l] to be

Therefore, we will let

If we evaluate constants A, B, and C at this value of alpha, we get

where we have defined To = zn,/c, and is the propagation delay for the zero-order mode. Since A is typically 0.005,

0

= 2 - 2A.

@ = 2 - € A , O < E < ~ .

A = To, B ‘V EAT,/^, C ‘V To/2

WALKER: A MODEL FOR THE a-PROFILE MULTIMODE OPTICAL FIBER CHANNEL: A LINEAR SYSTEMS APPROACH 1903

the above approximations for B and C are very good. Now returning to (6), we see that using the previous value of alpha gives a value of beta approximately equal to two. Then we can approximate (6) by

H ( f ) = rJA6exp [-j27rfC(6 + B/2C)'] d6. (7) 0

Now to evaluate (7); we start by making the substitution

p = 6 + B B / 2 C

which yields

H ( f ) = r J" ( p - B/2C) exp ( -j27rfCp2) dp cL1

Expanding H ( f ) , we get

where

YI (111 P Z ) = (1/2jwc) [exp (- jwCp1) - exp (-jwCpz)].

To evaluate the second integral, we make another substitution

= p(j27rfC)1/2.

Then we have for B < 0

~ ~ ( 1 1 ~ 1 2 ) = -(B/2C) (l/jwC)1/2exp (-1') d l . 1: Expanding, rearranging, and noting that C1 < 0 when B < 0, we get

YZ(&, t z ) = (~/4~)(1/j2f~)l/'[erf(~z) + erf(-&>].

Now H ( f ) becomes

H ( f ) = r{(l/(j4j7rfC))[exP (-j2KfPLI) - exp (-j27rfp:)] - (B/4C)(l / j2fC)l / ' . [erf(lz) + erf(-l1)11. (8)

Evaluating all constants and substituting into (8), we get

. [exp ( -jA2T07rf/4(t - 4/2)')] + ( E / ~ A ) ( l / j fTo)l /z [erf ( j 7 r f ~ ~ A ~ T ~ / 1 6 ) ~ ~ ~ + e r f ( j 2 ~ f t 2 A 2 T 0 / ( 1 6 / ~ 2 - S / E + 1)/16)'/']}

H ( f ) = {[(2 - t/2) sinc (AzT0(2 - t)7rf/4)]

. [exp (-j27rfTo(l - tzAz/32))]. (9) To determine the impulse response, we write (9) as

where

bz = E ~ A ~ T ~ ( ~ ~ / E ' - 8 / ~ + 1)/32

Ti = bz - bl = A2T0(4 - 2t)/8

and sin (x)

sinc (x) = -. X

Taking the inverse Fourier transform of (lo), we get

where

-1 t < O +1 t > O sgn (t) =

0 t < O { 1 t > 0 . U ( t ) =

Iv. FIELD mTENSITY TRANSFER FUNCTlON

The magnitude squared of the field transfer function is required in order to relate the input field intensity spectrum to the output field intensity spectrum. Therefore, we write (using (lo)),

1904 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 12, NO. 11, NOVEMBER 1994

where F* stands for the complex conjugate of F. In the following, we will sometimes refer to the magnitude squared response as the intensity transfer function. In the Appendix, we show that the intensity transfer function can be written in terms of the normalized frequency 52 as

where . L

I .

(1 = [(t - 4)2R/2~)1/2]

R = A2T,r f /4.

E(.) is the imaginary part of (.) and R(.) is the real part of (.). C(z) and S(z) are the Fresnel integrals and are defined as

C(z) = lz cos (7r?/2) d r

V. RESULTS Equation (13) is plotted in Fig. 2. Responses are shown as

functions of normalized frequency above the optical carrier frequency; responses below the optical carrier frequency are the mirror images of those shown. Normalized frequency responses are shown for values of E in the range from 0 to 4. We can see that the bandwidth (defined to be at the point where the magnitude squared response first reaches 1/2 of the response at R equals 0) is minimum where E equals 0, which corresponds to a equals 2; it reaches a maximum at E

equals 2, or a equals its optimum value [l]. The response at E equals 1 shows only a slight increase in bandwidth over that for E equals 0. While the suboptimum responses for E equal 3 and 4 are shown to have a greater bandwidth than responses at E equal 0 and 1, they both have considerable out-of-band ripple. In addition, Fig. 2 illustrates the very high sensitivity the frequency response has to very small changes in a near its optimum value, which are in effect, caused by changes in A. For example, for a percentage change in a approximately equal to A = 0.01, the transmission bandwidth of the fiber changes by a factor of 4.

The impulse response ( 1 1 ) is plotted in Fig. 3(a)-3(c). Again, the range of E is 0 to 4. The responses are shown to have a minimum width at t equals 2 (consistent with the maximum bandwidth point) and has a maximum width at epsilon equals 0. The impulse response shown in Fig. 3(a) is the same as

- 10’

Fig. 2. Magnitude squared frequency response versus normalized frequency.

that found by Gloge, et al. The responses at E equal 1, 3, and 4 show considerable time-spreading as compared to the optimum case.

Focusing on the normalized arrival and duration times for these responses, we see for E equal 0 the response starts at 1 (corresponds to the delay of the zero-order mode) and continues uniformly for A2/2 units above 1 . The response corresponding to E equals 2 starts where normalized time equals A2/8 units before 1 . This implies that some higher order modes are traveling faster than the zero-mode. Responses for E equals 3 and 4 move in the same direction as that for E equals 2, starting at 9A2/32 and A2/2 units before 1, respectively. For the final response, corresponding to the case where t equals 1 , we see that the arrival of the response begins A2/32 units before 1 and ends A2/4 units after 1 . In Figs. 3(a)-(c), analogous to the frequency response, the high sensitivity of time-spread to very small changes in a near its optimum value is very evident.

VI. CONCLUSION

In this paper, we have presented an alternative approach to the characterization of the frequency and impulse responses for the a-profile class of multimode optical fibers. The linear system approach has enabled us to demonstrate the very high sensitivity both transmission bandwidth and time-spread have to very small changes in a near its optimum value. In contrast, sensitivity to transmission bandwidth and time-spread is not very strong if a is in a small region about *A percentage points above or below the optimum value. The assumptions made in arriving at these characterizations are believed to be reasonable and will be adequate for modeling the multimode optical fiber channel under a variety of conditions in which the assumptions are satisfied. The exception to this may be the assumption of uniform power illumination of all propagating modes; however, when appropriate input lensing is employed, the assumption should be adequate for most system-level analyses.

We also believe that these results can be effectively em- ployed in modeling more complicated effects relating to mul-

1905 WALKER A MODEL FOR THE a-PROFILE MULTIMODE OPTICAL FIBER CHANNEL: A LINEAR SYSTEMS APPROACH

14/2-

1 2 2 - 2 2 1012- z

@A2

6/2 412

UA2

H 1 8/A2

-

-

- E=O -

16/21 16/A

1

E = I

(1+A?8) (1+A2/4) (1+3A%) t lTo

(b)

(C)

Fig. 3. equals three and four.

Normalized impulse response versus normalized propagation time for (a) epsilon equals zero and two (b); epsilon equals one; (c) epsilon

timode fiber transmission; for example, effects caused by variations in E as a function of fiber length; or perhaps, effects resulting when E is allowed to be a stochastic variable of fiber length. The author cites these examples to motivate further investigation in this area.

APPENDIX

A. Calculation of the Field Transfer Function

The input fields to the optical fiber can be represented as

where A, is the field amplitude. If the field is uniformly distributed over M propagating modes, and no modal mixing occurs, we can write the output field from the fiber as

where pm is the propagating constant for the mth mode, and iz is the fiber length. For real pm we can write

pmz = 27r f r ( m / M )

where 7(m/M) is the delay time for the mth mode. Then we can write the output field as

M

, 4 o ( t ) = ( A o / M ) exp {j2nf( t - r ( m / M ) ) ) . (16) m = O

From linear systems theory, we can write (16) as

, 4 o ( t ) = A O W ) exp (j2..ft> (17)

where M

~ ( f ) = exp { - j 2 7 r f 7 ( m / ~ ) ) ( 1 / ~ ) . (18)

For a large number of modes, i.e., as M approaches infinity, (18) can be approximated, using the theory of integration, as

m=O

1906 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 12, NO. 1 1 , NOVEMBER 1994

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B. Calculation of the Field Intensity Transfer Function

Starting with the field transfer function

H ( f ) = {[((2-t)/2)sinc(A2TO(2-t)7rf/4)] . [exp (-jA2T0n f /4(t - 4/2)’)] + ( t /4A)( l / j f T,)l/’[erf (j7r f E’A’T, /~~)~/ ’ + erf(j27rf~’A’T,/(16/~’ - S/E + 1)/16)’/’]} . [exp (-j27r fTo(l - ~’A’/32))]

and defining R = f / f o and f O = 4/7rA2T0, we have for the magnitude squared response

I ~ ( f ) l ’ = 1[(2 - c/2) sinc (2 - E)R] . exp [ - ~ R ( E - 4/2)’] + (~’7r/j64R)~/’[erf(jt~R/4)~/’ + erf{j(E - 4)/2)’~)’/’]1’.

Now we know from Euler’s identity that

exp (-jO(t - 4/2)’} = cos (E - 4/2)’R - j sin (E - 4/2)’R

and from [8] we get

e r f ( j 7 r ~ ’ / 2 ) ’ / ’ = (1 + ~ I ) { C ( Z ) - ~ s ( z ) ) where

C(z) = lz cos (7r7-’/2) d r

S(z) = Lz sin (7r7-’/2) d r

Now we define

(2 = [(t - 4)2R/27r]1/2.

Finally, for the magnitude squared response, we have

(H(S2)(2 = ((2 - ~ / 2 ) COS ((6 - 4)/2)’0) . sinc ((2 - E)R)

+ ((2 - ~ / 2 ) sin ((E - 4)/2)’R) . sinc ((2 - E ) O )

+ (27rE2/64fl)[C(Cd + c((2)1)2

+ (27rc2/640)[s(<1) + ~ ( ( 2 ) 1 } ~ .

ACKNOWLEDGMENT

The author wishes to acknowledge the following people for their support and assistance in the conduct of this research. First, he would like to thank Drs. B. O’Neal Jr. and A. Nilsson for their very helpful suggestions while conducting this research. Next, he wishes to thank Mr. W. Steltzenmuller, Dr. F. Noel, and Mr. P. Amigh for their support, which greatly contributed to the timely completion of this work. Finally, thanks go Mmes. Y. Sanders, V. Walker, and Mr. L. Chan for their considerable assistance in preparing this manuscript.

REFERENCES

[ I ] D. Gloge and E. A. I. Marcatili, “Multimode theory of graded-core fibers,” Bell System Tech. J . , vol. 52, pp. 1563-1578, Nov. 1973.

[2] R. M. Gagliardi and S . Karp, Optical Communications. New York Wiley, 1976.

[3] D. Marcuse, Theoty ofDielectric Waveguides. New York: Wiley, 1976. [4] Staff of the Bateman Project, Table of Integral Transforms, Vol. I. New

York: McGraw-Hill, 1954. [5] P. Henrici, Applied Computational Complexed Analysis, Vol. 2. New

York: Wiley, 1976. [6] G. P. Bois, Tables of Indejinite Integrals. [7] W. R. Bennet, Introduction to Signal Transmission. New York:

McGraw-Hill, 1970. 181 M. Abramowitz and I. Stegun, Eds. Handbook of Mathematical Func-

tions, 10th ed., National Bureau of Standards, Dec. 1972. [9] A. E. Taylor and W. R. Mann, Advanced Calculus. 2nd ed., Lexington,

MA: Xerox College Publishing, 1972.

New York: Dover, 1961.

Ernest L. Walker (S’6&M’68-SM’83) was born in Montrose, MS. He received the B.S. degree from the Indiana Institute of Technology, Fort Wayne, IN, the M.S. degree from Syracuse University, Syracuse, NY, and the Ph.D. degree from North Carolina State University, Raleigh, NC, all in electncal engineer- ing, in 1967, 1973, and 1982, respectively.

He has over 23 years of industrial experience in computer and communication systems, all with IBM Corporation. His most recent position before leaving IBM in 1990 was lead performance analyst

for internetworking products. He was also a member of the technical team which developed the IBM Token-ring Local Area Network. He is the holder of five U.S. patents, and several patent publications. Dr. Walker is currently with the Department of Electrical and Computer Engineering, University of Southwestern Louisiana, Lafayette, LA. His research interests are in the areas of communication theory, optical communications, communication networks, image and signal processing. and information storage and retneval.

Dr. Walker is a registered Professional Engineer in the states of North Carolina and West Virginia. He is a Director in the Pittsburgh Section of the IEEE, and holds membership in the communication and information theory societies. He is also a Member of the Society for Computer Simulation and Tau Beta Pi.