THE GAIN AND OPTIMAL LENGTH IN THE ERBIUM-DOPED FIBER AMPLIFIERS WITH 1480 NM PUMPING

354 _{IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 4, NO. 4, APRIL 1992}

The Gain and Optimal Length in the

Erbium-Doped Fiber Amplifiers

with

1480 nm Pumping

Min-Chum Lin and Sien Chi

Abstract-The approximate analytic expressions for the gain

and optimal length of the 1480 om-pumped erbium-doped fiber amplifier with an arbitrary dopant distribution are obtained when the saturation by ASE is neglected. Numerical calculations show that the valnes predicted from the analytic expressions of

the maximal gain and optimal fiber length as functions of input signal and pump powers are very accurate for narrow dopant distributions confined to the center of the fiber core.

INTRODUCTION

HE erbium-doped fiber can provide a broad-band, high-

T

gain, and low-noise amplification for optical signals of wavelengths near 1530 nm. In [l], an analytic solution of the

rate equations for an annual dopant distribution has been derived and the maximal gain at the optimal fiber length has been found, but the dependence of signal and pump powers on the fiber length are lost. In [2], the analytic expression for signal gain as an function of fiber length has been obtained. In this’letter, we will find the approximate analytic expres- sions for gain and optimal length of the erbium-doped fiber amplifier with an arbitrary dopant distribution. Numerical calculations

are

carried out for Gaussian dopant distribution and are compared with the analytic results.

THEORY

Regarding the erbium-ions doped in the single-mode fiber as homogeneously broadened two-level systems, we know that the population densities N , and N2 of the lower and upper levels, respectively, are determined by the rates of stimulated absorption and emission and spontaneous emission between the lower and upper states. When the maximal gain is less than 20 dB, the amplified spontaneous emission can be neglected [3]. Thus, at stationary conditions we obtain [4]

hv, Is

1

( u i N , - up’N2)- I P

+

(u,”N,

-

u,’N2)- _(la)

h V P or

Manuscript received November 12, 1991; revised January 10, 1992. This work was supported by National Science Council and Telecommunication Laboratories of R.O.C. under contract NSC-814417-E009-05.

The authors are with the Institute of Electro-Optical Engineering and

Center for Telecommunications Research, National Chiao Tung University, Hsinchu, Taiwan, Republic of China.

IEEE Log Number 9107076.

where Z; = h v p / r u i , Zs‘p = hvp/rup’, Z: = hv,/ru,“ and

Z: = hv, /TU,’. v, and up are the signal and the pump fre- quencies, respectively; h is Planck’s constant; r is the spontaneous emission decay lifetime; U,” and U,’ are the

stimulated absorption and emission cross sections of the signal beam, while up” and up’ are the stimulated absorption and emission cross sections of the pump beam, respectively; I, and Zp are the optical intensities of the signal and pump beams, respectively; the total dopant density distribution

Nt ( r ) = N , ( r )

+

N 2 ( r ) where N,, N , , and N2 are all as- sumed to be radially symmetric. We define Z,(z, r ) = P , ( z ) f , ( r ) and I P ( z , r ) = P p ( z ) f p ( r ) where P , ( z ) and

Pp( z ) are the signal and pump powers, and f,( r ) and f p ( r )

are the normalized signal and pump transverse intensity profiles, respectively. If efficient long-wavelength pumping is used to avoid excited-state-absorption effects [5], we can assume f , ( r ) =

f p ( r )

= f ( r ) for the following derivations. The signal and pump powers will evolve according to the following propagation equations:

OD

-

dP, = 2 7 r L Z,[u,’N2(r)

-

u,”N,(r)]rdr (2a)

dz

5

= ~ 2 7 r L OD Zp[up”Nl(r) - upe ”(r)] rdr. (2b)

dz

where the minus and plus signs in (2b) correspond to coprop- agating and counter-propagating pump waves, respectively. A.

Maximal Gain

Substituting (lb) into (2), we have

If we neglect the constant “1” in the denominators of (3) and

LIN AND CHI: GAIN AND OPTIMAL LENGTH IN ERBIUM-DOPED FIBER AMPLIFIERS 355

divide (3a) by (3b), we have

where

R

=

10”

N,( r)

f

(r)rdr

/

/0”

N,( r) rdr. The pumping efficiency

I

dP,/dPp

1

increases with

R

for all

P,

and

Pp.

For any length of the erbium-doped fiber, (4) can be inte- grated by parts to yield, for both copropagating and counter- propagating pumping

From

(3,

we can find R by measuring

Pyt,

Pyt

for given

P;’”

and

P;.

For the maximal gain,

dP,

/dPp

= 0, we have the output pump power

1

(6)

pout =

p ’ O p

R(u,’/u:Z$

- l/Z$) .

If the maximal gain is defined as G =

Pyt/P;’”,

then it can be calculated from the following equation:

pin vp

P;‘“

P . 0 ,

(7) B.

Optimal Fiber Length for Maximal Gain

By combining (la) and (2) and defining the confinement factor

r

= A/,”

N2(r)

f(r)rdr//;

N2(r)rdr

where A is the effective doped area, we have the propagation equation for the signal beam:

dP,

=

-P,

a,+ --

+-

--

hv,/P,”

-

dz

[

(

h

:

,

2

ht,

2)

]

(8)

where a, = 2 TU:/: N,( r)

f

(r)rdr is the absorption con- stant of the signal beam, and

P,’s

=

Ahv,/TI’(u:

+

U,’) is the intrinsic saturation power of the signal beam. Solving the differential equation (8), we obtain the output signal power at z = L p y = p,’“ pp” - pyt p;’” - py1

+

li

h v p hv,

-

exp

(

- a,L)exp (9) Without pump beam,

Pyt

= P;’”exp ( - a,L) for small input

signal power, and then a, =

log(P;’”/P?‘)/L.

When

P;’”

approaches to

PF,

P,””‘

= P,‘Sexp ( 1 - a,

L).

It implies that both absorption constant and intrinsic saturation power for the signal beam can be obtained from the monochromatic absorption measurement [2]. The optimal fiber length Lo, for

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 l.’O

WIDTH RA TI0 W / W o

Fig. 1. The exact (solid lines) and the approximate (dashed lines) maximal gains as functions of the width-ratio ( w / w,) .between Gaussian dopant distribution and transverse intensity profile for PF = 15 mw.

the maximal gain can be obtained by reorganizing (9), and

p , o p -

P;

P;’”

h V P

hv,

* ( G )

+

2

[ p o u t

+

- ( G - l ) ] ] . (10)

In summary, we can obtain the optimal fiber length for the maximal gain in the following way. At first, a piece of erbium-doped fiber with length

L

is chosen. From ( 5 ) , we can obtain

R

by measuring

P,““‘

and

P F

for given

PF

and

P;.

Subsequently we calculate

P

l

:

,

from (6), and solve G from (7) for arbitrary

Psi”

and

PT.

Then we can obtain a,

and

PJS

from the absorption measurement for the signal beam. Finally, the optimal fiber length can be calculated by substituting G into (10) for arbitrary

P;’”

and

P;.

The accuracy of the optimal fiber length calculated depends on that of the maximal gain calculated. The deviation of the optimal fiber length AL,, varies with the deviation of the

maximum gain AG as AL,, = - A G ( l / G

+

P;’”/Pss)/as.

CALCULATIONS AND DISCUSSION

To check the approximate expression of the maximal gain by numerical calculations, we take

f(r)

in Gaussian form,

f (r) = exp ( - r 2 / w,’)/ T

w,’

where

w o

is the spot size, and the effective core area

TWO’

= 35 pm2 [6].

N,(r)

is

also assumed to be in Gaussian form,

N,(r)

2:

exp ( - r 2 / w 2 ) / r w 2 . For the Al/P-silica erbium-doped fibers, typically we have parameters 7 = 10.8 ms, U,’ = 5.7

x cm2 for A, = 1530

nm; U,’ = 0.87 x IO-’’ cm2 and up” = 2.44 x cm2 for A, = 1480 nm [7]. a, is arbitrarily chosen to be 0.5 r n - l and then

P,”

is larger than 0.342 mw. We can compare the exact maximal gain calculated from (3) with that approxi-

mated by (7). Fig. 1 shows the exact (solid lines) and the approximate (dashed lines) maximal gains as functions of width ratio w /

w o

when the input pump power is fixed at 15

mw. At

w /

w o

= 0 . 4 , the difference ratios between approxi- cm2, and U,” = 6 . 6 x

356

“1

W/Wo=O. 4

0

INPUT PUMP POWER (mw)

Fig. 2. The exact (solid lines) and the approximate (dashed lines) maximal gains as functions of the input pump power for Gaussian dopant distribution and transverse intensity profile with width-ratio w / wo = 0.4.

mate and exact maximal gains (100% x

AG/G)

are

-2.56%, -1.962, -0.831%, and -0.221% for

P,”

=

0.001, 0.01, 0.1, and 1.0 mw, respectively; and then the difference ratios between approximate and exact optimal fiber

lengths (100% x AL,/L,) ,are less than 0.125%, 0.173%,

0.226%, and 0.178% for

P,”

= 0.001, 0.01, 0.1, 1 mw,

respectively. Fig. 2 shows that tlie maximal gain approxi- mated by (7) matches the exact maximal gain calculated from (3) very well for

P,“

>

0.001 mw and

:

P

>

5 mw when

w /

w o

= 0.4. For alumino-silicate glasses, it is easy to con-

fine the erbium-ions to the central 25 % of the fiber core [8].

IEEE PHOTONICS TECHNOLOGY LE’ITERS, VOL. 4, NO. 4, APRIL 1992

Therefore, the approximate analytic expressions are useful in real cases.

CONCLUSION

We have obtained approximate analytic expressions for the

maximal gain and optimal length of the erbium-doped fiber amplifier with an arbitrary dopant distribution. Numerical calculations show that

the

values predicted

from

the analytic expressions of the maximal gain and optimal fiber length as

functions of input signal and pump powers are very accurate for narrow dopant distributions confined to the center of the fiber core.

REFERENCES

M. Peroni and M. Tamburrini, “Gain in erbium-doped fiber ampli- fiers: a simple analytical solution for the rate equations,” Opt. Lett.,

A. A. M. Saleh, R. M. Jopson, J. D. Evancow, and J. Aspell,

“Modeling of gain in erbium-doped fiber amplifiers,” IEEE Photon.

Technol. Lett., vol. 2, pp. 714-717, 1990.

C. R. Giles and E. Desurvire, “Modeling erbium-doped fiber ampli-

fiers,” J. Lightwave Technol., vol. 9, pp. 271-283, 1991.

-

, “Propagation of signal and noise in concatenated erbium-doped fiber optical amplifiers,” J. Lightwave Technol., vol. 9, pp. R. I. Laming, S. B. Poole, and E. J. Tarbox, “Pumping excited-state absorption in erbium-doped fibers,” Opt. Lett., vol. 13, pp. R. 1. Mears, L. Rcekie, I. M. Jauncey, and D. N. hyne, “Low noise erbium-doped fiber amplifier ope* at 1.54 pm,” Electron. Lett.,

W. J. Miniscalco, “Erbium-doped glasses for fiber amplifiers at 1500

nm,” J. Lightwave Technol., v d . 9 , pp. 234-250, 1991. B. J. A i n s l i , “A review of the fabrication and properties of erbium- doped fibers for optical amplifuers,” J. Lightwave Technol., vol. 9, vol. 15, pp. 842-844, 1990.

147-154, 1991.

1084- 1086, 1988.

vol. 23, pp. 1026-1027, 1987.