International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 10, October 2013 ISSN 2319 - 4847
1
2
3
1
Physics Department, College of Education, Mustansiriyah University
2 Physics Department, College of Education, Mustansiriyah University
3 Physics Department, College of Science , Thi-Qar University
Fiber optical parametric amplifiers (FOPAs) based on four-wave mixing occurring inside optical fibers, can provide a large and flat gain over a wide bandwidth when designed suitably. In this paper, the coupled propagation equat ions of the dual-pumps
FOPA were solved analytically by considering the fiber as a concatenation sections, each one with very small length. A novel recursive relation of the resulted signal and idler waves was found. This relation may be determined the FOPA gain as a function to the input signal. Thereafter, the phase mismatch condition are analyzed in order to maximize the FOPA characteristics. An accuracy of the present solution coincides the numerical one and gives an excellent results within a few number of iterations. It was found that the center of FOPA with respect to zero wavelength is very important property to enhance the FOPA gain.
However, the parameters of chromatic dispersion and the location of amplifier center are analyzed in order to calibrate the dualpumps FOPA.
In addition to provide broadband and high gain, fiber optical parametric amplifiers (FOPAs) are spectrally flexible, they can operate with low noise features and they offer the possibility to achieve simultaneously wavelength conversion.
Indeed, it has been experimentally demonstrated that they can exhibit a gain bandwidth of more than 200 nm [1], a net black-box gain up to 49 dB and a conversion efficiency of 38 dB [2]. Moreover, it has been also shown that FOPAs can directly generate a broad and flat gain region by using two pump lasers [3] or multiple fibers with different group-velocity dispersions [4]. The latter case has been recently checked against experimental measurements that demonstrate a flattened gain bandwidth of 75 nm in a two fiber FOPA scheme [5]. However, a major limitation of FOPA performances lies in the high pumping level. In response to this limitation, highly nonlinear fibers that have a large nonlinear coefficient [1], [2], [6] have been proved to be the best candidates for FOPA.
For future broadband wavelength division multiplexing (WDM) requires amplifiers to compensate the loss of fiber and equalize the power of the various channels, the semiconductor optical amplifier (SOA), the erbium doped fiber amplifier
(EDFA), and Raman amplifier (RA) have been applied widely to provide a flat gain spectrum. But low loss windows
(1250-1650 nm) of all-wave fiber can’t be utilized fully, therefore it is very important to seek a new optical amplifiers operating outside of SOA, EDFA and RA gain bandwidth, and then a new type of broadband amplifier, FOPA is put forward, which utilizes four wave mixing (FWM) to amplify the signals and offers a broadband amplification at arbitrary wavelengths [7]-[12]. Furthermore, FOPA can also provide small noise penalties that are lower than the 3-dB quantum limit and is used as broadband wavelength converters, return to zero pulse generation, and all-optical sampling,.. etc [13].
Though simple single-pump FOPA could offer a gain bandwidth of more than 200 nm, the gain spectrum is not flat over the amplifier bandwidth but has a difference as high as 15 dB between the lowest and the highest value [14], which brings to the difficulty to equalize the power of the various channels in WDM systems. For this reason, much research has been done recently to deal with the lack of flatness of the gain spectrum for single-pump FOPA [15], [16]. However, singlepump FOPA has several inherent problems. On the one hand, the pump frequency overlaps the signal band, which makes it difficult to filter out the pump. On the other hand, when single-pump FOPA is operated in CW mode, the idler spectrum is broadened due to the required pump dithering, which degrades the bit error rate (BER) when FOPA is used as wavelength converter. Therefore dual-pump FOPA is introduced [17], [18], which results in at least three distinct parametric processes that can be balanced to create a broad and flat CW gain spectrum by suitable use of dispersion, unattainable by a single-pump FOPA [19].
In this paper, the FOPA with the case dual pumps is analyzed. Thereafter, the coupled propagation equations are solved analytically under a certain approximations that consider the pumps power much more than the signal and idler powers.
This approximation is very sensitive in order to neglect the pumps depletion. As a consequence, the coupled propagation equation are solved in analytic manner over a small distance of fiber. This solution are then generalized to include the total fiber length. Also, the phase matching condition is interpreted under the effects of the wavelength difference between the pumps and the amplifier center with respect to zero dispersion wavelength.

International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 10, October 2013 ISSN 2319 - 4847
We consider an electric field consisting of four continuous waves (CWs), at frequencies four distinct waves interact with each other under the condition w
1
 w
4
 w
2 w
1
through w
4
. In NFWM, the
 w
3
. Two of these, number 1 and 2, are in this work used as pumps, while 3 is a signal, also denoted s, that is to be amplified, and 4 is the idler, i, that arises when the signal is amplified. Under the assumption that the four waves are sufficiently separated in frequency and the input fields are parallel polarized, the nonlinear Schrödinger equation (NLSE) can be reduced to four coupled differential equations for the complex field amplitudes [20], [21] dA j dz


2 j
A j
 i  j
[| A j
|
2
 2 k
4
 1 ,
 k 
| j
A k
|
2
] A j
 i  j
 m 
A m
A

A
* n e i   m  n z (1)
Where j  1 , 2 , 3 , 4 ,  
 mn
 
 mn
  n
 

  n
,  j
are the nonlinearity factors,  j are the wavenumbers, and  j
are the attenuation factors. Note that; the last term represents all combinations that satisfy the condition m    n  j  0 .
To solve Equation.(1) analytically, many assumptions must be done, which are: the attenuation factors are assumed to be equal, the nonlinearity factors also may be assumed a same, and the pump powers remain much larger than the signal and idler powers at all times and that the power losses of the pumps is negligible. Under these assumptions, Equation.(1) may be obtained dA
1 dz


2
A
1
 i  |

A
1
|
2
 2 | A
2
|
2

A
1
( 2 ) dA
2 dz dA
3 dz dA
4 dz


2
A
2
 i  |

A
2
|
2


2
A
3


2
A
4
 2 i  |

A
1
|
2
 2 i  |

A
1
|
2
 2 | A
1
 | A
2
 | A
2
|
2
|
2

A
2
(3)
|
2

A
3
 i  A
1
A
2
A
*
4 e  i   z
(4)

A
4
 i  A
1
A
2
A
3
* e  i   z
(5)
The later set of equations completely determines the propagation dynamics of the optical signals along the fiber with loss and without pump depletion. Using the powers definition P j
 A j
2
, one may be obtained dA j dz


2
A j where k 
 i 

P j
 2 P k
 e
 z
A j
2 , 1 if
(6) j  2 , 1 . Using the transformations A j
 B j e
 z/2 into the last equation will get dB j
 i  [ P j
 2 P k
] e  z B j
(7) dz
It is a straightforward to solve Equation. (7) to obtain B j
( z )  B j
( 0 ) exp[ i  ( P j
 2 P k
) z eff
] , where z eff the effective length of the fiber. The amplitudes
 ( 1 e  z ) /  is
B j may be returned to A j yield A j
( z )  A j
( 0 ) e
 z / 2 exp[ i  ( P j
 2 P k
) z eff
] . Using these results into Eqs.(4) and (5), yields dA j


2
A j
 2 i  ( P
1
 P
2
) e
 z
A j
 2 i  P
1
P
2 e
 z e  i   z e
3 i  ( P
1
 P
2
) Z eff A

4
(8) dz
Here k= 4, 3 if j= 3,4 . Now, using the transformations
A j
 B j exp [-

2 z -
2 i  ( P
1

 P
2
) e  z ] (9) into Equation. (8) and simplifying the result, we get dB j dz
 2 i  P
1
P
2 e
 z e  iKz
DB
 k
(10) where K      ( P
1
 P
2
)   
L
  
NL is the total phase mismatch, and D  exp[ 4 i  ( P
1
 P
2
) /  ] . Using Equation. (10) and its complex conjugates, we get the second order generalized equation
Z   H Z   4 
2
P
1
P
2 e
2  z
Z
Where H    iK
 0 (11) for the cases Z  B
3
, B

4
. The latter two equations can
' t be solved analytically except for the case e
2  z  1 and this approximation is satisfied for small length, i.e.  z .
In our analysis, the fiber is considered as a concatenation sections, each one with very small length. As a consequence,
Equation. (11) has the solutions

International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 10, October 2013
B
3
(z)
B
4

(z)


C
1 e m
1
 z  C
2 e m
2
 z (12a)
C
3 e m
1
*
 z  C
4 e m
*
2
 z
(12b)
ISSN 2319 - 4847
Where m
1
 
1
2
(   iK )  g , m
2
 
1
2
(   iK )  g , g 
1
2
(   iK ) 2  16  2 P
1
P
2
.
Note that; C's are constants that must be determined and g represents the gain parameter. To determine the constants
C's, the initial conditions are applied at z  0 to get



B
3
B 
4
(
(

 z)  z)





 a b b  a 


 


B
3
B 
4
( 0 ) 
(0)


(13)
At this point we get a complete solution for one section with  z length. Equation. (13) can be written in general form as



B
3
B
4

(( m
(( m


1 )
1 )
 z
 z
)
)



 T m
 B
3


B

4
( m  z )
( m  z )




 a

 b b
 a



 


B
3
B
4

( m  z )
( m  z )



( 14 )
Here, the matrix T m
represents the transfer matrix of the m th section, which in fact does not depend on the section index.
Their elements are defined as a  e

(   iK
2
) z


 cosh( g  z ) 
(   iK )
2 sinh( g  z g
) 

 b  e

(   iK ) z
2 2 i  D P
1
P
2 sinh( g g  z )
The conversion of the amplitudes B to j
A , will obtain the recursive formula j



A
3
(( m
A

4
(( m


1 )
1 )

 z) 
 z)

 e    z / 2


 v u
* where m  0,1,..
N  1 , and v u *


 


A
3
A

4
( m  z)
( m  z)



(15) u  [cosh( g  z ) 
(   iK )
2 sinh( g g  z )
] e ix  z v  2 i  D P
1
P
2 sinh( g g  z ) e i ( x  y )  z x y
 
 
2
 4  ( P
1

3
2
 ( P
1
 P
2
 P
2
)
) m
Equation (15) represents the most important achievement in this paper as it illustrates an analytical solution of the 2-P FOPA in presence of attenuation. The conversion efficiency of any FOPA is the ratio of the output power of the conjugated signal and the input signal power, i.e. for the FWM-based FOPA it is given by G
4
( L )  P
4
( L ) / P
3
( 0 ) . Also, the input signal exhibits amplification (or attenuation) due to the FWM process. The signal gain is defined as the output signal power to the input signal power, i.e. G
3
( L )  P
3
( L ) / P
3
( 0 ) . In particular, when there is no idler field at the input of the fiber, the signal gain becomes
G
3
( L )  e
  L
| s |
2
 e
  L


 cosh
2
( gL ) 
(   iK )
2
4g
2 sinh
2
( gL )



(16 a )
G
4
( L )  e
  L
| r |
2
 e
  L



4  2 P
1
P
2 g
2 sinh
2
( gL )



(16 b )
The above equations show that, to achieve flat and broadband gain, two conditions must be met. Firstly, K should be near some constant with low ripple over large-signal wavelength range, that means  K /  (  w s
) must be small. Secondly, absolute value of this constant should be as small as possible, otherwise the gain may be too low to be practical. In a word,
K must be close to zero with low ripple as  w s
increases. Equation.(16) shows that the small-signal gain spectrum of
FOPA depends only on four quantities, namely  ,  P
1
P
2
,  
NL
, and   and hence where maximum gain can be achieved. The parameters
. The wavevector mismatch
 P
1
P
2
and knowledge of fiber dispersion since they depend only on the pump powers and  .
 
NL
  is determined by the dispersion of the type of fiber used and plays a crucial role in determining the shape of the gain spectrum; its effect is studied in preceding. The nonlinear wavevector mismatch  
NL
determines at which wavelengths K will vanishes,
can be calculated without any

International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 10, October 2013 ISSN 2319 - 4847
The signal and idler gain spectra are symmetric with respect to the center frequency. It is convenient to use w c
and as the two independent parameters, instead of w
1
and w
2
 w d
, and to get maximum parametric gain in Equation. (16 a ), the total phase mismatch K should be equal to zero or when      ( P
1
 P
2
) , and this occurs at signal frequencies that satisfy the well-known phase matching condition, [22]
    ( P
1
 P
2
)  0 (17) and the linear phase mismatch   is given by [23]
 

2
  s
  i
  p
1
  p
2
 
3
( w c
 w
0
) 2  ( 
 
2
[(  w s
)
2
 (  w d
)
2
] 

4
12
[(  w s
)
4
 (  w d
)
4
]
(18)
4
2 )( w c
 w
0
) 4
(19) where  s , i , p
1
, p
2 are the signal, idler, pump one and pump two propagation constants, respectively, w c
 ( w
1
 w
2
)
 w s
 w
3
 w c
,  w d
 ( w
1
 w
2
) 2  w c
 w
1
 w
2
 w c
. Equation. (18) may be
2 ,
  (  c
)  ( R
1
(
  c

0
 c
) 
  c
R
2
(

0
 c
)
2
)((
 c


3
 c

3
)
2
)  (
( 
C
2

B
(
2
B 2 )
2
)
)
2

R
2
6
((
 c


3
 c

3
)
4
 (
( 
C
2
B 2
 ( B 2 )
2
)
)
4
) (20) where R
1
 
3
( 2  c ) 3 , R
2
 ( 
4
/ 2 )( 2  c ) 4 ,   c
  c
 
0
, 
1
 
0
 B / 2 , 
2
 
0
 B / 2 , B  
2
 
1
is the bandwidth, and 
2 is the second-order dispersion, and 
3
, 
4 are the third- and forth-order dispersions respectively, generally provided by manufacturers, w is the zero-dispersion wavelength ZDWL .
0
Therefore, adjusting separately each the pump central wavelength, ZDWL and two pump wavelengths, the magnitude and shape of the gain spectrum can be optimized. The B term contributes only when two pumps are used and is independent of the signal and idler frequencies. This difference provides the main advantage of 2-P FOPA over 1-P FOPA as the term can be used to control the phase mismatch.
B
By properly choosing the pump wavelengths, it is possible to use this term for compensating the nonlinear phase mismatch  ( P 
1
P
2
) stemming from SPM and XPM. As a result, the total phase mismatch K can be maintained close to zero over a quite wide spectral range after the first term is made small by balancing carefully different orders of fiber dispersion. The importance of the
B term can be best seen by comparing the phase-matching parameter K for 1- and 2-P
FOPA. Equation. (3.79) shows that it is hard to maintain this phase-matching condition over a wide bandwidth in a 1-P
FOPA. This is because    0 when the signal wavelength approaches the pump wavelength, and hence K   ( P
1
 P
2
) .
The net result is that the signal gain is considerably reduced in the vicinity of the pump wavelength, and the gain spectrum exhibits a dip when 1-P FOPA is used. However, in the case of 2-P FOPA, by choosing the pump frequencies properly, B can be used to compensate for nonlinear contribution to phase mismatch.
In what follows, we will demonstrate how can get high, flat, broadband gain spectrum for any realistic set of experimental
FOPA parameters. Our simulation of the transmission is characterized by a large number of different parameters, most of which do not change from simulation to another. For completeness, we list a typical parameters set below, and maintain deviations from these values whenever they occur except when refer to that. We set   10 W
 1 km , P
1
 P
2
 1 .
2 W , which could be easily achieved with a CW pump in a highly nonlinear fiber,

4
= 1 × 10
-52 s
4 km
, the third-order dispersion 
3
= 0.1
× 10
-36 s
3
L  0.2km
, the fourth-order dispersion km
, attenuation coefficient   0.046Km
-1 , and
N  20 ,  z  L N , ZDWL 
0
 1550 nm and in all the following figures, we have constrained
Equation. (19) for parameters used .

2 to value given by
Figure (1) represent a compare between the suggested model and numerical solution of the gain spectrum for 2-P FOPA at many values of bandwidth B and frequency detuning   c
  c
 
0
, each shape represent the gain spectrum at fixed parameters, it is evident that the best gain has been terminated by fixing   c

0
shifting. Whereas the reiteration number in the suggested model is only one. There was no changing when the reiteration number is increasing and this is may be happen because the numerical solution contains truncation and rotation errors. We note from the figure a very high matching between the two solutions for all figure cases. Since, when bandwidth small  c
 
0 values, but for high B , the optimum solution will be at high  c

B is small, the optimum gain happens at

0
values. Anyway, the flattening rate decreasing with bandwidth increasing and vise versa. At last, the vertical lines represent pumps positions, whereas, it is clearly evident that gain spectrum is not identical, and that because of identical lost around the amplifier center.

International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 10, October 2013 ISSN 2319 - 4847
Figure(1): The gain spectrum of 2P- FOPA for different B , where each curve represents the optimum gain by selecting the required  c
 
0
.
Figure (2) represents the gain spectrum resulting for 2P-FOPA by the adoption of several values to the amplifier center shifting  c
 
0
and each shape has the range of bandwidth B between 0  180 nm . Since, it is evident from the figure the fact that  c the  c
 
0

 
0
 0 can not achieve broad bandwidth for all values of B as long as the optimum gain does not happen at
0 . With broadening   c

0
value we notice possibility to achieving broad gain at certain values of bandwidth
B . Accordingly, we expect a balance between B and  c
 
0
to achieve the ideal gain and that is what will explain in the following figures.
Figure(2): The gain spectrum of 2P-FOPA as a function of B and  c
 
0
Figures (3), and (4) represent the relationship between B and  c
 
0
for several values of 
3 and 
4
. The two figures represent a very accurate normalization for amplifier operation. In Figure (3) evident that, at constant 
4 with varying 
3
, this will increase or decrease the range of   c

0
for the same value of B . This change, which caused by 
3
, increases with increasing 
4
. In general, at all change will be occurs because of 
3

4
values, there is a value for  c
 
0
that corresponding certain B , at this value, no
changes. Either in Figure (4), varying 
4 with proven 
3 also will increase or reduce the range of  c
 
0
for the same value of B , and there is a value for  c
 
0
corresponding certain B does not happens any change upon it when changes 
4 value. The important point from the two figures is that the value of B in
Figure (3) is bigger than in Figure (4) and this gives a value of greater bandwidth at the expense of increasing the value

International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 10, October 2013 ISSN 2319 - 4847 of  c
 
0
and that lead to reduce the flatness value of the spectrum and vice versa for Figure (4). In general, to getting larger bandwidth it is better to use fixed value of 
4 against varying in 
4 value, while, to get the best flatness it is better to use a stability value of 
3
against varying 
4 value. Balancing between the optimum spectrum less difficult.

3 and 
4
, makes the possibility of obtaining
Figure(3): B as function of  c
 
0 for different values of 
3 and 
4
.
Figure(4): B as function of  c
 
0 for different values of 
3 and 
4
.
For small bandwidth B , the optimum gain happens at small   c

0 values, but for high B , the optimum solution will be at high  c
 
0
values. The flatness rate decreases by increasing bandwidth and vice versa. The balancing between B and
  c

0
may be achieved the ideal gain. To get larger bandwidth, it is better to use fixed value of 
4 against varying in 
3 value, while, to get the best flatness it is better to use a constant value of 
3
against varying 
4
value. Balancing of 
3 and 
4
may be used to valid the optimum gain spectrum.

International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 10, October 2013 ISSN 2319 - 4847
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