Ø< 0  ºü g Å õ m Í Ž Ò ÞW Ä] K ¡X ì Ä — ¤V R Ë Ž ì ŏ Œ

÷ m

Ǖ ¤ õ m Í ) m• ¤  ‹ ˜ m]  §T  ƒ º] K ¤c Ü R w Š • ¤ R  Ò Å] k ù • ¥4 Á S Ë  U ê sX N ËÅ k Ä; c" e8 ý ° Ë Ñø v ÚP Ê ] Ø Ç

X

Ø< 0  ºü g Å õ m Í Ž Ò ÞW Ä] K ¡X ì Ä — ¤V R Ë Ž ì ŏ Œ

ý

—

¡­ £ ¢

^∗

·  ™ »ø ¶ B< 

@

/½ ¨d  ¦a Ë :@ /† < Ɠ § „   / B N† < Æõ , € ª œ 702-101 (2007¸   10 Z 4 16{ 9  ~ à Î6 £ §)

‘ :

r  7 Hë  H\ " f  H F  g$ 3 Ä »\ " f F Gœ íé ß – ` O Û ¼_  + þ AI – Ð ” > rF ½ + É Ã º e ”   H µ 1 ߓ É r F  g_ " to — : r (optical soliton)_ 



1 l x`  ¦ l Õ ü t½ + É Ã º e ”   H — ¸4 S q– Ð z  ´Ã º x 9  ) ‡Ã º  ë ß – (Raman)† ½ Ó`  ¦ Ÿ í† < Ê   H “ ¦ à º q ‚  + þ A à »ø @` ç   ~ ½ Ó

&

ñ

d ” `  ¦ ƒ  ½ ¨ “ ¦ F  g_ " to — : r_  1 l x% i † < Æ& h  : £ ¤$ í `  ¦ s K  “ ¦  ô  Ç . — ¸4 S q_   © œ  ñ > à º[ þ t_  › ' a> d ” `  ¦ ë ß – 7

á

¤   H ì  r$ 3 & h  µ 1 ߓ É r F  g_ " to — : r K  ” > rF ½ + É Ã º e ”   H Ó ü to & h “   › ¸| [ þ t“     © œ, € 9 כ ¹ô  Ç F  g$ 3 Ä »_  7 á xÀ Ó, Õ

ªo “ ¦ F  g_ " to — : r`  ¦ µ 1 ÏÒ q t½ + É Ã º e ”   H œ íl „  § 4  ° ú כ`  ¦ ½ ¨Ù þ ¡ . s    œ íl › ¸| [ þ t`  ¦  6   x “ ¦ à ºu r Ð 3 xƒ  

½

¨\  ¦ : Ÿ x # Œ 1 l x% i † < Æ& h Ü ¼– Ð î ß –& ñ  ) a _ " to — : r_  „    0 p xô  Ç   © œ@ /\  ¦ ½ ¨Ù þ ¡ .

PACS numbers: 42.65.Tg, 42.81.Dp, 42.65.Sf

Keywords: “¦ ú q‚+þA ûø@`ç ~½Ó&ñd”, ëߖ†½Ó, Fg_"to—:r, úu rÓýtYUs‚

Hasegawaü < Tappert [1]\  _  # Œ s  : r& h Ü ¼– Ð F  g_ " to 

— :

rs  \ V| ÷ &% 3 “ ¦ Mollenauer1 p x [2]\  _  # Œ z  ´+ « >& h Ü ¼– Ð

½

©" î  ) a s Ê ê, F  g_ " to — : r“ É r €    o \  ¦ ì  rí ß –\  _ ô  Ç  + þ A_ 

’

<

Hz  ´ \ O s  @ /6   x| ¾ Ó & ñ ˜ Ð\  ¦ „  ² ú ˜½ + É Ã º e ”   H [ j@ / & ñ ˜ Є  

² ú

˜ B ^ ‰– Ð y Œ •F  g~ à Γ ¦ e ”   [3,4].  ú ª“ É r  ; Ÿ ¤`  ¦ t   H F  g _

"

to — : r_  1 l x% i † < Æ& h  : £ ¤$ í “ É r  µ 1 Ï& h -0 A © œ   › ¸ (SPM)† ½ Ó õ

 ç  H5 Å q• ¸ ì  rí ß – (GVD)† ½ Ó`  ¦ Ÿ í† < Ê   H q ‚  + þ A à »ø @` ç  

~

½

Ó& ñ d ”  (nonlinear sch¨odinger equation, NSLE)Ü ¼– Ð [ O " î s

 0 p x  . t ë ß –, _ " to — : r_   ; Ÿ ¤s  x  ïœ í\ " f & 7 ›ž Ð

œ

í“    â Ä º_  _ " to — : r „  5 Å x`  ¦ “ ¦ 9 l  0 A # Œ “ ¦ à º† ½ Ó _

 % ò † ¾ Ó, 7 £ ¤ 3 à º ì  rí ß –† ½ Ó (TOD), & # Q (Kerr) > à º_  q 

‚



+ þ A& h  ì  rí ß –† ½ Ó, Õ ªo “ ¦ Mitschkeü < Mollenauer [5] µ 1 Ï

|

ô  Ç Ä »• ¸  ë ß – í ß –ê ø Í\  _ ô  Ç NLSE _ " to — : r_   µ 1 Ï& h -Å Ò 

à º s 1 l x † ½ Ó`  ¦ Ÿ í† < Ê # Œ S X ‰ © œ÷ &# Q  ô  Ç . ´ ú §“ É r l ” > r

ƒ



½ ¨ [ þ t“ É r [6–10] s  Qô  Ç † ½ Ó[ þ t`  ¦ Ÿ í† < Êô  Ç “ ¦ à º à »ø @` ç



~ ½ Ó& ñ d ”  (higher-order nonlinear Schr¨odinger equation, HONLSE)_  + þ AI – Ð  6 £ §õ  ° ú  s   6   xô  Ç :

iE

ξ

+ α

1

E

τ τ

+ α

2

|E|

²

E = iα

3

E

τ τ τ

+iα

4

(|E|

²

E)

τ

+ iα

5

E(|E|

²

)

τ

, (1)

#

Œl " f " é ¶s  \ O   H E(ξ, τ )“ É r …  ;…  ;y     o   H „  l  © œ_ 

Ÿ

í| à ̂  `  ¦ _ p   9, ξü < τ   H " é ¶s  \ O   H / B Nç ß – x 9  r ç ß –\ 

@

/ô  Ç ¼ # p ì  r`  ¦ _ p  “ ¦, α

1

, α

2

, α

3

, α

4

, x 9  α

5

  H " é ¶s 

∗E-mail: [email protected]

\ O

  H GVD, SPM, TOD,  µ 1 Ï& h   â  o (self-steepening), x

9

 SRS† ½ Ós  y Œ •y Œ • K { © œ ) a . d ”  (1)\  @ /ô  Ç ì  r$ 3 & h “    Ž “ É r

“

¦w n   (dark-solitary wave), µ 1 ߓ É r“ ¦w n   (bright-solitary wave), Õ ªo “ ¦ µ 1 ߓ É r N> h_  _ " to — : r[ þ ts  ” > rF † < Ês  ƒ  ½ ¨  [

þ

t [6–10]_  # Œ — ¸4 S q_  > à º s _  ] j€  • › ¸| [ þ ts  ë ß –7 á ¤

÷

&  H  â Ä º µ 1 Ï| ÷ &% 3  . t ë ß –, ‰ & ³F  t _  ƒ  ½ ¨\ " f [6–

10], SRS† ½ Ó_  > à º 4 Ÿ ¤™ èà º ° ú כ`  ¦ 2 [½ + É Ã º e ”    H & h s 

“

¦ 9÷ &t  · ú §€ Œ ¤ :¤ æ ´, α

5

= ν

5

+ iµ

5

/ ‘ ט + N Ê Á  ™ Ê Á  ™ 

¡



´š ¿Ê Á  ï 5 Ñ/ ‘ ×£  · G ± ØÐ M  . ' Í   P : † ½ Ó ν

5

“ É r  1 l xà º\  _ 

”

>

r   H s e ”  > à ºs  9 { 9 ì ø Í& h Ü ¼– Ð B Ä º  Œ •“ É r ° ú כ`  ¦ t  9 > í ß –\ " f Á ºr | ¨ c à º e ”  . ô  Ǽ # , µ

5

† ½ ӓ É r F  g$ 3 Ä »\ " f



ë ß –õ & ñ \  _ ô  Ç q ‚  + þ A& h  ì  rí ß – \  _ ô  Ç > à º– Ð" f & 7 ›ž Ð

œ

í ` O Û ¼ „   \ " f ] X @ /– Ð Á ºr | ¨ c à º \ O   H † ½ Ós  .   

"

f, d ”  (1)\ " f µ

5

E(|E|

²

)

τ

† ½ Ós  ˜ Ð& ñ ÷ &# Q  ô  Ç . s   â Ä

º, Agrawalü < Headley [11]\  _  # Œ  7 H_   ) a& h s  e ”   H Ó

ü

to & h Ü ¼– Ð  8 & h ] X ô  Ç " é ¶s  \ O   H HONLSE ~ ½ Ó& ñ d ” `  ¦



6 £ §õ  ° ú  s  % 3   H .

iE

ξ

+ α

1

E

τ τ

+ α

2

|E|

²

E = iα

3

E

τ τ τ

− iα

4

(|E|

²

E)

τ

+α

5

E(|E|

²

)

τ

+ iα

6

E(|E|

²

)

τ

, (2)

#

Œl " f α

1

= −

¹₂

β

2

/|β

2

|, α

2

= N

²

= γP

0

T

²

/|β

2

|, α

3

= β

3

/(6|β

2

|T ), α

4

= 2N

²

/(ω

0

T ), α

5

= N

²

(T

R

/T ), α

₆

= ²α

₅

s  9, ô  Ǽ #  ² ¿ 1“  X < s Ä »  H SRS† ½ Ó_ 

>

à º B Ä º  Œ •l  M :ë  Hs  . ¢ ¸ô  Ç, Á º " é ¶ r ç ß – τ =

(t − z/v

g

)/T, ξ = (z|β

2

|)/T

²

, Õ ªo “ ¦ Á º " é ¶ „  l  © œ

-274-

1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65

−10

−5 0

5

(a)

λ ( µ m)

β₂(ps²/km) β₃(× 10⁻² ps³/km)

1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65

10¹

10²

(b)

λ (µ m) P

0

(watt)

Fig. 1. (a) The plot of β

2

(λ) and β

3

(λ) in Eq. (12) which are fitted to the experimentally measured data in Ref. [12] as the dispersion coefficients of the quadruple- clad fiber. The filled area represents the wavelength for β

₂

< 0 and β

₃

< 0 in 1.467µm < λ < 1.595µm.

(b) The peak power required to launch a solitary wave, P

0

= N

²

|β

2

|/(γT ), as a function of λ for N = 1, γ = 3 W

⁻¹

km

⁻¹

, and T = 50 fs.

E(ξ, τ ) = A(ξ, τ )/P

₀^1/2

, # Œl " f A(ξ, τ )  H …  ;…  ;y     o

  H „  l  © œ Ÿ í| à ̂  s  . # Œl " f   à º[ þ t v

g

, β

2

, β

3

, T , x 9  P

0

“ É r y Œ •y Œ •, ç  H5 Å q• ¸, GVD, TOD, ` O Û ¼_   ; Ÿ ¤ T

FWHM

≡ 1.763T , Õ ªo “ ¦ þ j“ ¦u   0 >° ú כ`  ¦
  · p . > Ã º T

R

“ É r 

_  î  rì ø Í  ”  1 l xà º ω

0

\ " f  ë ß –-s 1 p q Û ¼& 7 ˜á Ô! 3 _  l Ö  ¦ l

\  ¦
 ? / 9 ㍠ H q ‚  + þ A> à ºs   ( [ jô  Ç ? /6   x`  ¦ 0 A

# Œ ‚ à Г ¦ [4,11]). ‰ & ³F , α

6

= 0“    â Ä º d ”  (2)_  ì  r$ 3 & h 

“



 Ø  æ    (front) + þ AI _  “ ¦w n  K  Agrawalü < Headley [11]\  _ K " f ½ ¨K ”     e ” Ü ¼ 9 “ ¦w n  _  1 l x% i † < Æ& h  : £ ¤

$ í

s  à ºu & h Ü ¼– Ð ƒ  ½ ¨ ) a  e ”  . d ”  (2)_  µ 1 ߓ É r_ " to — : r K  _

 ” > rF   H þ j  H Hong [14]\  _  # Œ µ 1 ß) €& ’   HX <, Õ ª K 

 ” > rF  l  0 AK " f  H — ¸4 S q_  > à º[ þ t  s \  : £ ¤& ñ ô  Ç ] j ô



Ç › ¸| [ þ t“ É r œ íl  { 9   ` O Û ¼_    © œ, F  g$ 3 Ä »_  7 á xÀ Ó, Õ ª o

“ ¦ þ j“ ¦u   0 >ü < ° ú  “ É r Ó ü to & h “   ] jô  ÇÜ ¼– Ð Å Ò# Q”   .

t ë ß –, Hong [14]_   7 Hë  H\ " f  H à ºu & h “   ~ ½ ÓZ O Ü ¼– Ð ½ ¨ K

”   ì  r$ 3 K ü < Ó ü to & h “   › ¸|  \ " f # Q‹ "   1 l x`  ¦ ˜ Ðs 





H\  @ /ô  Ç ƒ  ½ ¨  H ”  ' Ÿ ÷ &t  · ú §€ Œ ¤Ü ¼ 9, s \  ¦ ‘ : r  7 Hë  H\ 

"

f Æ Ò½ ¨ “ ¦  ô  Ç .

 7

Hë  H_  ¢ - a„  $ í `  ¦ 0 A # Œ, €  $   7 Hë  H [14]\ " f µ 1 ϳ ð ) a µ

1

ߓ É r_ " to — : r_  ” > rF › ¸| \  @ / # Œ  A ü < ° ú  s  & ñ o  “ ¦



 ô  Ç . ¹ ¡ §f ” s   H _ " to — : r_  K \  ¦ ½ ¨ l  0 A # Œ d ”  (2)_  K

  6 £ §õ  ° ú  s  Å Ò# Q”   “ ¦ & ñ  

E(ξ, τ ) = E

0

sech (τ + χ ξ) e

^{i(K ξ−Ωτ )}

, (3)

Table 1. The coefficients corresponding to the four dif- ferent wavelengths toward the ZDW using the physi- cal parameters as N = 1, T = 50 fs, T

R

= 3 fs, γ = 3 W

⁻¹

km

⁻¹

, and ² = 0.1 which yield α

1

= 0.5, α

2

= 1.0, α

5

= 0.06, and α

6

= 0.006.

λ α

3

α

4

|E

⁰0

| P

0

(watt) 0.922λ

ZDW

−0.0026 0.0156 0.5163 595.8 0.975λ

ZDW

−0.0847 0.0495 0.5219 325.7 0.995λ

ZDW

−0.5374 0.0505 0.5239 76.4 0.999λ

ZDW

−2.7944 0.0507 0.5243 15.7

#

Œl " f E

0

  H 4 Ÿ ¤™ èà º  ; Ÿ ¤`  ¦ _ p ô  Ç . " é ¶s  \ O   H z  ´ Ã

º   à º[ þ t Ω, χõ  K“ É r î  rì ø Í  ”  1 l xà º ω

0

 Ò' _  s 1 l x ) a

”



1 l xà º, ç  H5 Å q• ¸– РÒ' _  s 1 l x ) a 5 Å q• ¸, Õ ªo “ ¦    o ) a   1

l

xà º\  ¦ y Œ •y Œ • _ p  ô  Ç  [6–8,10]. d ”  (3)\  ¦ d ”  (2)\  @ /{ 9 

# Œ > í ß – €  , E

0

x 9  Ω\  ¦  6 £ §õ  ° ú  s  % 3 >  ) a .

E

0

= ± s

−6α

3

(3 α

₄

+ 2 iα

₅

− 2α

₆

) , Ω = (3α

2

α

3

+ 3α

1

α

4

+ 2iα

1

α

5

− 2α

1

α

6

)

6α

3

(α

4

+ iα

5

) , (4)

#

Œl " f K(E

0

, Ω) = K

re

(E

0

, Ω) + iK

im

(E

0

, Ω)ü <

χ(E

0

, Ω) = χ

re

(E

0

, Ω) + iχ

im

(E

0

, Ω) [14]. _ " to — : rs  ” > r F

 l  0 AK " f K, χ, x 9  Ω ° ú כ[ þ ts  z  ´Ã º° ú כs  ÷ &# Q  Ù ¼

–

Ð, K

im

= 0ü < χ

im

= 0s  כ ¹½ ¨÷ & 9 s   H  6 £ §õ  ° ú  “ É r › ¸

|

d ” `  ¦ € 9 כ ¹– Ð ô  Ç .

(α

1

α

4

+ 3 α

2

α

3

)

²

· F(α

i

) = 0, (5)

(α

₁

α

₄

+ 3 α

₂

α

₃

)

³

· G(α

_i

) = 0, (6)

#

Œl " f F(α

i

) = (3α

4

− 2α

6

) · (....)   H α

i

_  4 Ÿ ¤™ èà º † < Êà º s

“ ¦ G(α

i

) = (α

4

− α

6

)(α

4

− α

6

+ iα

5

)

²

s  .   " f, d ”  (2)_  µ 1 ߓ É r_ " to — : rs  ” > rF  l  0 Aô  Ç › ¸| d ” `  ¦  6 £ §õ  ° ú   s



α

₁

α

₄

+ 3 α

₂

α

₃

= 0 ⇒ α

₄

= − 3 α

2

α

3

α

1

> 0, (7)

½

¨ >  ÷ &  HX < › ¸| d ” s  ë ß –7 á ¤   H s Ä »  H α

4

 0s   





 z  ´Ã ºs l  M :ë  Hs  . ¢ ¸ô  Ç α

2

= N

²

> 0 s Ù ¼– Ð, s   H α

3

/α

1

< 0 or β

2

β

3

> 0 (8)

`



¦ _ p ô  Ç . d ”  (7)Ü ¼– Ð Â Ò'  ½ ¨ô  Ç α

3

= −α

1

α

4

/3α

2

\  ¦ d

”

 (3)\  @ /{ 9  “ ¦ 4 Ÿ ¤™ èà º ”  ; Ÿ ¤`  ¦ F Gý a³ ð> – Ð
 ? /€  

E(ξ, τ ) = |E

₀⁰

| sech(τ + χξ)e

i(Kξ−Ωτ +φ)

, (9)

 ÷ & 9 # Œl " f

|E

₀⁰

| = s

2α

1

α

4

α

2

[(3α

4

− 2α

6

)

²

+ 4α

²₅

]

^1/2

, χ = − α

1

¡ α

42

+ 3 α

22

¢

3α

₂

α

₄

, K = 2α

1

α

22

3α

42

, Ω = α

2

α

4

, φ = −tan

⁻¹

µ 2α

5

3α

4

− 2α

6

¶

(10)

s

 . β

2

x 9  β

3

\  @ /ô  Ç ] j€  • › ¸| “ É r  6 £ §õ  ° ú   : ”  ; Ÿ ¤ s

 € ª œ_  à º s Ù ¼– Ð |E

⁰₀

|, β

2

< 0“ É r α

1

= −

¹₂

β

2

/|β

2

| > 0s  l

 M :ë  H\  $ í w n  “ ¦ ¢ ¸ô  Ç d ”  (8)\ " f β

2

β

3

> 0s l  M :ë  H

\

,  A ü < ° ú  “ É r › ¸| `  ¦ Ä »• ¸½ + É Ã º e ”  .

β

2

< 0 and β

3

< 0. (11) {

9

ì ø Í& h “   | 9 5 Å q+ þ A(graded-index) F  g$ 3 Ä »  H\ " f  H β

3

° ú כs 

†

½

Ó © œ € ª œÃ ºs  9 β

2

° ú כ“ É r 6 £ §Ã º [4]s l  M :ë  H\  0 A\ " f ½ ¨ô  Ç

›

¸| `  ¦ ë ß –7 á ¤½ + É Ã º \ O  . t ë ß –, d ”  (11)s  ë ß –7 á ¤| ¨ c à º e ” 





H B | 9 – Ѝ  H ¨ î ì  rí ß –(dispersion-flattened) F  g$ 3 Ä » e ” Ü ¼ 9 1.3-1.6 µm   © œ@ /\ " f  ×  æ (s ×  æ ¢ ¸  H  ×  æ) 9 þ tA ` ç

>

8 £ x`  ¦ t  9 ± ú “ É r ì  rí ß –> à º_  : £ ¤$ í `  ¦ ”    [4].   

"

f ‘ : r  7 Hë  H\ " f  H  A ü < ° ú  “ É r ì  rí ß – > à º° ú כ`  ¦  ×  æ-9 þ tA 

` ç

F  g$ 3 Ä » z  ´+ « >\ " f 8 £ ¤& ñ ô  Ç X <s ' \  ¦  6   x # Œ   © œ\ 

@

/ # Œ   H ô  Ç † < Êà º\  ¦  6   x ô  Ç  [12].

β

2

(λ) = −392.3 + 1140.2 λ − 1013.8 λ

²

+284.1 λ

³

(ps)

²

/km, β

3

(λ) = −5.3052 × 10

⁻⁵

λ

²

(1140.2 − 2027.6 λ

+852.3 λ

²

) (ps)

³

/km (12) Fig. 1(a)\ " f β

2

and β

3

 1 l xr \  6 £ §s  ÷ &  H   © œ

@

/\  ¦ ˜ Ð# ŒÅ Ò 9, 7 £ ¤, 1.467 µm < λ < 1.595 µm (G 0 >”  

% ò

% i ), ì  rí ß –s  0s  ÷ &  H ¿ º   © œ[ þ t (ZDWs) λ

_ZDW,1

= 1.315 µmü < λ

ZDW,2

= 1.595 µm\  ¦ ˜ Ð# Œï  r . Fig. 1(b)  H 

 © œ\  _ ” > r   H þ j@ / 0 > P

0

(λ) = N

²

|β

2

|/(γT

²

) (N = 1, γ = 3 W

⁻¹

km

⁻¹

, T

R

= 3 fs, and T = 30 fs) ZDWs\ 

"

f ¿ º> h_  F G™ è° ú כ`  ¦ ˜ Ð# ŒÅ ҍ  HX < s    © œ\ " f œ íl  ` O Û ¼

\



¦  Œ •1 l xr v l \  & h ] X † < Ê`  ¦ · ú ˜ à º e ”  . s ü < ° ú  “ É r & h “ É r  7 H ë



H [13]\ " f > h| ¾ ӝ ) a q ‚  + þ A à »ø @` ç   ~ ½ Ó& ñ d ”  (MNLSE)

`



¦  6   x # Œ  7 H_  ) a   e ”  . d ”  (7)`  ¦ ì  rí ß –> à º[ þ t– Ð

³ ð‰ & ³ €  

ω

0

= 2 β

2

(λ)

β

3

(λ) (13)

 ÷ &“ ¦ s \  ¦  6   x # Œ d ”  (9)\ " f _ " to — : rK  ” > rF 

l  0 Aô  Ç  ×  æ-9 þ tA ` ç F  g$ 3 Ä »\ " f_    © œ“ É r λ

sol

≡

0.922 λ

ZDW,2

' 1.475 µm – Ð ½ ¨ô  Ç  ( A \ " f λ

ZDW

≡ λ

ZDW,2

– Ð é  H ).   " f, ì  r$ 3 & h “   ~ ½ ÓZ O Ü ¼– Ð ½ ¨ô  Ç _ " to 

— :

rK   H ZDW (€  • 92 % of ZDW)   H% ƒ\ " f ” > rF † < Ê`  ¦ · ú ˜ Ã

º e ” Ü ¼
,  ×  æ-9 þ tA ` ç F  g$ 3 Ä »_  B | 9 õ  l  † < Æ& h “   כ ¹

™

è\  ¦ “ ¦ 9 €  , _ " to — : rs  ” > rF    H   © œ@ /\  ¦  8¹ ¡ ¤ ZDW

–

Ð ] X   H ½ + É Ã º e ” `  ¦  כ Ü ¼– Ð \ V © œ ) a .

Table 1“ É r Ó ü to & h “     à º[ þ ts  N = 1, T = 50 fs, T

R

= 3 fs, γ = 3 W

⁻¹

km

⁻¹

, s “ ¦ ² = 0.1{ 9 M : α

3

, α

4

, |E

₀⁰

|, Õ ª o

“ ¦ œ íl  þ j@ / 0 > P

0

° ú כs    © œs  ZDW– Ð ] X   H½ + É M :_ 

° ú

כ[ þ t`  ¦ ˜ Ð# Œï  r . Y >  t  < É ªp – Ðî  r  Òà º& h “   › ¸| d ” [ þ t, 7

£

¤ F(α

i

) = 0¢ ¸  H G(α

i

) = 0`  ¦  6   x €   d ”  (13)ü <  

 É

r   © œ › ¸| [ þ t`  ¦ ½ ¨½ + É Ã º e ” Ü ¼ 9 Table 2\  & ñ o  % i   ( 8  [ jô  Ç ? /6   x“ É r  7 Hë  H [14] ‚ à Л ¸).

s

 © œ\ " f  7 H_  ) a _ " to — : r_   1 l x`  ¦ s K  l  0 AK 

"

f, d ”  (2)\  ¦ ° ú ˜ f ” -> é ß – É Òo \  (split-step Fourier method) à ºu ~ ½ ÓZ O `  ¦ & h 6   x “ ¦ / B Nç ß –& h “   „     H Crank- Nicholson~ ½ ÓZ O `  ¦ & h 6   x “ ¦ Å Òl & h “    â > › ¸| `  ¦  6   x ô



Ç .  6   x ) a œ íl  _  › ¸| _  + þ AI   H E(ξ = 0, τ ) =

|E

₀⁰

| sech(τ )exp[i(φ − Ωτ )]s “ ¦ Ω x 9  φ  H d ”  (4)\ " f Å Ò

# Q”   .

Fig. 2a-d  H Table 1\  K { © œ÷ &  H œ íl  [ þ t_  à ºu  > í ß –





õ \  ¦ ˜ Ð# Œï  r . Fig. 2a\ " f ˜ Ѝ  H  ü < ° ú  s  d ”  (13)`  ¦ ë

ß

–7 á ¤   H œ íl  ` O Û ¼  H ¸ n u/ '  H-_ " to — : r ' Ÿ 1 l x`  ¦ ˜ Ðs €  " f ξ = 100ì  rí ß –  o  t , { 9 ì ø Í& h “   d ç / å J-9 þ tA ` ç z  ´o   F  g

$ 3

Ä »_   â Ä º
 
  H x  ïœ í  ; Ÿ ¤_  œ íl  \  e ” # Q" f_  TOD† ½ Ó x 9   ^ ‰- â  o † ½ Ó\  _ ô  Ç  + þ A_  ~ Õ ª Qf ”  [4] \ O  s

, î ß –& ñ ÷ &>  ”  ' Ÿ  ) a . _ " to — : rs  s    î ß –& ñ $ í `  ¦ ˜ Ðs   H s

Ä »  H α

1

, α

2

, α

3

, and α

4

ü < ° ú  “ É r > à º[ þ ts  d ”  (7)_  ] j€  •

›

¸| `  ¦ & ñ S X ‰ >  ë ß –7 á ¤ l  M :ë  Hs  . Fig. 2b-d  H   © œ

›

¸|  d ”  (13)s  0 Aì ø Í÷ &• ¸2 Ÿ ¤   © œ`  ¦ λ = 0.999 λ

ZDW

A á ¤Ü ¼

–

Ð    o ½ + É M : œ íl  _  „   \  ¦ ˜ Ð# Œï  r . Fig. 2bü < Õ ª_  ç

ß

–/ B G‚  \ " f ˜ Ð1 p ws   Å Ò €  •ç ß –_    © œ› ¸| _  0 Aì ø Ís  e ”   H

 â

Ä º, 7 £ ¤ λ = 0.975 λ

ZDW

, # Œ„  y  î ß –& ñ  ) a _ " to — : r_  „  5 Å x

`



¦ ˜ Ð# Œï  r . t ë ß –, Fig. 2c\ " f ˜ Ð1 p ws    © œs  U  ´# Qt 

€



 λ = 0.995λ

ZDW

„  5 Å x÷ &  H  _  ¡ óo  Òì  r\  ”  1 l x÷ &  H 

+ þ As 
 
  HX < s   H TOD† ½ Ó |α

3

| ∼ |β

3

|/|β

2

| >> 1s  β

2

→ 0ü < λ → λ

ZDW

{ 9 à º2 Ÿ ¤ t 
u >  & t l  M :ë  Hs  .



t } Œ •Ü ¼– Ð, Fig. 2d\ " f λ = 0.999 λ

ZDW

“    â Ä º, œ íl  

 Y >  ì  rí ß – U  ´s \  ¦ „    l  „  \  ¢ - a„  y  ™ èY > ÷ &# Q ! Q a

Ë

>`  ¦ · ú ˜ à º e ”   HX < B Ä º  H TOD° ú כ`  ¦  © œ W½ + É Ã º e ”   H “ ¦

à º† ½ Ó_  ° ú כ[ þ ts  ß ¼t  · ú §l  M :ë  Hs  .

Fig. 2\ " f_  _ " to — : r_   1 l x x 9  „   ÷ &  H  1 l x`  ¦ 7 á §



8  [ jy  s K  l  0 A # Œ, Fig. 3a-c\ " f  _  þ j“ ¦u ,

Table 2. Four solitary-wave solutions obtained by applying the auxiliary constraints, i.e., F(α

i

) = 0 or G(α

i

) = 0, which are used for the initial profiles for the numerical simulations in Fig. 4.

|E

0⁰

| Auxiliary constraint Wavelength constraint

I p

2α

1

α

4

/α

2

(α

²₄

+ 4α

²₅

)

^1/2

α

4

= α

6

6= 0 β

2

(λ)/β

3

(λ) = 1/(²T

R

)

II p

2α

1

/3α

2

α

5

= α

6

= 0 β

2

(λ)/β

3

(λ) = ω

0

III |E

0⁰

| = p

2α

1

α

4

/α

2

(9α

²₄

+ 4α

²₅

)

^1/2

α

6

= 0 β

2

(λ)/β

3

(λ) = ω

0

IV |E

0⁰

| = p

α

1

α

4

/α

2

α

5

3α

4

= 2α

6

6= 0 β

2

(λ)/β

3

(λ) = 2/(3²T

R

)

−100 −50 0 50 100

0 50 100

0 0.5 1

(a)

ξ τ

|U(ξ,τ)|

−1000 −50 0 50 100

50 100

−100 −50 0 50 100

0 50 100

0 0.5 1

(b)

ξ τ

|U(ξ,τ)|

−1000 −50 0 50 100

50 100

−100 −50 0 50 100

0 50 100

0 0.5 1

(c)

ξ τ

|U(ξ,τ)|

−1000 −50 0 50 100

50 100

−100 −50 0 50 100

0 50 100

0 0.5 1

(d)

ξ τ

|U(ξ,τ)|

−1000 −50 0 50 100

50 100

Fig. 2. Evolutions of the solitary-wave solution of Eq. (2) up to ξ = 100 using the coefficients in Table 1 when the wavelength constraint of Eq. (13) is violated from the exact value (a) λ = 0.922 λ

ZDW

to the perturbed values (b) λ = 0.975 λ

ZDW

, (c) λ = 0.995 λ

ZDW

, and (d) λ = 0.999 λ

ZDW

. Note that the instability occurs after λ > 0.975λ

_ZDW

due to huge increase in the TOD term. Note that the zero boundary condition on the left computational window for (c) and (d) is used to eliminate the radiation reentering into the right window. However, the periodic boundary conditions are used for (a) and (d).

_

"

to — : r_  FWHM (full-width at half maximum)  ; Ÿ ¤õ ,



6 £ §õ  ° ú  s  & ñ _  ) a \  -t  Q(ξ) ≡

Z

_∞

−∞

|E(ξ, τ )|

²

dτ / Z

_∞

−∞

|E(0, τ )|

²

dτ, (14)

\

 @ /ô  Ç „     o \  @ /ô  Ç † < Êà º– Ð y Œ •y Œ • Õ ª§ 4  . Fig. 3a\ 

"

f ˜ Ð1 p ws , λ = 0.922λ

ZDW

(solid line){ 9  M :ü < λ = 0.975λ

ZDW

(& h ‚  ){ 9 M : þ j“ ¦  0 >  H „    o  1 l xî ß – { 9 

&

ñ

 9 _ " to — : r_  FWHM\ " f_   ; Ÿ ¤“ É r Á ºr ½ + É Ã ºï  r_  כ

¹1 l x`  ¦ ˜ Ðs  9 Fig. 3a-bõ  { 9 u † < Ê`  ¦ · ú ˜ à º e ”  . ô  Ǽ # , z



´‚   x 9  & h ‚  Ü ¼– Ð ³ ðr  ) a \  -t [ þ t“ É r ξ = 100t & h \ " f

€



• 1 %& ñ • ¸ œ íl  ° ú כ˜ Ð  ×  ¦# QŽ  H . Fig. 3a-c\ " f  H,  



©

œs  λ = 0.995λ

ZDW

(& h - ‚  ) and λ = 0.999λ

ZDW

(U  ´

>

 = å S# Q”   ‚  ){ 9  M :, þ j“ ¦  0 >  H  o \      Ø Ô>  ×  ¦

0 10 20 30 40 50 60 70 80 90 100

0 0.5 1

P

p

( ξ ) (a)

λ=0.922λ_ZDW λ=0.975λ_ZDW λ=0.995λ_ZDW λ=0.999λ_ZDW

0 10 20 30 40 50 60 70 80 90 100

10 20 30

FWHM

(b)

0 10 20 30 40 50 60 70 80 90 100

0.6 0.8 1

ξ

Q( ξ )

(c)

Fig. 3. Evolutions of (a) the peak power P

p

(ξ) ≡ max(|E(ξ, τ )|

²

), (b) the pulse FWHM, and (c) the nor- malized total energy Q(ξ) for the wavelengths varying from the necessary constraint λ = 0.922λ

ZDW

toward the ZDW. Note that for λ > 0.975λ

ZDW

the normalized total energy goes beyond unity after ξ = 30 due to the instability caused by a huge increase in the TOD term and the sudden drops of the total energies for dotted and dot-dashed curves caused by the zero boundary con- ditions.

#

Q[ þ t 9 x 9  FWHM  ; Ÿ ¤s  7 £ x   H  כ `  ¦ ^  ¦ à º e ”  . s  ü

< ° ú  “ É r   õ \ " f, î ß –& ñ & h “   _ " to — : r`  ¦ Ò q t$ í l  0 AK " f

œ

íl  _    © œ“ É r d ”  (13)`  ¦ ë ß –7 á ¤K   t ë ß – €  •ç ß –_  0 A ì

ø

͝ ) a   © œ@ / 4λ ≈ 0.5λ

ZDW

\ " f _ " to — : r_  î ß –& ñ $ í `  ¦ S X ‰

“



 % i  .



6 £ §\ " f Table 2\ " f ì  rÀ ӝ ) a (I-IV) _ " to — : r 7 á xÀ Ó\  @ / ô



Ç Ã ºu & h “    1 l x`  ¦ ¶ ú ˜( R˜ Ð . Fig. 4“ É r y Œ • _ " to — : r[ þ t_ 

„



 ü < K { © œ ç ß –/ B G‚  `  ¦ ˜ Ð# Œï  r . ' Í   P : _ " to — : rI\  K  {

©

œ÷ &  H ] j€  •› ¸|  α

4

= α

6

, i.e., β

2

(λ)/β

3

(λ) = 1/(²T

R

)`  ¦ ë

ß

–7 á ¤   H  â Ä º, Fig. 4a  H €  •ô  Ç ¸ n u/ '  H-_ " to — : r_  ' Ÿ 1 l x`  ¦

˜

Ð# ŒÅ Ò 9 Fig. 5aü < b\ " f ˜ Ð1 p ws   0 >ü < FWHM  ; Ÿ ¤

•

¸ €  •ô  Ç Å Òl & h “      o\  ¦ ˜ Ð# Œï  r . t ë ß –, \  -t    



o  H  _  Á ºr ½ + É Ã ºï  rs  9, _ " to — : r1“ É r î ß –& ñ ÷ &>  „   H † d

`



¦ · ú ˜ à º e ”  . Fig. 4b  H MLSE\ " f β

2

(λ)/β

3

(λ) = ω

0

7

£

¤ λ = 0.922λ

ZDW

› ¸|  \ " f % 3 # Qt   H _ " to — : rII_  „  

−100 −50 0 50 100 0

50 100

0 0.5 1

(a) I

ξ τ

|U(ξ,τ)|

−1000 −50 0 50 100

50 100

−100 −50 0 50 100

0 50 100

0 0.5 1

(b) II

ξ τ

|U(ξ,τ)|

−1000 −50 0 50 100

50 100

−100 −50 0 50 100

0 50 100

0 0.5 1

(c) III

ξ τ

|U(ξ,τ)|

−1000 −50 0 50 100

50 100

−100 −50 0 50 100

0 50 100

0 0.5 1

(d) IV

ξ τ

|U(ξ,τ)|

−1000 −50 0 50 100

50 100

Fig. 4. Evolutions of the four different solitary-wave solu- tions in Table 2. (a) α

5

= α

6

6= 0 and (d) 3α

4

= 2α

6

6= 0 show the evolutions of the traveling solitary-waves. (c) α

5

= α

6

= 0 corresponding to Ref. [15] and (d) α

6

= 0 corresponding to the HONLSE in Ref. [11] show the evolutions of the stationary-solitary waves.



  © œI \  ¦ ˜ Ð# ŒÅ Ò 9, s   â Ä º• ¸ _ " to — : r_  þ j@ /  0 > x 9  FWHM  ; Ÿ ¤ (Fig. 5a-c)s  ξ = 100 t     o  _  \ O  6

£

§`  ¦ S X ‰“  ½ + É Ã º e ” “ ¦ î ß –& ñ  ) a „    H † d`  ¦ · ú ˜ à º e ”  .  7 H ë



H [11]_     : rõ   H ² ú ˜o , Table 2\ " f _ " to — : rIIIü < ° ú  “ É r K

 ” > rF † < Ê`  ¦ · ú ˜ à º e ” Ü ¼ 9, s   H α

6

= 0“   & h s  _ " to 

— :

rIIü < 7 á xÀ Ó  Ø Ô
   © œ\  @ /ô  Ç ] jô  Ç › ¸| “ É r { 9 u ô  Ç



. s   â Ä º Fig. 4c\ " f ¸ n u/ '  H-_ " to — : r_  ' Ÿ 1 l xs   8 y © œ

†

<

Ê`  ¦ ^  ¦ à º e ” Ü ¼ 9, Fig. 5a-b\ " f þ j@ /  0 >  H { 9 & ñ  t

ë ß – FWHM  ; Ÿ ¤_     o  H
p u`  ¦ · ú ˜Ã º e ” “ ¦, Fig. 5c\ 

"

f ˜ Ð# ŒÅ ҍ  H \  -t   H    o| ¾ Ós  p p  l  M :ë  H\  î ß –& ñ

 )

a _ " to — : r „    0 p x† < Ê`  ¦ · ú ˜ à º e ”  .  t } Œ •Ü ¼– Ð, Fig.

5d \ " f _ " to — : rIV_   1 l x`  ¦ ˜ Ð# ŒÅ ҍ  HX < s   H _ " to — : rIõ 

° ú

 “ É r î ß –& ñ  ) a „    s À Ò# Qf ” `  ¦ · ú ˜ à º e ”  . s  © œ_  à º u

r Ð 3 xƒ  ½ ¨\ " f α

6

† ½ ӓ É r ß ¼l   Œ •t ë ß – ZDW   © œ% ò % i 

\

" f _ " to — : r`  ¦ µ 1 ÏÒ q tr v l  0 AK " f  H ì ø Í× ¼r  “ ¦ 9÷ &# Q 

  H † ½ Óe ” `  ¦ · ú ˜ à º e ”  .

‘ :

r  7 Hë  H\ " f  H, F  g$ 3 Ä »\ " f F Gœ íé ß – ` O Û ¼_   1 l x`  ¦ l  Õ

ü

t½ + É Ã º e ”   H — ¸4 S q“   z  ´Ã º x 9  ) ‡Ã º  ë ß – (Raman)† ½ Ó`  ¦

Ÿ

í† < Ê   H “ ¦ à º q ‚  + þ A à »ø @` ç   ~ ½ Ó& ñ d ” _  µ 1 ߓ É r_ " to — : r K

 ” > rF ½ + É Ã º e ” 6 £ §`  ¦ d ”  (9)ü < ] jô  Ç › ¸| d ”  (7)õ  d ”  (11)\ " f ˜ Ð% i  . ] jô  Ç › ¸| d ” \ " f,  © œ  ñ > à º[ þ t_  › ' a>  d

”

`  ¦ ë ß –7 á ¤   H ì  r$ 3 & h  µ 1 ߓ É r F  g_ " to — : rK  ” > rF ½ + É Ã º e ” 





H Ó ü to & h “   › ¸| [ þ t“     © œ, € 9 כ ¹ô  Ç F  g$ 3 Ä »_  7 á xÀ Ó, Õ ª o

“ ¦ F  g_ " to — : r`  ¦ µ 1 ÏÒ q t½ + É Ã º e ”   H œ íl „  § 4  ° ú כ`  ¦ ½ ¨Ù þ ¡ .

Table 2\ " f ˜ Ð1 p ws  4t  µ 1 ߓ É r _ " to — : r[ þ ts  y Œ •y Œ •   É r

0 10 20 30 40 50 60 70 80 90 100

0 0.5 1

P

p

( ξ ) ^(a)

^III

III IV

0 10 20 30 40 50 60 70 80 90 100

5 10 15 20

FWHM

(b)

0 10 20 30 40 50 60 70 80 90 100

0.9 1 1.1

ξ

Q( ξ ) ^(c)

Fig. 5. Evolutions of (a) the peak power P

p

(ξ) ≡ max(|E(ξ, τ )|

²

), (b) the pulse FWHM, and (c) the nor- malized total energy Q(ξ) of Fig. 4 for the amplitudes and wavelength constraints specified in Table 2. In (c), Q(ξ) after ξ ≈ 50 is either slightly increasing or decreas- ing due to the weak instability occurring at the trailing edge of the traveling-solitary waves (solid and dashed curves) or the effective power loss term operating for the stationary solitary-waves (dot and dot-dashed curves).

]

jô  Ç › ¸| \ " f ” > rF † < Ê`  ¦ µ 1 ß+ À I . à ºu r Ð 3 xƒ  ½ ¨\  ¦ : Ÿ x 

#

Œ, _ " to — : rI-V[ þ ts  î ß –& ñ  ) a „    s À Ò# Q| 9  à º e ”   Ht 

\



¦ s K  % i  . : £ ¤y ,  ë ß –† ½ Ó α

6

_  % i ½ + Éõ  ß ¼l   H  Œ •t  ë

ß

– ZDW   © œ% ò % i \ " f _ " to — : r`  ¦ µ 1 ÏÒ q tr v l  0 AK " f  H ì

ø

Í× ¼r  “ ¦ 9÷ &# Q    H כ ¹™ èe ” `  ¦ ˜ Ð% i  .

P c

p 8 ý ò k >

‘ :

r ƒ  ½ ¨  H 2007¸   @ /½ ¨d  ¦a Ë :@ /† < Æ_  t " é ¶F K\  _ ô  Ç  כ {

9

m  .

Y c

p w Š à U Ø ”  ô

[1] A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973); 23, 1171 (1973).

[2] L .F. Mollenauer, R. H. Stolen and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).

[3] A. Hasegawa and K. Kodama, Solitons in Opti- cal Communications (Oxford University Press, New York, 1995.)

[4] G. P. Agrawal, Nonlinear Fiber Optics 3rd ed., (Aca-

demic Press, New York, 2001.)

[5] F. M. Mitschke and L. F. Mollenauer, Opt. Lett. 11, 659 (1986).

[6] K. Porsezian and K. Kakkeeran, Phys. Rev. Lett.

76, 3955 (1996).

[7] M. Gedalin, T. C. Scott and Y.B. Band, Phys. Rev.

Lett. 78, 448 (1997).

[8] D. Mihalache, N. Truta and L. C. Crasovan, Phys.

Rev. E 56, 1064 (1997).

[9] Z. Li, L. Li, H. Tian and G. Zhou, Phy. Rev. Lett.

84, 4096 (2000).

[10] S. L. Palacios, A. Guinea, J. M. Fern´andez-D´ıaz and R. D. Crespo, Phys. Rev. E 60, R45 (1999).

[11] G. P. Agrawal and C. Headley III, Phys. Rev. A 46, 1573 (1992).

[12] F. M. Sears, L. G. Cohen and J. Stone, J. Lightwave Technol. LT-2, 181 (1984).

[13] D. C. Calvo and T. R. Akylas, Phys. Rev. E56, 4757 (1997) and references therein.

[14] W. P. Hong, Z. Naturforsh 58a, 667 (2003).

[15] D. Anderson and M. Lisak, Phys. Rev. A 27, 1393 (1983).

[16] M. Florja´ nczyk and L. Gagnon, Phys. Rev. A 41, 4478 (1990).

[17] J. A. C. Heideman and B. M. Herbst, SIAM J. Nu- mer. Anal. 23, 485 (1986).

[18] M. F. S. Ferreira and S. C. V. Latas, Opt. Eng. 47, 1696 (2002).

[19] H. Tian, Z. Li and G. Zhou, Opt. Commun. 205, 221 (2002).

Dynamics of the Bright Solitary Waves in the Higher-order Nonlinear Schr¨ odinger Equation with Both Real and Imaginary Raman Terms

Woo-Pyo Hong

^∗

and Jong-Jae Kim

Catholic University of Daegu, Department of Electronics Engineering, Hayang 712-702 (Received 16 October 2007)

We investigate the dynamics of the bright solitary waves of the higher-order nonlinear Schr¨odinger equation (HONLSE) with both real and imaginary Raman terms, which can model an ultrashort pulse propagating through optical fibers. The analytic bright solitary-wave solutions are found under a constraint on the model coefficients, from which the physical parameters, such as the wavelength needed to launch the pulse, the types of optical fibers, and the required peak power, are obtained. Using the bright solitary waves as the initial profiles for numerical simulations, we show that dynamically stable or marginally stable pulse propagations can be achieved for certain ranges of the wavelengths.

PACS numbers: 42.65.Tg, 42.81.Dp, 42.65.Sf

Keywords: Higher-order nonlinear Schr¨odinger equation, Raman terms, Optical soliton, Numerical

∗E-mail: [email protected]