If we assume the s component field has a zero relative phase shift with respect to the phase lead of the f component field, then the exit fields Ef’ and Es

Appendix Ⅰ

The Relationship between the Elements of the Jones Matrix and a Waveplate

Suggest that a waveplate with retardation ∆ is orientated at the angle θ.

As shown in Fig. AⅠ(a), considering the orthogonal field components E_x and E_y impinging on a birefringence medium which has fast and slow axes (f and s) orientated at angle θ. The total field E_f, and E_s along the medium axis is therefore can be expressed as [A1]:

Ef = Ex sinθ + Ey cosθ (AⅠ-1) Es = Ex cosθ - Ey sinθ (AⅠ-2) These fields propagate through the medium with different velocities and possibly different attenuation. If we assume the s component field has a zero relative phase shift with respect to the phase lead of the f component field, then the exit fields E_f^’ and E_s^’ , as shown in Fig. Fig. AⅠ(b), are given by

Ef’ = Ef Tfe^jφ (AⅠ-3) Es’ = Es Ts (AⅠ-4) where φ is the phase lead ( for instance, λ/2 waveplate would have a phase lead of π ) and T_f and T_s are field amplitude transmission coefficients. With respect to the x and y axes, the output fields E_x^’ and E_y^’ are therefore given by

Ex’ = Es’ cosθ + Ef’ sinθ (AⅠ-5) Ey’ = Es’ cosθ – Ef’ sinθ (AⅠ-6) Substituting (AⅠ-1) to (AⅠ-4) into (AⅠ-5) and (AⅠ-6). By manipulating we obtain

⎥⎦

⎢ ⎤

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

+

−

−

= +

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

y x

s j

f s

j f

s j

f s

j f

y x

E E T

e T T

e T

T e

T T

e T E

E

θ θ

θ θ θ

θ

θ θ θ

θ θ

θ

ϕ ϕ

ϕ ϕ

2 2

2 2

' '

sin cos

cos sin cos

sin

cos sin sin

cos cos

sin (AⅠ-7)

By comparing equation AⅠ-7 with

⎥⎦

⎢ ⎤

⎣

⎥⎡

⎦

⎢ ⎤

⎣

=⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

y x

yy yx

xy xx

y x

E E J J

J J E

E

' '

(AⅠ-8)

Then the Jones matrix elements can be determined as:

Jxx＝Tfe^jφsin²θ＋Tscos²θ (AⅠ-9) Jxy＝Jyx＝(Tfe^jφ－Ts)cosθsinθ (AⅠ-10) Jyy＝Tfe^jφcos²θ＋Tssin²θ (AⅠ-11)

s θ

Y

E_y sinθ

Ey cosθ

E_x sinθ

Ex cosθ E_y

Ex

E_s^’ E_f^’

Ef’ sinθ X

Es^’ sinθ

Es’ cosθ E_f^’ cosθ

Y

θ

s

f f

X

(a) (b)

Fig. AⅠ Decomposition of field to derive the Jones matrix coefficients

Appendix Ⅱ

The Derivation of Power Intensity of a

Symmetric Resonator Laser Constructed of Fiber Loop Mirrors from Electrical Field

As shown in Fig. 2-6, the transmission coefficient, reflection coefficient, reflectance and transmittance from the discussion in chapter 2 can be expressed as [10]:

t = (1-2K)(1-γ)e^(-α+jβ)L (2-13) jr = 2j K 1−K (1-γ) e^(-α+jβ)L (2-14) R = r*r = 4 K(1-K) (1-γ)²e^-2αL (2-15) T = t*t = (1-2K)²(1-γ)²e^-2αL (2-16) Define the one round-trip gain parameter G as:

G= g×e^{(−αg+jβ)L2} (2-20) Define the one round-trip optical power transmittance coefficient of the WDM MUX and DeMUX as:

α_Wij =α_Wi×α_Wj (2-21) If the input amplitude of the electrical field is E_in, then we can find that the output amplitude of the electrical field E_out can be denoted as [8]:

( )^{2 5 5 ...}

3 3+ −

−

=tt E G rrtt E G tt rr E G

E_{out i j in}α_{Wij i j i j in}α_{Wij i j i j in}α_Wij (2-22)

in Wij

j i

Wij j i

out E

G r

r

G t

E t _{2 2}

1 α

α

= + (2-23) Then the output power intensity can be given as:

2 2 2 2 2

1 _{i j Wij} ⁱⁿ

Wij j i out

out E

G r

r

G t E t

I ×

= +

= α

α (2-24)

Now we derive every term in Eq. 2-24 by substituting Eq. 2-13 to Eq. 2-16 and Eq. 2-20 into Eq. 2-24. First, the value of E_in ² is equal to I_in. Then

* 2 2 2 2

2

2 2 1 1

1 ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟ +

⎟

⎠

⎞

⎜⎜

⎝

⎛

= +

+ rr G

G t t G

r r

G t t G

r r

G t t

Wij j i

Wij j i

Wij j i

Wij j i

Wij j i

Wij j i

α α α

α α

α (AⅡ-1)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟ +

⎟

⎠

⎞

⎜⎜

⎝

⎛

= +

+ ^{* * 2* 2}

* *

*

* 2 2

2

2 2 1 1

1 r r G

G t

t G r

r

G t t G

r r

G t t

Wij j i

Wij j i

Wij j i

Wij j i

Wij j i

Wij j i

α α α

α α

α (AⅡ-2)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+ +

= +

+ ^{2 2 * * 2 2 * * 4 4}

2 2

* 2 *

2 2 1

1 rr G r r G rr r r G

G t

t t t G

r r

G t t

Wij j j i i Wij

j i Wij

j i

Wij j j i i

Wij j i

Wij j i

α α

α

α α

α (AⅡ-3)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+ +

= +

+ ^{2 2 * * 2 2 4 4}

2 2 2

2 2 1

1 rr G r r G RR G

G T

T G

r r

G t t

Wij j i Wij

j i Wij

j i

Wij j i

Wij j i

Wij j i

α α

α

α α

α (AⅡ-4)

From Eq. 2-20,

) * ( )

2 (g e( ^{g j L2})(g e ^{g j L2})

G = × ^{−α + β} × ^{−α + β} (AⅡ-5) )

)(

( ^{( ) 2 ( ) 2}

2 _g j L _g j L

e g e

g

G = × ^{−α + β} × ^{−α − β} (AⅡ-6)

2 2

2

2 _gL

e g

G = ^{− α} (AⅡ-7)

4 2

4

4 _gL

e g

G = ^{− α} (AⅡ-8) Moreover, from Eq. 2-14,

] )

1 ( 1 2

][

) 1 ( 1 2

[ ) )(

(jr_i jr_j = j K_i −K_i −γ_i e^{(−αi+jβ)L1} j K_j −K_j −γ _j e^{(−αj+jβ)L3}

) ) (

( 1 3 _{1 3}

) 1 )(

1 ( ) 1 )(

1 (

4 _{i j i j i j} ^{L L j L L}

j

ir K K K K e e

r =− − − − − ^{− i + j +}

− γ γ ^{α α β}

) ) (

( 1 3 _{1 3}

) 1 )(

1 ( ) 1 )(

1 (

4 _{i j i j i j} ^{L L j L L}

j

ir K K K K e e

r = − − −γ −γ ^{−αi +αj β +} (AⅡ-9)

* *

* _j ( _{j i})

i r r r

r =

) ) (

(

*

* _j 4 _{i j} (1 _i)(1 _j)(1 _i)(1 _j) ^L1 ^L3 ^{j L}₁ ^L₃

i r K K K K e e

r = − − −γ −γ ^{−αi +αj − β +} (AⅡ-10)

From Eq. AⅡ-4 to Eq. AⅡ-10,

2 2

) ) (

(

) ) (

2 ( 2

* 2 *

2

]}

) 1 )(

1 ( ) 1 )(

1 ( 4

[

] )

1 )(

1 ( ) 1 )(

1 ( 4

{[

3 3 1

1

3 3 1

1

G

e e

K K

K K

e e

K K

K K G

r r G r

r

Wij

L L L j L j i

j i

j i

L L L j L j i

j i

j i Wij

j i Wij

j i

j i

j i

α

γ γ

γ γ α

α

α β α

α β α

+ + −

− + +

−

−

−

−

−

+

−

−

−

−

= +

2 2 ) ( ) (

) 2 (

2

* 2 *

2

]}

[

) 1 )(

1 ( ) 1 )(

1 ( 4

{

3 1 3

1

3 1

G e

e

e K

K K

K G

r r G r

r

Wij L

L j L L j

L L j i j i

j i Wij

j i Wij

j i

j i

α

γ γ α

α

β β

α α +

− +

+

−

+

−

−

−

−

= +

(AⅡ-11) Derived from Euler equation:

)) (

cos(

2 _{1 3}

) ( )

( _{1 3 1 3}

L L e

e^{jβ L+L} + ^{−jβ L+L} = β +

(AⅡ-12)

From Eq. 2-15,

) (

2 2

2 (1 )(1 ) 1 3

) 1 )(

1 (

16 _{i i i j j j} ^{L L}

j i

j

e i

K K

K K

R

R = − −γ − −γ ^{− α +α}

)

( 1 3

) 1 )(

1 ( ) 1 )(

1 (

4 _{i j i j i j} ^{L L}

j i

j

e i

K K

K K R

R = − − −γ −γ ^{−α +α} (AⅡ-13)

Substituting Eq. AⅡ-12, Eq. AⅡ-13 and Eq. AⅡ-7 into Eq. AⅡ-11, then ))

( cos(

2 ^{2 2 2} _{1 3}

2 2

* 2 *

2 ₂

L L e

R R g G

r r G r

r_{i j}α_Wij + _{i j} α_Wij = α_{Wij i j} ^{− αgL} β + (AⅡ-14) Substituting Eq. AⅡ-14, Eq. AⅡ-7 and Eq. AⅡ-8 into Eq. AⅡ-4, then

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+ +

= +

+ ^{− −}

−

2 2

2

4 4 4 3

1 2 2

2

2 2 2 2

2 2

)) (

cos(

2

1 1 _{Wij i j} ^L _{i j Wij} ^L

L Wij

j i

Wij j i

Wij j i

g g

g

e g R R L L e

R R g

e g T T G

r r

G t t

α α

α

α β

α

α α

α

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+ +

= +

+ ^{− −}

−

2 2 2 2 3

1 2 2

2

2 2 2 2

2 2

) (

)) (

cos(

2

1 1 ^{2 2}

2

L Wij

j i L

j i Wij

L Wij

j i

Wij j i

Wij j i

g g

g

e g R

R L

L e

R R g

e g T T G

r r

G t t

α α

α

α β

α

α α

α

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

− + +

= +

+ ^{− −}

−

] 1 )) (

[cos(

2 ) 1

1 ( ^{2 2 2 2 2 2 2} _{1 3}

2 2 2 2

2 2 _{2 2}

2

L L e

R R g e

g R

R

e g T T G

r r

G t t

L j i Wij

L Wij

j i

L Wij

j i

Wij j i

Wij j i

g g

g

β α

α

α α

α

α α

α

From the half angle formula:

2 2 cos sin²θ =1⁻ θ

then

⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎜⎜

⎝

⎛

+

− +

+ = ^{− −}

−

))]

2( ( [sin 4

) 1

1 (

3 1 2 2

2 2 2 2

2 2

2 2 2 2

2 2

2 2

2

L L e

R R g e

g R R

e g T T G

r r

G t t

L j i Wij L

Wij j i

L Wij

j i

Wij j i

Wij j i

g g

g

α β α

α α

α

α α

α

(AⅡ-15)

Therefore, substitute Eq. AⅡ-15 into Eq. 2-24 and we can get the output power intensity as:

L in

j i Wij

L Wij

j i

L Wij

j i

out I

L L e

R R g e

g R

R

e g T I T

g g

g

⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎜⎜

⎝

⎛

+

− +

= − −

−

))]

2( ( [sin 4

) 1

( ^{2 2 2 2 2 2 2 2} _{1 3}

2 2 2

2 2

2

α β α

α

α α

α

(2-25)