Influence of Polarization Transformation in Phase Conjugation of Polarization Multiplexed Data using Bragg-Scattering FWM in nonlinear SOAs
Aneesh Sobhanan Deepa Venkitesh




Influence of Polarization Transformation in Phase Conjugation of Polarization Multiplexed Data using Bragg-Scattering FWM in nonlinear SOAs
    Aneesh Sobhanan and Deepa Venkitesh^*
 Department of Electrical Engineering
 Indian Institute of Technology Madras, India-600036
 ^*deepa@ee.iitm.ac.in
    
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§ ABSTRACT

We theoretically investigate the suitability of the selection of pump frequency (ω_p)  with respect to the signal frequency (ω_s) for the best polarization insensitive phase conjugate generation through Bragg-scattering based four-wave mixing in nonlinear SOAs. We study the positive detuning (ω_s>ω_p) and negative detuning (ω_s<ω_p) scenario for the inherent polarization transformation of conjugate generation at both the ports of SOA. Conjugate generation is desired to be through a unitary transformation for its utility in further digital signal processing, and we prove that only positive detuning conditions result in unitary transformation.


§ INTRODUCTION

The mechanism of polarization insensitive optical phase conjugate (OPC) generation using Bragg-Scattering four-wave mixing (BS-FWM) in both the ports of the SOA in partially degenerate configuration is explained in our previous work <cit.>. We discuss the phase matching conditions that lead to efficient conjugate generation, along with the corresponding allowed and forbidden mixing processes <cit.>. In the same paper, we also provide experimental evidence of polarization insensitive phase conjugation with CW signals.  In this work, we present the specific polarization transformation properties of conjugate generation process in order to optimally allocate the relative frequency of the pump with respect to the signal. Unitary transformation is highly desirable for its utility in the most commonly used digital signal processing algorithms <cit.>.


§ ANALYTICAL FORMULATION

We consider polarization insensitive conjugate generation in an SOA through Bragg-scattering process. Polarization multiplexed signal is launched at the input port of the SOA and orthogonally polarized pumps are counter propagated (through input and output ports) in this configuration. Fig. <ref> shows the schematic of the propagation directions and the corresponding polarization of the pump and the signal in a scenario where the facet reflectivities of the SOA are also considered. 
Let the amplitudes of the orthogonally polarized pumps at the input of the SOA along the reference X and Y directions be A_P_X and  A_P_Y and that of the input signal  be represented as ([A_s1cos(ϕ)-A_s2sin(ϕ)]â_X+[A_s1sin(ϕ)+A_s2cos(ϕ)]â_Y), where A_s1 and A_s2 denotes the orthogonal components of the complex amplitudes of the signal electric field, which is rotated with respect to the reference X and Y directions through an angle ϕ as shown in Fig.<ref>. 
















Thus the electric fields corresponding to the orthogonally polarized degenerate pumps at the respective inputs of the SOA can be represented as,

    E⃗_P_x=A_P_xexp(j(ω_pt-ϕ_p))â_x,



    E⃗_P_y=A_P_yexp(j(ω_pt-ϕ_p))â_y,

and that of the polarization multiplexed signal at the input is given by,

    E⃗_S=exp(j(ω_st-ϕ_s)){[A_s1cos(ϕ)-A_s2sin(ϕ)]â_x+[A_s1sin(ϕ)+A_s2cos(ϕ)]â_y},


where ω_p and ω_s represent the pump and signal frequencies respectively.











We first consider the case of negative detuning (ω_s<ω_p). Considering non-forbidden phase conjugate generation processes in the Bragg-scattering FWM in <cit.>, the transfer matrix for the conjugate at both ports is given by, 


    [ E_cx; E_cy;      ]=A^2_P  exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))η[ (cosϕ ±sinϕ;  cosϕ ±sinϕ;       ][ A^*_s1; A^*_s2;        ]


Detailed derivation of Eqn.<ref> is given in Appendix. <ref>

Thus for ω_s<ω_p, the transfer matrix corresponding to the conjugate generation process is non-unitary.






We now consider the case of positive detuning. The transfer matrix of the conjugate, considering all the non-forbidden processes for output at both ports can be derived as,	






    [ E_cx; E_cy;      ]=A^2_P  exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))η[  sinϕ  cosϕ;  cosϕ -sinϕ;       ][ A^*_s1; A^*_s2;        ]

Detailed derivation of Eqn.<ref> is given in Appendix. <ref>

Thus, the transfer matrix in case of ω_s>ω_p is of the form of a unitary transformation.  η represents the conjugate conversion efficiencies in each case, which is in-turn decided by the strength of the Bragg grating that contributes to the specific conjugate generation. We derive with appropriate approximations that polarization transformation for the positive detuning case is unitary while that of the negative detuning case is non-unitary. Unitary transformation is a desirable condition for the polarization demultiplexing algorithm in DSP. Hence the positive detuning is desirable for Bragg scattering-FWM aided polarization insensitive phase conjugate generation, both at port 1 and port 2 of the SOA.



§ CONCLUSION

The transfer function of polarization insensitive conjugate generation using counter-propagating orthogonally polarized pumps using Bragg scattering FWM in SOA in port 1 and port 2 is studied. The comparison is done between two cases ω_p>ω_s and ω_s>ω_p. It is evident from this analysis that the optical phase conjugate process through Bragg-scattering FWM corresponds to a unitary transformation only for ω_s>ω_p and the corresponding transformation is non-unitary for ω_s<ω_p. The efficiency of the process with ω_s>ω_p is found to be smaller at least by 3 dB compared to that for ω_s<ω_p <cit.>. In spite of poorer efficiency, a positive detuning is preferred because of the unitary nature of the corresponding transformation, and  consequently avoiding the limitations posed by digital signal processing in handling non-unitary transformation. This work leads to an insight of the selection of pump and signals frequency for the polarization insensitive conjugate generation process through Bragg-scattering FWM in SOA. 


§ APPENDIX A- Ω_S<Ω_P

We consider all the non-forbidden processes that lead to phase conjugate generation. These processes are detailed in Tables. 3 & 4 of  <cit.> (reproduced here as Tables. 1 & 2 for clarity). Forbidden processes are represented in red font and allowed processes are represented in green font.





 §.§ Conjugate generation at port 1 of the SOA

The primary contributions for the generation of conjugates in port 1 are due to the scattering of the (a) reflected pump and counter propagating pump off the  grating formed due to the beat between reflected signal and the counter propagating pump (with grating vector k_p-k_s) and (b) co-propagating pump and reflected pump off the grating formed due to the beat between the co-propagating signal and counter propagating pump (with grating vector k_p+k_s). The facet reflectivities of the SOAs are typically small and hence the conjugate generated by scattering of reflected pumps can be neglected when compared to the efficiency of the conjugate generated by the transmitted pump.  The electric fields of conjugate generated through the remaining processes are written below, 


	
  * Conjugate generated in y-polarization due to the interaction between counter propagating pump (y-polarization) with the backward propagating grating (grating vector :k_p-k_s, with a phase mismatch k_c=0), represented as,
	
    η_1(E⃗_P_y.E⃗^*_S)E⃗_P_y   =   η_1[(A_P_yexp(j(ω_pt-ϕ_p))â_y).([A^*_s1cosϕ-A^*_s2sinϕ]â_x+[A^*_s1sinϕ+A^*_s2cosϕ]â_y)] 
          A_P_yexp(j(ω_pt-ϕ_p))â_y
    	
       =   η_1 A_P_y^2 exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s)) [A^*_s1sinϕ+A^*_s2cosϕ]â_y

	
	
	
	
  * Conjugate generated in x-polarization due to the interaction between co-propagating pump (x-polarization) with the backward propagating grating (grating vector :k_p+k_s, with a phase mismatch k_c=2(k_p-k_s)), represented as,
	
    η_2(E⃗_P_y.E⃗^*_S)E⃗_P_x   =   η_2[(A_P_yexp(j(ω_pt-ϕ_p))â_y).([A^*_s1cosϕ-A^*_s2sinϕ]â_x+[A^*_s1sinϕ+A^*_s2cosϕ]â_y)] 
          A_P_xexp(j(ω_pt-ϕ_p))â_x
    	
       =   η_2 A_P_y A_P_xexp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s)) [A^*_s1sinϕ+A^*_s2cosϕ]â_x

	

Here η_1 and η_2 represent the efficiency of each process.

The electric field of the Conjugate at Port 1, when orthogonal pumps carry equal power (A_P_x=A_P_y=A_P)


    E_C   =   η_1(E⃗_P_y.E⃗^*_S)E⃗_P_y+  	η_3(E⃗_P_y.E⃗^*_S)E⃗_P_x
       =    A^2_P exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))[A^*_s1sinϕ+A^*_s2cosϕ]
    [η_2â_x+η_1â_y]


    [ E_cx; E_cy;      ]=A^2_P  exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))[ η_2sinϕ η_2cosϕ; η_1sinϕ η_1cosϕ;         ][ A^*_s1; A^*_s2;        ]


    [ E_cx; E_cy;      ]=A^2_P  exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))η[ sinϕ cosϕ; sinϕ cosϕ;      ][ A^*_s1; A^*_s2;        ],

where η_1=η_2=η, which can be achieved by adjusting the power levels  of the counter-propagating pumps.



 §.§ Conjugate generation at port 2 of the SOA

The primary contributions for the generation of conjugates in port 2 are due to the scattering of the (a) reflected pump and co-propagating pump off the  grating formed due to the beat between co-propagating signal and pump (with grating vector k_p-k_s) and (b) counter-propagating pump and reflected pump off the grating formed due to the beat between the reflected signal and co-propagating pump (with grating vector k_p+k_s). The facet reflectivities of the SOAs are typically small and hence the conjugate generated by scattering of reflected pumps can be neglected while compared to the efficiency of the conjugate by the transmitted pump.  The electric fields of conjugate generated through the remaining processes are written below, 


	
  * Conjugate generated in x-polarization, due to the interaction between co-propagating pump (x-polarization) with the forward propagating grating (grating vector: k_p-k_s, with a phase mismatch k_c=0), represented as,
	
	
    η_3(E⃗_P_x.E⃗^*_S)E⃗_P_x   =   η_3[(A_P_xexp(j(ω_pt-ϕ_p))â_x).([A^*_s1cosϕ-A^*_s2sinϕ]â_x+[A^*_s1sinϕ+A^*_s2cosϕ]â_y)] 
          A_P_xexp(j(ω_pt-ϕ_p))â_x
    	
       =   η_3 A_P_x^2 exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s)) [A^*_s1cosϕ-A^*_s2sinϕ]â_x

	
	
  * Conjugate generated in y-polarization, due to the interaction between counter-propagating pump (y-polarization) with the forward propagating grating (grating vector: k_p+k_s), with a phase mismatch k_c=2(k_p-k_s)), represented as,
	
	
	
    η_4(E⃗_P_x.E⃗^*_S)E⃗_P_y   =   η_4[(A_P_xexp(j(ω_pt-ϕ_p))â_x).([A^*_s1cosϕ-A^*_s2sinϕ]â_x+[A^*_s1sinϕ+A^*_s2cosϕ]â_y)] 
          A_P_yexp(j(ω_pt-ϕ_p))â_y
    	
       =   η_4 A_P_xA_P_yexp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s)) [A^*_s1cosϕ-A^*_s2sinϕ]â_y

	


Here η_3 and η_4 represent the efficiency of each process.

The electric field of the conjugate at Port 2, when orthogonal pumps carry equal power (A_P_x=A_P_y=A_P), can be written as,


    E_C   =   η_3(E⃗_P_y.E⃗^*_S)E⃗_P_x+ 	 η_4(E⃗_P_x.E⃗^*_S)E⃗_P_y
       =    A^2_P exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))[A^*_s1cosϕ-A^*_s2sinϕ]
    [η_3â_x+(η_4â_y]


    [ E_cx; E_cy;      ]=A^2_P  exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))[  η_3cosϕ -η_3sinϕ;  η_4cosϕ -η_4sinϕ;          ][ A^*_s1; A^*_s2;        ]



    [ E_cx; E_cy;      ]=A^2_P  exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))η[  cosϕ -sinϕ;  cosϕ -sinϕ;       ][ A^*_s1; A^*_s2;        ],

where η_3=η_4=η, which can be achieved by adjusting the power levels of the counter-propagating pumps.






§ APPENDIX B- Ω_S>Ω_P

In this section, we consider positive detuning case. We consider all the non-forbidden processes that lead to phase conjugate generation. These processes are detailed in Tables. 1 & 2 of  <cit.> (reproduced here as Tables. 3 & 4 for clarity). Forbidden processes are represented in red font and allowed processes are represented in green font.






 §.§ Conjugate generation at port 1 of the SOA

The primary contributions for the generation of conjugates in port 1 are due to the scattering of the (a) reflected pump and counter propagating pump off the  grating formed due to the beat between co-propagating signal and pump (with grating vector k_s-k_p) and (b) co-propagating pump and reflected pump off the grating formed due to the beat between the co-propagating signal and counter propagating pump (with grating vector k_s+k_p). Considering low facet reflectivities of SOA, we assume that, the conjugate power due to reflected pumps are comparatively smaller and are hence neglected. The electric fields of conjugate generated through the remaining processes are written as, 



	
  * Conjugate generated in (y-polarization) due to the interaction between counter-propagating pump (y-polarization) with the forward propagating grating (grating vector: k_s-k_p, with a phase mismatch k_c=2(k_p-k_s)), represented as,
	
	
    η_1(E⃗_P_x.E⃗^*_S)E⃗_P_y   =   η_1[(A_P_xexp(j(ω_pt-ϕ_p))â_x).([A^*_s1cosϕ-A^*_s2sinϕ]â_x+[A^*_s1sinϕ+A^*_s2cosϕ]â_y)] 
          A_P_yexp(j(ω_pt-ϕ_p))â_y
    	
       =   η_1 A_P_xA_P_yexp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s)) [A^*_s1cosϕ-A^*_s2sinϕ]â_y

	
	
	
	
	
  * Conjugate generated in (x-polarization) due to the interaction between co-propagating pump (x-polarization) with the forward propagating grating (grating vector: k_s+k_p, with a phase mismatch k_c=2(k_p-k_s)), represented as,
	
	
    η_2(E⃗_P_y.E⃗^*_S)E⃗_P_x   =   η_2[(A_P_yexp(j(ω_pt-ϕ_p))â_y).([A^*_s1cosϕ-A^*_s2sinϕ]â_x+[A^*_s1sinϕ+A^*_s2cosϕ]â_y)] 
          A_P_xexp(j(ω_pt-ϕ_p))â_x
    	
       =   η_2 A_P_y A_P_xexp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s)) [A^*_s1sinϕ+A^*_s2cosϕ]â_x



The electric field of the conjugate at Port 1, when orthogonal pumps carry equal power (A_P_x=A_P_y=A_P)


    E_C   =   η_1(E⃗_P_x.E⃗^*_S)E⃗_P_y+  η_2(E⃗_P_y.E⃗^*_S)E⃗_P_x
       =    A^2_P  exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))
          [(η_2[A^*_s1sinϕ+A^*_s2cosϕ]â_x)+(η_1[A^*_s1cosϕ-A^*_s2sinϕ])â_y]



    [ E_cx; E_cy;      ]=A^2_P  exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))[  η_2sinϕ  η_2cosϕ;  η_1cosϕ -η_1sinϕ;          ][ A^*_s1; A^*_s2;        ]




    [ E_cx; E_cy;      ]=A^2_P  exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))η[  sinϕ  cosϕ;  cosϕ -sinϕ;       ][ A^*_s1; A^*_s2;        ],


where, η_1=η_2=η, which can be achieved by adjusting the power levels of the counter-propagating pumps.


 §.§ Conjugate generation at port 2 of the SOA

The primary contributions for the generation of conjugates in port 2 are due to the scattering of the (a) reflected pump and co-propagating pump off the  grating formed due to the beat between reflected signal and counter propagating pump (with grating vector k_s-k_p) and (b) counter-propagating pump and reflected pump off the grating formed due to the beat between the reflected signal and co-propagating pump (with grating vector k_s+k_p). Here again, we assume that the conjugation due to  reflected pump is negligible. The electric fields of conjugate generated through the remaining processes can be written as, 



	
  * Conjugate generated in (x-polarization) due to the interaction between co-propagating pump (x-polarization) with the backward propagating grating (grating vector: k_s-k_p, with a phase mismatch k_c=2(k_p-k_s)), represented as,
	
	
    η_3(E⃗_P_y.E⃗^*_S)E⃗_P_x   =   η_3[(A_P_yexp(j(ω_pt-ϕ_p))â_y).([A^*_s1cosϕ-A^*_s2sinϕ]â_x+[A^*_s1sinϕ+A^*_s2cosϕ]â_y)] 
          A_P_xexp(j(ω_pt-ϕ_p))â_x
    	
       =   η_3 A_P_y A_P_xexp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s)) [A^*_s1sinϕ+A^*_s2cosϕ]â_x

	

	
  * Conjugate generated in (y-polarization) due to the interaction between counter propagating pump (y-polarization) with the backward propagating grating (grating vector: k_s+k_p, with a phase mismatch k_c=2(k_p-k_s)), represented as,
	
	
    η_4(E⃗_P_x.E⃗^*_S)E⃗_P_y   =   η_4[(A_P_xexp(j(ω_pt-ϕ_p))â_x).([A^*_s1cosϕ-A^*_s2sinϕ]â_x+[A^*_s1sinϕ+A^*_s2cosϕ]â_y)] 
          A_P_yexp(j(ω_pt-ϕ_p))â_y
    	
       =   η_4 A_P_xA_P_yexp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s)) [A^*_s1cosϕ-A^*_s2sinϕ]â_y



Conjugate at Port 2, when orthogonal pumps carry equal power (A_P_x=A_P_y=A_P)

    E_C   =   η_3(E⃗_P_y.E⃗^*_S)E⃗_P_x+ 	 η_4(E⃗_P_x.E⃗^*_S)E⃗_P_y
       =    A^2_P  exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))
          [(η_3[A^*_s1sinϕ+A^*_s2cosϕ]â_x)+(η_4[A^*_s1cosϕ-A^*_s2sinϕ])â_y]


    [ E_cx; E_cy;      ]=A^2_P  exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))[  η_3sinϕ  η_3cosϕ;  η_4cosϕ -η_4sinϕ;          ][ A^*_s1; A^*_s2;        ]



    [ E_cx; E_cy;      ]=A^2_P  exp(j(2ω_p-ω_s)t-(2ϕ_p-ϕ_s))η[  sinϕ  cosϕ;  cosϕ -sinϕ;       ][ A^*_s1; A^*_s2;        ],

where, η_3=η_4=η, which can be achieved by adjusting the power levels of the counter-propagating pumps.
 

widest entry
	[1]Sobhanan Aneesh Sobhanan and Deepa Venkitesh, “Polarization-insensitive phase conjugation using single pump Bragg-scattering four-wave mixing in semiconductor optical amplifiers," Opt. Express 26, 22761-22772 (2018)
	[2]Ip Ezra Ip, Alan Pak Tao Lau, Daniel J. F. Barros, and Joseph M. Kahn, “Coherent detection in optical fiber systems," Opt. Express 16, 753-791 (2008).


