Abstract

The frequency dependence of the spur-free dynamic range (SFDR) in a modulator based on an injection-locked laser is analyzed. It is shown that as the modulation frequency approaches half of the locking range, the SFDR of the modulator approaches that of a standard Mach-Zehnder configuration. At low frequencies, the SFDR degrades by 2 dB for every octave of frequency increase.

© 2012 OSA

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References

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  1. C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
    [CrossRef]
  2. D. J. F. Barros and J. M. Kahn, “Optical modulator optimization for orthogonal frequency-division multiplexing,” J. Lightwave Technol. 27(13), 2370–2378 (2009).
    [CrossRef]
  3. “Spurious-free dynamic range,” Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/Spurious-free_dynamic_range , (accessed April 9, 2012).
  4. W. F. Egan, Practical RF system design (IEEE Press; Wiley-Interscience, 2003), pp. xxv, 386 p.
  5. N. Hoghooghi, I. Ozdur, M. Akbulut, J. Davila-Rodriguez, and P. J. Delfyett, “Resonant cavity linear interferometric intensity modulator,” Opt. Lett. 35(8), 1218–1220 (2010).
    [CrossRef] [PubMed]
  6. R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
  7. A. E. Siegman, Lasers (University Science Books, 1986), pp. xxii, 1283 p.
  8. N. Hoghooghi and P. J. Delfyett, “Theoretical and experimental study of a semiconductor resonant cavity linear interferometric intensity modulator,” J. Lightwave Technol. 29(22), 3421–3427 (2011).
    [CrossRef]
  9. R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
    [CrossRef] [PubMed]

2011 (1)

2010 (1)

2009 (2)

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

D. J. F. Barros and J. M. Kahn, “Optical modulator optimization for orthogonal frequency-division multiplexing,” J. Lightwave Technol. 27(13), 2370–2378 (2009).
[CrossRef]

2006 (1)

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

Ackerman, E. I.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

Akbulut, M.

Barros, D. J. F.

Bartal, G.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Betts, G. E.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

Cox, C. H.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Davila-Rodriguez, J.

Delfyett, P. J.

Gladden, C.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Hoghooghi, N.

Kahn, J. M.

Ma, R. M.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Oulton, R. F.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Ozdur, I.

Prince, J. L.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

Sorger, V. J.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Zentgraf, T.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Zhang, X.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

IEEE Trans. Microw. Theory Tech. (1)

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

J. Lightwave Technol. (2)

Nature (1)

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Opt. Lett. (1)

Other (4)

“Spurious-free dynamic range,” Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/Spurious-free_dynamic_range , (accessed April 9, 2012).

W. F. Egan, Practical RF system design (IEEE Press; Wiley-Interscience, 2003), pp. xxv, 386 p.

R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).

A. E. Siegman, Lasers (University Science Books, 1986), pp. xxii, 1283 p.

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Figures (4)

Fig. 1
Fig. 1

Resonant-cavity interferometric intensity (RCII) modulator.

Fig. 2
Fig. 2

Magnitude of the frequency response of the RCII modulator in the first-order approximation.

Fig. 3
Fig. 3

Improvement of the SFDR in the RCII modulator over a Mach-Zehnder (MZ) modulator as a function of modulation frequency.

Fig. 4
Fig. 4

Obtaining SFDR from the coordinates of the third-order spur intercept and the noise floor. SFDR is equal to 2/3 of the separation of the OIP3 from the noise floor.

Equations (55)

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dϕ dt =Δ ω 0 ω m sinϕ,
ϕ=arcsin( Δ ω 0 ω m ),
1 ω m dϕ dt = Δ ω 0 ω m sinϕ dϕ d t ˜ =Δ ω ˜ 0 sinϕ,
dϕ dt +sinϕ= Ω m sin( Ωt ),
ϕ( t+ 2π /Ω )=ϕ( t ).
ϕ( t )= n=0 a n sin( nΩt+ φ n ).
Ω n n a n cos( nΩt+ φ n ) + k n J k n ( a n )sin[ n k n ( nΩt+ φ n ) ] = Ω m sin( Ωt ),
Ω a 1 cos( Ωt+ φ 1 )+ J 1 ( a 1 )sin( Ωt+ φ 1 )+ J 1 ( a 1 )sin[ ( Ωt+ φ 1 ) ]= Ω m sin( Ωt ).
Ω a 1 cos( Ωt+ φ 1 )+2 J 1 ( a 1 )sin( Ωt+ φ 1 )= Ω m sin( Ωt ).
Ω a 1 = Ω m sin( φ 1 ) 2 J 1 ( a 1 )= Ω m cos( φ 1 ).
ϕ( t )= a 1 sin( Ωt+ φ 1 ),
sin[ ϕ( t ) ]=sin[ a 1 sin( Ωt+ φ 1 ) ]= k= J k ( a 1 )sin[ k( Ωt+ φ 1 ) ] .
sin[ ϕ( t ) ] k=1 1 J k ( a 1 )sin[ k( Ωt+ φ 1 ) ] =2 J 1 ( a 1 )sin( Ωt+ φ 1 ) A 1 sin( Ωt+ φ 1 ),
Ω a 1 = Ω m sin( φ 1 ) a 1 = Ω m cos( φ 1 ),
a 1 = Ω m 1+ Ω 2 .
A 1 =2J( a 1 ) a 1 = Ω m 1+ Ω 2 .
k 1 =±3, k 3 =0; and k 1 =0, k 3 =±1.
k 1 =±1, k 3 =0.
Ω a 1 cos( Ωt+ φ 1 )+3Ω a 3 cos( 3Ωt+ φ 3 )+2 J 1 ( a 1 ) J 0 ( a 3 )sin( Ωt+ φ 1 )+ 2 J 3 ( a 1 ) J 0 ( a 3 )sin( 3Ωt+3 φ 1 )+2 J 0 ( a 1 ) J 1 ( a 3 )sin( 3Ωt+ φ 3 )= Ω m sin( Ωt ),
Ω a 1 cos( Ωt+ φ 1 )+2 J 1 ( a 1 ) J 0 ( a 3 )sin( Ωt+ φ 1 )= Ω m sin( Ωt ) 3Ω a 3 cos( 3Ωt+ φ 3 )+2 J 3 ( a 1 ) J 0 ( a 3 )sin( 3Ωt+3 φ 1 )+2 J 0 ( a 1 ) J 1 ( a 3 )sin( 3Ωt+ φ 3 ).=0
a 1 Ω= Ω m sin( φ 1 ) 2 J 1 ( a 1 ) J 0 ( a 3 )= Ω m cos( φ 1 ) 3Ω a 3 +2 J 3 ( a 1 ) J 0 ( a 3 )sin( 3 φ 1 φ 3 )=0 2 J 3 ( a 1 ) J 0 ( a 3 )cos( 3 φ 1 φ 3 )+2 J 0 ( a 1 ) J 1 ( a 3 )=0.
ϕ( t )= a 1 sin( Ωt+ φ 1 )+ a 3 sin( 3Ωt+ φ 3 ).
sinϕ= Ω m sin( Ωt ) dϕ dt .
sinϕ= Ω m sin( Ωt )Ω a 1 cos( Ωt+ φ 1 )3Ω a 3 cos( 3Ωt+ φ 3 ).
sin[ ϕ( t ) ] A 1 sin( Ωt+ ϕ 1 )+ A 3 sin( 3Ωt+ ϕ 3 ),
A 3 =3Ω a 3 ,
A 1 = { ( Ω a 1 cos φ 1 ) 2 + [ Ω m Ω a 1 cos( π 2 + φ 1 ) ] 2 } 1/2 = [ ( Ω a 1 cos φ 1 ) 2 + ( Ω m +Ω a 1 sin φ 1 ) 2 ] 1/2 = [ ( Ω a 1 ) 2 + Ω m 2 +2Ω a 1 Ω m sin φ 1 ] 1/2 = [ ( Ω a 1 ) 2 + Ω m 2 ] 1/2 ,
a 1 Ω= Ω m sin( φ 1 ) a 1 = Ω m cos( φ 1 ) 3Ω a 3 + a 1 3 24 sin( 3 φ 1 φ 3 )=0 a 1 3 24 cos( 3 φ 1 φ 3 )+ a 3 =0.
a 1 = Ω m 1+ Ω 2 a 3 = a 1 3 24 1+ ( 3Ω ) 2 .
A 1 = Ω m 1+ Ω 2 , A 3 = Ω A 1 3 8 1+ ( 3Ω ) 2 = Ω Ω m 3 8 ( 1+ Ω 2 ) 3/2 1+ ( 3Ω ) 2 .
dϕ dt +sinϕ= Ω m sin( Ω 1 t )+ Ω m sin( Ω 2 t ).
ϕ( t )= n 1 , n 2 = a ( n 1 , n 2 ) sin[ ( n 1 Ω 1 + n 2 Ω 2 )t+ φ ( n 1 , n 2 ) ] = n a n sin( nΩt+ φ n ) ,
n nΩ a n cos( nΩt+ φ n ) + k n J k n ( a n )sin[ n k n ( nΩt+ φ n ) ] = Ω m [ sin( Ω 1 t )+sin( Ω 2 t ) ],
Ω 1 a ( 1,0 ) cos( Ω 1 t+ φ ( 1,0 ) )+2 J 1 ( a ( 1,0 ) )sin( Ω 1 t+ φ ( 1,0 ) )= Ω m sin( Ω 1 t ), Ω 2 a ( 0,1 ) cos( Ω 2 t+ φ ( 0,1 ) )+2 J 1 ( a ( 0,1 ) )sin( Ω 2 t+ φ ( 0,1 ) )= Ω m sin( Ω 2 t ).
a ( 1,0 ) = Ω m 1+ Ω 1 2 , a ( 0,1 ) = Ω m 1+ Ω 2 2 .
( 2 Ω 1 Ω 2 ) a ( 2,1 ) cos[ ( 2 Ω 1 Ω 2 )t+ φ ( 2,1 ) ]+2 J 1 ( a ( 2,1 ) )sin[ ( 2 Ω 1 Ω 2 )t+ φ ( 2,1 ) ] 2 J 2 ( a ( 1,0 ) ) J 1 ( a ( 0,1 ) )sin[ ( 2 Ω 1 Ω 2 )t+2 φ ( 1,0 ) φ ( 0,1 ) ]=0.
a ( 2,1 ) = a ( 1,0 ) 2 a ( 0,1 ) 8 1+ ( 2 Ω 1 Ω 2 ) 2 ,
A ( 2,1 ) = ( 2 Ω 1 Ω 2 ) a ( 1,0 ) 2 a ( 0,1 ) 8 1+ ( 2 Ω 1 Ω 2 ) 2 = Ω m 3 8( 1+ Ω 1 2 ) 1+ Ω 2 2 1+ ( 2 Ω 1 Ω 2 ) 2 .
A ( 1,2 ) = ( 2 Ω 2 Ω 1 ) a ( 0,1 ) 2 a ( 1,0 ) 8 1+ ( 2 Ω 2 Ω 1 ) 2 = Ω m 3 8( 1+ Ω 2 2 ) 1+ Ω 1 2 1+ ( 2 Ω 2 Ω 1 ) 2 .
A spur = Ω a 1 3 8 1+ Ω 2 = Ω Ω m 3 8 ( 1+ Ω 2 ) 2 ,
A 1 = Ω m 1+ Ω 2 = Ω Ω m 3 8 ( 1+ Ω 2 ) 2 = A spur .
IIP 3 (RCII) = 8 ( 1+ Ω 2 ) 3/2 Ω .
OIP 3 (RCII) =2 2 ( 1+ Ω 2 ) 1/4 .
SFDR (RCII) SFDR (MZ) = 2 3 20 log 10 ( 1+ Ω 2 ) 1/4 [ dB-Hz 2/3 ],
ω m = ω 0 Q I 1 I 0 ,
f( x )=sin( x ).
f( msin( Ω 1 t )+msin( Ω 2 t ) )=sin[ msin( Ω 1 t )+msin( Ω 2 t ) ].
f( x )=sin[ msin( Ω 1 t )+msin( Ω 2 t ) ] = k 1 , k 2 = J k 1 ( m ) J k 2 ( m )sin[ ( k 1 Ω 1 + k 2 Ω 2 )t ] 2 J 1 ( m ) J 0 ( m )sin( Ω 1 t )+2 J 0 ( m ) J 1 ( m )sin( Ω 2 t ) 2 J 2 ( m ) J 1 ( m )sin[ ( 2 Ω 1 Ω 2 )t ]2 J 1 ( m ) J 2 ( m )sin[ ( 2 Ω 2 Ω 1 )t ] 2 J 1 ( m ) J 0 ( m )[ sin( Ω 1 t )+sin( Ω 2 t ) ] 2 J 2 ( m ) J 1 ( m ){ sin[ ( 2 Ω 1 Ω 2 )t ]+sin[ ( 2 Ω 2 Ω 1 )t ] } ( m 3 8 m 3 )[ sin( Ω 1 t )+sin( Ω 2 t ) ] 1 8 m 3 { sin[ ( 2 Ω 1 Ω 2 )t ]+sin[ ( 2 Ω 2 Ω 1 )t ] },
m= 1 8 m 3 ,
OIP 3 (MZ) =2 2 ,
sin( n a n sin x n )= k n J k n ( a n )sin( kx ) ,
sin( n a n sin x n )= 1 2i [ exp( i n a n sin x n )exp( i n a n sin x n ) ] = 1 2i [ n exp( i a n sin x n ) n exp( i a n sin x n ) ].
exp( izsinϕ )= k= J k ( z ) e ikϕ ,
sin( n a n sin x n )= 1 2i [ n k n = J k n ( a n )exp( i k n x n ) n k n = J k n ( a n )exp( i k n x n ) ].
sin( n a n sin x n )= 1 2i [ k 1 = k 2 = k n = n J k n ( a n )exp( i k n x n ) k 1 = k 2 = k n = n J k n ( a n )exp( i k n x n ) ] = 1 2i [ k n J k n ( a n )exp( i k n x n ) k n J k n ( a n )exp( i k n x n ) ] = 1 2i { k [ n J k n ( a n ) ][ n exp( i k n x n ) ] k [ n J k n ( a n ) ][ n exp( i k n x n ) ] } = 1 2i { k [ n J k n ( a n ) ]exp( i n k n x n ) k [ n J k n ( a n ) ]exp( i n k n x n ) } = k [ n J k n ( a n ) ] 1 2i ( e ikx e ikx ) = k n J k n ( a n ) sin( kx ) ,

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