A nonlinear optoelectronic filter for electronic signal processing
Supplementary Information
William Loh1,2*, Siva Yegnanarayanan1, Rajeev J. Ram2 & Paul W. Juodawlkis1*
A
Transmission of two RF signals through a nonlinear optoelectronic filter
When two RF signals are transmitted through a nonlinear optoelectronic filter, the
behavior becomes asymmetric for the individual signals depending on their
amplitudes. The filter transmission for two RF inputs can be formulated by
considering the transfer function corresponding to the modulation-photodetection
process of a microwave-photonic link. The output photodetected voltage
V t  

 
P0R 

1- cos  0   v1 sin 1t   v2 sin 2t    
2 
V

 
(S.1)
can be expanded through a series of Bessel functions yielding11,13



cos  0   v1 sin 1t   v2 sin 2t   
V



  v1    v2 
nm0
cos  0 J 0 
 J0 

 V   V 
(S.2)


 v   v 
n

   1 2cos  0 J n  1  J m  2  cos   n1  m2  t  n  m even
n , m ³0 
 V   V 


 

-  1n1 2sin  0 J n   v1  J m   v2  sin   n1  m2  t  n  m odd

 V   V 
Here, P0 ,  , R , and V retain their definitions from equations (1) and (2) of the
main text, and  0 is defined as the modulator bias point. v1 ( v2 ) and 1 ( 2 ) denote
the amplitudes and frequencies of input 1 (2), respectively. Equation (S.1) is similar
to equation (2) of the manuscript but allows for operation at an arbitrary point along
the sinusoidal modulation response.
We are interested in determining the contributions to V  t  at the frequencies of
the two RF input signals. These contributions can be used to find the transmission
from RF input-to-output through the optoelectronic filter. With the help of
equation (S.2), the output voltages at 1 and 2 can be identified to be
 v   v 
V 1t   P0R sin  0 J1  1  J 0  2  sin 1t 
 V   V 
 v   v 
V 2t   P0R sin  0 J 0  1  J1  2  sin 2t 
 V   V 
(S.3)
corresponding to the cases of ( n  1 , m  0 ) and ( n  0 , m  1 ), respectively. We
would also like to determine the voltage at the frequency of the third-order
intermodulation product between inputs 1 and 2. In general, there are multiple
possible combinations of the third-order product, but only two of which generate
unwanted spurs within the band of 1 and 2 . Note that this statement implicitly
assumes 1 and 2 to be near each other in frequency. If we further assume the
scenario of one strong input ( v1 ) and one weak input ( v2 ), the dominant third-order
contribution can be identified to be
 v   v 
V 3t    P0R sin 0 J 2  1  J1  2  sin 3t 
 V   V 
(S.4)
where 3  21  2 .
Through division by the inputs at 1 or 2 , equations (S.3) and (S.4) can be
transformed into expressions describing the voltage transmission through the
optoelectronic filter. In general, the filter transmission depends on the amplitude
levels of both inputs. However, under the conditions that  v2 V  1 , the voltage
transmissions ( G i t  ) for i =1, 2, and 3 can be simplified to
P0R sin  0   v1 
J1 

v1
 V 
 P R sin  0   v1 
G  2 t   0
J0 

2V
 V 
G 1t  
G 3t   
 P0R sin  0
2V
(S.5)
 v 
J2  1 
 V 
From equation (S.5), it is clear that the saturation induced by v1 is asymmetric for the
inputs at 1 and 2 . Note that for the voltage component at 3 , no direct
transmission exists since the inputs to the filter were applied at the frequencies of 1
and 2 . In equation (S.5), we instead determine G 3t  through normalization of
equation (S.4) by the amplitude of the weaker input. G 3t  thus represents the
transmission from an input at 2 to an output at 3 resulting from a third-order
intermodulation product of the system.
B
Nulling of the stronger signal under amplitude modulation
The suppression of the stronger signal depends critically on its amplitude level at the
input to the optoelectronic filter. At the specific point where  v1 V  3.83 , the
Bessel response J1  v1 V  becomes identically zero for the stronger signal, nulling
its transmission. It is therefore intuitive that this nulling behavior becomes inhibited in
the presence of amplitude modulation where the amplitude level varies as a function
of time. We justify this statement now assuming amplitude modulation of the stronger
signal such that the total input to the optoelectronic filter can be represented as
Vin  t   v1 1  m cos  AM t   sin 1t   v2 sin 2t 
 v1 sin 1t   v2 sin 2t  
(S.6)
v1m
vm
sin  1   AM  t   1 sin  1   AM  t 
2
2
where m denotes the modulation index,  AM denotes the frequency of amplitude
modulation, and the rest of the variables retain their original definitions. After
expanding the first term of equation (S.6), it is clear that the amplitude modulation
consists of the two original inputs at 1 and 2 but also introduces two additional
sidebands spaced at a frequency  AM apart from 1 . These sidebands effectively
behave as additional inputs to the optoelectronic filter thus generating more Bessel
factors in the modulation-photodetection response. We can trace the contributions of
all four inputs to the output of the filter for the frequencies of 1 , 2 , 3  21  2 ,

 1   AM .
and  AM


 0   AM
At 1 , two of the contributions are 1 1  0  2  0   AM
and


1 1  0  2  1  AM
 1  AM
but are both zero when  v1 V  3.83 due to the
interaction
of
J1  v1 V  .
The
first
non-zero
contribution
results
from


3  1  0  2  1  AM
 1  AM
thus signifying that the output of the optoelectronic
filter can no longer be perfectly suppressed under amplitude modulation. Similarly,


the modulation sideband at  AM
receives contributions from 0  1  0  2  1  AM
and 2  1  0  2  1  AM , both of which are non-zero at  v1 V  3.83 . We note


 0   AM
that the only contribution to 2 is that due to 0  1  1 2  0   AM
, while


 0   AM
the dominant contribution to 3 results from 2  1  1 2  0   AM
.
Additional combinations of the inputs exist but will not be discussed further here.
C
Nulling of the stronger signal under phase modulation
The situation with phase or frequency modulation is dramatically different from that
of amplitude modulation. With phase or frequency modulation, the signal amplitude
remains fixed so that the condition J1  v1 V  can be continuously maintained. In
our analysis that follows, we explicitly treat the case of phase modulation noting that
the results can be readily extended to the case of frequency modulation. We consider
the situation where the stronger signal exhibits a variable phase (   t  ) so that the
total RF input can be expressed as
Vin  t   v1 sin 1t    t    v2 sin 2t 
(S.7)
Using equation (S.7) in place of the inputs for equation (S.1), we see that the net
effect on equation (S.2) is a replacement of 1t with 1t    t  . For a constant   t  ,
this replacement generates only a phase shift in the output of the optoelectronic filter.
When   t  is oscillatory as in the case of phase modulation, the filter nonlinearities
result in additional Bessel factors that multiply the expansion of equation (S.2).
If we assume   t  to be of the form 0 sin PM t  , we can identify the

 1  PM to be
optoelectronic filter outputs at 1 , 2 , 3  21  2 , and PM
 v   v 
V 1t   P0R sin  0 J1  1  J 0  2  J 0 0  sin 1t 
 V   V 
 v   v 
V 2t   P0R sin  0 J 0  1  J1  2  sin 2t 
 V   V 
v  v 
V 3t    P0R sin  0 J 2  1  J1  2  J 0  20  sin 3t 
 V   V 
v   v 

V PM
t    P0R sin  0 J1  1  J 0  2  J1 0  sin PM t 
 V   V 
(S.8)
where 0 represents the strength of the phase modulation and  PM represents the
modulation frequency. In contrast to the amplitude modulation case, equation (S.8)



shows that both the strong input ( V 1t  ) and its modulation harmonics ( V PM
t )
can be completely suppressed when  v1 V reaches 3.83. Note that the Bessel
functions of 0 indicate the depletion of the stronger signal by phase modulation,
redistributing the power into the modulation harmonics. This depletion applies also to
the voltage contribution at 3 , supplying power to the modulation sidebands around
3 (not evaluated in equation (S.8)). Equation (S.8) further shows that the
optoelectronic filter output at 2 is unaffected by the stronger signal’s phase
modulation (compare to equation (S.3)), which corresponds to intuition when the
modulation is viewed from the perspective of a frequency fluctuation.