Patent Publication Number: US-11050487-B2

Title: Method and apparatus for laser frequency control

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of priority to U.S. Patent Application Ser. No. 62/874,233 filed on Jul. 15, 2019 and entitled “Method and Apparatus for Laser Frequency Control”, the contents of which are incorporated herein by reference. 
    
    
     FIELD 
     The present disclosure generally relates to the field of optical networks and, in particular, to method and apparatus for laser frequency control. 
     BACKGROUND 
     Long-haul wavelength division multiplexing (WDM) optical networks are configured to transmit optical channel signals over a few hundred kilometers. However, the capacity of conventional optical channel in WDM networks is limited to about 200 Gigabits/s (Gbit/s). To increase the channel capacity beyond this limit and to take advantage of WDM interface, optical super-channel signals may be used. 
     An optical super-channel signal transmits data in optical super-channels. A super-channel is an optical channel that binds several subcarriers for unified transmission. Each subcarrier is a signal with a single-wavelength spectrum. The subcarriers of one super-channel can be processed simultaneously by optical networks hardware. Using subcarriers in the optical super-channel may considerably increase the transmission capacity and spectral efficiency of the WDM networks by reducing a guard band between the subcarriers within the super-channel. 
     In the conventional optical channel signal, a guard band is required between any two optical channels in order to mitigate an optical filtering effect and neighboring channel crosstalk. There is no optical filtering between subcarriers in the super-channel. As such, the guard band between subcarriers in an optical super channel can be allocated based on consideration of only the mitigation of cross talk between channels. If the crosstalk is low and subcarrier frequencies are stable, the bandwidth allocated to the guard band between the subcarriers can be reduced. However, each subcarrier is generated by a laser that may manifest an optical frequency drift. Such an optical frequency drift for a particular laser may cause a drift of the corresponding subcarrier towards a neighboring subcarrier in the optical super-channel signal. 
     Therefore, there is a need for improvements in laser frequency control techniques that compensate for problems related to laser drift issues that may cause channel crosstalk in the optical super-channel signal. 
     SUMMARY 
     An objective of the present disclosure is to provide a technique for controlling subcarrier frequencies in an optical super-channel signal in order to improve performance of an optical network. The apparatuses, methods and systems as disclosed herein permit controlling a difference between a first center frequency of a first optical subcarrier and a second center frequency of a second optical subcarrier of a super-channel optical signal. 
     In accordance with this objective, an aspect of the present disclosure provides a method for controlling a difference between a first center frequency of a first optical subcarrier and a second center frequency of a second optical subcarrier of an optical super-channel signal in an optical network. The method comprises: modulating the first optical subcarrier at a first optical side component frequency with a first side modulation frequency; modulating the second optical subcarrier at a second optical side component frequency with a second side modulation frequency, a difference between the first optical side component frequency and the second optical side component frequency being smaller than a difference between the first center frequency and the second center frequency. The method further comprises coupling together the modulated first and second optical subcarriers to obtain a modified optical signal; and monitoring a variation of the difference between the first center frequency and the second center frequency by detecting a radio-frequency (RF) power at a modulated beat frequency tone in the modified optical signal. The modulated beat frequency tone bears information of the first side modulation frequency and the second side modulation frequency. 
     The monitoring a variation of the difference between the center frequencies of the first and the second optical subcarriers may further comprise: converting the modified optical signal to an electrical signal; filtering the electrical signal by a first electronic filter to obtain a filtered electrical signal within a first bandwidth corresponding to the difference between the first optical side component frequency and the second optical side component frequency; squaring the filtered electrical signal; filtering again the filtered and squared electrical signal by a second electronic filter to obtain an RF signal at the modulated beat frequency tone; and measuring the RF power of the RF signal at the modulated beat frequency tone. 
     In at least one embodiment, a difference between the first optical side component frequency and the first center frequency of the first subcarrier may be maintained constant. In at least one embodiment, a difference between the second optical side component frequency and the second center frequency of the second subcarrier may be maintained constant. 
     The method may further comprise detuning at least one of the first center frequency and the second center frequency to maximize the RF power of the RF signal at the modulated beat frequency tone. 
     In some embodiments, a sum of a power of a first side optical component at the first side optical component frequency and a power of a second optical component at the second side optical component frequency may be less than 1% of the total power of the optical signal. 
     In some embodiments, the modulated beat frequency tone may be a difference between the first side modulation frequency and the second side modulation frequency. In some embodiments, the modulated beat frequency tone is a sum of the first side modulation frequency and the second side modulation frequency. 
     In accordance with another aspect of the present disclosure, there is provided an apparatus for controlling an optical super-channel signal in an optical network, the optical super-channel signal having a first optical subcarrier with a first center frequency and a second optical subcarrier with a second center frequency. The apparatus comprises: a first electro-optic modulator configured to: modulate the first optical subcarrier at a first optical side component frequency with a first side modulation frequency; a second electro-optic modulator configured to: modulate the second optical subcarrier at a second optical side component frequency with a second side modulation frequency, a difference between the first optical side component frequency and the second optical side component frequency being smaller than a difference between the first center frequency and the second center frequency. The apparatus also comprises a coupler configured to couple together the modulated first and second optical subcarriers to obtain a modified optical signal; and an RF power meter configured to measure an RF power of an RF signal generated from the modified optical signal, the RF power being measured at a modulated beat frequency tone. The modulated beat frequency tone bears information of the first side modulation frequency and the second side modulation frequency. 
     The apparatus may further comprise: a photodetector configured to receive the modified optical signal and to generate an electrical signal; a first electronic filter configured to filter the electrical signal to obtain a filtered electrical signal within a first bandwidth corresponding to a difference between the first optical side component frequency and the second optical side component frequency, the first electronic filter configured to significantly attenuate the electrical signal around the first side modulation frequency and the second side modulation frequency; and a squaring device configured to square the filtered electrical signal. 
     The apparatus may further comprise an amplifier configured to amplify the electrical signal after it was generated by the photodetector. The apparatus may further comprise a second electronic filter located between the squaring device and the RF power meter and configured to filter again the filtered and squared electrical signal within a second bandwidth having a center corresponding to the modulated beat frequency tone to obtain the RF signal. The squaring device may comprise a splitter and a multiplier. 
     In some embodiments, the modulated beat frequency tone may be a difference between the first side modulation frequency and the second side modulation frequency. In some embodiments, the modulated beat frequency tone is a sum of the first side modulation frequency and the second side modulation frequency. 
     In at least one embodiment, a difference between the first optical side component frequency and the first center frequency of the first subcarrier may be maintained constant. In at least one embodiment, a difference between the second optical side component frequency and the second center frequency of the second subcarrier may be maintained constant. 
     In accordance with another aspect of the present disclosure, there is provided a non-transitory computer readable medium with computer executable instructions stored thereon that, when executed by a processor, cause the processor to: filter a digital signal to obtain a filtered digital signal within a first bandwidth by attenuating the digital signal around a first side modulation frequency and a second side modulation frequency; square the filtered digital signal; filter again the filtered and squared digital signal to obtain a radio-frequency (RF) signal at a modulated beat frequency tone, the modulated beat frequency tone bearing information of the first side modulation frequency and the second side modulation frequency; and determine an RF power of the RF signal at the modulated beat frequency tone. The modulated beat frequency tone may be a difference between the first side modulation frequency and the second side modulation frequency. The modulated beat frequency tone may be a sum of the first side modulation frequency and the second side modulation frequency. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
       The features and advantages of the present disclosure will become apparent from the following detailed description, taken in combination with the appended drawings, in which: 
         FIG. 1A  depicts a block diagram of a WDM transmitter for generating an optical super-channel signal, in accordance with at least one non-limiting embodiment of the present disclosure; 
         FIG. 1B  illustrates a portion of the optical spectrum of the optical signal generated by the WDM transmitter depicted in  FIG. 1A ; 
         FIG. 1C  depicts a block diagram of logical blocks of a digital signal processor and a digital to analog converter (DAC) module of the transmitter of  FIG. 1A , in accordance with at least one non-limiting embodiment of the present disclosure; 
         FIG. 2  illustrates consequences of the drift of the optical center frequency of one of lasers in the transmitter of  FIG. 1 ; 
         FIG. 3  illustrates a portion of a spectrum of a modified optical signal, generated in accordance with various embodiments of the present disclosure; 
         FIG. 4  depicts a block diagram of logical blocks of a modified DSP and DAC module for generation of optical super-channel signal with side frequency components, in accordance with various embodiments of the present disclosure; 
         FIG. 5  depicts an illustration of three neighboring subcarriers having side frequency components, generated in accordance with various embodiments of the present disclosure; 
         FIG. 6  depicts an apparatus configured to detect a variation of a difference between two neighbouring side frequency components, in accordance with various embodiments of the present disclosure; 
         FIG. 7  illustrates a transfer function of a first electronic filter of the apparatus depicted in  FIG. 6 , in accordance with various embodiments of the present disclosure; 
         FIG. 8  depicts a flowchart illustrating a method for controlling a difference between a first center frequency of a first optical subcarrier and a second center frequency of a second optical subcarrier of an optical super-channel signal in an optical network, in accordance with various embodiments of the present disclosure; 
         FIG. 9  depicts a block diagram of an experimental set-up for confirmation of detectability of variation of a side component spacing, in accordance with various embodiments of the present disclosure; and 
         FIG. 10  depicts a power of a difference frequency tone measured using the set-up depicted in  FIG. 9 . 
     
    
    
     It is to be understood that throughout the appended drawings and corresponding descriptions, like features are identified by like reference characters. Furthermore, it is also to be understood that the drawings and ensuing descriptions are intended for illustrative purposes only and that such disclosures are not intended to limit the scope of the claims. 
     DETAILED DESCRIPTION 
     The instant disclosure is directed to systems, methods and apparatuses to address the deficiencies of the current state of the art. To this end, the instant disclosure describes systems, apparatuses and methods directed to controlling a difference between a first center frequency of a first optical subcarrier and a second center frequency of a second optical subcarrier of an optical super-channel signal in an optical network. 
     Referring now to the drawings,  FIG. 1A  depicts a block diagram of a WDM transmitter  50  for generating an optical super-channel signal  60 , in accordance with at least one non-limiting embodiment of the present disclosure.  FIG. 1B  illustrates a portion  100  of the optical spectrum of optical super-channel signal  60  generated by transmitter  50 . 
     It should be understood that each subcarrier of optical super-channel signal  60  can be used to carry an independent signal. Each of these subcarriers is associated with a control branch within transmitter  50 . Each of these branches comprises a transmitter digital signal processors (DSPs)  51 , a digital to analog converter (DAC)  52 , an electrical driver  54 , a laser  56 , and an electro-optical (EO) modulator  58 . As shown in  FIG. 1A , a plurality of branches are present, each of which corresponds to a respective super-channel subcarrier  111 ,  112 ,  113 ,  114 . As depicted in  FIG. 1A , in each branch, DSP  51  transmits a digital signal to a DAC  52 , which converts the digital signal to an analog signal. The digital signal is the data signal to be encoded for transmission. The EO modulator  58  may be modulated electrically by electrical drivers  54  in accordance with the analog signal output by DAC  52 . The output of laser  56  is optically modulated by EO modulator  58 . A coupler  65  may be used to combine separate subcarriers (e.g.  111 ,  112 ,  113 ,  114 , etc.) generated by each of the branches into one optical super-channel signal  60 . If each of the lasers is tuned to a different frequency, the coupling of the generated subcarriers can be a rather simple process. 
     It is contemplated that transmitter  50  may also comprise other devices, and the technology described herein is not limited to the embodiment illustrated in  FIG. 1A . 
       FIG. 1B  depicts two super-channels  110 ,  120 . Each super-channel  110 ,  120  comprises four subcarriers: four super-channel subcarriers  111 ,  112 ,  113 , and  114  in the first super-channel  110  and four super-channel subcarriers  121 ,  122 ,  123 ,  124  in the second super-channel  120 . 
     A guard band is typically required between neighboring super-channels  110 ,  120  and between neighboring subcarriers ( 111 ,  112 ,  113 ,  114  and  121 ,  122 ,  123 ,  124 ) in order to mitigate the effects of crosstalk. In order to improve spectral efficiency, a subcarrier guard band  130  between neighboring subcarriers in the same super channel (for example, between neighboring subcarriers  111  and  112 ) may be narrower than a super-channel guard band  135  located between neighboring super-channels  110 ,  120 . Typically, subcarrier guard band  130  between neighboring subcarriers may be on the order of a few GHz. 
     Referring again to  FIG. 1B , each subcarrier  111 ,  112 ,  113 ,  114  of super-channel  110  is generated by a different laser  56 . Each laser  56  has a laser optical frequency which corresponds to a center frequency of the corresponding subcarrier. For example, in  FIG. 1B , subcarrier  111  has a center frequency f 1  (illustrated by an arrow). Also in  FIG. 1B , subcarrier  112  has a center frequency f 2  (also illustrated by an arrow). Thus, the laser associated with subcarrier  111  would have a laser optical frequency of f 1  and the laser associated with subcarrier  112  would have a laser optical frequency of f 2 . 
     Due to changes in temperature during operation, differences between an operating temperature and the temperature at which a calibration was performed, and other such effects, the laser optical frequency may drift with time. The drift of the laser optical frequency over time may be as high as +/−2.5 GHz. Such frequency drift may significantly contribute to an undesired variation of frequency spacing between the subcarriers. An additional guard band between subcarriers is usually introduced to take into account such laser frequency drift. This guard band may be defined on the basis of a worst case drift from two different lasers (e.g. each of the lasers drifting towards the other). This increased guard band reduces the spectral efficiency of the system. The technique described herein may help to reduce the bandwidth allocated to guard bands between the subcarriers attributable to accommodating the laser frequency draft. The technique described herein permits reducing the variation of spacing between frequencies of the lasers that generate the subcarriers and to improve the relative stability of subcarriers in the super-channel. 
     The technique described herein permits reduce variation of the frequency difference |f 2 −f 1 | between two neighboring subcarriers (also referred to herein as a “subcarrier spacing”). 
       FIG. 2  illustrates consequences of the drift of the optical frequency of one of lasers  56  in transmitter  50 . 
     A portion  200  of the spectrum of optical signal  60  contains two neighboring subcarriers  111  and  112  of super-channel  110 . First subcarrier  111  has a subcarrier center frequency f 1 , and second subcarrier  112  has a subcarrier center frequency f 2 . 
     In  FIG. 2 , scenario  201  illustrates first and second subcarriers  111 ,  112  when they have optimal desired subcarrier spacing. The difference between optical frequencies f 1  and f 2  is approximately equal to the desired frequency difference Δf d . The desired variation of a subcarrier spacing (f 1 -f 2 ) may be, for example, a few MHz. 
     Scenario  202  illustrates first and second subcarriers  111 ,  112  when they are located too far from each other. In such case, the difference Δf 1  between optical frequencies f 1  and f 2  is larger than Δf d . Therefore, a portion of the spectrum is wasted. 
     Scenario  203  illustrates first and second subcarriers  111 ,  112  when they are located too close to each other, so that optical frequency difference Δf s  between first and second subcarriers  111 ,  112  is less than Δf d . In this case, the crosstalk between first and second subcarriers  111 ,  112  is higher than desired. 
     In accordance with at least one embodiment of the present disclosure, a modified optical signal is generated in order to control the subcarrier spacing. 
       FIG. 3  illustrates a portion  300  of a spectrum of the modified optical signal, generated in accordance with at least one embodiment of the present disclosure. The modified optical signal contains first and second side frequency components  331 ,  332 . The first side frequency component  331  has frequency f A . The second side frequency component  332  has frequency f B . In some embodiments, first and second side frequency components  331 ,  332  may be generated on neighboring edges of first and second subcarriers  311 ,  312 . As is described herein below, these first and second side frequency components  331 ,  332  are used to monitor the subcarrier spacing (f 1 −f 2 ) between first and second subcarriers  311 ,  312 . 
     In at least one embodiment, a frequency difference Δf 1A  between optical center frequency f 1  of first subcarrier  311  and frequency f A  of first side frequency component  331  is maintained as constant. The optical center frequency f 1  of first subcarrier  311  may drift away from the desired frequency due to the drift of the laser  56  that generates first subcarrier  311 . The frequency f A  of first side frequency component  331  also follows the drift of the subcarrier frequency f 1  of first subcarrier  311 . Therefore, by monitoring the difference between frequencies f A , f B  of side frequency components  331 ,  332  of neighboring subcarriers  311 ,  312 , it is possible to monitor the subcarrier spacing. 
       FIG. 3  illustrates the drift of subcarrier  312 , in accordance with at least one embodiment of the present disclosure. For simplicity of the illustration, center frequency f 1  of first subcarrier  311  is maintained constant in  FIG. 3 . It is contemplated that center frequencies of any subcarrier of one super-channel may increase or decrease. 
     Scenario  301  illustrates first and second subcarriers  311 ,  312  when they have optimal desired subcarrier spacing. Scenario  302  illustrates first and second subcarriers  311 ,  312  when they are located too far from each other. In scenario  302 , when subcarrier center frequency f 2  decreases, the frequency difference between subcarrier center frequencies f 1 , f 2  of first subcarrier  311  and second subcarrier  312 , respectively (i.e. subcarrier spacing) increases. The frequency difference Δf AB  between frequencies of first and second side frequency components  331 ,  332  also increases. 
     When the center frequency f 2  increases (scenario  303 ), the subcarrier spacing (f 1 −f 2 ) decreases. The frequency difference Δf AB  between frequencies of first and second side frequency components  331 ,  332  also decreases. 
     Referring now to  FIG. 1C , which depicts a block diagram of logical blocks of DSP  51 , DAC module  52  of transmitter  50 , and electrical driver  54  of  FIG. 1A , in accordance with at least one non-limiting embodiment of the present disclosure. There are four data streams for dual-polarization QAM transmitter  50 . The logical blocks of DSP  51  may comprise: encoding block  72 , bit-to-symbol mapping block  74 , header insertion block  76 , pulse shaping block  78 , and pre-distortion block  79 . It should be noted that DSP  51  may have other configuration and structure. DAC module  52  has four DAC units  80 . 
     As illustrated in  FIG. 1C , DSP  51  transmits four V(t) signals to DAC module  52 , one V(t) signal for each DAC unit  80 . These four V(t) signals are: two signals in X-polarization V XI (t), V XQ  (t), and two signals in Y-polarization V YI  (t), V YQ  (t). I and Q correspond to in-phase and quadrature phase signals, respectively. 
     The desired optical output with optical subcarrier may be obtained when V XI (t), V XQ  (t), V YI (t), V YQ  (t) are applied to EO modulator  58  and EO modulator  58  is an in-phase quadrature phase MZM (IQ-MZM). 
     The resulting field may be described as:
 
 E   X0 ( t )= V   XI ( t )+sqrt(−1)* V   XQ ( t )  (1)
 
     The equation (1) is for X polarization of the field. The field in Y polarization can be described with a similar equation. 
     The transmitter  50  with DSP  51  and DAC module  52  of  FIG. 1C  transmits optical super-channel signal illustrated in  FIG. 1B . 
     In at least one embodiment, in order to generate the modified optical signal containing subcarriers  311 ,  312  with side frequency components  331 ,  332 , a side frequency component generation logical block is added in DSP  51  of transmitter  50 . 
       FIG. 4  depicts a block diagram of logical blocks of a modified DSP  451  and DAC module  52  for generation of optical super-channel signal with side frequency components, in accordance with at least one non-limiting embodiment of the present disclosure. To generate modified optical signals  311 ,  312  with side frequency components  331 ,  332 , DSP  51  in transmitter  50  is replaced with modified DSP  451  of  FIG. 4 . 
       FIG. 4  illustrates how the side frequency component with amplitude modulation is applied in X polarization. Mathematical operations of adding are applied in digital domain. The following term (corresponds to block  491   a  in  FIG. 4 ) is added to V XI (t) signal in X-polarization at adder  492   a:  
 
 A   A  cos(2πΔ f   1A   t )cos(2π f   A   m   t ).  (2)
 
     Therefore, DSP  451  transmits to DAC module  52  the I-component in X polarization of V(t) signal that can be written as follows:
 
 V   XI ( t )+ A   A  cos(2πΔ f   1A   t )cos(2π f   A   m   t )  (3)
 
     The following term (corresponds to block  491   b  in  FIG. 4 ) is added at to V XQ (t) signal in X-polarization at adder  492   b:  
 
 A   A  sin(2πΔ f   1A   t )cos(2π f   A   m   t ).  (4)
 
     DSP  451  transmits to DAC module  52  the Q-component in X polarization of V(t) signal that can be written as follows:
 
 V   XQ ( t )+ A   A  sin(2πΔ A1   t )cos(2π f   A   m   t )  (5)
 
     Adder  492   b  adds term (4), which corresponds to block  491   b , to V XI (t) signal in X polarization, the Q-component in X-polarization of V(t) 
     Optionally, similar terms may be added to corresponding V(t) components in Y polarization. 
     As a result of using DSP  451  in transmitter  50 , the optical signal has an optical side component frequency f A  with a side frequency component modulation frequency f A   m . 
     Referring also to  FIG. 3 , the frequency difference f 1A  between subcarrier center frequency f 1  of first subcarrier  311  and optical side component frequency f A  of first side frequency component  331  may be maintained very accurately, because the side frequency component is generated digitally in DSP  451 . An error may be due to a clock error in transmitter  50 . However, such clock error may be less than 20 parts per million (ppm). For example, for frequency difference f 1A =34 GHz, the maximum error may be 0.68 MHz. 
     The first side frequency component  331  having optical side component frequency f A  is modulated at first modulation frequency f A   m . A second side frequency component  332  with optical side component frequency f B  is modulated at second modulation frequency f B   m . It is contemplated that other subcarriers may have side frequency components that are modulated with corresponding side frequency component modulation frequencies f i   m . 
     Each side optical frequency component  331 ,  332  has a very narrow spectrum, which is determined by the modulation frequency f A   m , f B   m . The modulation frequency f A   m  may be between a few MHz and a few tens MHz. In at least one embodiment, the power of the optical frequency components  331 ,  332  is much smaller compared to the total power of the optical super-channel signal. In some embodiments, the power of the optical frequency components  331 ,  332  may be, for example, less than 1% of the total power. 
     The side component frequencies f A  and f B  are in the range of C-band which is between about 191 and about 196 terahertz (THz). A side component spacing f A −f B  may be between 0 and a few GHz. In some embodiments, modulation frequencies f A   m  and f B   m  of first and second side frequency components  331 ,  332  are in the range of a few MHz or a few tens of MHz, and the side component spacing f A −f B  is in the range of a few hundred MHz. 
       FIG. 5  is an illustration of a spectral portion  500  with three neighboring subcarriers  511 ,  512 ,  513  having side frequency components  531 ,  532 ,  533 ,  534 , in accordance with various embodiments of the present disclosure. In some embodiments, one subcarrier  512  may have two side frequency components  532 ,  533 . 
     Referring back to  FIG. 3 , to transmit information over the fiber link, each of the two subcarriers  311 ,  312  is modulated with data (using MZMs  471   a ,  472   b  depicted in  FIG. 4 ). After data modulation, the electrical fields of first and second subcarriers  311 ,  312  can be written as follows:
 
 E   1 ( t )= E   10 ( t )exp(− i 2π f   1   t ),  (6)
 
 E   2 ( t )= E   20 ( t )exp(− i 2π f   2   t ),  (7)
 
where E 10 (t) and E 20 (t) are the baseband complex electrical fields.
 
     One can assume that the frequency difference between subcarrier optical center frequency f 1  of first subcarrier  311  and optical side component frequency f A  of side frequency component  331  is positive: Δf 1,A =f 1 −f A &gt;0. One can also assume that a difference between subcarrier optical center frequency f 2  of second subcarrier  312  and optical side component frequency f B  of second side frequency component  332  is negative: Δf 2B =f 2 −f B &lt;0. The electrical fields of first and second subcarriers  311 ,  312  may be derived as follows:
 
 E   1 ( t )=[ E   10 ( t )+ A   A  exp( i 2πΔ f   1A   t )]exp(− i 2π f   1   t ),  (8)
 
 E   2 ( t )=[ E   20 ( t )+ A   B  exp( i 2πΔ f   2B   t )]exp(− i 2π f   2   t ),  (9)
 
     In some embodiments, side frequency component  331  (with optical side component frequency f A ) is located in a vicinity of an edge of the subcarrier spectrum of first subcarrier  311 . However, side frequency component  331  may have any frequency within the bandwidth of the corresponding subcarrier  311  (e.g., within the bandwidth Δf 1  of first subcarrier  311 ). 
     In some embodiments, for convenience, the differences between the central subcarrier frequencies and side component frequencies may be equal: |Δf 1A |=|Δf 2B |. Referring again to  FIG. 3 , one may derive that a controlling difference between side component frequencies of two neighboring side frequency components  331 ,  332  is:
 
 f   A   −f   B =( f   1   −f   2 )−(|Δ f   1A   |−|Δf   2B |).  (10)
 
     In equation (10), f 1 −f 2  is a target subcarrier spacing between two neighboring subcarriers  311 ,  312 . The target subcarrier spacing f 1 −f 2  may be pre-determined, for example, from the design of the optical link. If (|Δf 1A |+|f 2B |) is known, a difference between two neighbouring side frequency components f A −f B  is maintained constant at a desired value in order to maintain the target subcarrier spacing f 1 −f 2  constant. 
       FIG. 6  depicts an apparatus  600  configured to detect the difference between side component frequencies f A f B  of two neighbouring side frequency components, in accordance with various embodiments of the present disclosure. 
     It should be noted that, although the apparatus is depicted for two subcarriers, it may be configured to detect subcarrier spacings for pairs defined between any number of subcarriers. In some embodiments, one subcarrier may be selected to be a master subcarrier and the other subcarriers may be slaves. The slave subcarriers may track (follow) the master subcarrier. 
     The apparatus  600  comprises a signal generator  601  and a controller  602 . The signal generator  601  comprises DAC converters  52 , electrical drivers  54 , lasers  56  and EO modulators  58 , which have been described above. 
     The DSPs  451  were discussed and depicted in further details in  FIG. 4 . The DSPs  451  cause the optical signal with subcarriers  311 ,  312  to have side frequency components  331 ,  332  (illustrated in  FIG. 3 ). The side frequency components  331 ,  332  at f A  and f B  are modulated at side modulation frequencies f A   m , f B   m . Two subcarriers  311 ,  312  with side frequency components  331 ,  332  are coupled by a coupler  670 . The output optical signal of coupler  670  comprises a first subcarrier at frequency f 1  with a first side frequency component at f A  and a second subcarrier at frequency f 2  with a second side frequency component at f B . 
     At the output of coupler  670 , one portion of light  672  in the output optical signal may be transmitted to the optical link (not depicted), while another portion of light  673  may be tapped to controller  602  in order to determine the side component spacing f A −f B , as described below. 
     The tapped light  673  is converted to photocurrent by a photodetector (PD)  674 . The PD  674  is a low-pass PD and may have a bandwidth that is larger than the designed spacing |f A −f B |. In some embodiments, the photocurrent may be amplified by a transimpedance amplifier (not depicted). 
     In at least one embodiment, frequencies of each of side frequency components  331 ,  332  relative to subcarrier frequencies f 1  and f 2 , |Δf 1A |=|Δf 2B |, may be chosen such that a side component spacing f A −f B  is equal to the filter center frequency of a first electronic filter  676  in  FIG. 6 . In at least one embodiment, the first electronic filter  676  is a bandpass filter with a narrow passband, as described herein. 
     The optical electric field for side frequency components  331 ,  332 , E A  and E B , respectively, may be written as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
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                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               f 
                               B 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               f 
                               B 
                               m 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     where A A  and A B  are the amplitudes, and t is time. 
     When side frequency components  331 ,  332  are transmitted together, the optical intensity is:
 
 I ( t )=| E   A ( t )+ E   B ( t )| 2 ,  (13)
 
     Taking into account equations for optical electric field for side frequency components  331 ,  332  of equations (11)-(12), one may re-write the intensity as follows: 
     
       
         
           
             
               
                 
                   
                     I 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         A 
                         A 
                         2 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           cos 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               f 
                               A 
                               m 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         A 
                         B 
                         2 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           cos 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               f 
                               B 
                               m 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         A 
                         A 
                       
                       ⁢ 
                       
                         A 
                         B 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                               π 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     f 
                                     A 
                                   
                                   - 
                                   
                                     f 
                                     B 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               f 
                               A 
                               m 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               f 
                               B 
                               m 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     The photocurrent and voltage V PD (t) generated by PD  674  are proportional to optical intensity I(t) of equation (14):
 
 V   PD ( t )≈ I ( t ).  (15)
 
     It should be noted that the equations provided herein assume that side frequency components  331 ,  332  have the same polarization, which can be provided by optical design of MZMs  472   a ,  472   b.    
     The first two terms of equation (14) separately contain information about side frequency component  331  and side frequency component  332 . The last term of equation (14) contains the side component spacing (f A −f B ). However, the information cannot be extracted by detecting side frequency components  331 ,  332 , the tone corresponding to their sum f A   m +f B   m  or to their difference f A   m −f B   m , because the fast changing beat factor cos(2π(f A −f B )t) averages these tones out. 
     In order to detect side frequency components  331 ,  332 , or the tones corresponding to their sum or difference, it is necessary to have a non-zero factor cos(2π(f A −f B )t) after averaging. In some embodiments, this may be achieved by a squaring operation. It should be noted that other operations may also be used to obtain a non-zero factor cos(2π(f A −f B )t) after averaging. 
     In at least one embodiment, the first two terms of equation (14) accounting for base-band beating can be neglected if a bandpass filter is applied to the signal. The bandpass filter may be designed such that the only term of equation (14) that may pass the bandwidth of the bandpass filter is the last term of equation (14). 
     Referring again to  FIG. 6 , the electrical signal generated by the PD  674  is filtered by first electronic filter  676 . The first electronic filter  676  is a narrow bandpass analog electronic filter. 
       FIG. 7  is an illustration of the transfer function H of first electronic filter  676  of  FIG. 6 , in accordance with at least one embodiment of the present technology. 
     A filter center frequency f f  of first electronic filter  676  may be set to a specific frequency chosen such that it corresponds to a desired difference between f A  and f B . The center frequency f f  of first electronic filter  676  is chosen to be significantly higher than f A   m  and f B   m , so that only the beat term f A −f B  can pass through the first electronic filter  676 . 
     For example, the center frequency f f  of first electronic filter  676  may be 300 MHz. The passband of first electronic filter  676  may be a few MHz. 
     When the optical signal passes through first electronic filter  676 , the base-band f A   m  and f B   m  tones are significantly attenuated. Referring to equation (14), the first electronic filter  676  is configured to pass through only the last (third) term of equation (14), which is: 
     
       
         
           
             
               
                 
                   2 
                   ⁢ 
                   
                     A 
                     A 
                   
                   ⁢ 
                   
                     A 
                     B 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         
                           π 
                           ⁡ 
                           
                             ( 
                             
                               
                                 f 
                                 A 
                               
                               - 
                               
                                 f 
                                 B 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         t 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           f 
                           A 
                           m 
                         
                         ⁢ 
                         t 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                             1 
                             2 
                           
                           ⁢ 
                           
                             f 
                             B 
                             m 
                           
                           ⁢ 
                           t 
                         
                         ) 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     The difference between side component frequencies, i.e. side component spacing f A −f B , determines the beat term. The beat term corresponding to side component spacing f A −f B  passes through first electronic filter  676 . 
     The locking range is determined by filter parameters of first electronic filter  676  such as, for example, center frequency, bandwidth, roll-off, etc. In at least one embodiment, the transfer function H of first electronic filter  676  acts as an etalon for optical wavelength locker with a much higher finesse. 
     As depicted in  FIG. 6 , after passing through first electronic filter  676 , the intensity is then squared by a squaring device  678 . The squaring device  678  is configured to perform a square operation on the photocurrent (intensity of the optical signal). For example, the squaring device  678  may comprise a multiplier. In some embodiments, the square operation may be performed by a splitter and the multiplier. 
     After squaring the last term of the equation (14), the squared voltage that is obtained at the output of the squaring device  678  is: 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                          
                         
                           H 
                           ⁡ 
                           
                             ( 
                             
                               
                                 f 
                                 A 
                               
                               - 
                               
                                 f 
                                 B 
                               
                             
                             ) 
                           
                         
                          
                       
                       2 
                     
                     ⁢ 
                     
                       A 
                       A 
                       2 
                     
                     ⁢ 
                     
                       A 
                       B 
                       2 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         cos 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                             1 
                             2 
                           
                           ⁢ 
                           
                             f 
                             A 
                             m 
                           
                           ⁢ 
                           t 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         cos 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                             1 
                             2 
                           
                           ⁢ 
                           
                             f 
                             B 
                             m 
                           
                           ⁢ 
                           t 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Equation (17) may be rewritten as follows: 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                          
                         
                           H 
                           ⁡ 
                           
                             ( 
                             
                               
                                 f 
                                 A 
                               
                               - 
                               
                                 f 
                                 B 
                               
                             
                             ) 
                           
                         
                          
                       
                       2 
                     
                     ⁢ 
                     
                       A 
                       A 
                       2 
                     
                     ⁢ 
                     
                       A 
                       B 
                       2 
                     
                     ⁢ 
                     
                       
                         1 
                         + 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 f 
                                 A 
                                 m 
                               
                               ⁢ 
                               t 
                             
                             ) 
                           
                         
                       
                       2 
                     
                     ⁢ 
                     
                       
                         1 
                         + 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 f 
                                 B 
                                 m 
                               
                               ⁢ 
                               t 
                             
                             ) 
                           
                         
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     In at least one embodiment, the term corresponding to the difference frequency f A   m −f B   m  may be detected. It should be noted that in some embodiments, a sum frequency term f A   m +f B   m  may also be detected. 
     The power is therefore proportional to the following term, which one may detect
 
 I   2 ( t )=2| H ( f   A   −f   B )| 2   A   A   2   A   B   2  cos(2π( f   A   m   −f   B   m ) t )  (19)
 
where H(f) is an amplitude transfer function of first electronic filter  676 .
 
     The detected power of the difference frequency tone measured at frequency f A   m −f B   m  is:
 
 P∝A   A   2   A   B   2 ( f   A   −f   B ).  (20)
 
     Referring again to  FIG. 6 , after the squaring operation performed by squaring device  678 , the optical signal passes through a second electronic filter  680 . The second electronic filter  680  is configured to filter radio-frequency (RF) power of the RF signal about a modulated beat frequency tone. 
     It should be noted that, in at least one embodiment, both first electronic filter  676  and second electronic filter  680  are narrow bandpass filters. The second electronic filter  680  may have a very narrow pass band, such as several kilohertz (kHz). The first electronic filter  676  is configured to pass through the beat term f A −f B , while the second electronic filter  680  is configured to pass through the modulated beat frequency tone. In some embodiments, the modulated beat frequency tone is a difference between the first side modulation frequency and the second side modulation frequency |f A   m −f B   m | (also referred to herein as a “difference frequency”). In some other embodiments, the modulated beat frequency tone is a sum of the first side modulation frequency and the second side modulation frequency (f A   m +f B   m ). 
     Referring to equation (19), in some embodiments, correlation may be used to detect the beat term. The controller  602  may have other devices that can generate another cosine function cos(2π(f A   m −f B   m )t) and multiply the power expressing by equation (19). This additional multiplication operation may permit detecting the power of the beat term f A −f B , and therefore monitor the beat term f A −f B . 
     In some embodiments, an RF peak power detector may be used to detect the modulated frequency difference |f A   m −f B   m |. 
     As depicted in  FIG. 6 , an RF power meter  682  measures the power of the modulated frequency difference |f A   m −f B   m |. 
     From equation (20), one can understand that the RF power received by the RF power meter  682  is maximized when the difference between side component frequencies of side frequency components  331 ,  332  is lined up to the center of the second bandpass filter  680 . By maximizing the RF power at modulated frequency difference f A   m −f B   m , one may maintain constant the difference between the side component frequencies, i.e. side component spacing, f A −f B . It should be noted that by maximizing the RF power at the modulated frequency difference f A   m −f B   m , one may lock the side component spacing, f A −f B  to the center frequency of first bandpass filter  676 . 
     A control algorithm may be used to control the optical frequencies of lasers  456   a ,  456   b , so that the RF power detected by RF power meter  682  is maximal. In some embodiments, dithering of the laser frequency may be used to provide tuning direction information for controlling the optical frequencies of lasers  456   a ,  456   b  based on the power detected by RF power meter  682 . 
     In some embodiments, the modulated beat frequency tone is a sum of the modulation frequencies of side frequency components, f A   m +f B   m . The RF peak of the sum f A   m +f B   m , may be detected by RF power meter  682  and used to control the optical frequencies  456   a ,  456   b  in a similar manner. In such embodiments, first electronic filter  676  is configured to filter a sum of the modulation frequencies of side frequency components, f A   m +f B   m , and second electronic filter  680  is configured to filter RF peak of the sum of first side modulation frequency and second side modulation frequency, f A   m +f B   m . 
     In the embodiment depicted in  FIG. 6 , analog components are used to monitor the modulated beat frequency tone. In other embodiments, the monitoring of the modulated beat frequency tone may be performed in a digital domain. An analog-to-digital convertor (ADC) (not depicted) may convert the signal generated by PD  674  to the digital domain using. Then, filtering and other operations as discussed herein may be performed by digital processing in order to extract the RF power of modulated beat frequency tone. 
     In at least one embodiment, a non-transitory computer readable medium with computer executable instructions stored thereon may be provided. The computer executable instructions may be executed by a processor. First, the processor may filter the digital signal to obtain a filtered digital signal within a first bandwidth by attenuating the digital signal around a first side modulation frequency and a second side modulation frequency. Then, filtered digital signal is squared and then filtered again to obtain the RF power at the modulated beat frequency tone. 
       FIG. 8  depicts a flowchart illustrating a method for controlling a difference between a first center frequency f 1  of a first optical subcarrier and a second center frequency f 2  of a second optical subcarrier of the optical super-channel signal in the optical network, in accordance with various embodiments of the present disclosure. 
     At step  802 , the first optical subcarrier is modulated at a first optical side component frequency f A  with a first side modulation frequency f A   m . The difference between the first optical side component frequency and the first center frequency of the first subcarrier, Δf 1A =f 1 −f A , is maintained constant. 
     At step  804 , the second optical subcarrier is modulated at a second optical side component frequency f B  with a second side modulation frequency f B   m . The difference between the second optical side component frequency and the second center frequency of the first subcarrier, Δf 2B =f 2 −f B , is maintained constant. In at least one embodiment, the difference between the first optical side component frequency and the second optical side component frequency is smaller than a difference between the first center frequency and the second center frequency. 
     At step  810 , the modulated first and second optical subcarriers are coupled together to obtain a modified optical signal. 
     At step  812 , the modified optical signal may be converted to electrical signal. 
     At step  814 , the electrical signal may then be filtered by a first bandpass electronic filter. The first electronic filter has a bandwidth corresponding to a difference between the first optical side component frequency and the second optical side component frequency, i.e. side component spacing, f A −f B . 
     At step  816 , the filtered electrical signal may be squared. 
     At step  818 , the squared electrical signal may be filtered by the second electronic filter to obtain the RF signal at the modulated beat frequency tone (f A   m +f B   m ) or (f A   m −f B   m ). The modulated beat frequency tone bears information of the first side modulation frequency and the second side modulation frequency by being either a sum (f A   m +f B   m ) or a difference f A   m −f B   m  of the first side modulation frequency and the second side modulation frequency. 
     At step  820 , the RF power is detected at the modulated beat frequency tone in the modified optical signal. 
     At step  830 , a variation of the difference between the first center frequency f 1  of the first optical subcarrier and the second center frequency f 2  of the second optical subcarrier is monitored. As discussed above, the RF power, monitored at step  820 , is related to the side component spacing f A −f B . The monitored RF power is smaller if the side component spacing f A −f B  is away from the desired value. To reduce deviation of the difference between the first center frequency f 1  of the first optical subcarrier and the second center frequency f 2  of the second optical subcarrier (i.e. f 1 −f 2 ) from the desired value, at least one of the first center frequency f 1  and the second center frequency f 2  may be tuned. 
     At step  840 , at least one of the first center frequency f 1  and the second center frequency f 2  may be detuned to maximize the RF power of the RF signal at the modulated beat frequency tone. 
       FIG. 9  depicts a block diagram of an experimental set-up 900 for confirmation of detectability of the side component spacing f A −f B . One laser  902  was used to generate two laser lines with known frequency spacing. Laser spectrum  903  of laser  902  is schematically illustrated in  FIG. 9 . 
     The output of a continuous wave (CW) laser  902  passes through in-phase quadrature phase MZM (IQ-MZM)  904 . The IQ-MZM  904  is driven by a driver  906  and a DAC  908  to provide a frequency shift with envelope modulation. The IQ-MZM  904  modulates the laser output at 2.929 MHz and 3.9063 MHz. Two laser lines  905  at frequencies f A , f B  are schematically illustrated in  FIG. 9 . The optical signal is converted to photocurrent at PD  974  which has a low pass bandwidth of about 400 MHz. 
     The photocurrent then passes through a bandpass filter  976  with a bandwidth of about 50 MHz centered at about 300 MHz. The filtered signal is then squared by squaring device  678  and received by a scope  985 . The power of the difference frequency tone, f A   m −f B   m , which is 3.9063−2.929=0.9773 MHz is then measured as a function of the spacing between 2 laser lines f A  and f B . 
       FIG. 10  depicts a power of the modulated frequency difference tone (beat tone), |f A   m −f B   m |, measured at the scope  985  of the set-up of  FIG. 9 . The frequency offset between two laser tones f A , f B  is swept between 150 MHz and 450 MHz. The power of the modulated frequency difference tone at 0.9773 MHz depends on the separation between frequencies f A , f B  and has a maximum when the laser frequency separation is around 300 MHz. The modulated frequency difference tone |f A   m −f B   m | was measured to be 0.9773 MHz for each laser frequency spacing f A −f B . 
     In  FIG. 10 , curve  1010  depicts a squared transfer function of the passband filter  976 . Dots depict measured power of difference frequency tone |f A   m −f B   m |. 
     The set-up and measurements of  FIG. 9-10  illustrate that the measured power of difference frequency tone at f A   m −f B   m  may be indeed expressed by equation (20). The results illustrate that monitoring of f A   m −f B   m  may help to monitor the deviation of the subcarrier spacing f 1 −f 2  from a desired value. 
     The described herein technique for controlling of the difference between center frequencies of optical subcarriers of an optical super-channel signal does not require an optical etalon. Furthermore, the second electronic filter acts as an RF frequency discriminator and provides a high resolution on the order of MHz. The RF components used in the technique as described herein may be low-speed, which can help to reduce the cost of the transmitter. The technique as described herein may help to lock the center frequencies of the subcarriers at a desired frequency separation. Moreover, such technique may allow for decreasing and increasing, on demand, of the frequency separation between the subcarriers. 
     It is to be understood that the operations and functionality of the disclosed methods and apparatuses may be achieved by hardware-based, software-based, firmware-based elements and/or combinations thereof. Such operational alternatives do not, in any way, limit the scope of the present disclosure. 
     It will also be understood that, although the inventive concepts and principles presented herein have been described with reference to specific features, structures, and embodiments, it is clear that various modifications and combinations may be made without departing from such disclosures. The specification and drawings are, accordingly, to be regarded simply as an illustration of the inventive concepts and principles as defined by the appended claims, and are contemplated to cover any and all modifications, variations, combinations or equivalents that fall within the scope of the present disclosure.