WHITE PAPER. Filter Bandwidth Definition of the WaveShaper S-series Programmable Optical Processor

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1 WHITE PAPER Filter andwidth Definition of the WaveShaper S-series 1 Introdction The WaveShaper family of s allow creation of ser-cstomized filter profiles over the C- or L- band, providing a flexible tool for indstrial or research laboratories andpass filter profiles can be created with a bandwidth ranging from 1 Hz to 5 THz, in 1 Hz steps However, pon measring the spectral response of the filter, sers can be confsed by the bandwidth definition, which conforms to neither a fll-width half-maximm (FWHM) nor a 1/e measrement Frthermore, an inspection of the shape of a nominally flat-top filter shows that there is a finite slope to the roll-off edge of the filter This can limit the adjacent channel extinction, particlarly for very narrow filter bandwidths What does the specified bandwidth refer to - and is it constant for all devices and channels? This white paper aims to provide the reader with an nderstanding of how the Finisar WaveShaper can generate spectral filters that are defined as having a bandwidth, First, a theoretical treatment of the filter shape is presented, with a prediction of how the filter bandwidth shold be specified Second, experimental reslts are shown which confirm the theoretical analysis Finally, these findings are smmarized and clearly stated for the reader Theory In this section, an expression for the defalt, flat-top filter shape prodced by the WaveShaper family of Programmable Optical Filters is presented, along with an analysis to appropriately define the filter bandwidth specification It shold be noted that this analysis is derived for a simple flat-top filter response with no additional amplitde or phase shaping 1 Prediction of filter shape An ideal optical filter of bandwidth,, shold look like a rectanglar fnction, from -/ to / That is, an ideal channel shape can be expressed as 1, / f / R( f ) = (1), otherwise This rectanglar fnction is generated on the liqid crystalon-silicon (LCOS) chip, which sits in the Forier plane of a 4-f imaging system Propagation throgh the optical system can be redced to an optical transfer fnction that is composed of a relatively narrow assian spectrm, given by L f σ ( f f ) / ( ) Ae, = () A is the spectrm amplitde, f o is the filter center freqency and σ is the spectral variance of the assian spectrm The spectral variance is related, throgh the standard deviation, to the 3 d bandwidth of a assian spectrm by the following relation: W3 d σ = (3) ln The bandwidth of the optical transfer fnction assian is roghly 1 Hz, which sggests that the standard deviation is eqal to 4466 Hz The filter shape generated is then given by the convoltion of (1) and (), or S( f ) = R( f ) L( f ) + = R( f ) L( f f ) df Sbstitting (1) and () into (4), and setting f o =, A = 1, the integral becomes (4) ( f f ) /σ = (5) S( f ) e df In order to evalate this integral, a variable sbstittion is reqired Redefining the variable to be eqal to allows (5) to be rewritten as f f, σ = (6) ( ) σ, 1 S = e d (7) Page 1

2 APPLICATION NOTE: Filter andwidth Definition of the WaveShaper S-series Normalized power (d) Hz 4 Hz 6 Hz 8 Hz 1 Hz -5 Figre 1: The filter shape, S(f), calclated for different filter bandwidths, Freqency offset (Hz) 1 = = f σ f σ Evalation of (7) reqires se of the error fnction, which is commonly defined as π erf( a) e dt (8) a t = (9) Using (9), the integral in (7) can be evalated, reslting in the final form of the channel shape generated by the WaveShaper, given by S f σ π f f σ σ ( ) = erf erf (1) Examples of this filter shape are shown in Figre 1, S(f) calclated for several different vales of and an optical transfer fnction bandwidth of 1 Hz The net effect of convoltion with a assian is to smooth ot the sharp featres of the rectanglar fnction, which makes the otpt spectrm narrower than the original rectanglar profile at the top and wider at the bottom, necessitating the existence of a point in the spectrm that crosses the rectanglar profile bondary This crossover point marks the relative power level at which the actal filter shape has a bandwidth, Figre : Identifying the location of the crossover point, the filter shape (red) crosses the ideal rectanglar window (ble) Defining the channel bandwidth specification In the above analysis, the desired filter profile is a rectanglar fnction with bandwidth, The actal filter shape generated, however, is narrower at the top part of the channel, and spills ot of the rectanglar window at the bottom, even thogh the channel is labeled as a bandpass filter with bandwidth, To clearly define the bandwidth of the actal filter shape, it is instrctional to look at the power level at which S(f) crosses the rectanglar fnction bondary This is illstrated in Figre, the crossover point, is identified as the point the generated filter shape crosses the ideal rectanglar window If the crossover point, P cx, occrs at a constant power level, this power level can be sed to strictly define the bandwidth of the filter response The power ratio at which P cx occrs, for a given filter bandwidth,, is given by P cx S ( ) S(), = (11) S(f) was defined in the previos section Using (1), this can be evalated and simplified to P cx erf( ) =, (1) erf ( ) σ = (13) Page

3 APPLICATION NOTE: Filter andwidth Definition of the WaveShaper S-series erf( )/erf( /) / Figre 3: Error fnction ratio trend as the ratio of filter bandwidth to optical transfer fnction bandwidth increases Previosly, a relation between the 3 d bandwidth of the assian filter,, and s was stated in (3), which is sed here to express as = ln (14) Hence, an expression for the crossover point, P cx, is fond in terms of the ratio between the filter bandwidth,, and the innate optical transfer fnction bandwidth, It shold be noted that this only holds for a assian spectrm Frther conclsions can be drawn by considering the natre of the error fnction, which tends to ½ as increases This is illstrated in Figre 3 the ratio of error fnctions is calclated as / is increased Clearly, for large ratios of /, the ratio of error fnctions tend to one; conservatively speaking, it appears safe to conclde that when the desired filter response is at least 3 times larger than the optical transfer fnction bandwidth, the crossover point occrs at for, P (15) This translates to, on a decibel scale, a level that is -6 d down from the filter peak This is illstrated in Figre 4, which plots the expression given in (1) as a fnction of the actal filter bandwidth For vales of that are at least three times larger than, it is evident that the crossover point is located at, roghly, -6 d from the filter peak To reiterate, when a WaveShaper Programmable Optical Processor is set to prodce a rectanglar fnction of cx 1 Crossover point level (d) Figre 4: Location of crossover point with respect to actal filter bandwidth bandwidth, which is at least three times larger than the optical transfer fnction bandwidth, the otpt filter response will have, roghly, a 6 d bandwidth of If the bandwidth is less than three times the optical transfer fnction bandwidth, the bandwidth is defined at the crossover point indicated in Figre 4 3 Experimental Reslts In order to compare the presented theory to measred reslts, prodction data from a range of WaveShaper s were extracted and analyzed Normalized power (d) andpass filter bandwidth (Hz) predicted measred Freqency offset (Hz) Figre 5: Comparison between predicted and measred channel shape for a 5 Hz filter response Page 3

4 APPLICATION NOTE: Filter andwidth Definition of the WaveShaper S-series Normalized power (d) port 1 port port 3 port 4 Normalized power (d) port 1 port port 3 port Freqency (THz) Figre 6: For port operation with 1 Hz channel spacing comparing (solid) measred and (dashed) predicted data One 5 Hz channel, from a WaveShaper 1S, was taken to compare to the predicted filter shape shown in (1) Measred reslts are compared to this predicted channel shape in Figre 5 The spectra match reasonably well, with the exception of a small broadening on one side of the experimental data, which is attribted to the asymmetric beam shape throgh the device To compare the otpt of a mltiport device with filter profiles sent to niqe ports, a WaveShaper 4S was set to otpt adjacent 1 Hz bandpass filters to each port The measred data is shown in Figre 6 compared to predicted channel shapes, as derived in the previos section It shold be noted that an optical transfer fnction bandwidth of 1 Hz was sed in these calclations, which takes into accont the resoltion bandwidth of the optical spectrm analyzer The predicted model prodces an excellent fit compared to the nominal 1 Hz bandpass filter responses, with a slight broadening of the measred data from port 3 From (1), the predicted crossover point of a 1 Hz bandpass filter, sing an optical transfer fnction bandwidth of 1 Hz, is -33 d From the measred spectra in Figre 6, the average crossover point occrred at -338 d Similarly, the behavior of 5 Hz channels can be predicted with the presented models A WaveShaper 4S was therefore set to otpt adjacent 5 Hz channels to niqe ports, and the measred data was compared with predicted spectra, with the reslts shown in Figre Freqency (THz) Figre 7: For port operation with 5 Hz channel spacing, comparing (solid) measred and (dashed) predicted data As the generated filter bandwidth of 5 Hz was mch larger than the optical transfer fnction, the experimental data shows an even better fit to theory The crossover point of the measred spectra occrred at an average of -598 d; in comparison, the predicted spectra have crossover points at -6 d Occrances samples Avg: -6 d StdDev: 41 d Channel crossover point (d) Figre 8: Srvey of crossover points for six interleaved devices, reslting in 58 samples To draw a conclsion abot the location of the crossover point of a WaveShaper filter response, a relatively large sample size was reqired Six WaveShaper processors were measred; each device was set to prodce 5 Hz interleaver patterns, odd and even, sliced into distinct channels, and the crossover point was Page 4

5 APPLICATION NOTE: Filter andwidth Definition of the WaveShaper S-series measred for each slice This reslted in a total of 58 measrements, giving an appropriate sample size to determine statistical data abot the location of the crossover point of a WaveShaper bandpass filter The crossover point analysis is shown in Figre 8, tablated in histogram format For all samples, the mean crossover point location occrs at -6 d, with a standard deviation of 41 d; the standard deviation is acceptable considering the resoltion of the measrement data The simple mathematical model presented in this white paper gives accrate agreement with measred spectral data, allowing accrate definition of the crossover point, confirmed by statistical data 4 Conclsion This paper has presented a theoretical analysis of the bandpass filter shape generated by the WaveShaper family of s, given as the convoltion between a rectanglar profile and a narrow assian spectrm ood matching between theory and experimental data was fond, confirming the accracy of this approach Additionally, it was predicted that the crossover point between the otpt filter response and the initial rectanglar profile of bandwidth wold occr at roghly - 6 d, if the width of the rectanglar profile was greater than 3 Hz For filter profiles narrower than 3 Hz, the crossover point was fond to be given by P cx erf( ) =, (16) erf ( ) Acknowledgements: This white paper was written by Cibby Plikkaseril and was made possible with the assistance of Michaël Roelens, Jeremy olger, Simon Poole and Steve Frisken = ln (17) This was confirmed by experimental data; over 58 measrements for prodction devices were sed to bild a statistical evalation of the location of the crossover point of 5 Hz bandpass filter responses The mean crossover point of the sample set was fond to be -6 d, with a standard deviation of 41 d 1389 Moffett Park Drive Snnyvale, CA 9489 Tel: Fax: waveshaper@finisarcom 1 Finisar Corporation All rights reserved Page 5 Finisar is a registered trademark WSPR 3/1