Introduction The WaveShaper family of Programmable Optical Proce ssors allow creation of usercustomized filter profiles o ver the or band providing a fleible tool for industria l or research laboratories Bandpass filter profiles can be creat ed wi ID: 23748 Download Pdf

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Introduction The WaveShaper family of Programmable Optical Proce ssors allow creation of usercustomized filter profiles o ver the or band providing a fleible tool for industria l or research laboratories Bandpass filter profiles can be creat ed wi

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Page 1 WHITE PAPER Filter Bandwidth Definition of the WaveShaper S-series Programmable Optical Processor 1. Introduction The WaveShaper family of Programmable Optical Proce ssors allow creation of user-customized filter profiles o ver the (- or )- band, providing a fle+ible tool for industria l or research laboratories. Bandpass filter profiles can be creat ed with a bandwidth ranging from 10 -Hz to 5 THz, in 1 -Hz st eps. However, upon measuring the spectral response of th e filter, users can be confused by the bandwidth definition, which conforms to neither a full-width

half-ma+imum (FWH0 ) nor a 1/e measurement. Furthermore, an inspection of th e 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 ad3acent channel e+tinction, particularly for very narrow fi lter 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 understanding of how the Finisar WaveShaper Programmable Optical Processor can generate spectra l filters that are defined as having a bandwidth, . First, a theoretical

treatment of the filter shape is presen ted, with a prediction of how the filter bandwidth should be sp ecified. Second, e+perimental results are shown which confir m the theoretical analysis. Finally, these findings are s ummarized and clearly stated for the reader. 2. Theory In this section, an e+pression for the default, fla t-top filter shape produced by the WaveShaper family of Programmable Optical Filters is presented, along wi th an analysis to appropriately define the filter bandwid th specification. It should be noted that this analysi s is derived for a simple flat-top filter

response with no addit ional amplitude or phase shaping. 2.1 Prediction of filter shape An ideal optical filter of bandwidth, , should loo6 li6e a rectangular function, from -B/2 to B/2 . That is, an ideal channel shape can be e+pressed as 1, / 2 / 2 ( ) 0, otherwise. B f B R f - ˆ ³ (1) This rectangular function is generated on the li7ui d crystal- on-silicon ()(OS) chip, which sits in the Fourier p lane of a 4-f imaging system. Propagation through the optical sys tem can be reduced to an optical transfer function that is composed of a relatively narrow -aussian spectrum, given by 2 2 ( ) /2

( ) , f f L f Ae - - (2) where is the spectrum amplitude, is the filter center fre7uency and is the spectral variance of the -aussian spectrum. The spectral variance is related, through the standard deviation, to the 3 dB bandwidth of a -aus sian spectrum by the following relation9 2 2ln 2 dB BW (3) The bandwidth of the optical transfer function -aus sian is roughly 10 -Hz, which suggests that the standard de viation is e7ual to 4.2466 -Hz. The filter shape generated is then given by the con volution of (1) and (2), or ( ) ( ) ( ) ( ) ( ) . S f R f L f R f L f f df +¯ -¯ = * ´ ´ ´ = - (4)

Substituting (1) and (2) into (4), and setting = 0, = 1, the integral becomes 2 2 ( ) /2 ( ) . f f S f e df - - (5) In order to evaluate this integral, a variable subs titution is re7uired. Redefining the variable to be e7ual to f f (6) allows (5) to be rewritten as ( ) 2 2 , S u e du (=)

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APP)I(ATION NOTE9 Filter Bandwidth Definition of t he WaveShaper S-series Programmable Optical Processor Page 2 Figure 1: The filter shape, S(f), calculated for di ffere t filter ba dwidths, B. where - - (?) Evaluation of (=) re7uires use of the error functio n, which is commonly defined as

erf( ) . a e dt (@) Asing (@), the integral in (=) can be evaluated, re sulting in the final form of the channel shape generated by th e WaveShaper Programmable Optical Processor, given by 2 2 ( ) 2 erf erf . 2 2 B B f f S f s p s s - - - = - (10) E+amples of this filter shape are shown in Figure 1 , S(f) calculated for several different values of and an optical transfer function bandwidth of 10 -Hz. The net effe ct of convolution with a -aussian is to smooth out the sh arp features of the rectangular function, which ma6es t he output spectrum narrower than the original rectangu lar profile at

the top and wider at the bottom, necessi tating the e+istence of a point in the spectrum that crosses t he rectangular profile boundary. This Bcrossover point ” mar6s the relative power level at which the actual filter shape has a bandwidth, . Figure 2: Ide tifyi g the locatio of the crossover poi t, where the filter shape (red) crosses the ideal recta gular wi dow (b lue). 2.2 Defining the channel bandwidth specification In the above analysis, the desired filter profile i s a rectangular function with bandwidth, . The actual filter shape generated, however, is narrower at the top pa rt of the

channel, and spills out of the rectangular window a t the bottom, even though the channel is labeled as a ban dpass filter with bandwidth, . To clearly define the bandwidth of the actual filte r shape, it is instructional to loo6 at the power level at whic h S(f) crosses the rectangular function boundary. This is illustrated in Figure 2, where the crossover point, is identifi ed as the point where the generated filter shape crosses the ideal rectangular window. If the crossover point, c( , occurs at a constant power level, this power level can be used to strictly define the bandwidth of the

filter response. The power ratio at which c( occurs, for a given filter bandwidth, , is given by ( ) (0) cx (11) where S(f) was defined in the previous section. Asing (10), this can be evaluated and simplified to ( ) erf( ) 2erf cx (12) where (13) -50 -40 -30 -20 -10 Normalized power (dB) -100 -50 0 50 100 Frequency offset (GHz) 20 GHz 40 GHz 60 GHz 80 GHz 100 GHz

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APP)I(ATION NOTE9 Filter Bandwidth Definition of t he WaveShaper S-series Programmable Optical Processor Page 3 Figure 3: Error fu ctio ratio tre d as the ratio o f filter ba dwidth to optical tra sfer fu ctio ba

dwidth i creases. Previously, a relation between the 3 dB bandwidth o f the -aussian filter, , and s was stated in (3), which is used here to e+press as 2 ln 2 . (14) Hence, an e+pression for the crossover point, c( , is found in terms of the ratio between the filter bandwidth, , and the innate optical transfer function bandwidth, . It should be noted that this only holds for a -aussian spectrum. Further conclusions can be drawn by considering the nature of the error function, which tends to as increases. This is illustrated in Figure 3 where the ratio of error fu nctions is

calculated as B/B is increased. (learly, for large ratios of B/B , the ratio of error functions tend to oneE conservatively spea6ing, it appears sa fe to conclude that when the desired filter response is a t least 3 times larger than the optical transfer function ban dwidth, the crossover point occurs at for , . G cx B B P (15) This translates to, on a decibel scale, a level tha t is -6.02 dB down from the filter pea6. This is illustrated in F igure 4, which plots the e+pression given in (12) as a funct ion of the actual filter bandwidth. For values of that are at least three times larger than

, it is evident that the crossover point is located at, roughly, -6 dB from the filter pea6. To reiterate, when a WaveShaper Programmable Optica l Processor is set to produce a rectangular function of Figure 4: -ocatio of crossover poi t with respect to actual filter ba dwidth. bandwidth , which is at least three times larger than the optical transfer function bandwidth, the output fil ter response will have, roughly, a 6 dB bandwidth of . If the bandwidth is less than three times the optical transfer function bandwidth, the bandwidth is defined at the crossover point indicated in Figure 4. 3.

E+perimental Results In order to compare the presented theory to measure d results, production data from a range of WaveShaper Programmable Optical Processors were e+tracted and analyzed. Figure 5: /o0pariso betwee predicted a d 0easured cha el shape for a 51 G23 filter respo se. 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 erf( )/erf( /2) 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 B/B -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 Crossover point level (dB) 0 10 20 30 40 50 60 70 80 90 100 Bandpass ˛lter bandwidth (GHz) -50 -40 -30 -20 -10 Normalized power (dB) -60 -40 -20 0 20 40 60 Frequency offset (GHz)

predicted measured

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APP)I(ATION NOTE9 Filter Bandwidth Definition of t he WaveShaper S-series Programmable Optical Processor Page 4 Figure 6: Four port operatio with 11 G23 cha el s paci g co0pari g (solid) 0easured a d (dashed) predicted data. One 50 -Hz channel, from a WaveShaper 1000S, was ta 6en to compare to the predicted filter shape shown in ( 10). 0easured results are compared to this predicted cha nnel shape in Figure 5. The spectra match reasonably well, with the e+cepti on of a small broadening on one side of the e+perimental da ta, which is attributed to the

asymmetric beam shape th rough the device. To compare the output of a multiport device with fi lter profiles sent to uni7ue ports, a WaveShaper 4000S w as set to output ad3acent 10 -Hz bandpass filters to each port. The measured data is shown in Figure 6 compared to pred icted channel shapes, as derived in the previous section. It should be noted that an optical transfer function bandwidt h of 12 -Hz was used in these calculations, which ta6es into account the resolution bandwidth of the optical spe ctrum analyzer. The predicted model produces an e+cellent fit compa red to the nominal 10 -Hz

bandpass filter responses, with a slight broadening of the measured data from port 3. From ( 12), the predicted crossover point of a 10 -Hz bandpass filt er, using an optical transfer function bandwidth of 12 -Hz, i s -3.03 dB. From the measured spectra in Figure 6, the average crossover point occurred at -3.3? dB. Similarly, th e behaviour of 50 -Hz channels can be predicted with the presen ted models. A WaveShaper 4000S was therefore set to out put ad3acent 50 -Hz channels to uni7ue ports, and the measured data was compared with predicted spectra, with the results shown in Figure =. Figure

5: Four port operatio with 51 G23 cha el s paci g, co0pari g (solid) 0easured a d (dashed) predicted data. As the generated filter bandwidth of 50 -Hz was muc h larger than the optical transfer function, the e+pe rimental data shows an even better fit to theory. The crosso ver point of the measured spectra occurred at an average of - 5.@? dBE in comparison, the predicted spectra have crossover points at -6.02 dB. Figure 6: Survey of crossover poi ts for si( i terl eaved devices, resulti g i 526 sa0ples. To draw a conclusion about the location of the cros sover point of a WaveShaper

Programmable Optical Processo r filter response, a relatively large sample size was re7uired. Si+ WaveShaper processors were measuredE each devic e was set to produce 50 -Hz interleaver patterns, odd and even, sliced into distinct channels, and the crossover po int was -20 -15 -10 -5 Normalized power (dB) 193.95 193.96 193.97 193.98 193.99 194.0 194.01 194.02 Frequency (THz) port 1 port 2 port 3 port 4 -20 -15 -10 -5 10 Normalized power (dB) 193.9 193.95 194.0 194.05 194.1 194.15 Frequency (THz) port 1 port 2 port 3 port 4 10 20 30 40 50 60 70 80 90 Occurances -8.0 -7.5 -7.0 -6.5 -6.0 -5.5

-5.0 Channel crossover point (dB) 528 samples Avg: -6.2 dB Std.Dev: 0.41 dB

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APP)I(ATION NOTE9 Filter Bandwidth Definition of t he WaveShaper S-series Programmable Optical Processor Page 5 measured for each slice. This resulted in a total o f 52? measurements, giving an appropriate sample size to determine statistical data about the location of th e crossover point of a WaveShaper bandpass filter. The crossover point analysis is shown in Figure ?, tabulated in histogram format. For all samples, the mean cros sover point location occurs at -6.2 dB, with a standard d eviation of

0.41 dBE the standard deviation is acceptable consi dering the resolution of the measurement data. The simple mathematical model presented in this whi te paper gives accurate agreement with measured spectr al data, allowing accurate definition of the crossover point, confirmed by statistical data. 4. (onclusion This paper has presented a theoretical analysis of the bandpass filter shape generated by the WaveShaper f amily of Programmable Optical Processors, given as the convolution between a rectangular profile and a nar row -aussian spectrum. -ood matching between theory and e+perimental data

was found, confirming the accurac y of this approach. Additionally, it was predicted that the crossover p oint between the output filter response and the initial rectangular profile of bandwidth would occur at roughly - 6 dB, if the width of the rectangular profile was g reater than 30 -Hz. For filter profiles narrower than 30 -Hz, t he crossover point was found to be given by ( ) erf( ) 2erf cx (16) where 2 ln 2 . (1=) This was confirmed by e+perimental dataE over 52? measurements for production devices were used to bu ild a statistical evaluation of the location of the cross over point of 50

-Hz bandpass filter responses. The mean crossove r point of the sample set was found to be -6.2 dB, with a s tandard deviation of 0.41 dB. Acknowledgements: This white paper was written by (ibby Puli66aseril and was made possible with the assista nce of 0ichaFl Roelens, Geremy Bolger, Simon Poole and Ste ve Fris6en. 13?@ 0offett Par6 Drive Sunnyvale, (A @40?@ Tel.9 H1-40?-54?-1000 Fa+9 H1-40?-541-613? waveshaperIfinisar.com http9//www.finisar.com/optical-instrumentation J2012 Finisar (orporation. All rights reserved. Finisar is a registered trademar6. WSPR 03/12

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