Filter Bandwidth Definition of the
WaveShaper S-series Programmable
Optical Processor
Filter Bandwidth Definition of the WaveShaper S-series of Programmable Optical Processors
Section 1: Introduction
The WaveShaper family of Programmable Optical Processors allow creation
of user-customized filter profiles over the C- or L- band, providing a flexible tool for
industrial or research laboratories. Bandpass filter profiles can be created with a bandwidth ranging from 10 GHz to 5 THz, in 1 GHz steps. However, upon measuring the
spectral response of the filter, users can be confused by the bandwidth definition,
which conforms to neither a full-width half-maximum (FWHM) nor a 1/e measurement. Furthermore, 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, particularly for very narrow filter bandwidths. What does the specified
bandwidth refer to - and is it constant for all devices and channels?
1 Introduction
2 Theory
3 Experimental Results
4 Conclusion
This white paper aims to provide the reader with an understanding of how the
Finisar WaveShaper Programmable Optical Processor can generate spectral filters that
are defined as having a bandwidth, B. First, a theoretical treatment of the filter shape is
presented, with a prediction of how the filter bandwidth should be specified. Second,
experimental results are shown which confirm the theoretical analysis. Finally, these
findings are summarized and clearly stated for the reader.
Page 2
This document aims to provide the reader with an understanding of how the
Filter Bandwidth Definition of the WaveShaper S-series of Programmable Optical Processors
Finisar WaveShaper generates spectral filters that are defined as having a bandiwidth, B. First, a theoretical treatment of the channel shape is presented, with
a prediction of how the channel bandwidth should be specified. Second, experimental results are shown which confirm our theoretical analysis. Finally, these
findings are summarized and clearly stated for the reader.
Theory
Section 2: Theory
In this section, an expression for the channel shape produced by the Finisar WaveIn this section,along
an expression
the default,
filter shapedefining
produced by
Shaper tunable filter is presented,
with anforanalysis
offlat-top
appropriately
the WaveShaper family of Programmable Optical Filters is presented, along with an
the filter bandwidth specification. It should be noted that this analysis is de-
analysis to appropriately define the filter bandwidth specification. It should be noted
rived for a simple that
flat-top
filter isresponse
by the
WaveShaper,
with no
this analysis
derived for produced
a simple flat-top
filter response
with no additional
additional amplitude
or phase
shaping.
amplitude
or phase
shaping.
2.1 Prediction of filter shape
Prediction of channel shape
An ideal optical filter of bandwidth, B, should look like a rectangular function,
An ideal optical filter
of bandwidth,
B,ideal
should
look
like
from -B/2
to B/2. That is, an
channel
shape
canaberectangular
expressed as function,
from −B/2 to B/2. That is, an ideal channel shape can be expressed as
R(f ) =
1, −B/2 ≤ f ≥ B/2
0,
otherwise.
(1)
This rectangular function is generated on the liquid crystal-on-silicon (LCOS)
chip, which sits in the Fourier plane of a 4–f imaging system. Propagation
through
THEORY
This rectangular function is generated on the liquid crystal-on-silicon (LCOS)
chip, which sits in the Fourier plane of a 4-f imaging system. Propagation
through the
THEORY
optical system can be reduced to an optical transfer function that is composed of a
system relatively
can be reduced
to anspectrum,
optical transfer
narrow Gaussian
given by function that is composed
1
the optical
the
system
can be
reducedspectrum,
to an optical
transfer
of optical
a relatively
narrow
Gaussian
given
by function that is composed
of a relatively narrow Gaussian spectrum, given by
2
2
L(f ) = Ae−(f −fo ) /2σ ,
(2)
−(f −fo )2 /2σ 2
,
(2)
L(f ) = Ae
where A is the spectrum amplitude, fo is the filter center frequency and σ 2 is
2
where
is the
spectrum amplitude,
the filter
center
frequency
is 2the
center
frequency
and
σ
is specwhere
A isvariance
the spectrum
amplitude,
fo is thefo isfilter
the
spectral
of Athe
Gaussian
spectrum.
The
spectral
varianceand
is σrelated,
tral variance of the Gaussian spectrum. The spectral variance is related, through the
the
spectral
of deviation,
the Gaussian
spectrum.
The spectral
is spectrum
related,
through
thevariance
standard
to the
3 dB bandwidth
of avariance
Gaussian
standard deviation, to the 3 dB bandwidth of a Gaussian spectrum by the following
through
the standard
deviation, to the 3 dB bandwidth of a Gaussian spectrum
by the following
relation:
relation:
by the following relation:
BW3dB
.
(3)
σ= √
BW
2 3dB
2 ln 2
σ= √
.
(3)
2 2 lnfunction
2
The bandwidth of the optical transfer
Gaussian is roughly 10 GHz,
The suggests
bandwidth
of the
the standard
optical transfer
function
roughly 10 GHz,
which
that
deviation
is equalGaussian
to 4.2466is GHz.
The bandwidth of the optical transfer function Gaussian is roughly 10 GHz,
whichThe
suggests
that the
standard
is equalistothen
4.2466
GHz.
filter shape
generated
by deviation
the WaveShaper
given
by the convolution
which suggests that the standard deviation is equal to 4.2466 GHz.
filter(2),
shape
of The
(1) and
or generated by the WaveShaper is then given by the convolution
of (1) and (2), or
S(f ) = R(f ) ∗ L(f )
S(f ) = R(f ) ∗ L(f )
+∞
= +∞ R(f ) L(f − f ) df .
−∞
R(f ) L(f − f ) df .
=
−∞
(4)
(4)
Page 3
Normalized
Normalized
powerpower
(dB) (dB)
.
(3)
σ= √
The bandwidth of the optical transfer
Gaussian is roughly 10 GHz,
2 2 function
ln 2
The
bandwidth
of
the
optical
transfer
function
Gaussian
is
roughly
10
GHz,
BW3dBof Programmable Optical Processors
Filter Bandwidth Definition of the WaveShaper S-series
.
(3)
σdeviation
= √ function
which
suggests
that of
thethe
standard
is equal
to 4.2466 is
GHz.
The
bandwidth
optical
transfer
Gaussian
roughly 10 GHz,
which suggests that the standard deviation
2 2 lnis2equal to 4.2466 GHz.
Thesuggests
filter shape
by the
WaveShaper
is then
givenGHz.
by the convolution
which
thatgenerated
the standard
deviation
is equal
to 4.2466
The
shape
generated
by the
WaveShaper
is then
givenisbyroughly
the convolution
The filter
bandwidth
of the optical
transfer
function
Gaussian
10 GHz,
of (1)
and
(2),
or
The
filter
shape
generated
by
the
WaveShaper
is
then
given
by
the
convolution
The
filter
shape
generated
is
then
given
by
the
convolution
of
(1) and (2), or
of
(1) and
(2), or
which
suggests
that the standard deviation is equal to 4.2466 GHz.
of (1) and (2), or
The filter shape generated by the WaveShaper is then given by the convolution
of (1) and (2), or
S(f ) = R(f ) ∗ L(f )
S(f ) = R(f ) ∗ L(f )
S(f ) = R(f ) ∗ L(f )
Prediction of channel shape
(4)
+∞
(4)
+∞
S(f
)
=
R(f
)
∗
L(f
)
(4)
Prediction of channel shape
R(f ) L(f − f ) df .
=
= −∞
+∞ R(f ) L(f − f ) df .
= −∞ R(f ) L(f − f ) df .
(4)
+∞
=
0,
A
=
1,
the
integral
Substituting (1) and (2) into−∞
(4), and
setting
f
20
GHz
o
into
and
Substituting
(1) and Substituting
(2) =
into (4),
and
setting
= =1,1, the
theintegral
integral
(1) R(f
and
(2)
(4),
= 0,
becomes
) L(f
− ff)osetting
df= . 0,fo A
40 AGHz
0
becomes
−∞
60
GHz
=
0,
A
=
1,
the
integral
Substituting
(1)
and
(2)
into
(4),
and
setting
f
20
o
becomes
80
40 GHz
0
becomes
100
GHz
60 GHz
B/2
Substituting
(1) and (2) into (4),
f = 0, A80=GHz1, the integral
B/2 and setting
-10
−(f −f )22 /2σ 22 o S(f ) = B e−(f −f ) /2σ df .
(5)
100 GHz
df .
(5)
S(f ) = − B/2/2 e
becomes -10
2
2
− B/2 −(f −f ) /2σ
e
df .
(5)
S(f ) =
−aBB/variable
In order-20to evaluate this integral,
substitution is required. Redefining
/2
2
In order to evaluate this integral, a variable
substitution
is required. Redefining
2
−(f integral,
−f )2 /2σa
InS(f
order
to evaluateethis
variable
substitution is required. Redefining
df
.
(5)
)
=
the In
variable
u
to
be
equal
to
-20
order to
thisto
integral,
a variable substitution is required. Redefining
the variable
u evaluate
to bethe
equal
− B/2 to
variable
u to be equal
the variable-30 u to be equal to
In order to evaluate this integral, a variable
f − f substitution is required. Redefining
−f,
(6)
u = f√
-30
(6)
u= √
2σ ,
the variable u to be equal to
f
−
f
2σ
-40
u= √
,
(6)
allows (5) to be rewritten as
2σ
allows (5)
to be rewritten as
f −f
-40
(6)
u = √ ,
allows (5)
to
be
rewritten
as
√ 2σuu22 −u2
-50
allows (5) S(u)
to be rewritten
= 2√2σas e−u2 du,
(7)
S(u)
= 2√2σ u1u2 e du,
(7)
allows (5)
-50 to be rewritten as
2
-100
-80
-60
-40
-20
0
60
80
100
−u 40
u1 20
S(u) =
2
2σ
du,
(7)
e
Frequency offset (GHz)
where -100
u
u
1
2
-80
-60
-40
-20 √
0
20
40
60
80
100
where
−u2
Frequency
offset
(GHz)
S(u)
=
2
2σ
e
du,
(7)
Figure
1
–
The
filter
shape,
S(f
),
calculated
for
different
where
where
u1
filter bandwidths, B.
Figure 1 – The filter shape, S(f ), calculated for different
filter bandwidths, B.
where
2
2
2
2
−f
√
− B/22σ
−f
(8)
u1 = B/2√− f
u2 = √ 2σ .
(8)
B/2 2σ
−f
.
u2 = √
2σ
Evaluation of (7) requires
use of the error function, which is commonly defined
Evaluation of (7) requires use of the error function, which is commonly defined
u1 =
− B/2
as Evaluation of (7)asrequires use of the error function, which is commonly defined
√
a
as
π
−t2
erf(a)
=
e
dt.
(9)
√2
0 a
π
2
erf(a) =
e−t dt.
Using (9), the integral in (7)
resulting in the final form(9)
of
2 can be evaluated,
0
the Using
channel
shape
generated
WaveShaper,
given
(9),
the integral
in by
(7)the
canFinisar
be evaluated,
resulting
inby
the final form of
WaveShaper,
− B givenby
the channel shape generated
√ by theBFinisar
/2 − f
/2 − f
− erf √
(10)
S(f ) = σ 2π erf √
.
B/2 2σ
− B/22σ
√
−f
−f
√
√
− erf
.
(10)
S(f ) = σ 2π erf
Examples of this filter shape are shown
2σ in Figure 1, S(f
2σ) calculated for several
different
values
B filter
and an
optical
transfer
of 10 GHz.
The
Examples
of of
this
shape
are shown
in function
Figure 1,bandwidth
S(f ) calculated
for several
net effectvalues
of convolution
a Gaussian
is function
to smooth
out the sharp
of
different
of B and with
an optical
transfer
bandwidth
of 10 features
GHz. The
the
rectangular
function,with
which
makes theis output
spectrum
than the
net effect
of convolution
a Gaussian
to smooth
out thenarrower
sharp features
of
original
rectangular
profilewhich
at themakes
top and
at the
bottom,narrower
necessitating
the
Page 4
the
rectangular
function,
thewider
output
spectrum
than the
Normali
-30
Filter Bandwidth Definition of the
WaveShaper S-series of Programmable Optical Processors
-40
-50
-100
-80
-60
-40
-20
0
20
40
Frequency offset (GHz)
60
80
100
20 GHz
40 GHz
60 GHz
80 GHz
100 GHz
0
Figure 1 – The filter shape, S(f ), calculated for different
filter bandwidths, B.
Normalized power (dB)
-10
-20
−f
2σ
B/2 − f
.
u2 = √
2σ
-30
u1 =
-40
-50
-100
-80
-60
-40
− B/2
√
-20
0
(8)
20
40
60
80
100
Frequencyfunction,
offset (GHz)
Evaluation of (7) requires use of the error
which is commonly defined
as
Figure 1 The filter shape, S(f), calculated for different filter bandwidths, B.
√
a
π
2
erf(a) =
e−t dt.
(9)
2
Using (9), the integral in (7) can be0 evaluated, resulting in the final form of the
channel
generated
the WaveShaper
Programmable
Optical
Processor,
Using (9),
the shape
integral
in (7)by can
be evaluated,
resulting
in the
final given
form of
by
the channel shape generated by the Finisar WaveShaper, given by
B
−B
/2 − f
/2 − f
√
√
S(f ) = σ 2π erf
− erf
.
2σ
2σ
√
(10)
Examples of this filter shape are shown in Figure 1, S(f ) calculated for several
different values ofExamples
B andofan
transfer
function
of for
10several
GHz.difThe
thisoptical
filter shape
are shown
in Figurebandwidth
1, S(f) calculated
values of B and
transferis
function
bandwidth
10 GHz.
The features
net effect of
net effect offerent
convolution
withanaoptical
Gaussian
to smooth
outofthe
sharp
of convolution with a Gaussian is to smooth out the sharp features of the rectangular
the rectangular function, which makes the output spectrum narrower than the
function, which makes the output spectrum narrower than the original rectangular
original rectangular
profile at the top and wider at the bottom, necessitating the
profile at the top and wider at the bottom, necessitating the existence of a point in the
existence ofspectrum
a pointthat
in the
spectrum
that crosses
the rectangular
profile
crosses
the rectangular
profile boundary.
This “crossover
point”boundary.
marks
the relative power level at which the actual filter shape has a bandwidth, B.
3
Page 5
THEORY
Filter Bandwidth Definition of the WaveShaper S-series of Programmable Optical
Processors
THEORY
Figure 2 – Identifying the location of the crossover point,
where the WaveShaper filter shape (red) crosses the ideal rectFigure
Identifying
the location
of the crossover
where thepoint,
filter shape (red) crosses the
Figure
2 –2 Identifying
the location
of thepoint,
crossover
angular
window
(blue).
ideal rectangular
window
(blue.
where the WaveShaper filter shape
(red) crosses
the
ideal rectangular window (blue).
Defining the channel bandwidth specification
2.2 Defining the channel bandwidth specification
In
the above the
analysis,
the desired
filter profile specification
is a rectangular function with bandDefining
channel
bandwidth
width, B. The actual filter shape generated by the WaveShaper, however, is narIn the above analysis, the desired filter profile is a rectangular function with bandIn the
analysis,
the spills
desiredout
filterof
profile
a rectangular function
rower at the top part of
theabove
channel,
and
the isrectangular
windowwith
at
width, B. The actual
filterB.shape
generated
by
the WaveShaper,
however,
istop
narbandwidth,
The
actual
filter
shape
generated,
however,
is
narrower
at
the
part of
the bottom, even though the channel is labeled as a B bandwidth filter.
rower at the top the
part
of
the
channel,
and
spills
out
of
the
rectangular
window
at
channel, and spills out of the rectangular window at the bottom, even though the
To clearly define
the bandwidth
of the filter
WaveShaper
filterB. shape, it is instrucchannel
as a bandpass
the bottom, even thoughis labeled
the channel
is labeledwith
as bandwidth,
a B bandwidth
filter.
tional to look at the power level at which S(f ) crosses the rectangular function
To clearly define the
bandwidth
of bandwidth
the WaveShaper
filter
shape,it isitinstructional
is instruc-to
To clearly
define the
of the actual
filter shape,
boundary. This is illustrated in Figure 2, where the crossover point, Pcx , is identhe power
level
which S(f)
crosses
the rectangular
function boundary.
This is
tional to look at look
the atpower
level
atatwhich
S(f
) crosses
the rectangular
function
tified as the pointillustrated
where in
the
generated
filter
shape
from
the
WaveShaper
crosses
Figure 2, where the crossover point, is identified as the point where the
boundary. This is illustrated
in Figure 2, where the crossover point, Pcx , is idengenerated
filter shape crosses the ideal rectangular window.
the ideal rectangular
window.
tified as the point where the generated filter shape from the WaveShaper crosses
constant
powerpower
level,level,
thisthispower
If the crossover point,
cx , occurs
If the P
crossover
point,atPcxa
, occurs
at a constant
power level
level can
the ideal rectangular window.
be useddefine
to strictly
define
the bandwidth
of the
filter response.filter response.
can be used to strictly
the
bandwidth
of the
WaveShaper
If the crossover point, Pcx , occurs at a constant power level, this power level
a given
filter
B,B,isis given
givenby
The power ratio atThe
which
Pratio
cx occurs,
power
at whichfor
Pcx occurs,
a givenbandwidth,
filter bandwidth,
can be used to strictly define
the bandwidth
of theforWaveShaper
filter response.
by
The power ratio at which Pcx occurs, for a given filter bandwidth, B, is given
by
S(−B/2)
,
(11)
Pcx =
S(0)
S(−B/2)
, Using (10), this can be evalu(11)
=
P
cx
where S(f ) was defined in the previous
section.
S(0)
where S(f) was defined in the previous section. Using (10), this can be evaluated and
ated and simplified
to
simplified
to in the previous section. Using (10), this can be evaluwhere S(f ) was
defined
ated and simplified to
erf (γ)
,
(12)
Pcx =
2 erf (γ/2)
erf (γ)
,
(12)
Pcx =
where
2 erf (γ/2)
where
4
B
γ=√ .
2σ
B
γ=√ .
2σ
(13)
(13)
4
Page 6
In the aboveofanalysis,
the desired filter
profile
a rectangular function
with
bandFilter Bandwidth Definition
the WaveShaper
S-series
ofisProgrammable
Optical
Processors
width, B. The actual filter shape generated by the WaveShaper, however, is narrower at the top part of the channel, and spills out of the rectangular window at
the bottom, even though the channel is labeled as a B bandwidth filter.
To clearly define the bandwidth of the WaveShaper filter shape, it is instrucDefining the channel bandwidth specification
tional to look at the power level at which S(f ) crosses the rectangular function
boundary. This is illustrated in Figure 2, where the crossover point, Pcx , is iden1.5
tified as the point where the generated filter shape from the WaveShaper crosses
1.5
the ideal rectangular window.
1.4
If the crossover point, Pcx , occurs at a constant power level, this power level
1.4
1.3
1.2
1.1
erf( )/erf( /2)
erf( )/erf( /2)
1.3 to strictly define the bandwidth of the WaveShaper filter response.
can be used
The power
ratio at which Pcx occurs, for a given filter bandwidth, B, is given
1.2
by
1.1
1.0
0.9
0.8
0.7
1.0
0.6
0.9
Pcx =
0.5
1.0
S(−B/2)
,
S(0)
1.5
2.0
2.5
(11)
3.0
3.5
4.0
4.5
5.0
where S(f ) was defined in the previous section.B/BUsing (10), this can be evaluG
0.8
ated and simplifiedFigure
to 3
0.7
Error function ratio trend as the ratio of filter bandwidth to optical transfer function
bandwidth increases.
0.6
where
Pcx =
0.5
1.0
where
1.5
2.0
2.5
erf (γ)
,
2 erf (γ/2)
3.0
B/BG
3.5
(12)
4.0
4.5
5.0
Figure 3 – Error function ratioB trend as the ratio of Wave√ . transfer function bandShaper filter bandwidth γto=optical
2σ
width increases.
4
(13)
Previously, a relation between the 3 dB bandwidth of the Gaussian filter, BG, and
Previously, a relation
between
theis3used
dBhere
bandwidth
σ was stated
in (3), which
to express of
γ asthe Gaussian filter, BG ,
and σ was stated in (3), which is used here to express γ as
√
B
γ = 2 ln2
.
BG
(14)
Hence, an expression for the crossover point, Pcx , is found in terms of the ratio
Hence, an expression for the crossover point, Pcx, is found in terms of the ratio
between the WaveShaper filter bandwidth, B, and the innate
optical transfer
between the filter bandwidth, B, and the innate optical transfer function bandwidth,
bethis
noted
thatforthis
only holds
for a Gaussian
function bandwidth,
BG . It
BG. It should
beshould
noted that
only holds
a Gaussian
spectrum.
spectrum.
Further conclusions can be drawn by considering the nature of the error funcFurther conclusions
cantends
be drawn
considering
the nature
of the
error
tion, which
to ½ as γ by
increases.
This is illustrated
in Figure
3 where
the funcratio of
1
/2 as γ isincreases.
illustrated in Figure 3, where the
tion, which tendserror
to functions
calculated asThis
B/BGisis
increased.
γ
ratio of erf(γ)/erf( 2 ) is calculated as B/BG is increased.
Clearly, for large ratios of B/BG , the ratio of error functions tend to one; conservatively speaking, it appears safe to conclude that when the WaveShaper filter
response is at least 3 times larger than the optical transfer function bandwidth,
the crossover point occurs at
for B BG ,
Pcx ≈ 12 .
(15)
This translates to, on a decibel scale, a level that is -6.02 dB down from the
channel peak. This is illustrated in Figure 4, which plots the expression given in
Page 7
0.6
Filter Bandwidth Definition0.5of the WaveShaper S-series of Programmable Optical Processors
1.0
1.5
2.0
2.5
3.0
3.5
B/BG
4.0
4.5
5.0
Figure 3 – Error function ratio trend as the ratio of WaveShaper filter bandwidth to optical transfer function bandwidth increases.
-3.5
Previously, a relation between the 3 dB bandwidth of the Gaussian filter, BG ,
and σ was stated in (3), which is used here to express γ as
Crossover point level (dB)
-4.0
√
B
.
γ = 2 ln2
BG
-4.5
(14)
-5.0
Hence, an expression for the crossover point, Pcx , is found in terms of the ratio
between the WaveShaper filter bandwidth, B, and the innate optical transfer
function bandwidth, BG . It should be noted that this only holds for a Gaussian
-5.5
-6.0
spectrum.
Further conclusions can be drawn by considering the nature of the error funcBandpass filter bandwidth (GHz)
tion, which tends to 1/2 as γ increases. This is illustrated in Figure 3, where the
Figure 4 Location
crossover
point with respect to actual filter bandwidth.
as B/BGof is
increased.
ratio of erf(γ)/erf( γ2 ) is calculated
Clearly, for large ratios of B/BG , the ratio of error functions tend to one; confor large
of B/BG, thethat
ratio of
error functions
tend to one;filter
conservaservatively speaking, itClearly,
appears
saferatios
to conclude
when
the WaveShaper
tively speaking, it appears safe to conclude that when the desired filter response is at
response is at least 3 times larger than the optical transfer function bandwidth,
-6.5
0
10
20
30
40
50
60
70
80
90
100
least 3 times larger than the optical transfer function bandwidth, the crossover point
the crossover point
occurs
occurs
at at
for B BG ,
Pcx ≈ 12 .
(15)
This translates to, on a decibel scale, a level that is -6.02 dB down from the
This translates to, on a decibel scale, a level that is -6.02 dB down from the filter
channel peak. This
is illustrated in Figure 4, which plots the expression given in
peak. This is illustrated in Figure 4, which plots the expression given in(12) as a function of the actual filter bandwidth. For values of B that are at least three times larger
than BG, it is evident that the crossover point is located at, roughly, -6 dB from the filter
5
peak.
To reiterate, when a WaveShaper Programmable Optical Processor is set to produce a rectangular function of bandwidth B, which is at least three times larger than
the optical transfer function bandwidth, the output filter response will have, roughly,
a 6 dB bandwidth of B. If the bandwidth B is less than three times the optical transfer
function bandwidth, the bandwidth is defined at the crossover point indicated in
Figure 4.
Page 8
Filter Bandwidth Definition of the WaveShaper S-series of Programmable Optical Processors
Section 3: Experimental Results
predicted
measured
0
Normalized power (dB)
-10
-20
-30
-40
-50
-60
-40
-20
0
Frequency offset (GHz)
20
40
60
Figure 5 Comparison between predicted and measured channel shape for a 50 GHz filter
response.
In order to compare the presented theory to measured results, production data
from a range of WaveShaper Programmable Optical Processors were extracted and
analyzed.
One 50 GHz channel, from a WaveShaper 1000S, was taken to compare to the
predicted filter shape shown in (10). Measured results are compared to this predicted
channel shape in Figure 5. The spectra match reasonably well, with the exception of
a small broadening on one side of the experimental data, which is attributed to the
asymmetric beam shape through the device.
To compare the output of a multiport device with filter profiles sent to unique
ports, a WaveShaper 4000S was set to output adjacent 10 GHz bandpass filters to each
port. The measured data is shown in Figure 6 compared to predicted channel shapes,
as derived in the previous section. It should be noted that an optical transfer function
bandwidth of 12 GHz was used in these calculations, which takes into account the
resolution bandwidth of the optical spectrum analyzer.
Page 9
Filter Bandwidth Definition of the WaveShaper S-series of Programmable Optical Processors
5
port 1
port 2
port 3
port 4
Normalized power (dB)
0
-5
-10
-15
-20
193.95
193.96
193.97
193.98
193.99
Frequency (THz)
194.0
194.01
194.02
Figure 6 Four port operation with 10 GHz channel spacing comparing (solid) measured and
(dashed) predicted data.
The predicted model produces an excellent fit compared to the nominal 10 GHz
bandpass filter responses, with a slight broadening of the measured data from port
3. From (12), the predicted crossover point of a 10 GHz bandpass filter, using an optical transfer function bandwidth of 12 GHz, is -3.03 dB. From the measured spectra in
Figure 6, the average crossover point occurred at -3.38 dB.
Similarly, the behaviour of 50 GHz channels can be predicted with the presented models. A WaveShaper 4000S was therefore set to output adjacent 50 GHz channels
to unique ports, and the measured data was compared with predicted spectra, with
the results shown in Figure 7.
5
port 1
port 2
port 3
port 4
Normalized power (dB)
0
-5
-10
-15
-20
193.9
193.95
194.0
194.05
Frequency (THz)
194.1
194.15
Figure 7 Four port operation with 50 GHz channel spacing, comparing (solid) measured and
(dashed) predicted data.
Page 10
Filter Bandwidth Definition of the WaveShaper S-series of Programmable Optical Processors
90
80
70
528 samples
Avg: -6.2 dB
Std.Dev: 0.41 dB
Occurances
60
50
40
30
20
10
0
-8.0
-7.5
-7.0
-6.5
-6.0
Channel crossover point (dB)
-5.5
-5.0
Figure 8 Survey of crossover points for six interleaved devices, resulting in 528 samples.
As the generated filter bandwidth of 50 GHz was much larger than the optical
transfer function, the experimental data shows an even better fit to theory. The crossover point of the measured spectra occurred at an average of -5.98 dB; in comparison,
the predicted spectra have crossover points at -6.02 dB.
To draw a conclusion about the location of the crossover point of a WaveShaper
Programmable Optical Processor filter response, a relatively large sample size was
required. Six WaveShaper processors were measured; each device was set to produce
50 GHz interleaver patterns, odd and even, sliced into distinct channels, and the crossover point was measured for each slice. This resulted in a total of 528 measurements,
giving an appropriate sample size to determine statistical data about the location of
the crossover point of a WaveShaper bandpass filter.
The crossover point analysis is shown in Figure 8, tabulated in histogram format.
For all samples, the mean crossover point location occurs at -6.2 dB, with a standard
deviation of 0.41 dB; the standard deviation is acceptable considering the resolution of
the measurement data.
The simple mathematical model presented in this white paper gives accurate
agreement with measured spectral data, allowing accurate definition of the crossover
point, confirmed by statistical data.
Page 11
Filter Bandwidth Definition of the WaveShaper S-series of Programmable Optical Processors
Section 4: Conclusion
Conclusion
CONCLUSION
CONCLUSION
Conclusion
This paper has presented
a theoretical analysis of the WaveShaper output bandThis paper has presented a theoretical analysis of the bandpass filter shape
pass filter shape,generated
given asby the
convolution
between
a rectangular
profile given
and as
a
the WaveShaper
family
of Programmable
Optical Processors,
This paper has presented
a
theoretical
analysis
of
the
WaveShaper
output
bandconvolution
between a matching
rectangular profile
and atheory
narrow and
Gaussian
spectrum. Good
narrow Gaussian the
spectrum.
Adequate
between
experimental
pass filter shape,matching
given as
the
convolution
between
a
rectangular
profile
a
between theory and experimental data was found, confirming theand
accuracy
data was found.
this approach.
narrow Gaussian of
spectrum.
Adequate matching between theory and experimental
Additionally, it was predicted that, for a reasonably small Gaussian spectrum,
data was found.
Additionally, it was predicted that the crossover point between the output filter
the crossover point between
the output filter response and the initial rectangular
response
and the initial
rectangular
profile of bandwidth
would occur
at roughly -6
Additionally, it was predicted
that,
for a reasonably
small BGaussian
spectrum,
profile of bandwidth
B
would
occur
at
roughly
-6
dB,
if
the
width
of
the
rectangular
dB, if the width of the rectangular profile was greater than 30 GHz. For filter profiles
the crossover point
between the output filter response and the initial rectangular
profile was greater
than than
30 GHz.
Forcrossover
filter profiles
30 GHz, the
narrower
30 GHz, the
point wasnarrower
found to be than
given by
profile of bandwidth B would occur at roughly -6 dB, if the width of the rectangular
crossover point was found to be given by
profile was greater than 30 GHz. For filter profiles narrower than 30 GHz, the
crossover point was found to be given byerf (γ)
,
(16)
Pcx =
2 erf (γ/2)
erf (γ)
where
,
(16)
Pcx =
2 erf (γ/2)
where
√
B
where
γ = 2 ln2
.
(17)
BG
√
B
This was confirmed by experimental
data; . over 528 measurements for (17)
proγ = 2 ln2
BG
duction devices were used to build a statistical evaluation of the location of the
This was confirmed by experimental data; over 528 measurements for procrossover point of 50 GHz bandpass filter responses. The mean crossover point
duction devices were used to build a statistical evaluation of the location of the
the sample set -6.2 dB,This
with aconfirmed
standard
deviation of 0.41 dB.
by experimental data; over 528 measurements for produccrossover point of 50 GHzwas
bandpass filter
responses. The mean crossover point
tion devices were used to build a statistical evaluation of the location of the crossover
the sample set -6.2
dB, with a standard deviation of 0.41 dB.
point of 50 GHz bandpass filter responses. The mean crossover point of the sample set
was found to be -6.2 dB, with a standard deviation of 0.41 dB.
Acknowledgements: This white paper was written by Cibby Pulikkaseril and was
made possibleThis
with the
assistance
of Michaël
Roelens, by
Jeremy
Bolger,
Simon Poole and
Acknowledgements:
white
paper
was written
Cibby
Pulikkaseril
Steve Frisken.
and was made possible with the assistance of Michaël Roelens, Jeremy Bolger,
Acknowledgements: This white paper was written by Cibby Pulikkaseril
Simon Poole and Steve Frisken.
and was made possible with the assistance of Michaël Roelens, Jeremy Bolger,
Simon Poole and Steve Frisken.
Page 12