Analog-to-digital conversion is today a key technol-ogy in a wide range of applications. Some applica-tions, including radio astronomy, advanced radarsystems, and spread spectrum communication, re-quire extreme bandwidths. In many of these cases thespeed of the analog-to-digital conversion is a limitingfactor: The progress of electronic ADCs has been rel-atively slow compared with the progress in digitalsignal processing.1 To eliminate this bottleneck,a large number of widely different photonic ADCschemes have been proposed, whereby an analog elec-trical baseband input signal in some way is trans-ferred onto an optical carrier and optical interferenceor nonlinearity is used to facilitate the quantizationand/or binary encoding of the signal. Categorizing theschemes according to the means of applying the inputsignal, both quantizers that use optical amplitudemodulation,2–4 phase modulation,5–9 and frequencymodulation10–14 as well as acousto-optical modula-tion of a deflection angle,10,15 have been studied.
The analysis in this paper applies to any photonicADC system in which the analog input signal is con-verted to an optical wavelength. After this conversionthe signal is binary encoded by some kind of filterarray. Fiber-based Bragg gratings and thin-film fil-ters as well as Mach–Zehnder filters have been sug-gested,14 but the setup considered here is the oneillustrated in Fig. 1. In this case a spectrometerlikesetup is used to convert the optical wavelength into aspatial position. To perform the digitization of thisanalog spatial position, one divides the spectrumplane into a number of sharply defined spatial re-gions. In each of these regions there is a diffractiveoptical element (DOE) that splits and redirects thebeam onto the corresponding detectors, and the out-put currents from these detectors are then thresh-olded by electronic comparators. Note that all filtersmentioned are linear (passive). Thus the equations inthis paper also apply to nonspectrometerlike variantswith equivalent parameters. Note also that Subsec-tions 2.A and 2.B below further apply to other beam-steerer-based ADCs in that they provide a detailedanalysis of the acceptable spot size on the DOE array.In the light of recent demonstrations of ultrahigh-speed �16.25 GHz� beam-steering devices,16 this ap-proach could indeed be attractive.
Signal sampling in tunable-laser-based ADC sys-tems can be performed in a number of ways. One couldperform the sampling optically by modulating the in-tensity of the light, or one could do it electronically. Inthe latter case the sampling could be performed before
The authors are with Chalmers University of Technology, Fo-tonik, MC2, Kemivägen 9, Gothenburg, 41296 Sweden. J. Stigwall’se-mail address is [email protected].
Received 13 June 2005; revised 12 December 2005; accepted 16December 2005; posted 30 January 2006 (Doc. ID 62735).
the tuning of the laser by use of a sample-and-hold(S&H) or a similar circuit; it could also be accom-plished by sampling the detector currents after thetuning in parallel by use of an array of such circuits, inwhich case the tuning section of the laser would bedriven directly by the input signal. One benefit of usingoptical sampling is that potentially the power dissipa-tion could be lowered substantially because power-hungry S&H would be avoided and the temporal jitterfrom optical sampling could be further be reduced by 2orders of magnitude compared with that from elec-tronic sampling.17 Note also that the analog binaryencoding of the signal at the detectors in effect lowersthe demand on signal-to-noise ratio (SNR). Thus onecan use considerably less precise S&H when the de-tector currents are sampled and held than when theactual analog input signal is sampled and held.
In this paper we are concerned mainly with aninput signal applied directly to the tuning of the laser(including the optically sampled case), but the oppo-site case is also studied for comparison. When theinput signal is applied directly to the tuning sectionthere will be a chirp-induced trade-off between spec-tral resolution and bandwidth, and it is this trade-offthat is the main topic of this paper. Furthermore, wealso calculate the required spectral resolution by tak-ing the comparator’s accuracy and noise into account.
2. Theory
To calculate the limits on bandwidth and resolutionof a spectrometerlike ADC we must consider two pa-rameters: To start with, we need to know the maxi-mum spot size in the spectrum plane that can betolerated (i.e., the lower limit on spectral resolution)with a certain amount of noise and a certain compar-ator accuracy. Then, to stay below this maximumspot size, we calculate the limit on FM tuning speed.In the end, a relation that connects maximum reso-lution, bandwidth, and noise tolerance is obtained.
During the following calculations we consider onlyone noise term that sums up the effects of all tradi-tional noise sources as well as the comparator’s am-biguity. Note also that we consider the maximuminstantaneous relative amplitude of this noise term,not the average noise-to-signal ratio. A statisticalnoise analysis is presented in Subsection 2.B withwhich we can relate this measure to the electrical
SNR of the detector outputs and the probability of atwo bit error.
A. Required Spectral Resolution
Ideally, the broadening (or spot size) should be zero.In that case, the transitions from one digital level tothe next will be infinitely sharp and the power ontothe detectors will go from zero to one and vice versainstantly. Consequently, a noise intensity almost aslarge as the signal intensity could be accepted.
In reality, the broadening will of course be non-zero. Adding a finite amount of noise implies thatthere will be a region about each transition wherethe binary output value of one or more detectors willnot be clearly defined. If a Gray-encoded binarycode is used, the possible error within these regionswill remain smaller than one digital level, that is,as long as only one bit is ambiguous. To take fulladvantage of the encoding scheme and maximizethe bandwidth we define acceptable broadening asthe maximum amount of broadening under whichonly one bit is ambiguous. We also note that it is theleast-significant bit (LSB) that will be most criticalbecause its bit-flip frequency is the highest. In aGray-encoded binary the LSB changes value at ev-ery second digital increment, so we allow the LSB tobe ambiguous 50% of the time; as the other bits willhave slightly smaller ambiguity regions, this will en-sure that only one bit is ambiguous at any time.
The binary pattern for the LSB of a Gray-encodedquantizer, regardless of the number of bits, will con-sist of two zeros, two ones, two zeros, etc. To preventedge effects, we think of the code as circular; i.e., wepad the binary encoding DOE array with identicalcopies. Furthermore, we assume that the DOEs areequally efficient and that no light misses the detec-tors owing to extra diffractive broadening when onlya part of the beam hits the “one” area. Using anappropriate DOE array design algorithm, we can alsoneglect the effect of interelement phase mismatchwhen the spot straddles two DOEs, as shown in Ref.18. Making these assumptions, we can calculate theamount of power that falls onto the LSB detector bysimply integrating the intensity of the spot in thespectrum plane over all DOE elements that yieldLSB � 1. That is, to get the power onto the LSBdetector as a function of spot position (or wave-length), one should convolve the LSB bit pattern withthe spot intensity function.
Let us start by defining the LSB bit pattern as afunction of position x along the spectrum plane wherethe DOEs and thus the digital values are spacedby �x:
qLSB�x� � mod�ceil��x��x��2�, 2�, (1)
where mod(x, y) is the y modulus of x and ceil(x)rounds to the next integer larger than x. In otherwords, the LSB bit pattern is a unit height squarewave with period �LSB � 4�x with a zero-to-one tran-sition for increasing x at x � 0. It is this transition
Fig. 1. Schematic setup of a spectrometerlike quantizer whenthe input signal is controlling the wavelength of a tunable laser.The wavelength is in turn converted into a spatial position in thespectrum plane by the grating and the focusing lens and thenencoded into binary form by an array of beam-splitting diffractiveoptical elements.
that we shall study in closer detail in what follows,but we must first define the spot profile, which isassumed to be Gaussian with the normalized inten-sity profile
Is�x� ��2
wx��exp�2x2
wx2 , (2)
where wx is the 1�e2 radius of the beam (i.e., wherethe optical intensity has dropped to �14% of its max-imum value). Knowing the intensity profile of thebeam, we can calculate the optical intensity receivedby the LSB detector as a function of the beam positionby convolving it with the bit pattern qLSB:
gLSB�x� � qLSB�x� � Is�x�. (3)
To analyze gLSB we first decompose qLSB into aFourier series with components at spatial frequencieskx � 2�n��LSB � �n�2�x:
qLSB�x� �12 �
2� �
n�1, 3, 5, . . .
� 1n sinn�x
2�x. (4)
To get gLSB�x� we evaluate Is at the same spatialfrequencies:
Is�kx� � ��Is�x�� � exp�kx2wx
2
8 � exp��2n2wx2
32�x2 .
(5)
Now we can calculate gLSB�x� by multiplying qLSB�x�and Is�kx� in the spatial frequency domain (i.e., a con-volution in the spatial domain):
gLSB�x� �12 �
2� �
n�1, 3, 5, . . .
� 1n sinn�x
2�x� exp�
�2n2wx2
32�x2 . (6)
1. Large Broadening, Sinusoidal ApproximationTo estimate the acceptable broadening wx with a cer-tain amount of noise Jnoise, we assume here thatwx��x is large. Under this assumption the higherspatial frequency components of gLSB will be stronglyattenuated, and we thus need only to take the first�n � 1� component into account. That is, we make theapproximation
gLSB�x� �12 �
2�
sin �x2�xexp�
�2wx2
32�x2�
12 � A1 sin �x
2�x gLSB
sine�x�. (7)
If the LSB is ambiguous only when |x|��x 1�2, we know that it is well determined at leasthalf of the time, and thus we evaluate gLSB at x��x� 1�2 to get the limits on the relative noise ampli-tude:
Jnoisesine gLSB
sine �x � �x�2� � 1�2
� A1 sin�
4��22 A1 �
�2�
exp��2wx
2
32�x2. (8)
Setting, for instance, wx � 2�x (i.e., the beam’s 1�e2
radius is twice the DOE array element’s width) weget a limit on the relative noise of Jnoise
sine 0.13. Thissituation is plotted in Figs. 2(a) and 2(c). We alsocalculate the inverse function wx
sine�Jnoise�, which rep-resents the limit on the broadening for a given noiselevel:
wxsine
4�x� �2 ln �2
�Jnoise�1�2
. (9)
As we shall see in Subsection 2.A.3, the sinusoidalapproximation is valid when wx��x 1.1.
2. Small Broadening, Gaussian ApproximationAnother important case is that when Jnoise is rela-tively large, thus requiring wx to be small. Underthese circumstances, in which many spatial fre-quency components are present, the sinusoidal ap-proximation in relation (7) is not accurate. TheGaussian approximation
gLSB�x� � 1 �12 exp��
122x
wx�
122� gLSB
Gauss�x� (10)
happens to be much better and will be used insteadwhen wx is small (for x � 0). The limit on Jnoise in thiscase evaluates to
Fig. 2. (a), (b) Spot intensity profiles Is(x). (c) LSB bit functionqLSB(x) (dashed line) together with the resultant LSB output,gLSB(x) � qLSB(x) R Is(x) (solid curves with shaded noise regions) forwx � 2�x. Here the sinusoidal approximation is practically iden-tical to the exact solution. (d) gLSB(x) with acceptable noise regionsfor wx � 0.5�x, together with the Gaussian approximation(dashed–dotted curve).
See Figs. 2(b) and 2(d) for plots of Is and gLSB forwx � 0.5�x. In this case, Jnoise 0.48; i.e., the noiseamplitude is allowed to be 96% of the signal ampli-tude.
3. Accuracy of the ApproximationsAs can be seen from Fig. 3, the Gaussian approxi-mation underestimates the exact expression for gLSBat x � �x�2 within 0.4% when wx 1.1�x (exceptfrom a very small overestimation, around wx
� 0.5�x). Above wx � 1.1�x the approximation inexpression (7) (the sinusoidal approximation) can beused, in which case gLSB is again underestimatedwith a similar precision ��0.6%�. Note that, as gLSB isunderestimated, we make a slightly pessimistic un-derestimation of the limits on Jnoise and wx.
Calculated Jnoise values for a number of wx��x ra-tios are as follows: 0.48 at 0.5; 0.22 at 1.5; 0.13 at 2;0.028 at 3; and Jnoise � 0.0032 at wx/�x � 4.
B. Two-Bit Error Probability
In the subsection above we lumped all types of noiseand uncertainty into one single parameter, i.e., opti-cal intensity noise amplitude Jnoise. Here we presenta more accurate model in which we calculate thetwo-bit error probability in the read-out code as afunction of spot size and detector output SNR. Duringthese calculations we still lump all noise sources intoone, but we model the noise distribution by a normal(i.e., Gaussian) distribution X such that the electricaloutput amplitude from detector k can be written as
gk�x� � gki�x� � X � qk�x� � Is�x� � X, (13)
where gki�x� is the ideal, noise-free output amplitude
and X has the density function
fx�y� �1
��2�exp�
y2
2�2, (14)
where � is the standard deviation of the distribution.By definition, the noise power is equal to the vari-
ance, �2, and we approximate the signal power of g�x�by its maximum value, 0.52. The electrical SNR isthus
SNR � 0.25��2. (15)
Indeed, the LSB signal amplitude will be reducedconsiderably because of broadening, but the most-significant bit signal will retain almost its full ampli-tude, making this definition of SNR meaningful.
Because of the long tails of the noise distribution,there is always a risk of a two-bit error in the read-outbinary code (remember, one bit is allowed to be un-determined). The error is, however, not easily quan-tifiable because the value of a Gray-encoded wordwith two-bit errors is not related in a simple way tothe correct value. Thus we consider just such mea-surements all to be erroneous and define a maximumerror probability P as the probability of a second bitbeing erroneous. For each bit the probability of a zeroread-out is
Pk0�x� � p�gk�x� 0.5� � �0.5 � gk
i�x�� �, (16)
where �x� � �1�2�erfc��x��2� is the standard nor-mal distribution. The probability of a one is obviouslyPk
1 � 1 � Pk0, and the erroneous read-out (zero or
one) is always the one with lower probability. Thuswe can write the probability of error as
Pkerr�x� � min�Pk
0, Pk1� �
12 � �12 � Pk
0�x��. (17)
For each sample there is one bit that is allowed tobe noisy and N � 1 bits that are not allowed to be. Ifbit s is the bit that is allowed to be noisy we can writethe total probability of another bit being erroneous as
P�x� � 1 � �k�s
�1 � Pkerr�x�� (18)
and the average probability as
P ��xmin
xmaxP(x) d xxmax � xmin
. (19)
By numerically performing the convolution gki�x�
� qk�x� � Is�x� for all values of k and x we get anaccurate value of P. Performing this calculation for a
Fig. 3. Error of the two approximations shown, at x � �x�2, asfunctions of wx.
number of values of wx��x and of the SNR, we ob-tained the contour plot shown in Fig. 4. The numer-ically calculated curves of constant P have also beencompared with the SNR calculated from the approx-imate expressions for Jnoise in inequalities (8) and (11)as SNR � 0.52��Jnoise�2�2 (i.e., the noise is assumed tohave an average amplitude of Jnoise�2). This curvefollows the numerically calculated curves closely forlarge wx��x, and we can see that it corresponds to aprobability of a two-bit error from 10�2 to 10�3. For asmaller spot size the analytical calculations, how-ever, underestimate the required SNR by a few deci-bels. The reason for this discrepancy is that thenonswitching bits �k � s� in these cases all contributebecause they have similar probabilities of error. Theanalytical calculations that do not take into consid-eration the number of bits therefore cannot predictthe two-bit error probability. For larger spot sizes theprobability of error is dominated by the bit that isabout to switch value or has just done so, reducing thedependency on N.
C. Chirp-Induced Spectral Broadening
In a wavelength-based ADC there are two factorsthat limit the spectral and hence the ADC resolutionof the system. Naturally the resolution can never bebetter than the fixed-wavelength resolution of thespectrometer, i.e., the transform limit. If the inputsignal controls the wavelength directly, the resolu-tion will also be reduced by the fact that the wave-length chirps over the length of the grating.
In what follows, we describe the resolution by thebroadening in the spectrum plane. This in our setupis in fact a spatial spot-size broadening as in Subsec-tion 2.A, but we consider it in the frequency domainby noting that the mapping from frequency to posi-tion in the spectrum plane is linear. We shall thus be
able to translate back to spatial broadening easily inSubsection 2.D when we later calculate the limits onoverall ADC resolution.
Let us first consider an optical carrier with an in-stantaneous frequency:
��t� � �0 ���
2 sin�2�ft�, (20)
where �0 � c��0 is the center optical carrier fre-quency, �� is the tuning range, and f is the inputfrequency.
In effect, the grating in a spectrometer setup col-lects light from a slice of several points in timethrough the beam and weighs the sum by the beamamplitude on each point of the grating (Fig. 1). For anoptical beam with a Gaussian spatial intensity profilethe optical Fourier transform that is performed bythe grating–lens combination will thus be apodizedin time by the same Gaussian function. With a 1�e2
temporal intensity radius of wt the (nonnormalized)temporal apodization function in optical amplitudecan be written as
a�t� � exp�t2
wt2 (21)
and the transform-limited 1�e2 broadening in the fre-quency domain is thus
w� trans �1
�wt. (22)
The chirp limitation, however, is simply the extentof the tuning range that fits inside the temporal apo-dization window. Obviously, the largest chirp occursat t � 0 as the sine in Eq. (20) passes zero. Weconsider only this extreme case because this is wherewe get the maximum broadening. At this point thechirp is
��
�t � ���f, (23)
and the chirp-induced broadening is thus
w� chirp ���
�t wt � ���fwt. (24)
As the two broadening mechanisms are indepen-dent and Gaussian in shape, we know that the totalbroadening will also be Gaussian in shape, with the1�e2 intensity radius
w� tot � �wv trans2 � wv chirp
2 �1�2
� ��1��wt�2 � ����fwt�2�1�2. (25)
Fig. 4. Required detector output SNR versus beam radius. Two-bit error probability P for N � 6 bits (solid curves). The SNRcalculated from Jnoise is also plotted (dotted curve).
See Figure 5 for an example plot of the broadeningas a function of the temporal apodization window size(i.e., beam width).
D. Optimum Apodization Window
Now the optimum temporal apodization window size,which for a continuous wave is determined by thebeam width incident onto the grating (see Fig. 1) andin the optically sampled case by the pulse duration,can be calculated by minimizing w� tot. Skipping thesquare root, we have
�
�wt�wv trans
2 � wv chirp2 � � 0, (26)
giving
2�2��2f 2wt �2
�2wt3 � 0 (27)
)
wtopt �
1�
�1���f. (28)
Note that the broadenings are also equal at thispoint:
wv transopt �
1�wt
� ���f � wv chirpopt . (29)
The minimum spectral broadening is thus
wv totmin � �2��f. (30)
E. ADC Resolution
Now that we know the minimum spectral broadeningfor a certain input frequency as well as the maximumacceptable broadening of the spot size when noise is
taken into account, we can calculate the maximumresolution as a function of input frequency and viceversa. First we recall that the spectral digital levelspacing that corresponds to the spatial DOE spacing,�x, is ���m when the number of resolved levels is m;i.e., the relative optical frequency is related to posi-tion as
� ���
m�x x. (31)
Thus we get from Eq. (30) that
w� � wx
��
m�x � �2��f, (32)
and the number of resolvable levels is
m ���
2fwx
�x, (33)
where wx can be approximated as either wxsine if
wx � 1.1�x [inequality (9)] or wxGauss if wx 1.1�x
[inequality (12)]. Let us also solve the expressions forf and ��:
f ��
2m2wx
�x2
, (34)
�� � 2m2f �xwx
. (35)
We can also solve the expression for Jnoise by usingthe approximations in inequalities (8) and (11): ifwx � 1.1�x,
Jnoisesine
�2�
exp��2m2f16�� ; (36)
if wx 1.1�x,
JnoiseGauss
12 �
12 exp��
12 1
m���
2f �12
2�. (37)
Note that the case just covered, in which the inputsignal is applied directly to the tuning section, alsoapplies to the case of optical sampling. In a systemsimilar to the one described in Ref. 19, one could usean agile wavelength converter to tune the wavelengthof an incident sampling pulse train directly. In thatcase, the apodization window, wt, equals the 1�e2
half-duration of the pulses.
F. Sample Rate Limit Using a Sample-and-Hold Circuiton the Input
If a S&H circuit is inserted before the tuning stage,the bandwidth will be set by this S&H circuit and we
Fig. 5. Maximum spectral broadening as a function of temporalapodization window wt. Transform-limited (dashed curve), chirp-limited (dotted curve), and total (solid curve) broadenings are plot-ted for f � 1 GHz and �� � 20 GHz.
shall instead calculate the maximum sample rate, M,of the optical quantizer. In practice, this rate is oftenlimited by the laser tuning speed but will for smalltuning ranges also be fundamentally transform lim-ited.
Now the chirp is no longer a problem, and thebroadening will be transform limited only by the fi-nite length of the sampling period. With a steadywavelength during �t � ��M for each sample, where� is the usable fraction of each sampling period, weestimate wt to be
wt ��t2 �
�
2M, (38)
giving the transform-limited broadening
w� �1
�wt�
2M��
. (39)
As before, the maximum acceptable broadening isset by the noise amplitude, and we get
w� � wx
��
m�x �2M��
, (40)
where wx as before is approximated by either wxsine if
wx � 1.1�x [inequality (9)] or wxGauss if wx 1.1�x
[inequality (12)]. The number of resolvable levels is
m ����
2Mwx
�x . (41)
Let us also solve the expressions for M (samplingrate) and �� (tuning range):
M ����
2mwx
�x, (42)
�� �2mM
��
�xwx
. (43)
The most important thing to note here is that m inthese two inequalities is not squared as it is in ine-qualities (34) and (35).
G. Noise Sources
One of the most important sources of noise at giga-hertz frequencies is the random detection noise stem-ming from optical shot noise as well as thermalelectric noise. Taking these two factors into accountyields as the electrical SNR of a simple photodiodereceiver
SNR �Psig
Pshot � Ptherm, (44)
with Psig � R2Pin2 , Pshot � 2q�RPin � Id��f, and Ptherm
� 4�kBT�RL�Fn�f,20 where R is the responsivity of thedetector, Pin is the optical power to each detector, q isthe electron charge, Id is the dark current, �f is thebandwidth of the detector, kB is Boltzmann’s con-stant, T is the temperature, RL is the load resistance,and Fn is the amplifier noise figure. As the signal is tobe thresholded at half-amplitude, Pin here is the av-erage optical power.
As we shall see, the required powers will for mostsetups be low enough to make the shot noise negligi-ble. Considering thermal noise only, the required op-tical power per detector is then simply
Pin �2R �SNR�kBT�RL�Fn�f�1�2. (45)
Note that �f here is the bandwidth of the signal atthe detector. If the sampling of the signal is per-formed before the detection, this will for a NyquistADC equal the maximum analog input frequency fmax.If the signal, however, is not sampled before the de-tectors, the frequency at the LSB detector will bemuch higher. Assuming a triangular wave input sig-nal at fmax, the LSB frequency will be �m times ashigh as the input frequency (m�2 periods going up,m�2 periods going down):
fLSB � mfmax � 2Nfmax. (46)
Another important source of noise is the fluctua-tion that is due to nonuniformities in the output ofthe DOE array. In Ref. 18 it was shown that DOEarrays can be designed with an overall maximumuniformity error below 10%. With no other noisepresent this would, according to the ratios that wecited in Subsection 2.A.3, permit a spot size slightlylarger than wx � 2�x. This of course also assumes ahigh-quality beam profile.
Although the tuning-induced intensity fluctuationscan be rather large, it is possible to cancel this noiseby using a balanced comparison with a DOE arraythat simultaneously generates the complementarybinary word. Ordinary relative intensity noise canalso be canceled in this way.
3. Discussion
Calculations of the performance of wavelength-basedADCs were presented in Refs. 10, 11, and 13–15. Allexcept Ref. 11, however, deal only with the case inwhich the analog input signal is sampled and heldbefore the tuning. If the input signal, however, isapplied directly to the tuning section to avoid theneed to use electronic sampling, the temporal–spectral trade-off presented in this paper [expres-sions (26)–(30)] has to be taken into account. Thistrade-off was quickly covered in Ref. 11, resulting inan expression identical to inequality (34) but with aconstant spot size wx��x � 1 (i.e., a spot diametertwice the DOE spacing). In that paper, a flat-toppedbeam equal in width to the DOE spacing was, how-ever, assumed.
Although we cannot seem to find the equationsreferred to in the Conclusions section of Ref. 14, wesee that the maximum conversion rate values plottedin their Fig. 6 agree almost exactly with inequality(42) in this paper if � � 1550 nm and � � 0.5,wx��x � 0.5 is assumed. Equation (12) of Ref. 13 alsoagrees within reasonable values of � and wx if oneassumes that the 300 THz optical carrier frequency isa misprint and should be replaced by the tuningrange, ��.
In the research reported in Refs. 10 and 15 thetuning was imposed externally by an acousto-optical modulator, which modulated the transversephase distribution of the beam to deflect it. Consid-ering that an acousto-optical modulator could aswell be used as a tunable Bragg grating in fact toshift the wavelength of the light, it is not surprisingthat the resultant resolution–bandwidth equationsare virtually identical to inequality (42); Eq. (5) ofRef. 15 is exactly identical if � � 1 and wx��x� 0.5 are assumed and if m is replaced by N � 1. Thisslight difference is due to a different definition ofthe frequency span. Here the span corresponds to thewhole DOE array width, not to the distance from thecenter of the leftmost to the center of the rightmostelement.
Despite the fact that the calculations in Ref. 15were performed for a S&H input signal, the experi-ments were conducted with a continuous wave RFinput signal. As the resolution was rather low �3 bits�there was, however, a coincidental agreement be-tween experiment and theory in which the theorygave a maximum frequency of 1.25 MHz and the ex-periment suggested 1 MHz. Using the parametersgiven in Ref. 15 (�� � 21.9 MHz and wx��x � 0.5)with a resolution of 1 bit, we found for m � 2 levels(because the most-significant digit was measured)f � 0.68 MHz, in good agreement with the experi-mental value considering that inequality (34) is notentirely accurate for the most-significant bit.
As we have seen from this paper, spot sizes sub-stantially larger than the size of the DOEs can in factbe used without dramatically increasing the de-mands on detection SNR, and, furthermore, the max-imum modulation frequency in the directly tunedcase increases as the square of the spot diameter.Looking at Fig. 4, we believe that it is fair to say thatwx��x � 2 is a good trade-off between tuning speedand noise sensitivity. At this point, the maximummodulation frequency is increased by a factor of 16,while the electrical detection SNR must be elevatedby �10 dB for the lower 2 bit error rates comparedwith the small-spot limit �wx��x 0.5�. In Fig. 6 therelationships among maximum frequency, bits of res-olution, and minimum tuning range are plotted forthis spot size �wx��x � 2�.
Let us try putting some figures into the resultingexpressions of the previous sections. Setting �� 1.5 �m and �� � 100 nm, we get �� � c����2
� 13 THz. Using reasonably good comparators, wecan let wx��x � 2 (Jnoise 0.13; electrical SNR,
�20 dB). With m � 26 � 64 we now get f 6.5GHz in the direct-driven case [inequality (34)]. TheLSB bandwidth in the direct-driven nonsampled casewill then be slightly more than 400 GHz [Eq. (46)],and, assuming that R � 1 A�W, RL � 50 �, Fn � 2,and T � 300 K, the required average optical powerper detector is 0.16 mW [Eq. (45)]. If optical samplingis used, the optimal pulse duration is �1 ps [Eq. (28)],and the bandwidth requirement at all detectors willbe reduced to the sampling rate. The average opticalpower requirement will also be reduced accordingly(depending on the sampling rate). If the signal, how-ever, is sampled and held before the tuning of thelaser (with � � 0.5), the maximum sampling rate isM � 327 gigasamples/s and the required opticalpower is 0.10 mW per detector at this sampling rate.Because this sampling rate is hardly feasible whenthe present electronic S&H technologies are used,there is thus a large margin that can be used toreduce the required tuning range, decrease the spotsize, increase the resolution of the ADC, or some com-bination of these.
Although �� � 100 nm and f � 6.5 GHz tunablelasers have not yet been demonstrated, we are hope-ful that there will eventually be semiconductor lasers(or wavelength converters) available with similarspecifications. Lavrova et al., for instance, have dem-onstrated wavelength switching with a 30 nm tuningrange and a 2.5 GHz switching rate.19 As it seems,the switching in Ref. 19 is fast enough to switch thewavelength between two 10 Gbit�s return-to-zerosymbols, which would imply that the maximumswitching frequency is already considerably higherthan 2.5 GHz. Unless the tuning range can be ex-tended, this however does not enable directly tunedADC conversions to be made at higher frequenciesunless the resolution is also lowered; the maximumfull amplitude input frequency at 6 bits of resolution,wx��x � 2, � � 1.5 �m, and �� � 30 nm, is 1.95 GHz[inequality (34)].
4. Conclusions
An analytical model for the resolution, bandwidth,and noise tolerance of wavelength-based photonic
Fig. 6. (a) Maximum frequency f versus tuning range �� (or ��)for a range of resolutions N (number of bits), directly driven with-out S&H. (b) Sample rate M versus tuning range for a range ofresolutions with � � 0.5, with S&H. In both cases, wx��x � 2 and�0 � 1550 nm.
analog-to-digital converters has been presented andcompared with accurate numerical calculations aswell as with previous calculations and experiments.The model allows calculations of Gaussian-apodizedspectral filter arrays with various resolutions to bemade, and it shows that the filters do not necessarilyhave to resolve each digital level perfectly if thesignal-to-noise ratio is slightly elevated. In the spec-trometerlike scheme that uses an array of diffractiveoptical elements, this means that the spot can beconsiderably larger than the individual DOEs. Inturn, the relaxed demand on spectral resolution in-creases the predicted bandwidth of directly tunedwavelength-based ADC systems by at least an orderof magnitude, making 6 bit, 2 GHz bandwidth opti-cally sampled wavelength-based ADCs feasible withcomponents that are available today.
This research was supported by the Chalmers Cen-ter for High-Speed Electronics and Photonics (Swed-ish Foundation for Strategic Research) as well as theSwedish Research Council.
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