## Abstract

Frequency shift keyed (FSK) modulation formats are well-suited to deep space links and other high loss links. FSK’s advantage comes from its use of bandwidth expansion. I.e., FSK counteracts power losses in the link by using an optical bandwidth that is greater than the data rate, just as pulse position modulation (PPM) does. Unlike PPM, increasing FSK’s bandwidth expansion does not require increased bandwidth in electronic components. We present an FSK modulator whose component count rises logarithmically with the bandwidth expansion. We tested it with four-fold bandwidth expansion at 5 and 20 Gbit/s. When paired with a pre-amplified receiver, the required received power was about 4 and 5 dB from the theoretical best for such receivers. We also tested the FSK transmitter with a photon counting receiver.

© 2013 OSA

## 1. Introduction and background

We expect optical communications to play an important role in free-space communications links with extremely high power losses, like links to earth from deep space [1]. Optics’ greatest advantage comes from the short wavelengths of optical beams compared to RF beams, greatly reducing diffraction: optical power is concentrated into a much smaller area after long propagation.

Another advantage comes from the large bandwidths available in optical components and channels – 4 THz in the gain region of erbium-doped fiber amplifiers (EDFAs). We can use this bandwidth to obtain higher data rates and to improve receiver efficiency. For example, compare the two modulation formats in Fig. 1 . The top of Fig. 1 depicts simple ON/OFF keyed (OOK) modulation with a ½ mark ratio. Each symbol period codes a single bit of data. In contrast, pulse position modulation (PPM) transmits several bits of data with each symbol. The bottom of Fig. 1 illustrates (PPM), which is more photon-efficient but less bandwidth-efficient. In PPM, the optical energy is concentrated into a shorter pulse, requiring more bandwidth, and creating a large fraction of dead time. The timing of this short pulse then gives several bits of data: in Fig. 1, each pulse can occupy any of 8 timing slots to produce ${\mathrm{log}}_{2}(8)=3$ bits of data per symbol.

Under ideal circumstances, a photon counting PPM receiver could operate with arbitrarily few photons per data bit, down to even a tiny fraction of a photon on average for each data bit [2]. For example, a PPM signal with $M$ slots per symbol and $N$ photons per symbol could give up to $N/{\mathrm{log}}_{2}\left(M\right)$ photons per bit; given unlimited bandwidth, we could make $M$ arbitrarily large. Of course, bandwidth is limited, and the first limitation we reach is the electronic bandwidth required by the receiver to measure the arrival time of the PPM pulses. Assuming a data rate of $b$ bits/s, OOK requires ~$b$ Hz of bandwidth in the receiver. On the other hand, PPM requires ~$bM/{\mathrm{log}}_{2}\left(M\right)$ Hz of bandwidth. For large $M$, this can strain electronic bandwidths, even at modest data rates (bandwidth is not the only non-ideality that could harm PPM performance; others include poor transmitter extinction, sources of background light, and high transmitted peak powers, which lead to nonlinear distortions in the transmitter).

One solution to the electronic bandwidth problem is to use a different orthogonal modulation format: frequency shift keying (FSK), shown in Fig. 2 . In FSK, bits are coded by the carrier wavelength of the pulse rather than its position in time. So, by transmitting 1 carrier wavelength at a time, chosen from 8 carrier wavelengths, we can code ${\mathrm{log}}_{2}(8)=3$ bits of data per symbol. FSK’s power performance is the same as PPM; we are simply slicing data in frequency rather than time. But, the electronic bandwidth requirements are much lower than in PPM: $b$ data bits/s requires only ~$b/{\mathrm{log}}_{2}\left(M\right)$ Hz of electronic bandwidth, where $M$ is the number of carrier wavelengths. For a general discussion of receiver sensitivity and modulation formats and their relationship to optical and electrical bandwidth, see [3].

Unfortunately, designs for optical FSK transmitters have not yet reached a mature commercial state. Many possible FSK transmitter designs, however, do exist in the literature. Perhaps the simplest FSK transmitter directly tunes a laser to different optical frequencies, but these transmitters generally do not have >GHz symbol rates and they rarely use FSK orders beyond about $M=2$ [4–11]. In another class of FSK transmitters, a source generates a comb of optical frequencies, and then a tunable filter selects one at a time. MEMS based optical filters can have excellent extinction ratios, but cannot tune at rates far beyond 1 GHz [12]. Micro-ring optical filters can be tuned by a GHz RF signal, but usually over a narrow range of optical wavelengths, and often with a low extinction ratio [13]. One integrated FSK transmitter uses fast-tuning lithium niobate modulators: a single optical frequency enters the integrated device, which then generates a comb of optical frequencies from the single input frequency. The same integrated device then selects one frequency in the comb for transmission. The comb of frequencies, however, is closely spaced at a few tens of GHz [14, 15]. Our design is also a lithium niobate based device, but can operate on a widely spaced comb of optical frequencies. In our FSK transmitter, for *M*-ary FSK, the number of modulators grows only by ${\mathrm{log}}_{2}\left(M\right)$, a favorable scaling for size, weight, and insertion loss. In experiment, we show that this design performs close to the theoretical optimum at data rates of 5 Gbit/s and 20 Gbit/s with 4-ary FSK (2 bits per symbol).

## 2. Capacity, bandwidth expansion, and power efficiency

In this section, we quantify how power efficient FSK compares with OOK. Important to this subject is the concept of bandwidth expansion. Roughly, bandwidth expansion (BWE) is the Hz of optical bandwidth required to transmit 1 bit/s of data. We can expand bandwidth in two different ways. First, we can choose a modulation format, like PPM, that uses extra bandwidth, as described in the previous section. Second, we can add error correction bits, displacing information bits and reducing the overall data rate. Let $B$ be the optical bandwidth of the transmitted signal, let $C$ be the link’s ideal data capacity in bits/s, and let $Q<C$ be the true data rate. Then $B/C$ is the minimum BWE required to close the link and $B/Q$ is the true BWE. An OOK link with no error coding has $Q=b$ bits/s and $B\approx b$ Hz, so its bandwidth expansion is ~1. If we add rate ½ error coding, half the OOK symbols are dedicated to error code bits. Thus, for $Q=b$ bits/s, $B\approx 2b$ Hz, and the bandwidth expansion is ~2. In FSK, bandwidth expansion occurs due to both error code bits and to the many carrier frequencies used. So, in 8-ary FSK at 10 Gsymbol/s, there are 8 different carrier frequencies, each using ~10 GHz of bandwidth for a total of 80 GHz of required optical spectrum. If extra spacing is used between the 8 carrier frequencies to improve extinction, then greater than 80 GHz of bandwidth is required.

For a good assessment of FSK’s advantages, we need to be able to quantify how this use of expanded spectrum improves power performance. Figure 3 shows the energy efficiency of an OOK communications system with a ½ mark ratio [16, 17]. We assume that the receiver is an ideal photon counter and that there is no background noise in the communications channel; all noise comes from the discrete and random nature of photon counting. The vertical axis is the energy per bit (dB scale) needed to maintain zero errors, assuming the use of an ideal Shannon code. The horizontal axis shows the bandwidth expansion in Hz of bandwidth per bit/s of capacity (i.e., $B/C$). Expanding the bandwidth by adding error coding quickly reduces the energy required per bit. This effect, however, saturates, and performance is never better than 1 photon per bit.

We can avoid this saturation by using PPM or FSK modulation. By increasing the order of PPM or FSK (the $M$ parameter above), we increase the energy efficiency without limit. Figure 4 shows the energy efficiency per bit for PPM and FSK modulation using an ideal photon counting receiver with no background in the communications signal [16, 17]. The envelope of these curves gives the optimal performance at any given bandwidth expansion, and it approximately equals $1/{\mathrm{log}}_{2}\left(\text{BWE}\right)$.

## 3. FSK transmitter design

FSK can provide large gains in data rate and energy efficiency compared to OOK. The most obvious FSK transmitter using lithium niobate modulators requires $M$ laser sources for the carrier frequencies, and $M$ modulators to modulate those lasers. If instead a single tunable filter could select one of the $M$ frequencies, the transmitter could cost less, be smaller, and use less power. Unfortunately, no commercial filter tunes at a ~10 GHz rate over 100s of GHz with ~10 GHz of bandwidth. Something similar, however, does exist. Figure 5 shows a lithium niobate Mach-Zehnder tunable filter. By making one arm longer than the other, we can create an interferometric filter that passes every other carrier wavelength. Changing the voltage on an electrode in the other arm allows us to select which half to pass [18]. By cascading filters with different length imbalances in the two arms, each filter stage can block half of the remaining frequencies. So we need only ${\mathrm{log}}_{2}\left(M\right)$ modulators rather than $M$. Figure 6 shows a schematic of this kind of FSK transmitter.

We tested this design using electro-optic modulators with 20-GHz RF bandwidth, V_{π} of 5 V at 20 GHz, and DC extinction ratio of 25 dB. Each stage of the transmitter used a separate electro-optic modulator, each custom produced by a commercial supplier. Our source was a bank of 4 DFB lasers from 192.2 to 192.5 THz, although lasers that produce many wavelengths at regular spacing exist in C-band [19]. The 1st stage filter had a single electrode, as shown in Fig. 5, and a filter periodicity of 100 GHz. The 2nd stage filter had a filter periodicity of 200 GHz and had two electrodes of two different lengths to tune it to one of 4 different operating voltages. A single electrode would have required the generation of a precise 4-level RF drive, which is difficult to do. With 2 electrodes, we were able to use two binary RF drives.

## 4. FSK transmitter performance

An FSK transmitter must produce high extinction between ON and OFF wavelengths. This becomes very important for large $M$ because even small powers in each of the $M-1$ OFF wavelengths can sum to significant power. Optical amplifiers are average power limited, so that power in these OFF wavelengths reduces the gain for the ON wavelength. As seen in Fig. 6, the last modulator in an M-ary PPM system must be able to precisely tune to one of M wavelengths. We show our tuning method in Fig. 7 . This modulator has 3 electrodes for an 8-ary FSK signal, each driven by a binary RF signal. The electrodes have different lengths so that the binary RF signal driving the longest one tunes the filter to either the middle of the 4 longer wavelengths or the middle of the 4 shorter wavelengths. Then the next longest electrode tunes to the upper or lower half of the remaining wavelengths, and so on. Making sure the filters tune precisely requires adjustment of the DC bias on the electrodes and of the RF levels on each electrode. We adjusted the DC bias with automatic bias control and adjusted the RF levels manually to reach optimal performance. One could use a single electrode modulator instead, with an M-level RF drive, but producing a multi-level RF signal with high precision is difficult. Good extinction requires each RF drive to have enough bandwidth to produce a high fidelity binary signal.

Unfortunately, the dynamic extinction ratio is difficult to measure, especially in an FSK signal. One method is to take a time-frequency spectrogram [3, 20]. Figure 8 shows a spectrogram of the 4-FSK transmitter operating at 2.5 Gsymbol/s. We collected the data by passing the transmitted signal through a tunable 3-GHz Fabry-Perot filter. We then measured the filtered signal on a 20-GHz photodiode. By sweeping the optical filter over the full range of FSK wavelengths, we obtained a spectrogram of the transmitted FSK signal. Figure 8 shows that the extinction ratio of each channel is always >20 dB, which corresponds to an ER-induced receiver penalty of less than ~0.5 dB.

We tested the bit error rate (BER) performance of the 4-FSK transmitter with the pre-amplified receiver described in [21]. An ideal pre-amplified receiver amplifies the FSK signal, separates and detects the 4 wavelengths, and chooses the wavelength with the largest signal, producing two bits of data for each received 4-ary symbol. Our receiver is a simplified sub-optimal version and is shown in Fig. 9 . The pre-amplified FSK signal is separated into the 4 wavelengths using an arrayed waveguide grating (AWG) followed by a Fabry-Perot filter with peaks at the 4 wavelengths. These 4 wavelengths are then mixed in a mesh of couplers. At the output of this mesh, the summed power of wavelengths 1 and 2 is compared with the summed power of wavelengths 3 and 4, giving the most significant bit (MSB). Then the summed power of wavelengths 1 and 3 is compared with the summed power of wavelengths 2 and 4, giving the least significant bit (LSB).

Figure 10
shows the measured bit error probability of this system. Our transmitter produced an FSK signal with a symbol rate of 2.5 GHz, a data rate of 5 Gbit/s, and it used a 2^{7}-1 pseudorandom data pattern (all the components in our setup have good response at low frequencies, so longer patterns should perform just as well; our main concern is with high frequency response and rise time). Compared with the theoretically optimal pre-amplified system (solid red curve), our FSK system required between 3.3 and 4.1 dB more received power than a system with a theoretically optimal pre-amplified receiver. Reference [21] presents results for a similar experiment with 8-ary FSK at a 2.5 GHz symbol rate. That paper achieved a power penalty of 1.5 dB, better than our 3.3 to 4.1 dB penalty. We can account for most of this difference. First, the extinction ratio in [21] was >30 dB, better than our 20 dB. Poor extinction diverts gain in the transmitter away from the intended signal wavelength, and it also makes it harder for the receiver to choose the correct wavelength. Our 20 dB extinction leads to ~1 dB of additional penalty [22]. Second, our filtering was different. In [21], the signal pulse and the filter profile were both approximately Gaussian, producing excellent matching. Our filter was a cascade of two Fabry-Perot filters, which gives a double-Lorentzian profile. This mismatch leads to at least 0.3 dB of additional penalty [22]. That still leaves us with approximately 0.5 to 1.3 dB of extra penalty. Much of this penalty is probably due to the variability in the peak pulse intensities in Fig. 8, a problem that is absent from the parallel setup of [21]. Some additional penalty may be due to differences in EDFA noise figure and other variability in components.

We also tested the error probabilities at a symbol rate of 10 GHz, giving a data rate of 20 Gbit/s, shown in Fig. 11 . This time, our FSK system required from 4.4 to 7.5 dB more received power. The additional penalty was partly caused by a reduced extinction ratio in each channel at the higher rate, which was typically ~20 dB, but sometimes closer to 15 dB.

## 5. Photon counting receivers

As seen in Figs. 10 and 11, photon counting receivers can in theory perform better than pre-amplified receivers, and this improvement is very great at large $M$. To explore this potential, we tested the FSK signal on a receiver using superconducting nanowire photon counters instead of optically pre-amplified photodiodes [23, 24]. We had to test at a lower rate – 100 MHz – because of these detectors’ long reset times.

Our goal in this experiment was not simply to measure the error rate of an uncoded FSK link. After all, we chose photon counters to construct the most power efficient system, and the most efficient system would also use forward error correcting (FEC) codes. We did not have the resources and equipment needed to construct a receiver capable of decoding an FEC in real time. Nonetheless, we can infer ideal coded performance, measured by the number of photons received per bit received, from easily measured statistics: 1) the probabilities of each wavelength being transmitted (in our case, all wavelengths were equally likely); and 2) the transition probabilities that a pulse transmitted on wavelength *k* was received as a pulse on wavelength *i*, or was erased altogether. Table 1
shows the probabilities that we measured. We took real time oscilloscope traces of the FSK transmitter’s output. We know the transmitted data pattern, so, from these scope traces, we can measure how often our received signal matches the transmitted one.

From these probabilities, it is easy to calculate the mutual information, $I\left(X;Y\right)$, between the transmitted signal, $X$, and the received signal, $Y$. According to Shannon, the mutual information gives the data capacity of the link when using an ideal error correcting code, measured in number of data bits received per symbol transmitted [25]. The next two paragraphs explain the origins of $I\left(X;Y\right)$. The reader may skip them to see the final expression for $I\left(X;Y\right)$ and calculations.

We can easily express and understand the mutual information in terms of the information entropy, which measures the number of bits of uncertainty carried by a random variable, $X$: $H\left(X\right)=-{\displaystyle {\sum}_{i}{p}_{i}{\mathrm{log}}_{2}\left({p}_{i}\right)}$, where $X$ is a random variable of all possible transmitted symbols, and ${p}_{i}$ is the probability that $X$ takes on value ${x}_{i}$. In the simplest case, each symbol can take on one of two equally possible wavelengths, so ${p}_{i}$ equals ½ for $i=1,2$, and $H\left(X\right)=1$ bit per symbol. This matches our intuition: a signal that is in one of two equally possible states should carry 1 bit of uncertainty (i.e., information). For an 8-ary FSK system, we expect that the transmitted signal carries 3 bits of uncertainty, and indeed ${p}_{i}=1/8$ for $i=1\text{to}\text{8}$, giving $H\left(X\right)=3$ bits.

Of course, so far we only know the information in our transmitted signal, $X$, but we want to know what information we can obtain from our received signal, $Y$. $Y$may be corrupted by noise: if we transmit ${x}_{i}$, we want to receive ${y}_{i}$, but may receive ${y}_{k}$ instead. One useful quantity to consider is $H\left(X|Y={y}_{k}\right)=-{\displaystyle {\sum}_{i}{p}_{i|k}{\mathrm{log}}_{2}\left({p}_{i|k}\right)}$, where ${p}_{i|k}$ is a conditional probability that $X={x}_{i}$ given that we receive $Y={y}_{k}$. Ideally, ${p}_{i|k}=1$ when $i=k$ and is 0 otherwise, which means we receive exactly the information that was sent. If so, then $H\left(X|Y={y}_{k}\right)=0$. Thus $H$, as a measure of uncertainty, is telling us that knowing the received signal, $Y={y}_{k}$, means that we know the transmitted signal, $X$, with certainty (i.e., with 0 uncertainty), which is the goal of any communications system. In reality, there is some chance that the signal is received incorrectly, or is erased all together, so that ${p}_{i|k}$ is not as ideal as above. Now, we’re interested in all possible received signals, $Y$, so we take the expected value of $H\left(X|Y={y}_{k}\right)$, averaged over all $k$, to obtain a new quantity, $H\left(X|Y\right)=E\left[H\left(X|Y=y{}_{k}\right)\right]$. Similar to $H\left(X|Y={y}_{k}\right)$ above, $H\left(X|Y\right)$ measures, in bits per symbol, the remaining uncertainty in $X$ after measuring $Y$. A communications receiver tries to remove all uncertainty in $X$, so that it can say confidently from a measurement of $Y$ what the value of $X$ was. Thus we would like $H\left(X|Y\right)$ to be as close to 0 as possible. From this, we arrive at the definition for mutual information: $I\left(X;Y\right)\equiv H\left(X\right)-H\left(X|Y\right)$. If $H\left(X|Y\right)=0$, we know $X$ with certainty from our measurement of $Y$. Thus we get $I\left(X;Y\right)=H\left(X\right)$, the maximum possible number of bits per symbol. When $H\left(X|Y\right)>0$, some of the transmitted bits will have to be error correcting bits, reducing the number of information bits, $I\left(X;Y\right)$, we get per symbol. From measurements of the probabilities, like ${p}_{i|k}$, we can calculate$I\left(X;Y\right)$.

Figure 12 shows a schematic of the experiment we used to measure FSK link’s data capacity when using photon counting detectors. The measured FSK signal was collected and stored on a digital oscilloscope. We then processed the stored data to measure the probabilities required to calculate the link’s data capacity, shown in Table 1. From our discussion above and a little algebra, the number of bits per symbol in a link with optimal error correction is

As mentioned before, the detectors we used, though fast for photon counters, limited us to 100 Msymbol/s. Higher symbol rates are possible by using arrays of photon counters to counteract the reset time [26]. Such arrays could detect FSK signals at Gsymbol/s rates. Moreover, we could add carrier wavelengths to expand the data rate further – each doubling of the number of carriers adds a bit of data per symbol. In contrast, the same process in PPM requires a doubling of the number of time slots, and thus a doubling of electronic rates.

## 5. Conclusions

Optical FSK communications systems can make possible high data rate communications over data links with high power losses and can do so without requiring electronic components to have bandwidths much greater than the data rate. Optical FSK systems have not yet been deployed in operational links, but this paper shows progress toward a practical transmitter. We have presented a transmitter that is capable of very high order FSK without impractical requirements on the number of components. We have demonstrated it with a 4-ary FSK signal at 5 Gbit/s and 20 Gbit/s with ~4-dB and ~5-dB power penalties compared to the theoretical best when detected by a pre-amplified receiver. We also demonstrated it with a photon counting receiver at 100 Mbit/s and 2-ary FSK, yielding a total power penalty 10.4 dB, of which only 2.4 dB was unaccounted for by losses in the optics.

There are many opportunites for more development. The transmitter reported here would benefit from a higher extinction ratio, which becomes very important at large $M$, and this might be achieved using more sophisticated bias control circuits for the EO modulators. Shortening RF paths through more integration could reduce size, lower RF losses, and improve stability. Also, integrating and packaging these devices are necessary for fielding. There is also the question of the best FSK receiver design, which has not been the focus of this paper. Many of our tests have been with pre-amplified receivers, which are comparatively easy to build, but which add noise. FSK’s greatest gains occur at large $M$ with strong FEC and photon counting receivers incorporating arrays of photon counting detectors, such as high pixel count APD arrays or superconducting nanowire single photon detector arrays. Without optical amplification, the receiver will need very low losses in its train of components. Finally, there is a tradeoff between the number of spatial modes on the detectors (which can be engineered, e.g. using adaptive optics) and the optical filtering capabilities of the receiver. So, there are many considerations in the design of FSK systems, but the great advantages of optical FSK in high loss links will drive the search for solutions.

## Acknowledgment

This work was sponsored by the Assistant Secretary of Defense for Research and Engineering under Air Force Contract #FA8721-05-C-0002. Opinions, interpretations, recommendations and conclusions are those of the authors and are not necessarily endorsed by the United States Government.

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