Abstract:
A random number generator includes a light source emitting light at a first frequency, an optical unit including an optical component configured to receive light at the first frequency and emit light at a second frequency, and a measurement unit configured to receive light at the second frequency, and generate a random output value related to a phase parameter of the light at the second frequency.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Patent Application No. 61/672,171 filed Jul. 16, 2012, the disclosure of which is incorporated herein by reference in its entirety. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     This invention was made with Government support under contract number N00014-10-1-0281 awarded by the Office of Naval Research. The Government has certain rights in this invention. 
    
    
     BACKGROUND 
     Random numbers may be used in cryptography, computer simulations, data storage, and secure data transfer, among other applications. Truly random numbers are desirable for such applications. Software-based random number generators (RNG) are not completely random and are not immune to attack. It is therefore desirable to have truly random physical RNGs not dependent on software. 
     SUMMARY 
     Random quantum phenomena may be used as the basis for a physical RNG. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  illustrates an example optical RNG. 
         FIG. 1B  illustrates phase versus gain in a second-order degenerative optical parametric oscillator (OPO). 
         FIG. 2A  illustrates an example optical RNG including dual OPOs. 
         FIG. 2B  illustrates phase information related to signals in an OPO. 
         FIG. 3  illustrates an example random sequence generated by an optical RNG. 
         FIG. 4A  illustrates an example of an n-OPO optical RNG. 
         FIG. 4B  illustrates sequences of signal and pump pulses from an OPO. 
         FIG. 5  illustrates an example RNG including an optical parametric generator (OPG). 
         FIG. 6  illustrates an example random number sequence generated by an optical RNG. 
         FIG. 7A  illustrates a relationship between clock speed and randomness in an OPO. 
         FIG. 7B  illustrates randomness in an OPO at one example clock speed. 
         FIG. 7C  illustrates randomness in an OPO at another example clock speed. 
         FIG. 7D  illustrates randomness in an OPO at another example clock speed. 
         FIG. 8  illustrates test results for one example optical RNG. 
     
    
    
     DETAILED DESCRIPTION 
     A physical RNG may be implemented using one or more OPOs or OPGs. The RNG is based on the inherently random quantum-mechanical phase of outputs generated through a down-conversion process in a nonlinear optical component, related to the randomness in the phase of quantum noise 
     At the quantum level, a nonlinear optical component of some implementations converts an input photon oscillating at one frequency to multiple output photons oscillating at frequencies other than the frequency of the input photon, and with random phase related to random quantum noise. The sum of the frequencies of the multiple output photons has a relationship to the frequency of the input photon. In an nth-order parametric down-conversion process, there are n output photons generated from one input photon. This feature of a nonlinear process is the basis for an implementation of the RNG using an OPO or and OPG, which includes a nonlinear crystal. 
     At the signal level, when an input light wave impinges on an nth-order nonlinear crystal in some implementations, the crystal passes part of the input wave, and converts the rest of the input wave into n output waves, where the sum of the frequencies of the n output waves has a relationship to the frequency of the input wave. This feature of a nonlinear crystal is the basis for an implementation of the RNG using an OPO or an OPG, which include a nonlinear crystal. 
     The conversion of light at one frequency to light at a lower frequency is sometimes referred to as optical parametric down conversion. 
     An OPO of some implementations includes a nonlinear optical crystal and an optical resonator. In an nth-order OPO, input light (“the pump”) received at the crystal is converted into n crystal outputs of known frequency. The resonator in the OPO is designed to resonate at a selected one of the crystal output frequencies, which is herein referred to as the signal. The light in the OPO, including the signal, repeatedly loops through the resonator and crystal, and each time, the intensity at the signal frequency is amplified. The OPO eventually reaches a stable state at the signal frequency if the inherent gain of the OPO is at least equal to the inherent loss of the OPO. 
     The stable OPO signal has a phase relative to the pump. Each time that the OPO is started, it will stabilize with a randomly-acquired phase. Each restart of the OPO thus results in generation of a new signal with random phase. The random phase of a signal is used to determine a discrete random number in an RNG. 
       FIG. 1A  illustrates an overview of an RNG  100  including a light source  110 , an optical unit  120 , and a measurement unit  130 . Light from source  110  is converted to optical information within optical unit  120 , which is used by measurement unit  130  to determine a discrete random number. 
     Light source  110  may be any source of light in any spectrum, and the light may be broadband or narrowband light. Light source  110  may be a continuous-wave source or a pulsed source. Light source  110  in some implementations is a laser. 
     Light source  110  includes optional shutter  115 . When pumping an OPO (i.e., generating a pump for the OPO) for example, shutter  115  may selectively occlude light source  110  so as to pump the OPO for a limited time or for a limited number of pulses. Shutter  115  may be opened for a first amount of time to allow pumping of the OPO then closed for a second amount of time to occlude light source  110  and thereby stop pumping the OPO. Shuttering allows for restarting of the OPO without residual signal information so that the OPO may stabilize with random phase. With no pumping, after a time (dependent on the characteristics of the OPO and source  110 ), light circulating in the OPO will decay below the noise floor of the OPO. 
     If light source  110  is pulsed, and the time between pulses is long enough to allow the signals to decay below the noise floor, then shutter  115  may be omitted. Further, in one implementation including an OPG instead of an OPO, as will be described below, shutter  115  may be omitted. 
     Optical unit  120  outputs a signal to measurement unit  130 , to determine phase information which may then be used to determine a random number. Phase information may be a phase parameter such as a phase of a signal with respect to the pump, or a phase difference between two or more signals. Optical unit  120  and measurement unit  130  are described in detail below with respect to several implementations of RNG  100 . 
     For the case in which optical unit  120  includes a degenerate second-order OPO, a crystal converts a pump into two outputs, a signal and an idler at the same frequency equal to half the pump frequency. The signal and idler can be considered to be indistinguishable from each other, and are referred to together as the signal. A resonator is designed to resonate at the signal frequency. The stable wave generated by the OPO has one of two opposite phases (i.e., phases separated by π), and the phase acquired is a random result due to quantum noise. Thus, the bi-stable random phase may be used as the basis for generating random-valued bits in a binary number. Examples of nonlinear crystals include MgO-doped periodically poled lithium niobate, BaB 2 O 4 , and KNbO 3 . 
       FIG. 1B  illustrates the incremental amplitude gain in a second-order degenerate OPO as a function of the relative phase between the pump and the signal. Maximum amplification of the signal occurs when the relative phase is zero or pi, where the energy flows from the pump into the signal, and maximum attenuation of the signal occurs when the relative phase is pi/2, where the energy flow is from the signal to the pump. If the maximum gain exceeds the loss in the resonator, depending on the zero-point fluctuations of the signal modes, the OPO will stably oscillate with the phase of either zero or pi. None of the design parameters favors oscillation in one or the other phase in some implementations. 
       FIG. 2A  illustrates an RNG  200 , including a light source  210 , an optical unit  220 , and a measurement unit  230 . 
     Light source  210  is illustrated as a pump unit  212  and an optical switch  215  with a clock  216 . Pump unit  212  may be continuous wave or pulsed. Optical switch  215  is opened or closed according to clock  216 , to allow or block pumping of optical unit  220 , respectively. 
     Optical unit  220  includes a first OPO  221  (labeled “OPO1”), a second OPO  222  (“labeled OPO2”), a path adjuster  223 , a beam splitter  224 , a beam splitter  227 , a reflector  225  and a reflector  226 . 
     OPO  221  and OPO  222  are matched OPOs, with matched nonlinear optical crystals and matched resonators. Pump unit  212  output (as modified by switch  215 ) is directed to beam splitter  224  and split into separate pumps for OPO  221  and OPO  222 . The pump for OPO  221  is directed to OPO  221  by reflector  225 . If switch  215  is kept open long enough, the output of each OPO  221  or  222  stabilizes to a signal of a known frequency and random phase. 
     Path adjuster  223  is provided to adjust the optical length of the path, extending from OPO  222  to beam splitter  227 , to match the optical length of the path through OPO  221  to beam splitter  227 . Optical path length as used in this document relates to the time needed for light to traverse the length of the path. Reflector  226  directs the output of OPO  221  toward beamsplitter  227 . Beamsplitter  227  adds the signals from OPO  221  and OPO  222 . 
     In the case in which OPO  221  and OPO  222  are matched degenerate second-order OPOs, the signals generated by OPO  221  and OPO  222  both have a frequency that is half of the pump frequency and a phase with respect to the pump of either zero or pi (π). 
       FIG. 2B  illustrates field waveforms for two OPO signals as related to the field of a pulsed pump, where one OPO signal is zero-phase and the other OPO signal is pi-phase with respect to the pump. The field waveforms of the OPO output signals are similar with respect to amplitude versus time, but phase is different by pi. Adding similar waveforms, both with zero-phase, results in a zero-phase waveform. Adding similar waveforms, both with pi-phase, results in a pi-phase waveform. Adding similar waveforms, one with zero-phase and one with pi-phase, results in one waveform canceling the other in the ideal case, such that the sum is zero. In an actual system, noise in the signals may not cancel, and therefore the sum will generally not be zero. An RNG may be implemented with OPOs that generate output signals at randomly zero-phase and pi-phase. 
     Referring again to  FIG. 2A , measurement unit  230 , which includes a photodetector  232 , receives the sum of signals from beamsplitter  227 . Photodetector  232  measures intensity of light received but does not recognize phase of the light. Therefore, zero-phase and pi-phase will affect photodetector  232  in the same manner. Photodetector  232  outputs a high signal for both zero-phase and pi-phase signal sums, indicating that the outputs of both OPOs were in phase with each other, and outputs a low signal for a signal sum in which the amplitudes canceled. Other detectors may be used instead of photodetector  232 , such that the phase may be read directly from a signal. 
     Measurement unit  230  may use the output of photodetector  232  to determine a random number. As the output of photodetector  232  is either high or low, it can be directly converted to a binary representation by comparison to a threshold. For example, if the output of photodetector  232  was above a threshold, a binary value of “one” may be assigned, and if the output of photodetector  232  was below the threshold, a binary value of “zero” may be assigned. Alternatively, the binary values may be assigned in opposite manner, such that above the threshold is assigned a “zero” and below the threshold is assigned a “one”. As measurement unit  230  assigns a value to each of a sequence of the signal sums, a sequence of binary numbers results. 
       FIG. 2B  illustrated the use of a pulsed pump. A continuous wave pump may used instead, by opening switch  215  for a time long enough to generate a stable signal, then closing switch  215  for a time long enough to decay the signal in OPO  221  or OPO  222 . The OPO signal will then be continuous for a time instead of pulsed as illustrated in  FIG. 2B . 
       FIG. 3  illustrates a graph plotting time in milliseconds versus output power of a photodetector, such as photodetector  232  in RNG  200 . Output power is shown scaled such that the maximum value is one. A clock signal, such as from clock  216  of RNG  200 , is also plotted on the graph as a dotted line square wave, where a high level of the clock square wave indicates that optical switch  215  is open, and a low level indicates that optical switch  215  is closed. Photodetector output power is plotted in solid lines on the graph. A threshold value may be established between the noise level (approximately 0.2 on the graph) and maximum (1 on the graph), such that during each clock high pulse, a binary number is assigned to the sum of the OPO signals as described above. For example, at the clock pulse starting at 1 ms, the photodetector output power is approximately a “one” on the graph, and a logic value of “one” has been assigned because the power is above a threshold. 
     Over time, a sequence of clocked pumping as described results in the generation of a sequence of binary numbers, which are random. Therefore, the system with parallel OPOs  221  and  222  as described with respect to  FIG. 2A  forms RNG  200 . 
     A variation of RNG  200  is to maintain OPO  221  with a consistent acquired zero- or pi-phase signal by pumping OPO  221  continuously, while switching OPO  222  on and off to generate a sequence of signals each with random phase, and comparing the consistent phase signal of OPO  221  with the random phase signal of OPO  222  to generate a random number. Alternatively, OPO  221  may be replaced with any consistent phase source at the signal frequency. 
       FIG. 4A  illustrates an RNG  400  based on the random quantum physical properties of an OPO. At a high level, RNG  400  may operate in a similar way as RNG  200 : pumping of two matched optical unit results in two signals with similar waveform and random phase, the signals are summed, the sum is applied to a photodetector, and the output power of the photodetector is used to assign a binary number to the signal, where the binary number is random. One distinction between RNG  200  and RNG  400  is that the single OPO of RNG  400  may operate as if it were multiple OPOs. 
     RNG  400  includes a light source  410 , an optical unit  420 , and a measurement unit  430 . Light source  410  includes a pump  412 , an optical switch  415 , and a clock  416 , which operate together in a manner similar to the description of light source  210  of  FIG. 2A . 
     Optical unit  420  includes an OPO  425  with a nonlinear crystal  426  and a resonant cavity. A resonant cavity may be a microresonator, linear resonator, ring resonator, disk resonator, waveguide, or fiber based resonator, for example. Resonator size can vary. In the implementation of  FIG. 4A , the resonant cavity includes reflective surfaces (“RS”) M1-M8, each of which may be fully or partially reflective, and may be partially reflective on one surface and fully reflective on an opposite surface. Optical path length of a resonant cavity may be adjusted in a control loop to compensate for system changes. For example, in  FIG. 4A , a piezo transducer  450  may be controlled by a controller  452  to move RS M4 back and forth in response to a difference (i.e., “error”) between the phase of the transmitted and received pump. The error signal may be generated, for example, by modulating a frequency on the transmitted pump frequency and filtering the modulated frequency out of the received pump in filter  454 . Intensity of the filtered error signal may be measured in a photodetector  456  and provided to controller  452 . 
     OPO  425  further includes an output coupler  427 . Output coupler  427  is a beam splitter that passes part and reflects part of any received light. 
     Light source  410  pumps light into OPO  425  through RS M1 to RS M2 and through crystal  426 . At least a portion of the energy of the pump is converted in crystal  426  to multiple outputs. The outputs and the remaining portion of the pump circulate through the resonant cavity of OPO  425  by reflection off of the RS sequence M3-M4-M5-M6-M7-M8-M1-M2 followed by propagation again through crystal  426 , and so on. 
     To generate a signal at a magnitude large enough to be detected over the noise in the system, energy is added to OPO  425  until the signal stabilizes at a randomly-acquired phase. This may be accomplished, for example, by pumping several pulses. 
     A single OPO may be implemented by OPO  425 . To implement a single OPO using OPO  425  with pulsed pumping, the resonant cavity is arranged such that the roundtrip optical path length of the resonant cavity is equal to the time between pump pulses. Optical path length in this context means the time required for the light to traverse the path. A sequence of pulses is applied to OPO  425 , and each new pulse is introduced at approximately the same time as the existing signal begins its next loop of the resonant cavity upon reflection from RS M1. For example, the number of pulses required may be equal to ten. 
     An n-OPO may be implemented in OPO  425  by arranging the resonant cavity such that the roundtrip optical path length is equal to n times the time between pump pulses. Operation of a 2-OPO implemented using OPO  425  in pulsed mode is described next. 
     OPO  425  is labeled “Twin OPO” in  FIG. 4A  to indicate that it may be implemented as a 2-OPO, which is effectively two matched degenerate second-order OPOs sharing the same crystal and resonant cavity. Roundtrip optical path length of the cavity is twice the time between pump pulses. When a first pump pulse is introduced to the Twin OPO, crystal  426  generates a first signal at half the frequency of the pump, which begins to circulate in the resonant cavity. Note that “signal” in the second-order degenerate OPO is the indistinguishable combination of the “signal” and “idler”. The first signal travels an optical path length equal to the time between pump pulses before the second pump pulse is introduced to the Twin OPO. Crystal  426  generates a second signal at half the frequency of the pump, which also begins to circulate in the resonant cavity. The second signal travels a distance equal to the time between pump pulses by the time a third pump pulse is introduced. The third pump pulse serves to augment the first signal, as the first signal and the third pump pulse arrive at crystal  426  at approximately the same time. Similarly, a fourth pump pulse serves to augment the second signal. 
     It can be seen that in the second-order degenerative Twin OPO, the first, third, fifth, and so on signals build to form an “odd” OPO signal, and the second, fourth, sixth, and so on signals build to form an “even” OPO signal. “Odd” and “even” is used in this manner merely as nomenclature to distinguish the two OPO signals and does not describe any feature of the two signals themselves. The odd and even OPO signals do not interact with each other. Therefore, the Twin OPO is effectively two OPOs sharing the same components. 
       FIG. 4B  illustrates how the pump pulses relate to the odd and even OPO signals. Pulses augmenting the odd OPO signal are labeled OPO1, and pulses augmenting the even OPO signal are labeled OPO2. 
     Referring again to  FIG. 4A , output coupler  427  reflects part of the odd and even OPO signals (with some residual portion of the pump) to measurement unit  430  by way of reflective surfaces (RS)  440  and  442 . One or both of RS  440  and  442  may be included in optical unit  420  or in measurement unit  430 . 
     Measurement unit  430  receives the odd and even OPO signals in interferometer  435 . The even OPO signal lags the odd OPO signal by an amount of time equal to the time between pump pulses. To sum the odd and even OPO signals, the odd OPO signal is delayed in interferometer  435  by a time equal to the lag. Interferometer  435  includes two optical path arms  436  and  437  where the difference between the roundtrip optical paths of arms  436  and  437  is equal to the time that the even OPO signal lags the odd OPO signal. The output of interferometer  435  is a summed signal. 
     Alternatively to using interferometer  435  to introduce a delay, a reference OPO may be used to generate a signal to sum with the either, or both of, the odd and even OPO signals. 
     Measurement unit  430  filters the summed signal to remove residual pumps, and then measures the intensity of the summed signal using detector  438 , such as a photodetector or camera. The intensity of the summed signal may then be converted to binary values, as described above. 
     The Twin OPO described with respect to  FIG. 4A  is one example of an n-OPO. Another example is a 3-OPO (“Triplet OPO”), in which the roundtrip optical path length of the Triplet OPO is three times the period between pump pulses, thus, three unrelated signals are generated within the Triplet OPO, separated by the time between pump pulses. The three signals circulate in the Triplet OPO concurrently, and each acquires a random phase. If using a degenerate second-order optical crystal, the random phase acquired will be zero-phase or pi-phase. 
     Higher-order OPOs may be constructed similarly as described with respect to the Twin OPO and Triplet OPO. 
     One advantage to using the n-OPO described with respect to RNG  400  is that multiple random-phase signals may be stabilized or decayed within the resonator in a time that exceeds the stabilization or decay time of a single OPO by a short time. 
     As a comparison, a 2-OPO (Twin OPO) may be significantly faster than the dual OPO of  FIG. 2A  because of the decreased stabilization and decay times. Additionally, the 2-OPO uses the same cavity and crystal thereby avoiding matching of two separate OPOs as is the case with the dual OPO of  FIG. 2A . 
     OPOs have been described thus far because the OPO provides for amplification of the down-converted signal to a detectable level, thereby reducing the power requirement on the light source and/or reducing the requirements on nonlinear interaction strength. Other implementations not requiring the amplification of an OPO are also within the scope of this disclosure. 
       FIG. 5  illustrates an RNG  500  using an OPG that does not require the use of an OPO. RNG  500  includes a light source  510 , an optical unit  520 , and a measurement unit  530 . Light source  510  is similar to light sources described above, and is not further separately described. 
     Optical unit  520  is an OPG including a nonlinear optical crystal  525 , or alternatively another nonlinear optical component, that converts a pump into multiple outputs at possibly different frequencies. A sequence of starts and stops of crystal  525 , such as through pulsing the pump, results in a corresponding sequence of instances for each of the multiple outputs. Each instance has a random phase. 
     Interferometer  532  includes two unequal-length optical arms, such that the difference between the roundtrip optical path lengths of the two arms is equal to the time between sequential instances of outputs at a selected frequency. With this construction, an instance of an output at the selected frequency is delayed and compared with the next instance of that output, so that the phases of two sequential instances of that output are compared in interferometer  532 . 
     The output of interferometer  532  may be subsequently filtered by filter  534  to pass light at the selected frequency. As previously described, the intensity of the light from interferometer  532  may be measured by photodetector  536 , and used to assign a random number. 
     In an alternative construction of RNG  500 , a selected output sequence of crystal  525  is combined with a reference signal and supplied directly to photodetector  536 . In another alternative construction, multiple outputs of crystal  525  may be combined together and supplied directly to photodetector  536 . 
       FIG. 6  illustrates an example of random number generation for an optical unit  120  in which the output of optical unit  120  is compared to itself after a delay. Each Signal represents a restart of the crystal or OPO in optical unit  120 , and is therefore not related to the Signal before or following. The first Output shown is based on the previous Signal not shown, and represents a logic value of ‘1’. The second Output is based on the sum of the second Signal (phase pi) and the first Signal (phase zero). Since the second and first Signal phases are different, the second Output represents a logic value of ‘0’. The third Output is based on the sum of the third Signal (phase zero) and the second Signal (phase pi). Since the third and second Signal phases are different, the third Output represents a logic value of ‘0’. The fourth Output is based on the sum of the fourth Signal (phase zero) and the third Signal (phase zero). Since the fourth and third Signal phases are the same, the fourth Output represents a logic value of ‘1’. The analysis is the same for the rest of the Outputs. 
     In some implementations, for example an implementation in which signal phase is one of two known phases, a heterodyne detector or a high-resolution spectrometer may be used instead of one of the measuring units described, to detect slight differences of frequency that occur between different phase states. 
     Note that, although the descriptions above describe two signals being compared, it is also within the scope of the concept described to compare three or more signals by, for example, adding a parallel OPO, or adding delay arms in the interferometer for sequential Signals. 
     An RNG has been described that uses the random quantum noise of a nonlinear crystal to generate signals with random phase. An RNG may compare a signal with random phase to one or more other signals with random phase, and the phase information between the random phase signals, which is also random, is used as the basis for assigning a discrete value. Alternatively, an RNG may directly measure phase of a signal to determine phase information. 
     Therefore, random quantum noise is used to generate a random number. 
     An OPO used in an RNG may be, but is not necessarily, degenerate. In the case of a non-degenerate (for example, type II) OPO, the resulting random phase is not discrete, but is rather continuous. The pump may be pulsed or continuous wave. 
     An OPO or OPG implementation may use third-order or higher parametric down-conversion. A degenerate third-order parametric down-conversion process will result in three discrete phase states of zero (0), 2π/3, and 4π/3, which can be used, for example, for ternary random number generation. 
     An optical unit with a third-order non-linearity may be used for degenerate four-wave mixing, in which the third-order non-linearity is used to generate a signal and an idler, and the sum of the frequencies of the signal and idler is equal to twice the frequency of the pump, such that one of the signal or idler is at a frequency greater than the pump frequency. 
     The described RNG concept allows random number generation with no need for electronic or computer post processing on a generated bit sequence. The described RNG may be implemented with all-optical operation, it is fast, it is truly random, it is robust, and it has no requirement for photon counting. 
     Several implementations of the all-optical RNG have been described, including, for example: using either continuous wave or pulsed light sources; either summing two signals or interfering a signal output from the optical unit with the next output; using degenerate or non-degenerate OPOs; using either OPOs or an OPG; implementing two or more OPOs either physically separately or sharing one OPO cavity; and utilizing either discrete or continuous random phases. Other options include alternative kinds of nonlinear processes, resonator selection, and light source selection, among others. Additionally, other optical components may be used instead of a crystal. For example, an optical fiber may be used. The variety of options available lead to many combinations for implementation of the RNG. Other implementations will be apparent from the discussions above and the claims. 
     An OPO-based quantum RNG can be implemented using one or more on-chip X-3 OPO, allowing for CMOS compatibility. Additionally, using micro- and nano-resonators, high-speed all-optical quantum RNGs with multi gigabit-per-second (Gbps) rates are possible. 
       FIG. 7A  illustrates how randomness, measured by the rate of bit flip in a 100-kb long sequence, can break as a function of clock speed. The maximum bit-rate supported by an OPO-based RNG such as described depends on the turn-on and turn-off dynamics of the OPO. At the end of each clock cycle, the intracavity field should decay to the quantum noise level, or the residual field from the previous state will seed oscillation of the next state.  FIGS. 7B, 7C, and 7D  illustrate output samples for three points on the curve of  FIG. 7A , illustrating that when an intracavity reference exists in the form of residual photons from the previous clock cycle, the randomness of the sequence breaks. 
     Faster bit rates, at least in the Gbps range, are expected to be achievable using pumps with higher repetition rates in combination with shorter OPO cavities. 
     Further, an OPO may be operated closer to threshold, where the build-up time is longer than the decay time. In this case, oscillation may not reach steady-state, but relative phase may still be measured with a sufficiently sensitive detector to assign a bit value, decoupling maximum RNG speed from cavity decay time. Potentially, the cavity could be eliminated and a single pass parametric down conversion used with a speed as high as the repetition rate but requiring either a sensitive detection system or a relatively high peak power. 
     An Example and Test Results 
     In one system used to evaluate the OPO-based RNG concept, a pulse-pumped Twin OPO was used in which two identical OPOs operate in the same ring resonator with the roundtrip optical path length equal to twice the time between pump pulses. The OPO was pumped by a 1560-nm mode-locked Er-fiber laser (Menlo Systems C-fiber, 100 MHz, 70 fs, 300 mW) where the beam was conditioned by a mode-matching telescope for efficient pumping of the OPO. The resonator was a 6-m ring cavity. The pump pulses were converted to two independent signals. These two temporally separated signals had half the repetition rate of the pump, the same polarization and spectral properties, and experienced the same optical paths. An unequal arm interferometer was used to measure the relative phase states of the two signals by interfering them temporally. 
     The cavity optics included one pair of concave mirrors (M2 and M3) with ROC=50 mm and six flat mirrors, five of which (M4-M8) were gold coated with a material exhibiting approximately 99% reflection. A single dielectric mirror (M1) was used to introduce the pump, which had 90% transmission for the pump and more than 99% reflection in the 2.8-4 mm range. Mirror M1 had a ‘chirped’ design of dielectric layers to compensate the dispersion of the nonlinear crystal. Broadband gain centered around 3.1 mm was provided by 1-mm long MgO-doped periodically poled lithium niobate (MgO:PPLN) crystal. The poling period is 34.8 mm for broadband type-0 (e=e+e) phase matching at a temperature of 32° C. The crystal was cut such that the mid-IR beam propagated perpendicularly to the poling domains when the beam entered at the Brewster angle. The beam waist for the signal in the crystal was approximately 10 mm. 
     Mirrors M2 and M3 were set to 5-degree angles of incidence to compensate the astigmatism caused by the Brewster angled crystal and allow stable resonances in the 6-m long cavity. 
     The output was extracted with a pellicle beam splitter (OC) having approximately 8% reflection over a broad bandwidth. The filters were AR coated Ge substrates to block the pump and allow the mid-IR signal to pass. Oscillation occurred when signal and idler waves were brought into degenerate resonance by fine-tuning the cavity length with the piezo stage of M4. Three resonances occurred separated by approximately 1.5 mm of roundtrip optical path length, corresponding to half of the signal central wavelength. Continuous operation of the OPO was obtained by locking the optical path length to track the center of the strongest resonance using a dither-and-lock scheme. 
     The Twin OPO started oscillating at a pump average power of about 120 mW, and the maximum mid-IR output power was 4 mW, with signal spectrum centered at 3.1 mm and the pump centered at 1.56 mm. As a test of the randomness of the Twin OPO output, complementary stable fringe patterns at the output of the interferometer were obtained when the beams in the arms were slightly angled vertically, where blocking and unblocking the pump resulted in random toggling between these two patterns. To capture a bit stream, however, the beam angles were well-aligned in the interferometer and a photodetector was used at the output, while an Acousto-Optic Modulator (AOM) caused periodic restarting of the Twin OPO. A binary sequence was extracted from the interferometer output as described above. A sequence of 1 billion bits was taken with this method, and the output proved to be random, with an average of 0.5000. 
     To verify statistical randomness, a series of tests developed by the National Institutes of Standards and Technology (NIST) were performed, and the summary of results are presented in  FIG. 8 . The 1-Gb sequence passed all the NIST statistical tests indicating it is random with 99% confidence. 
     To maintain randomness for the test system described, the turn off time should be long enough to allow the intracavity power to decay from the steady state level, about 1 W, to the quantum noise level of one photon per mode, which is about 1 mW. Noise level is P noise =hν*Δν, where hν is the photon energy at the central signal wavelength of 3.1 mm, and Δν is the OPO bandwidth at 3-dB level, estimated to be approximately 10 THz. The intensity decay time of the OPO can be estimated using: 
               τ   off     =     T       2   ⁢           ⁢     δ   E       -     2   ⁢           ⁢     δ   E     ⁢         P   off       P   th                     
where δ is the electric-field fractional round-trip loss, P th  is the pump power at threshold, P off  is the pump power at the “off” state, and T is the cavity roundtrip time. In the presence of the AOM, the OPO threshold measured before M1 is increased to 190 mW because of pulse broadening in the AOM. The pump power at the off state is 168 mW, and intracavity power loss (2 δE) is estimated to be 0.27 resulting in the 1/e intensity decay time of 1.2 ms. Hence the minimum turn-off time required for decaying from steady state power to quantum noise level is about 17 ms corresponding to a maximum clock speed of approximately 30 kbps.
 
     In the example used in the test, the clock speed was slow enough (and OPO “on”-time long enough) that oscillation built to a steady-state level. The build-up of the tested RNG was much faster than decay due to the low (approximately 23%) modulation depth of the AOM that biases the pump slightly below threshold during the “off” phase of the clock, resulting in a much longer decay time than if the OPO were un-pumped. The clock rate was limited by the time it took for the intensity to decay below noise, which is 10 to 20 times longer than the 1/e cavity decay time when the OPO is pumped well over threshold and allowed to reach steady state.