quantum key distribution ( qkd ) is a technique that allows two parties ( alice and bob ) to generate a secret key despite the computational and technological power of an eavesdropper ( eve ) who interferes with the signals .together with the vernam cipher , qkd can be used for unconditionally secure data transmission .the basic ingredient of any qkd protocol is the distribution of _ effective _ quantum states that can be proved to be entangled .the first complete scheme for qkd is that introduced by bennett and brassard in ( bb for short ) . in a quantum optical implementation of this protocol , alice encodes each random bit into the polarization state of a single - photon .she chooses for her encoding one of two mutually unbiased bases , e.g. either a linear or a circular polarization basis . on the receiving side ,bob measures each photon by selecting at random between two polarization analyzers , one for each possible basis .once this phase is completed , alice and bob use an authenticated public channel to process their correlated data in order to obtain a secret key .this last procedure , called _ key distillation _, involves , typically , postselection of data , error correction to reconcile the data , and privacy amplification to decouple the data from eve .a full proof of the security for the whole protocol has been given in .after the first demonstration of the feasibility of this scheme , several long - distance implementations have been realized in the last years ( see and references therein ) . however , these practical approaches differ in many important aspects from the original theoretical proposal , since that demands technologies that are beyond our present experimental capability .especially , the signals emitted by the source , instead of being single - photons , are usually weak coherent pulses ( wcp ) with typical average photon numbers of or higher .these pulses are described by coherent states in the chosen polarization mode .the quantum channel introduces considerable attenuation and errors that affect the signals even when eve is not present .finally , the detectors employed by the receiver have a low detection efficiency and are noisy .all these modifications jeopardize the security of the protocol , and leads to limitations of rate and distance that can be covered by these techniques .a positive security proof against all individual particle attacks , even with practical signals , has been given in .more recently , a complete proof of the unconditional security of this scheme in a realistic setting has been achieved .this means that , despite practical restrictions , with the support of the classical information techniques used in the key distillation phase , it is still possible to obtain a secure secret key .the main limitation of qkd based on wcp arises from the fact that some pulses contain more than one photon prepared in the same polarization state .now eve is no longer limited by the no - cloning theorem since in these events the signal itself provides her with perfect copies of the signal photon .she can perform the so called _ photon number splitting _ ( pns ) attack on the multi - photon pulses .this attack provides eve with full information about the part of the key generated with the multi - photon signals , without causing any disturbance in the signal polarization .together with an optimal eavesdropping attack on the single - photon pulses , the pns attack constitutes eve s optimal strategy .this result is stated for a conservative definition of security . in this paradigm, it is commonly assumed that some flaws in alice and bob s devices ( e.g. the detection efficiency and the dark count probability of the detectors ) , together with the losses in the channel , are controlled by eve , who exploits them to obtain maximal information about the shared key .  in this paperwe analyze a different scenario .we impose constraints on eve s capabilities , and we are interested in the influence that this effect has on her best strategy .it is necessary to distinguish this work from earlier ones : here we consider a more relaxed definition of security than the one in .in particular , we study the situation where eve is not able to manipulate alice and bob s devices at all , but she is limited to act exclusively on the quantum channel ( see e.g. ) . the main motivation to consider this scenario is that from a practical point of view it constitutes a reasonable description of a realistic situation , where alice and bob can limit eve s influence on their apparatus by some counterattack techniqueshowever , this scenario has not been analyzed thoroughly .see appendix [ why ] for a discussion of the articles by gilbert and hamrick . in discussions within the scientific communityone often hears the hope that it is sufficient to consider the pns attack , but this time taking into account the finite detection efficiency of bob s detectors . as a resultthe loss of a photon in the pns attack reduces the probability to detect the remaining signal with the inefficient detectors and less multi - photon signals contribute to the final key .this suggests higher available rates .however , the analysis of this scenario is rather subtle , as we will show in this paper .note that a first counter example against that believe is contained in showing that the unambiguous state discrimination attack of can outperform the adaptation of the photon - number splitting attack of in the discussed scenario of limited eavesdropping capabilities .this result applied to signals containing at least three photons .we show that already two - photon processes allow for improved eavesdropping in the restricted scenario .  with our article, we point out the difficulty of analysing the scenario where bob s detection efficiency can not be manipulated . for thiswe refer to the standard bb protocol , where in the first part only the raw bit rate ( before the key distillation phase ) is monitored but not the number of coincidence detections .we construct two specific eavesdropping strategies which do not subtract photons from all the multi - photon pulses , and that are more powerful than the pnsattack for some relevant regimes of the observed error rate .they are based on specifically chosen cloning attacks .the results obtained here do not constitute a complete analysis of eve s optimal attack under these restriction , they introduce a new class of eavesdropping strategies that become relevant only in this scenario .our results clearly show that a simple extension of the pns attack in this scenario fails to deliver security . in an extended version of the protocol , where alice and bob can access the complete photon number statistics of the arriving signal , we find that the advantage of the cloning attacks is not as evident , but requires a deeper analysis .the paper is organized as follows . in sec .[ sec : tools ] we describe in more detail the scenario we consider here .this includes the signal states and detection methods employed by alice and bob together with the technologies assumed for eve . in sec .[ sec : pns ] we introduce the complete pns attack , and we analyze a particular process that is part of this attack and involves only single - photon signals and two - photon signals . in sec .[ sec : clon ] we introduce two more processes that do not subtract photons from the pulses .they are based on cloning machines operating only on two - photon pulses . in sec .[ sec : comp ] , these two processes are compared with the pns process . we show that , for some relevant regimes of the observed error rate in the sifted key , the two processes based on cloning machines provide eve with more information than the pnsprocess .the extended version of the protocol , where alice and bob use the full statistics at their disposal to detect eve , as introduced in is briefly consider in sec .[ sec : stat ] . finally , sec .[ sec : conc ] concludes the paper with a summary .alice uses wcp signal states that are described by coherent states with a small amplitude .this corresponds to the description of a dimmed laser pulse .we consider otherwise a perfect implementation of the signal states .the coherent state is given by with being the creation operator for one of the four bb polarizations modes .however , usually there is no reference phase available outside alice s lab , and the state that bob and eve see is not a coherent state , but the phase - averaged form of the signal , .this results in an effective signal state which is a mixture of fock states with a poissonian photon - number distribution of mean .it is described by the density matrix where the state denotes the fock state with photons in one of the four bb polarization states .we consider that bob employs the active detection setup shown in fig .[ detector ] .   and ) of the measurement as desired .the polarization analyzer consists of a polarizing beam splitter ( pb ) and two ideal detectors .the pb discriminates the two orthogonal polarized modes .detection efficiencies are modeled by a beam splitter ( bs ) of transmittance .[ detector ] ]  it consists of a polarization analyzer and a polarization shifter which effectively changes the polarization basis of the subsequent measurement .the polarization analyzer has two detectors , monitoring each one output of a polarizing beam splitter .these detectors are characterized by their detection efficiency .they can be described by a combination of beam splitters of transmittance and ideal detectors .this model can be simplified further by considering that both detectors are equal .in this situation , it is possible to attribute the losses of both detectors to a single - loss beam splitter which is located after the transmission channel .we assume that the detectors can not distinguish the number of photons of arrival signals , but they provide only two possible outcomes : click `` ( at least one photon is detected ) , and ' ' no click " ( no photon is detected in the pulse ) .the action of bob s detection device can be characterized by two povm , one for each of the two polarization basis used in the bb protocol .each povm contains four elements : , and .the outcome of the first operator , , corresponds to no click in the detectors , the following two povm operators , and , give precisely one detection click ( these are the desired measurements ) , and the last one , , gives rise to both detectors being triggered .if we denote by the state which has photons in one mode and photons in the orthogonal polarization mode with respect to the polarization basis , the elements of the povm for this basis are given by where .the detectors show also noise in the form of dark counts which are , to a good approximation , independent of the signal .note that the observed errors can be though as coming from a two - step process : in the first step the signals are changed as they pass eve s domain in the quantum channel , in the second step random noise from the detector dark counts is added .if we assume that the second step can not be influenced by eve , then alice and bob can infer the channel error rate , that is assumed to be due to eavesdropping , from their data and their knowledge of the detector performance .this means that only this reduced channel error rate needs to be taken into account in the privacy amplification step .   as discussed before, we allow eve to have at her disposal all technology allowed by quantum mechanics , but she is limited to use it exclusively on the quantum channel .this assumption has two consequences on eve s possible eavesdropping strategies that are vital for the security analysis of the next sections .in particular , the detection efficiency of bob s detectors is fixed and eve can not influence it to obtain extra information . moreover , since we consider that the noise of the detectors is independent of the signals entering them , eve can not make use of the dark counts to increase her information .in the photon - number splitting attack eve performs a quantum non - demolition measurement of the _ total _ number of photons of each signal .whenever she finds that a signal contains two or more photons , she deterministically takes one photon out .the remaining photons are then forwarded to bob .the photons in eve s hand will reveal its signal polarization to eve if she waits with her measurement until she learns the polarization basis during the key distillation phase . if the loss of the channel is strong enough , eve can block all the single - photon pulses and forward only the remaining photons of multi - photon signals by a lossless channel ; on these signals she can obtain the whole information . in this situation no secure key can be generated .when the loss is not high enough for this , then eve can block only a fraction of the single - photon signals , but she can perform some optimal eavesdropping attack on the remaining single - photon pulses . moreover , the whole process can be adapted such that it mimics the photon number statistics of a lossy channel in typical situations .  when bob usesa detection setup with ideal detectors , or eve can manipulate their efficiency such as , then the pnsattack constitutes eve s optimal strategy .the reason is that in this case all signals that provide eve with full information about the key ( multi - photon pulses ) contribute for the raw key .if the detectors have a detection efficiency which eve can not change , we find that with certain probability the multi - photon signals can also contribute to vacuum events in the detection process .in this situation , there are regimes where the pnsattack is still eve s optimal eavesdropping strategy .this happens when the loss in the channel is sufficiently high such that the number of non - vacuum signals expected to _ arrive _ at bob s detection device is smaller than the number of multi - photon signals . herewe consider the regime where this is not the case .this means that eve needs to compensate the effect of the undetected multi - photon signals by increasing the number of single - photons signals that is sent to bob .this fact reduces the effectiveness of the pns attack , and one might consider the existence of better strategies for eve .we focus in a particular combination of processes that are contained in the extended pns attack , now with imperfect detectors .this combination includes only some two - photon processes ( with probability ) and some one - photon processes ( probability ) from the whole eavesdropping strategy .it is represented in fig .[ pns ] .the pulse contains two photons , eve takes on photon out of it , and she sends the remained one photon to bob . in the case of single - photon signals ( probability ) , eve performs an optimal eavesdropping attack ( oa ) on these pulses .[ pns ] ]  the objective is to obtain eve s maximum information on this combination of processes given a particular disturbance in the signals . for that , we employ the concept of mutual information given by shannon . under this definition, it has been proven that the optimal attack on single - photon signals ( oa ) , i. e. the one that provides eve maximum information about the raw key , coincides with the optimal individual attack on these signals .this optimal individual attack has been introduced by fuchs _ et ._ in . in the symmetric strategy , every qubit signal sent by alice is transformed into the mixed state .the disturbance represents the error rate in the sifted key within the chosen signals ; it is not the overall observed error rate .the connection between the two error rates is made in section [ observed ] . for a given value of ,eve s maximum information in this attack is given by  where the function is defined as . with this result , now it is straightforward to obtain eve s maximum information in the pns process of fig .[ pns ] , as a function of and ,   in the next section we introduce two more combination of processes that have the same input signals as those of fig .[ pns ] . then , in sec . [ sec : comp ] we show that these new processes provide eve more information than the pns process , for some relevant regimes of .moreover , the raw bit rate of all the processes can be selected to be the same .this means that the substitution of the combination of processes of fig .[ pns ] by any of the new combinations leads to a better eavesdropping strategy in terms of shannon information .another possible eavesdropping alternative for eve is not to reduce the number of photons in the signal as in the pns attack . instead, she can interact with the signals via a photon - number conserving interaction of a probe system with the signal photons . then , after the information about the polarization basis is publicly revealed , eve can obtain information about the key by measuring her probe . in this attack ,multi - photon signals maintain their photon - number and can therefore contribute with higher probability to a `` click '' event .therefore , the fraction of the single - photon signals in this attack can be decreased . in principle, one would like to optimize this type of attack over all possible probes and their interaction .however , for the sake of simplicity , we restrict our analysis to the case of two particular interactions representing cloning machines .this is motivated by the fact that the optimal individual attack for single - photon pulses coincides with the optimal phase - covariant cloning machine .these cases already proof our point .consider the process represented in fig .[ clo ] .  .in the case of a single - photon pulse ( probability ) , she blocks it.[clo ] ]  the input signals are the same of the process of fig .[ pns ] . buthere eve employs an asymmetric cloning machine for all two - photon pulses , while she blocks all the single - photon pulses .the parameter can be selected such that the processes of fig .[ pns ] and fig .[ clo ] have the same raw key rate .this is done in sec .[ sec : comp ] , when we compare them .we consider two particular asymmetric cloning machines that have been proposed by ac ' in _ et al . _ in .they generalize the asymmetric cloning machines introduced in to the case .but before introducing these cloning machines and study the performance of the process of fig .[ clo ] for each of them ( strategies a and b below ) , we introduce a qubit representation for the two - photon pulses emitted by alice that is used in the next subsections .the set of two - photon signals employed in the bb protocol span a three dimensional hilbert space. they can be represented in the _ symmetric _ subspace of two - qubits , which contains the signal states , and , where .   in this attack eveuses an asymmetric universal cloning machine .it takes as an input state two copies of an unknown one - qubit state , plus a two - qubit probe .its unitary transformation is defined by  where the operator ( for ) with the usual pauli operators , the states , , and . in the output , the state of the first two qubits belong to the symmetric subspace of two - qubits signals , and correspond to the two photons that are sent to bob .the third and fourth qubit constitute the probe that is kept by eve .next we calculate the information that eve can obtain on alice s signal as part of a sifted key by measuring her probe after the public announcement of basis .eve s probe for the signals , and , after applying the cloning machine , is given by and respectively , where  .note that in the subspace spanned by the two - qubit states and eve can discriminate between and perfectly . in the orthogonal subspace spanned by and , however , the states and present a non - vanishing overlap . in this subspace , eve s maximum information is given by  .this means that eve s maximum information in this cloning machine can be written as a function of as this expressions holds also for the signals of the other polarization basis , so that it denotes also the total shannon information over all signals .  here , and also in the next subsection, we consider that double click events are not discard by bob , but they contribute to the raw key . every time bob obtains a double click , he just decides randomly the bit value .the second cloning machine we consider is a phase - covariant cloning machine .the unitary transformation of this cloning machine is given by  where can be any state in the symmetric 2-qubit hilbert space , and is the unitary transformation  has the same form as but interchanging zeros and ones on the right hand side of eq .( [ v ] ) , and .the first two output qubits of this cloning machine belong again to the symmetric subspace of two - qubits signals and correspond with the two photons which are sent to bob , while the other two qubits constitute eve s probe .  following the same argumentation used in strategy a ,when alice sends the signals ,  , the state of eve s probe is given by or , depending on the particular state chose by alice .the states and can be written in the basis , , , as respectively , where the coefficients , and are complicated functions of the parameter . the exact expression of these coefficients is given in appendix b. to calculate eve s maximum information in this case , we can decompose eve s optimal measurement again in two steps .first , she performs a projection measurement onto the two orthogonal subspaces spanned by , , and , , respectively .this measurement reduces the optimization problem in the whole space , to discriminate in each subspace between two equiprobable one - qubit states whose density matrices have the same invariants .this problem was solved by levitin in .the maximum of the mutual information in each subspace is given by , where represents the trace of the product of the two states , and is the determinant of their density matrices . using the expressions for eve s maximum information in each subspaces , one can obtain eve s maximum information in the cloning machine as a function of the coefficients , and .it is given implicitly by the disturbance in this case has the form again , due to symmetry with respect to the polarization bases , eq .( [ i_clon2 ] ) holds also for the total average shannon information .the processes represented in fig . [ pns ] and in fig . [ clo ] , for both cloning machines , gives a symmetric detection pattern .that is , if bob measures the signals in the same basis chosen by alice when preparing the states , then the probability of obtaining a correct result , a wrong result , or a double click is the same for all the signals .otherwise the outcomes corresponding to events with one detection click are completely random . for a fair comparison of the pns process and the two cloning processes, we need to assure that the raw bit rate in bob s detectors is the same for all of them , .the left - hand - side of the equation is the number of clicks of the pns process , while the right - hand - side is the number of clicks expected in fig .[ clo ] for both cloning machines .this means that .if we include this value in eq .( [ eqpns ] ) , eve s maximum information in the pns process is now written as this expression can now be directly compared with eq .( [ i_clon1 ] ) and eq .( [ i_clon2 ] ) .the results are plotted in fig .[ res ] and show regimes of for which   ( solid ) .the lower line corresponds to , while the upper line corresponds to .universal cloning machine , strategy a ( dotted ) .phase - covariant cloning machine , strategy b ( dashdot).[res ] ]  the process based on cloning machines provides eve with more information than the pns process .note that eve s maximum information in the cloning processes is independent of .this fact comes from the matching condition for the raw bit rate . in the pns process ,as expected , when approaches one , also approaches one , since the pns attack is the optimal strategy for eve s in the case of ideal detectors . in the phase - covariant cloning machine of strategy b, eve s maximum information never reaches one .the reason is that in this particular cloning machine none of the two qubits kept by eve can reach a fidelity one with respect to the input state . for low values of ,this cloning machine gives eve more information than the universal cloning machine of strategy a. from the perspective of cloning machines this fact is not surprising .the fidelity achievable in the clones depends always on the set of allowed input states . as more information about the input setis known , better the input states can be cloned .the phase covariant cloning machine exploits the fact that the input states are _ equatorial _ qubits .that is , the component of their bloch vector is zero .the cloning machine of strategy a , however , is designed to clone _ any _ input qubit with the same fidelity .   in this sectionwe obtain the relationship between the disturbance , which appears in fig .[ res ] , and the overall observed error rate which is measured in the experiment .this relationship can be established by an analysis of the pns attack alone , which includes here the optimal eavesdropping on single photon signals , as before .the probability that a multi - photon signal undergoes the pnsattack and then is detected by bob s detection device is given by ,\ ] ] where is the photon - number distribution of the signal states emitted by alice , given by eq .( [ sigalice ] ) . on the other hand , the expected click rate at bob s side has the form where is the transmission efficiency of the quantum channel .the total loss in of the quantum channel is given by . from and one can obtain the probability that single - photon signals contribute to the raw key .it is given by the multi - photon pulses does not introduce any error in the sifted key .this means that after substituting the values of and into eq .( [ ed ] ) we finally obtain this result is illustrated in fig .[ figed ] for some typical values of and .   versus the disturbance as a function of the loss in db of the quantum channel .the mean photon number is and is in this example.[figed ] ]  when the loss in the channel increases , the observed error rate for each value of , as expected , decreases .the regime that we consider here corresponds with a value of the loss in the channel such as eve can perform a pns attack on all the multi - photon pulses , but .the first condition requires which provides an upper bound for the transmission efficiency of the channel }{\mu\eta_{det}}\ ] ] the second constraint implies a lower bound for }{\mu\eta_{det}}\ ] ] the meaning of this condition is to guarantee that we are not in a regimen where the pns attack is still eve s optimal strategy . when and , we obtain for the lower and upper bound and , respectively .the process based on cloning machines become more powerful than the pns process for lower values when the loss is high .the typical value of the observed error rate in the experiments is around if we consider only errors in the quantum channel .therefore , it is interesting to see how the value of the disturbance change as a function of the loss in the channel when we impose to be .this is illustrated in fig .[ figuno ] .we find , in combination with fig .[ res ] that for losses higher than the pns attack is clearly no longer optimal for these typical parameters .   as a function of the loss in db of the quantum channel for a fix value of the observed error rate . the mean photon number is and is .[ figuno ] ]  it is worth to point out that when the losses in the channel are small , but still inside the interval imposed by eq .( [ int1 ] ) and eq .( [ int2 ] ) , the eavesdropping attack which includes the cloning machine can be made more powerful than the pns attack even for a lower value of than the one given in eq .( [ ed2 ] ) .the reason is that , although in this situation eve can not discard too many single - photon pulses , she can re - distribute the errors from the single - photon processes into the two - photon processes . to this end, she increases her intrusion via the cloning attack on the two - photon signals , while she reduces her intrusion on the single - photon signals .the exploitation of this effect is beyond the scope of this article .in the previous sections we consider the case of the standard bb protocol , where only the raw bit rate is monitored . here , we briefly discuss the case of the extended version of the protocol , where alice and bob uses the full statistics at their disposal to detect eve .  in this scenario , it is straightforward to see that the processes of fig .[ pns ] and fig .[ clo ] are not equivalent .the pnsprocess of fig .[ pns ] can never produce a double click in bob s detectors , while the process of fig .[ clo ] presents always a non - vanishing probability of producing a double click , independently of the basis that bob uses for his measurement . in fact , the pns attack never produces a double click event when bob chooses for his measurement the same basis that alice used when preparing the signals .this means that , in principle , alice and bob might employ this information to discard any eavesdropping strategy that includes the cloning process .however , if we consider a real implementation of the protocol , then the situation is not so simple .the reason is that the quantum channel is not just lossy , but presents a misalignment that introduces errors in the signals . as a resultwe have that any multi - photon signal has a non - zero probability of providing a double click , independently of the basis used by bob in his measurement .this means that eve must adapt the pns attack such that it reproduces the expected misalignment in the channel .otherwise her attack would be detected .in particular , eve have to introduce some noise in the signals that are sent to bob . in the case of single - photon pulses , this can be achieved by sending the signals through a depolarizing channel of appropriate parameters .this is precisely the effect of the symmetric oa introduced in sec .[ sec : pns ] .therefore , in these pulses , eve can always get information from this extra noise .the multi - photon pulses , however , gives already eve full information about the key , and she can not exploit the noise she needs to introduce to get more information from the single - photon pulses .  the eavesdropping attack which includes the cloning machine can also be adapted such that it reproduces the statistics that is expected from a realistic channel .however , the question whether it remains more powerful than the pns attack , in this scenario , requires a deeper analysis .if we consider the situation where eve performs a pns attack on the pulses that contain more than two photons , and the misalignment in the channel is sufficiently strong for eve to get full information from the cloning process , then eve can obtain as much information as with the pns attack .if the misalignment is smaller , it seems that the strategy that combines the cloning process with the pns attack on the remaining multi - photon pulses can not be more powerful than the pns attack .the reason is that the adapted version of the pns attack still contains processes that do not produce any double click in bob s detectors , independently of the basis that bob uses for his measurement . to compensate this effect, eve has to subtract more than one photon from the multi - photon pulses , such that she creates processes that do not produce double clicks .but now the effectiveness of the complete strategy decreases , since the probability that the signals which provide eve full information about the key ( multi - photon pulses ) contribute to the raw key decreases .although this fact constitutes a handicap of the eavesdropping strategy that combines the cloning process of fig .[ clo ] with the pns attack on the rest of the pulses , it might be of relative importance in practice .double clicks are rare events that have a very small probability to occur , and the statistical fluctuations in the channel , together with the effect of dark counts in bob s detectors , makes the detection of eve s presence not easy . moreover, eve might also use a mixed strategy that combines probabilistically the pns process of fig .[ pns ] and the cloning process of fig .[ clo ] , such as her attack remains still more powerful than the pns attack , while making her detection even more difficult .in an ideal quantum optical implementation of qkd , the sender use single photons to encode the information he transmits . however , current experiments are not based on single photon sources , but they are usually based on weak coherent pulses ( wcp ) with a low average photon number .also the detectors employed by the receiver are not perfect , but have a low detection efficiency and are noisy .this fact , together with the loss in the quantum channel , limits the distances that can be covered by these methods . in this scenario , it is tempting to assume that the loss in the detectors can not be changed by eve in order to increase the covered distance , while the pns attack , like in the case of a conservative definition of security , still constitutes eve s optimal strategy . in this paperwe disprove this belief for the case of the standard bb protocol , where only the raw bit rate ( before the key distillation phase ) is monitored .we constructed two specific eavesdropping strategies which include processes that do not subtract photons from the pulses , and that are more powerful than the pns attack for some relevant regimes of the observed error rate .these strategies are based on the use of cloning machines . a complete analysis of eve s optimal attack in this situation is still missing . in the extended version of the bb protocol , where alice and bob consider the full statistics at their disposal , the situation is not as straightforward , and a deeper security analysis of this scenario is required .the authors wish to thank a. acn , a. dolinska , p. van loock and ph .raynal for very useful discussions , and especially k. tamaki for his critical discussion of this article .we also like to thank gerald gilbert for the clarifying discussion of his investigations .this work is supported by the dfg under the emmy noether programme and the network of competence qip of the state of bavaria .an investigation of the scenario where eve can not improve bob s detectors has been undertaken in .we believe that this investigation is incomplete so far .the authors claim that they have performed an thorough analysis of the situation where bob s detection efficiency can not manipulated by alice , together with the restriction of an individual attack. the authors define individual attack radically different from the usual terminology that is used in the analysis of quantum key distribution : they refer to attacks that act on _ photons _ individually , rather than_ signals_. in practice this means that the authors consider optimal attacks on single photon signals , which can be implemented by attaching a probe to a single photon , on the other hand , they disallow attaching a probe to a two - photon signal , since that would mean to interact with the two photons coherently. in fact , they state that such manipulations would be possible only when quantum computers become available .  of course , it is not unusual to start with some assumptions about restrictions of eavesdropping strategies .for example , investigation of an individual attack scenario ( now referring to the standard definition that relates to the signal pulses ) has proven to be very powerful since the analysis can be performed easily and the resulting parameters for privacy amplification and the secure key rate corresponds roughly to the subsequently derived values that assure security again all attacks , including coherent attacks on all signals . from this experiencethe individual attack derives its role as a first step investigation of the performance and security analysis of qkd schemes .the relationship between individual and coherent attacks has been strengthened by the results of wang .  in the scenario considered by can not see an equivalent role . another motivation to investigate restricted scenarios might be the technological challenge of different eavesdropping strategies. however , the technological difference between attaching a probe to a single photon as compared to attaching a probe to a two - photon signal is not evident .clearly , these questions do not invalidate the obtained results .however , in our point of view , the authors of are inconsistent in describing the restrictions of their considered eavesdropping attacks .they claim that as a consequence of their restriction they need to consider only three types of attacks :  1 .direct attacks in which eve can unambiguously determine the signal state by a direct measurement of the signal .this corresponds to the unambiguous state discrimination attack in .( requires at least photons . ) 2 .indirect attacks , which is precisely the pns attack that extract one photon from the signal .( requires at least photons . )3 .  combined attacks which performs the indirect and the direct attack .( requires therefore at least photons , and is in the analysis later on shown to be an inferior attack . )it is left open how these categories emerge and why this should be a complete description . as first point of criticism notethat the authors apply for the direct attack the results of that provide the performance for optimal unambiguous state discrimination measurements .can we implement this attack by acting individually on photons ?a second point of criticism is that to perform these attacks eve needs to know the number of photons in each pulse .if one thinks of photons as distinguishable particles in a pulse , this might be easy . in a proper quantum optical description ,however , these type of counting mechanisms , which do not disturb the signal , will require in all experience the same level of interaction between a probe and the total signal , as does a general eavesdropping attack on the signal .so far , we pointed at inconsistencies that do not endanger the security statement derived in .these attacks overestimate eve s capabilities as compared to the initial restriction that require individual attacks on photons . however , the categorization by gilbert and hamrick left out possible attacks .those attacks still operate on individual photons only . as an examplelet us consider a two - photon pulse . according to the only attack we need to consider is the pnsattack .instead , let eve perform a direct measurement on the photons , for example in the sense of a minimum - error measurement. of course , the error will be non - zero , since on a two - photon state in the bb84 polarizations one can not perform successfully unambiguous state discrimination .however , optimal eavesdropping on single - photon signals also results in some errors .another attack would be to separate the two photons .then one can attack probes to both photons and try to combine the photons again in bob s detection apparatus , e.g. by sending them to bob in close sequence so that bob does not notice that they have been separated .moreover , similar attacks are omitted for higher photon numbers .these examples question the completeness of the proposed classification of eavesdropping attacks in .note that after receiving an advance copy of this manuscript the authors of revised their work , acknowledging the incompleteness of their analysis .this means that we have to treat the classification as an assumption that only those three classes are of relevance .this includes the assumption that for the two - photon pulse the pns attack is optimal in their restricted scenario . as a consequence , a security claim for an experimental implementation of qkdshould not be based on this analysis , as done in , since it underestimates eve s ability .nevertheless , within the three investigated classes of eavesdropping attacks , gilbert and hamrick have been able to show that the unambiguous state discrimination attack can be more effective for eve than the photon - number splitting attack for signals containing three or more photons .in this appendix we provide the exact expressions for the coefficients , and that are introduced in eq .( [ coefa ] ) , as a function of the angle .     99 s. wiesner , sigact news * 15 * , 78 ( 1983 ) . c. h. bennett and g. brassard , proc .conference on computers , systems and signal processing , bangalore , india ( ieee press , new york , 1984 ) , 175 .g. s. vernam , trans . of the aiee* 45 * , 295 ( 1926 ) .m. curty , m. lewenstein and n. ltkenhaus , quant - ph/0307151 .n. ltkenhaus , applied phys .b * 69 * , 395 ( 1999 ) .d. mayers , j. of acm * 48 * , 351 ( 2001 ) .p. w. shor and j. preskill , phys .lett . * 85 * , 441 ( 2000 ) .e. biham , m. boyer , p. o. boykin , t. mor and v. roychowdhury , in _ proceedings of the thirty second annual acm symposium on theory of computing _ , new york , usa ( acm press , new york , 2000 ) , 715 ( 2000 ) .lo and h. f. chau , science * 283 * , 2050 ( 1999 ) . c. h. bennett , f. bessette , g. brassard , l. salvail and j. smolin , j. cryptology * 5 * , 3 ( 1992 ) .c. marand and p. d. townsend , opt . lett . * 20 * , 1695 ( 1995 ) .d. s. bethune , m. navarro and w. p. risk , applied opt .lp * 41 * , 1640 ( 2002 ) .r. j. hughes , j. e. nordholt , d. derkacs and c. g. peterson , new j. phys . * 4 * , 43 ( 2002 ) .d. stucki , n. gisin , o. guinnard , g. ribordy and h. zbinden , new j. phys ., * 4 * 41 ( 2002 ) .b. huttner , n. imoto , n. gisin and t. mor , phys .a * 51 * , 1863 ( 1995 ) .g. brassard , n. ltkenhaus , t. mor and b. c. sanders , phys .lett . * 85 * , 1330 ( 2000 ) .n. ltkenhaus , phys . rev .a * 61 * , 052304 ( 2000 ) .h. inamori , n. ltkenhaus and d. mayers , quant - ph/0107017 .w. k. wootters and w. h. zurek , nature * 299 * , 802 ( 1982 ) .note that by a public communication protocol that differs from the bb84 announcement of the bases , eve no longer obtains full information about those bits .see and v. scarani , a. ac ' in , n. gisin , and g. ribordy , quant - ph/0211131 .m. duek , m. jahma , and n. ltkenhaus , phys .a * 62 * 022306 ( 2000 ) .g. gilbert and m. hamrick , quant - ph/0009027 .n. ltkenhaus , phys .a * 59 * , 3301 ( 1999 ) .b. yurke , phys .a * 32 * , 311 ( 1985 ) .alternatively , one can also characterize bob s detection setup using only one povm which already includes the action of the polarization shifter . for instance, any attempt by eve to shift the wavelength of the signals into a region of a higher detection efficiency of bob s detectors can be blocked by the use of a narrow frequency filter .n. ltkenhaus and m. jahma , new j. of phys .* 4 * , 44 ( 2002 ) . x .-wang , quant - ph/0110089 . c. a. fuchs , n. gisin , r. b. griffiths , c .- s .niu and a. peres , phys .rev a * 56 * , 1163 ( 1997 ) .d. bruss , m. cinchetti , g. m. dariano and c. macchiavello , phys .a * 62 * , 012302 ( 2000 ) . a. ac ' in , n. gisin and v. scarani , quant - ph/0302037 .n. j. cerf , phys .lett . * 84 * , 4497 ( 2000 ) . c .- s .niu and r. b. griffiths , phys .a * 58 * , 4377 ( 1998 ) .when alice uses the basis the situation is equivalent .l. b. levitin , in _ quantum communications and measurements _ , v. p. belavkin , o. hirota and r. l. hudson , eds . , ( plenum press , new york , 1995 ) , p. 439 . c. a. fuchs , proc . of the fourth international conference on quantum communication ,measurement , and computing , 1998 , ( kluwer academic , new york , 2000 ) , p. 11 . if alice uses the other basis , the situation is again equivalent .but note that for this cloning machine one has to change the notation , and use the states , instead of , , where and . n. gisin , g. ribordy , w. tittel and h. zbinden , rev . mod . phys . * 74 * , 145 ( 2002 ) . g. gilbert and m. hamrick , algorithmica * 34 * , 314 ( 2002 ) ; quant - ph/0106033 .

Abstract: quantum key distribution with the bb84 protocol has been shown to be unconditionally secure even using weak coherent pulses instead of single - photon signals . the distances that can be covered by these methods are limited due to the loss in the quantum channel ( e.g. loss in the optical fiber ) and in the single - photon counters of the receivers . one can argue that the loss in the detectors can not be changed by an eavesdropper in order to increase the covered distance . here we show that the security analysis of this scenario is not as easy as is commonly assumed , since already two - photon processes allow eavesdropping strategies that outperform the known photon - number splitting attack . for this reason there is , so far , no satisfactory security analysis available in the framework of individual attacks .


diode lasers have come to play a fundamental role in experimental physics .their low cost , efficiency , and small size , among other things , have made them a standard part of many experiments . while many turn - key extended - cavity diode laser ( ecdl ) systems are presently available , they are typically much more expensive than the cost of the parts required to build a custom system , and they are not easily serviced when problems arise .also , it is difficult to modify them for special purposes , and they do not provide the extreme stability and narrow line - widths needed for many applications . as such , many researchers continue to design and build custom ecdls . in this paperwe discuss the design and performance of a precision current driver , one of the key components in any ecdl system .the laser current controller we present was designed to meet the specifications needed for the master laser of an optical frequency standard , one of the most demanding ecdl applications .optical frequency standards require a laser with a line - width of a few khz or less .to achieve this , these lasers need to have the current stabilized to within a few tens of nano - amperes rms over relevant time scales , and typically must be locked to a high - finesse cavity . to simplify the lock to the cavity , it is desirable for the laser current driver to have a modulation input with enough bandwidth to allow feedback through the current driver .this reduces complexity and reduces the chance of destroying the diode when compared to applying feedback directly to the diode .our current driver has several unique design features .the noise and stability of our current driver are as good or better than any laser current driver we are aware of .the current driver also has a modulation input with a 3 db roll off above 20 mhz .this is much larger than any of the commercial drivers we have seen , and it maintains its high stability and low noise even while the high - bandwidth modulation circuit is active and connected to the driver s output .the driver is also relatively small in size and features remote programming , allowing the driver to be placed near the laser to remove the negative effects of long cables .this design is inexpensive enough to be used as a reliable lab standard or in student lab experiments .our design builds upon the hall - libbrecht current driver , a precision current controller design which has been used extensively in atomic physics labs .we made several design improvements and updated components .the most radical improvement is the use of a precision digital - to - analog converter ( dac ) , rather than a potentiometer , to program the current set point .this eliminates a major source of noise and instability .this also makes it possible to program the current set point remotely , allowing the current driver to be placed in close proximity to the laser diode .this can significantly reduce current noise at the laser , and will lower phase shifts when using the modulation input .the use of a dac also allows much more accurate control of the current set point , and makes it possible to return to a previous current set point with extremely high precision .our driver utilizes surface mount technology , allowing us to use the most modern components and reducing inductance and noise pick - up from longer pin leads .in addition to explaining our improvements to the hall - libbrecht circuit we present a much more detailed description and a more comprehensive and careful performance analysis of our design than was given in .this discussion illuminates both our improvements as well as subtleties present in the hall - libbrecht driver which were not discussed in .in figure [ cap : schematic ] we have broken the schematic of the circuit into several sections , which we will address separately .as presented , the circuit is configured to drive cathode grounded lasers , but we have also constructed anode - grounded supplies with a few straightforward modifications .although not indicated on the schematic , each op - amp is powered by the filtered voltages and , with a pair of capacitors connected between power and ground near each power pin of each op - amp .surface - mount ceramic capacitors and in value are used for the op - amps in the `` modulation input '' sub - circuit , the quad op - amp in the `` differential buffers '' sub - circuit , and the op - amp used for the differential amplifier in the `` current monitor '' sub - circuit . for the power lines for the other three op - amps a ceramic capacitor and a electrolytic capacitorthe circuit is laid out on a 6.125 x 4.25 inch printed circuit board ( pcb ) that is electrically isolated from the cast aluminum box it is mounted in .all of the resistors , with the exception of the ones specifically discussed in the sections below , are standard 1/8 watt thick - film chip resistors with a temperature coefficient of ppm/ in a 1206 surface mount package .all of the non - polar capacitors are ceramic capacitors in a 1206 surface mount package .other components are discussed later .the heart of the current driver is the `` current regulation '' sub - circuit .this sub - circuit is a fairly standard current source design , but components have been chosen very carefully for optimal performance . in this sub - circuit a regulated voltage , labeled in fig.[cap : schematic ] , is used to drive current through a precision sense resistor , labeled , to the output of the driver .the amount of current flowing out to the laser diode can be determined from the voltage drop across .one side of the sense resistor is connected to the inverting input of an ad8671 precision op - amp , and the output of the op - amp is connected to the gate of an nds0605 mosfet .negative feedback in this circuit causes the mosfet gate voltage to change , modifying the current flowing through the mosfet , until the voltage at the output of the sense resistor is equal to the set voltage , , applied to the non - inverting input of the op - amp . in this waythe op - amp forces the current to remain fixed at a value of regardless of the load impedance . due to the mosfet s gate capacitance ,the resistors and capacitor on the line between the op - amp and the mosfet are needed to reduce the bandwidth of the circuit and prevent oscillations . as was done in ,an inductor is placed after the mosfet to limit high - frequency current noise and to de - couple the current regulation electronics from any intentionally - applied current modulation .the sense resistor is a key component which needs to be considered carefully stability of the current driver can be no better than the stability of this resistor .the value for should be chosen to provide a large enough voltage drop for accurate current regulation at the target current , but small enough such that the voltage drop across this resistor does not reduce the compliance voltage of the driver below what is necessary to power the laser diode .when driving a red diode laser to a maximum current of 100 ma , we used a pair of 100 ohm vishay smr3d precision resistors in parallel for a total resistance of 50 ohms .these parts are surface - mount components for reduced noise pickup and have a low temperature coefficient of just 2 ppm/ . for higher / lower current applications , smaller/ larger resistance values could be used . in applications that do nt require the highest stability , resistors with higher temperature coefficients may be used .the op - amp we used was the ad8671 from analog devices .similar to the lt1028 used in the original hall design , this op - amp has voltage noise levels comparable to the johnson noise of a 50 ohm resistor , the ad8671 having slightly better current noise and slightly poorer voltage noise than the lt1028 . in practice , as well as in numerical spice models, we found that the ad8671 was less prone to oscillate in this circuit than the lt1028 , requiring less severe bandwidth limiting to make the circuit stable .the ad8671 chip also has a _ much _ higher input impedance ( both common and differential mode ) than the lt1028 with comparable input capacitance. this property limits current drift due to thermal or other changes in the leakage current into the op - amp inputs .in addition , the ad8671 does not require any sort of trim or compensation to achieve low bias and low overshoot in our application .although the ad8671 has a somewhat smaller gain - bandwidth product ( gbp ) than the lt1028 used in the original hall design ( 10 mhz vs 50 mhz ) , the filtering capacitors and resistors needed to stabilize the circuit effectively remove any additional bandwidth that would be gained by using the lt1028 .furthermore , the inherent low noise of the circuit and the use of the de - coupling inductor makes higher bandwidth in this part of the circuit unnecessary .it is worth noting here that we have used the ad8671 op - amps throughout most of the circuit .the hall design used lt1028 op - amps only in two " critical " locations , with different op - amps being used in the modulation input circuit and the monitoring circuit . since both of these circuits connect directly to the current output , it made sense to us to take advantage of the low noise and high input impedance of the ad8671 .though more expensive than many lower performance op - amps , they are still a relatively inexpensive part of this circuit .  for the inductorwe choose a api delevan series s1210 ferrite core shielded surface - mount inductor .the nds0605 mosfet was chosen for its low on - state resistance and a relatively large current carrying capacity in a small form factor .it also has a small input capacitance , small switching delays , and low gate - body leakage .   the set - point voltage , , which is applied to the non - inverting input of the op - amp in the `` current regulation '' sub - circuit is generated by a precision digital to analog converter ( dac ) .using a precision dac rather than a manual potentiometer removes a significant source of noise and drift in the circuit .in addition , it affords much greater accuracy and repeatability .we did an extensive search to find a dac which would optimize long - term stability .the chip we settled on is the 16-bit ad5541 .this chip comes in several performance grades .all grades have the same stability , temperature coefficient , and repeatability , but differ in the absolute accuracy of the output .we chose the `` j '' grade , mainly because it was carried by our regular electronics supplier , but lower cost / lower performance and higher cost / higher performance grades are also available .for applications where the noise and stability of the current driver are not a great concern , we have included pads on the pcb where a traditional potentiometer can be used in place of the dac .the ad5541 chip is specified to have a zero - code temperature coefficient of 0.05 ppm/ and a gain error temperature coefficient of 0.1 ppm/ , both of which are significantly less than the temperature coefficient of the sense resistors discussed in sec .[ sub : currentregulation ] .the accuracy of the `` j '' grade chip is guaranteed to be at least as good as times the least significant bit ( lsb ) , or 2 parts in 65,535 . for our test setupwe used a 50 ohm sense resistor resulting in a maximum current output of 100 mv .this corresponds to an accuracy of , similar to its measured accuracy discussed in sec .[ sub : accuracyandrepeatability ] .  the output of the dac can be set anywhere from the voltage of its analog ground pin ( labeled in fig . [cap : schematic ] ) up to the voltage applied to its reference input . as was done with the analog potentiometer used in , the ground pin of the dac in our circuitis not connected directly to board ground .there are two reasons why this is necessary .the first is that the output current of the circuit does not depend on the set point voltage relative to board ground , but relative to . as such ,if the voltage drifts relative to board ground , we want to drift with it such that remains constant .the second is that the dac requires the voltage on its reference to be no greater than above the voltage of its analog ground pin . in order to adjust the output current of the device allthey way down to zero 0 , it must be possible to make at least as large as .this imposes a lower limit on the voltage applied to the analog ground pin .  in our circuit applied directly to the dac s reference pin , and a voltage of ( labeled in fig.[cap : schematic ] ) is applied to the dac s ground pin , giving the current driver an output current which can range from 0 to .the voltage , is generated using an lm399 zener diode connected to , which creates an extremely stable 6.95 v drop .the resistors and make a voltage divider to lower this 6.95 v difference closer to the required 5 volts .we then buffer the voltage with an op - amp .any drift in the ratio of and will result in a drift in the current set point , but there is much less sensitivity to thermal drift in these resistors than in the sense resistor because only the ratio , and not the absolute resistance matters .both resistors are made of the same material and are in close proximity such that they will have similar thermal drifts .for these two resistors we use vishay tnpw0805 precision resistors which have an accuracy tolerance of 0.1% and a thermal coefficient of 25 ppm/ .the resistor in this sub - circuit is a vishay tnpw04027501dt9 precision , low - temperature coefficient resistor .we expect , however , that using a standard resistor in this location would not affect the performance of the current driver .similar to what was done in , a pair of low - pass filters reduce high frequency noise in . because the output current of the circuit depends not on the difference of from board ground , but on the difference between and , these filters ( as well as the low pass filter in the current regulation sub - circuit ) are tied to rather than ground .the `` power filter '' sub - circuit generates stable , filtered voltages for the rest of the circuit .this sub - circuit is nearly identical to the one in .we have added some additional filter capacitors , changed some component values to optimize performance , and added two protection diodes between the three pins of the lm317 voltage regulator , as recommended by the manufacturer .we have also used surface - mount components wherever possible to reduce noise and stray inductance .  in this part of the circuit ,power lines are first filtered by a series of capacitors and inductors to provide the positive and negative voltages and which are used to power all of the op - amps in the circuit .the filtered power is then regulated with the lm317 adjustable voltage regulator to generate the extremely stable voltage labeled .this voltage is used to drive current to the laser diode in the `` current regulation '' sub - circuit and to generate the set - point voltage in the `` digital set point '' sub - circuit as described above .the transistor below the lm317 is part of an enable / slow start mechanism discussed in .this soft - start feature helps protect the laser diode from damage , and in the case of a power failure it also allows the digital controller time to reset the current set point before turns fully on after power is restored .the regulated output level of the lm317 is nominally equal to when the _ enable _ switch is grounded , the current driver is `` disabled . '' in this case the resistor is placed in parallel with both the impedance of the transistor and with the series resistance of the resistor labeled and the 1 k resistor near the switch .this effectively reduces the value of in eq.[eq : vreg ] . with a proper choice of resistor values it is possible to make insufficiently large to forward bias the laser diode when the switch is in this position .  when the switch is opened , if not for the 1 capacitor , current would immediately cease to flow through .this would cause the voltage at the gate of the transistor to suddenly equal the voltage at the emitter of the transistor , shutting off current through the transistor as well , restoring the voltage regulator to its full output value as if the soft - start circuitry was not present .the presence of the 1 capacitor prevents the current flowing through from abruptly stopping .instead , the current continues to flow as the capacitor charges , and the voltage rises on the gate of the transistor slowly .this causes the voltage regulator s output to gradually increase to its final value .because of the non - linear nature of the transistor , the start time is considerably larger than just the time constant of the resistor and the 1 capacitor . using a value of and a mmbt2907a transistor in our setup, we measured a start - up time constant of 11.5 ms .the capacitors are low - leakage radial aluminum electrolytic capacitors , and the capacitors are surface - mount tantalum capacitors . while developing this circuit the 1.5 resistor in the upper right - hand corner of the schematic overheated , probably due to increased current draw when our lt1028 op - amps oscillated or saturated . while diagnosing this problem , we installed a vishay wsc-1 series 1 watt surface - mount wire - wound power resistor .we have experienced no such difficulties since we began using ad8671 op - amps , and have since replaced this resistor with a standard 1/8 watt metal film resistor . although we do nt expect the choice of resistor to affect performance, one should note that the wire - wound resistor was still installed while taking the data presented in sec .[ sec : results ] .the `` modulation input '' sub - circuit is the same one used and described in .the key components in this circuit are the upper op - amp and the two resistors labeled and ( the lower op - amp is part of a balancing circuit that we describe later ) .the non - inverting input of the upper op - amp is connected directly to the output of the current driver , which is at a voltage .the op - amp is wired to generate a gain of , such that it generates a voltage of at its output .the output of the op - amp is connected to the current driver s output through a resistor , labeled in fig .[ cap : schematic ] .the voltage drop across this resistor is equal to , and so the current flowing through this resistor is . a second identical resistor , labeled in fig .[ cap : schematic ] connects the output of the laser driver to the modulation input . if a voltage is applied to this input , the current flowing through this resistor is .the current which is added to the output of the laser driver is just the difference in these two currents , .this injected current inevitably results in a change of the output voltage of the current driver , but the op - amp constantly corrects its output to keep it equal to as changes .in addition to the modulation current , an additional bias current will be injected by this sub - circuit if the resistors are not equally matched or if the gains in the two amplifiers are not correct .for example , if and differ by some small fraction , the modulation current injected into the laser will be the bias term ( the second term in the above equation ) will cause current measurements based on the voltage drop across to be inaccurate . and if resistances change in time , due to thermal effects for example , this bias current will change , affecting the stability of the current driver . as such , these resistors need to be well matched and have a fairly low temperature coefficient .all of the resistors in the `` modulation input '' sub - circuit except for the impedance matching resistor on the input are vishay tnpw0805 surface - mount thin - film chip resistors with a resistance tolerance of and a temperature coefficient of 25 ppm/. with this accuracy , the bias term in eq.[eq : imodb ] should be no larger than when driving a typical red laser diode ( for which ) . because the resistors and have the same composition and are physically close to each other , such that they should be at similar temperatures , we would nt expect temperature to have a large effect on the matching of the two resistors .accordingly , the fact that these resistors have a much larger temperature coefficient than the current sense resistor discussed in sec .[ sub : currentregulation ] should have no significant effect on the stability of the current driver .the lower op - amp in this sub - circuit forces to 0 v when nothing is connected to the modulation input ( alternatively , this part of the circuit could be left off , and the modulation input could be shorted to ground or the `` modulation input '' sub - circuit completely disconnected from the current output when not in use ) .this op - amp generates a voltage of which is connected to the modulation input through two more resistors , labeled and .it is easy to see that , if these are perfectly matched to the other two resistors , the voltage at the modulation input will float to zero when nothing is connected to it . if they are not exactly balanced , an additional bias current will be injected into the output of the current driver when the modulation input is floating .given the precision of these resistors , we would expect this bias to be on the order of and we would nt expect that temperature drift should be an issue .although not discussed in , we have come to realize that this design allows modulation speeds greater than the bandwidth of the op - amps . at very high modulation frequenciesone would expect that the op - amp would not be able to follow the changes in induced by the oscillating modulation current . in this limit , if we assume that the current driver is connected to a load with impedance , that the output of the upper op - amp in this sub - circuit settles to twice the time - averaged output voltage , and that , eq . [ eq : imodb ] becomes if , then this equation is nearly the same as the low frequency limit in eq .[ eq : imodb ] .if the op - amp is modeled as an ideal op - amp with a low - pass filter with a time constant of on its input , it can easily be shown that changes monotonically from it s low - frequency value of to this high frequency limit as the modulation frequency increases . using this model ,the phase shift of goes to zero in both the low- and high - frequency limits , with the largest phase shift , occurring at a frequency of , being equal to once again , if , the maximum phase shift in this simplified model will be very small .the `` current monitor '' sub - circuit generates a voltage proportional to the output current of the driver so that the output can be monitored .the first op - amp in this sub - circuit buffers the voltage at the low - voltage side of the sense resistor with a high input impedance op - amp .this decouples the current monitor output from the current regulation sub - circuit so that measurements can be made without significantly affecting the output of the current driver .the second op - amp subtracts this voltage from to produce a voltage relative to the board s ground equal to .because the output of the current monitor sub - circuit is referenced to board ground , attaching a meter to this output could potentially introduce ground noise .op - amp `` c '' in the `` differential buffers '' sub - circuit eliminates this problem , decoupling the meter s ground from the circuit .similarly , the other three op - amps in this sub - circuit decouple the current driver s ground from the grounds of the three digital input lines which program the dac ( labeled , , and on fig .[ cap : schematic ] ) .these three op - amps are also used to re - reference the digital inputs to the ground pin of the dac ( which is not board ground ) .the circuit can be controlled by any device capable of generating a 16-bit serial ttl signal as well as a ttl clock and enable signal . to collect some of the data presented in this paper, we used a digital output card on a computer for this purpose .we have also developed a dedicated digital control circuit based on a pic18f4550 microprocessor .this is a 40-pin usb programmable microprocessor with 35 input / output pins .it is programmed to read in the signal from a digital encoder , and subsequently output a value to the dac . the micro - controller maintains stability and eliminates noise pickup by only outputting values to the dac when the user changes the current set point .the pic18f4550 also includes a built - in analog to digital converter ( adc ) which we use to read back and display .   to protect the laser diode, it is often desirable to limit the maximum output current of the driver . an easy way to dothis is by changing the resistance of the sense resistor , .because the dac can only generate a set - point voltage as low as , the maximum output of the current driver will be .however , because the modulation current from the modulation input sub - circuit does not pass through the sense resistor , it is possible to exceed this maximum current by applying a large enough voltage to the modulation input .  as discussed in , another way to limitthe output current is by reducing the voltage .the compliance voltage of the current driver is determined by the impedance of the mosfet and the inductor in the current regulation sub - circuit and the value of .if is the maximum current for the laser diode , and if , , and are the voltage drops across the laser diode , the mosfet , and the inductor ( respectively ) at this current , then if is set to the current driver output will not be able to exceed .the above voltage drops can be estimated from the specifications of the components , or the ideal value for can be determined empirically by setting to its lowest value and gradually increasing until the output current equals .because an inductor decouples the modulation input sub - circuit from the current regulation sub - circuit , a high - frequency signal on the modulation input could cause the current driver to output more than . at low modulation frequenciesthe output current will never exceed , regardless of the amplitude of the voltage applied to the modulation input .the maximum current that can be generated without risking damage to the current driver is 127 ma ( limited by the s1210 inductor ) . for higher current applications , we designed our circuit boards to accommodate both surface - mount and leaded components at a few critical points , allowing a wide range of components to be used .for example , we have realized a high current driver for a tapered amplifier by replacing the following components : the s1210 inductor ( rated for 127 ma ) as well as the two inductors in the " power filter "sub - circuit were replaced with leaded 125 inductors ; the nds0605 mosfet ( with a maximum rated current of 180 ma ) was replaced with an irf9z14 mosfet in a to-220 package which can output 6.7 a ; the surface - mount lm317 regulator ( with a current limit of 1 a ) was replaced with an lm1084-adj regulator in a to-220 package , which is capable of 5 a ; the 1.5 ohm resistor in the " voltage regulation " sub - circuit was replaced with a leaded 1 , 5 w power resistor ; and the sense resistor was realized with two 5 , 10 w leaded power resistors placed in parallel for a combined resistance of 2.5 . due to the larger gate capacitance of the mosfet , we also had to replace the capacitor in the " current regulation " sub - circuit with a 100 nf capacitor to keep the circuit from oscillating .although these substitutions increase the noise and reduce the stability of the circuit , they allow an output of 2 a ( limited intentionally by our choice of to match the capacity of our tapered amplifier ) .we did a series of tests to characterize the performance of our current driver . for each of these tests ,the sense resistor consisted of two 100 resistors in parallel , for a sense resistance of .to characterize both the long - term stability and the high frequency noise of the current driver , we used it to drive a laser diode in series with a 2.5 precision resistor with a current of 74.5 ma .the current was determined by measuring the voltage drop across the resistor with a 6.5 digit multimeter . from this data we extracted the spectral density of the current noise , shown in figure [ cap : noise](a ) .the intrinsic noise of our measuring apparatus was determined by taking a similar measurement but with the current driver disconnected from the laser/2.5 resistor . for most of the spectrum ,our driver is at or near the detection limit of our equipment , and at all frequencies it is at least an order of magnitude smaller than the measurements presented in .  ]the long term drift in the output current is shown in fig .[ cap : noise](b ) .this data was collected in the same manner as the data used to produce fig .[ cap : noise](a ) , but with a current set point of 74.8 ma . during thismeasurement thermistors were used to measure the temperature at the location of the current sense resistor ( labeled in fig.[cap : schematic ] ) , the dac , and the box that houses the current driver .the mean temperatures at these locations was 40.6 , 40.5 , and 22.9 respectively .as seen in fig .[ cap : noise](c ) , the drift in current and temperature are highly correlated .this data suggests a temperature coefficient of , precisely what would be expected given the temperature coefficient of the sense resistor .   the use of a digital set point results in very good accuracy , linearity , and repeatability .we tested the linearity of the set point by connecting our current driver to a laser diode in series with a 10 precision resistor .we then determined the actual current produced by the driver at various digital set points by measuring the voltage drop across the resistor .the deviation of the actual current from a best - fit line is shown in fig .[ cap : accuracy](a ) .the linearity is within the lsb specification for the dac chip we used . as discussed in sec .[ sub : currentsetpoint ] , similar chips with higher or lower linearity are available .lsb points specified in the dac s data sheet . plot ( b ) show the rms repeatability of the current output , measured as a function of the digitally programmed set point .[ cap : accuracy ] ]  the set point repeatability of our circuit is much better than the absolute accuracy . to test the repeatability ,the current set point was set digitally , and the actual current was measured .the digital set point was then changed momentarily to significantly increase the output current .the digital set point was then reset to its original value , and the actual current was re - measured .this was done ten times at each digital set point , and the rms deviation of those ten measurements was calculated .the rms deviation is plotted as a function of the digital current set point in fig .[ cap : accuracy](b ) . as shown in this figure , the set - point repeatability is about 40 na rms .we measured the modulation response of the current driver by terminating the output with a 50 resistor and comparing the amplitude of a sine wave applied to the modulation input of the current driver input to the amplitude of the sinusoidal voltage modulation generated across this resistor .the output amplitude was then normalized to the expected output of to produce the curve shown in fig .[ cap : temporal](a ) .we found the 3 db point to be over 20 mhz , with an attenuation of less than 4 db out past 100 mhz .we found , however , that the frequency response was severely degraded when driving reactive loads .as discussed in sec .[ sub : currentmodulation ] , this is a much better frequency response than one might naively expect based on the bandwidth of the op - amps .response for the signal applied to the modulation input .the dashed line is the theoretical high frequency response assuming ideal components and transmission lines given by eq .[ eq : hfmodlimit ] . in ( b ) the transient response of the current driver to a total load change of roughly 3%the black line is the current through the load , and the gray line is the impedance of the load .[ cap : temporal ] ]   the transient response of the current driver was studied by using it to drive current through a 10 resistor in series with a vzn1206 mosfet .the transistor gate voltage was suddenly changed to vary the overall impedance of the load seen by the current driver .the current was determined from the voltage across the resistor , and the time - varying impedance was calculated by dividing the voltage drop across the entire transistor - resistor circuit by the current found in the resistor .the measurements are shown in fig .[ cap : temporal](b ) .the distance between the peaks in the two traces is 14 ns , indicating a very fast initial response to the impedance change .after 100 ns , the impedance had settled down to a constant value . by fitting the current signal beyond this time to an exponential, we extracted a time constant for the driver s response to small load changes of 70 ns .our laser diode current driver design , an improvement upon the hall - libbrecht design , has extremely good long - term stability , low noise , fast modulation response , and fast response to load impedance changes .the implementation of a digital set - point circuit and the use of modern surface - mount chips has resulted in more than an order - of - magnitude improvement in noise and stability .it also allows for extremely precise repeatability , high accuracy , and the ability to control the current set point remotely without degrading the performance of the current driver .the device out - performs any commercial device we are aware of , yet is simple and inexpensive enough to be used as a `` lab standard '' for both highly sensitive as well as less sensitive diode laser applications .

Abstract: we present a low - noise , high modulation - bandwidth design for a laser current driver with excellent long term stability . the driver improves upon the commonly - used hall - libbrecht design . the current driver can be operated remotely by way of a micro - processing unit , which controls the current set point digitally . this allows precise repeatability and improved accuracy and stability . it also allows the driver to be placed near the laser for reduced noise and for lower phase lag when using the modulation input . we present the theory of operation for our driver in detail , and give a thorough characterization of its stability , noise , set point accuracy and repeatability , temperature dependence , transient response , and modulation bandwidth .