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Is there a version of Serre's modularity conjecture for projective representations?

Serre's modularity conjecture asserts that a continuous odd irreducible representation $$\overline{\rho} : G_\mathbb{Q} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$$ must be modular, in the sense that there is some eigenform $f$ so that $\overline{\rho} \sim \overline{\rho}_f$, and moreover gives formulae for the weight and level of $f$.

Is there a version for representations $\overline{\rho} : G_\mathbb{Q} \rightarrow \mathrm{PGL}_2 (\mathbb{F}_q)$? If so, what is known?

• Not a complete answer, but you might find certain theorems in chapter 4 of math.leidenuniv.nl/scripties/BosmanPhD.pdf usefull. In particular it contains some (references to) results of when a representation to PGL_2 can be lifted to one of GL_2 after which you can use Serre's modularity conjecture.
– M.D.
Apr 7 '14 at 19:38
• See also homepages.warwick.ac.uk/~maseap/Johan/papers/polynomials.pdf. Serre's modularity conjecture is a theorem (Khare, Wintenberger $2009$). Apr 7 '14 at 20:12

The theorem 1.3 in "Lifting Irreducible Galois Representations" by Fakhruddin, Khare and Patrikis applies to this group, it will show that under some refined conditions $$\bar{\rho}$$ lifts to $$\rho:G_{\mathbb{Q}}\rightarrow \text{PGL}_2(W(\mathbb{F}_q))$$ which is ramified at finitely many primes and potentially semistable at all primes (and of course is irreducible too). I am not sure if such a representation should come from an odd representation $$\tilde{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(W(\mathbb{F}_q))$$, but if it does, then it comes from an eigenform (by the Fontaine-Mazur, theorem of Taylor/Kisin/Emerton).

• Thanks, that's a really helpful answer! Nov 9 '18 at 13:11

View $$\overline{\rho}$$ as a representation

$$\overline{\rho}: G_{\mathbb{Q}} \rightarrow \mathrm{PGL}_2(\overline{\mathbb{F}}_p).$$

The obstruction to lifting this to a Galois representation with values in $$\mathrm{GL}_2(\overline{\mathbb{F}}_p)$$ lies in $$H^2(\mathbb{Q},\overline{\mathbb{F}}^{\times}_p)$$, but this vanishes by a theorem of Tate (here we use the fact that we have extended scalars so that the coefficients are divisible).

Now consider the lifted representation

$$\rho: G_{\mathbb{Q}} \rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p).$$

If the image of complex conjugation in $$\overline{\rho}$$ is non-trivial, then the image in the lift will be non-scalar, and vice versa. Hence Serre's conjecture implies that all such "odd" (complex conjugation is non-trivial) representations come from modular ones.

Some extras: using the fact that the class number of $$\mathbb{Q}$$ is one, one can twist this representation so that it is unramified at all primes where $$\overline{\rho}$$ itself is unramified. In fact, examining Tate's theorem more closely, one can even find a lift such that, given any local lift $$\varrho$$ (a representation of the decomposition group $$D_q$$ of $$\mathbb{Q}$$) of $$\overline{\rho}|_{D_q}$$, one may insist that

$$\rho |_{I_q} = \varrho |_{I_q}.$$

By this result, working out the invariants $$N(\rho)$$ and $$k(\rho)$$ associated to $$\overline{\rho}$$ is a purely local problem. In particular, there is indeed a corresponding projective version of Serre's conjecture (now a theorem, of course!)

Note that, in general, finding a lift will require increasing the coefficient field from $$\mathbb{F}_p$$ to some finite extension. (If the corresponding class in $$H^2(\mathbb{Q},\mathbb{F}^{\times}_p)$$ is non-zero, there will never be a lift valued in $$\mathrm{GL}_2(\mathbb{F}_p)$$. Of course all lifts are twists of each other.)

One warning: for local reasons, it might be the case that any such lift $$\varrho$$ is more ramified at $$q$$ than $$\overline{\rho}|_{D_q}$$ is (this only happens if the latter is ramifed, of course, since unramified representations always have unramified lifts). This happens, for example, if the projective image of $$D_q$$ has image the Klein $$4$$-group $$K$$. (The projective image associated to an abelian representation will be projectively equivalent to $$1 \oplus \chi$$ and so have cyclic projective image. Hence the lift of any projective faithful representation of $$K$$ will be non-abelian. However, if the lift came from an extension with no more ramification, it would have to be abelian.)

One reference for some of these remarks is Serre's paper on weight one modular forms which I think is from some point in the '70s. (Serre is working with complex projective representations, but the arguments are pretty much the same.)

(Ah, I now realize this question is from 2014, sorry.)

• I'm not sure why the action on $\bar{\mathbb{F}}_p^{\times}$ in $H^2$ will be trivial can you explain.
– user130124
Nov 8 '18 at 18:11
• Thanks for a really detailed answer. Yes it's from 2014 but still interesting to me! Nov 9 '18 at 13:12


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Main

Energy exchange with the environment is an inescapable feature for any physical system. As such, the fundamental assumption of Hermiticity, as convenient as it may be, for example, in the theoretical description of quantum systems, necessarily constitutes an approximation. Historically, non-Hermitian characteristics tended to be neglected as a matter of course, while the utility of their twofold appearance as gain and loss was primarily seen in their capacity for mutual cancellation. Yet, sparked by the ground-breaking work of Bender and Boettcher on parity-time (PT) symmetry10 and its subsequent adaptation to optical settings11, a slew of works12 have revealed fascinating features arising from the complex interplay of gain and loss, for example, in non-orthogonal eigenmodes13, peculiar transport properties14, loss-induced transparency15 and unidirectional invisibility16 as well as exceptional points16,17 with enhanced response18, PT-symmetric lasers19,20 and even light-funnelling21. However, although gain and loss are readily incorporated into classical optics as a complex-valued refractive index profile, the consequences of changing the number of particles pose fundamental constraints on how these concepts can be brought to bear on genuine quantum states of light22. Fortunately, inadvertently introducing additional quantum noise can be avoided by entirely passive configurations. For example, quasi-PT-symmetric loss distributions in directional couplers have been shown to systematically accelerate the onset of fully destructive two-photon quantum interference23. Along similar lines, lossy beamsplitters7, opaque scattering media8, non-unitary metasurfaces24 or photonically implemented quantum decay processes9 may enhance two-photon coincidence rates for indistinguishable photon pairs, yielding a peak instead of the conventional dip in the Hong–Ou–Mandel (HOM) experiment, which in Hermitian systems would be indicative of fermionic anti-bunching behaviour.

In this Letter we experimentally demonstrate that, even if loss-induced anti-bunching in the form of a HOM peak is observed, the photons’ bosonic characteristics of destructive quantum interference—a HOM dip—can nevertheless be retained by an appropriate choice of the observation basis. To this end, we implement non-Hermitian two-port couplers based on femtosecond laser-written birefringent waveguides25, where polarization oscillations of photons play the role of continuous coupling between orthogonal polarization bases,6 and extended ancillary arrays provide the desired amounts of loss26,27 through polarization-sensitive coupling28. Notably, indistinguishable photons are known for being able to partially evade lossy domains in unison29, which affects the transition probability of photon pairs depending on their degree of indistinguishability, resulting in an apparent enhancement of the contribution of indistinguishable photon pairs (Fig. 1a). Regardless of the number of transmitted photon pairs however, bunching can be enforced for indistinguishable photons through the choice of the observation basis. As outlined in Fig. 1b, bunched photon pairs cannot be detected via coincidence measurement in specific bases, allowing for a suppression of their registered coincidences Cind compared to the baseline Cdis of distinguishable photon pairs that remain unaffected (Fig. 1c). We harness a combination of these two effects to induce a continuous transition from enhancement (that is, a HOM peak) in one polarization basis set to full suppression (a HOM dip) in its counterpart that has been rotated by 45° (Fig. 1d), whereas the conventional HOM dip as a sign of bosonic two-photon interference on a balanced beamsplitter remains strictly basis-independent (Fig. 1e). To gain more detailed insights into these dynamics, we study the HOM patterns for representative basis orientations associated with visibilities of v = Cind/Cdis − 1 < 0 (that is, the conventional HOM dip) to peaks (ν > 0). Furthermore, by experimentally probing the influence of the paths taken by photon pairs through the sample, we show that HOM interference actually remains unaffected by the specific phases accumulated by their constituent particles, revealing an entirely new approach to systematically tailor photon coincidence statistics beyond techniques based on selective transmission24.

Our setting is based on birefringent waveguides, which can be described with a pair of propagation constants, $$\beta_{\rm{H}}$$ and $$\beta_{\rm{V}}$$, associated with photons polarized horizontally and vertically, in line with the structure’s principal axes (Fig. 2a). As demonstrated recently, such waveguides can be employed as a directional coupler in polarization space6. Viewed through this lens, they promote a periodic, coherent transfer of excitation between the diagonal state $${|{{{\mathrm{D}}}}\rangle }={ (|{{{\mathrm{V}}}}\rangle + |{{{\mathrm{H}}}}\rangle )}/{\sqrt 2}$$ and the anti-diagonal one $${|{{{\mathrm{A}}}}\rangle }= {(|{{{\mathrm{V}}}}\rangle - |{{{\mathrm{H}}}}\rangle )}/{\sqrt 2}$$ upon propagation along the longitudinal coordinate z. Although both $${|{{{\mathrm{D}}}}\rangle}$$ and $${|{{{\mathrm{A}}}}\rangle}$$ evolve with the mean propagation constant $${\bar \beta }= {{(\beta _{{{\mathrm{V}}}}} + {\beta _{{{\mathrm{H}}}})}/{2}}$$, the rate of coupling between them is governed by the mismatch $$\Delta \beta = (\beta_{\rm{V}} - \beta_{\rm{H}})/2,$$ that is the strength of birefringence. To impose dissipation to the system, we employ (spatial) coupling to ancillary waveguide arrays placed in its vicinity (Fig. 2b). Although no actual absorption is introduced, the presence of these extended arrangements of similarly birefringent waveguides provides a flexible means to drain light from the states $${|{{{\mathrm{H}}}}\rangle}$$ and $${|{{{\mathrm{V}}}}\rangle}$$ with polarization-dependent coupling strengths28 CH and CV. Once photons have tunnelled out of the target waveguide, the comparatively stronger coupling within the arrays immediately conveys the photons to the sides by means of ballistic discrete diffraction27. Although the resulting polarization-sensitive loss rates γH and γV indeed selectively affect the evolution of the initial amplitudes $${\alpha }_{{{{\mathrm{H}}}}/{{{\mathrm{V}}}}}$$ ($${\alpha }_{{{\mathrm{H}}}}^{2} + {\alpha }_{{{\mathrm{V}}}}^{2} = {1}$$) of a single-photon state

$${|\psi (z)\rangle} = {\alpha }_{{{\mathrm{H}}}}{\rm{e}}^{ - {\gamma }_{{{\mathrm{H}}}}z} {\rm{e}}^{ - {{{\mathrm{i}}}}{\beta }_{{{\mathrm{H}}}}z}{|{{{\mathrm{H}}}}\rangle} + {\alpha }_{{{\mathrm{V}}}} {\rm{e}}^{ - {\gamma }_{{{\mathrm{V}}}}z} {\rm{e}}^{ - {{{\mathrm{i}}}}{\beta }_{{{\mathrm{V}}}}z}{|{{{\mathrm{V}}}}\rangle}$$
(1)

when described as a superposition of the polarization eigenstates $${|{{{\mathrm{H}}}}\rangle}$$ and $${|{{{\mathrm{V}}}}\rangle}$$, the mean loss rate $${\bar \gamma} = {{({\gamma }_{{{\mathrm{H}}}} + {\gamma }_{{{\mathrm{V}}}})}/{2}}$$ represents the balanced global loss rate imposed on the states $${|{{{\mathrm{D}}}}\rangle}$$ and $${|{{{\mathrm{A}}}}\rangle}$$. In turn, the difference in the individual loss rates $${\Delta \gamma} = {{({\gamma }_{{{\mathrm{H}}}} - {\gamma }_{{{\mathrm{V}}}})}/{2}}$$ appears as the imaginary part in their coupling rate (Fig. 2c).

As shown in Fig. 3a, horizontally and vertically polarized excitations of the target waveguide are indeed attenuated at markedly different rates, as confirmed by tight-binding simulations. Notably, the Zeno dynamics26 in tight-binding lattices do not yield a purely exponential decay of quantum states, and the resulting deviations at short propagation distances30 can be accounted for by defining effective loss rates ɣH/V and their respective mean $$\bar \gamma$$ and mismatch $$\Delta \gamma$$ (see Supplementary Information for details). Figure 3b shows the observed classical intensity distribution excitations after a propagation length of z = 15 cm, where a fraction of 48.3% of vertically polarized light has remained in the target waveguide (corresponding to a loss rate of γV = 0.02433 cm−1), while the substantially larger coupling in the horizontal polarization component was able to reduce the respective transmission to 4.5% (ɣH = 0.1035 cm−1).

To study the impact on HOM dynamics, a polarization-duplexed two-photon state, with one photon each in the diagonal and anti-diagonal states, was synthesized from a spontaneous parametric downconversion (SPDC) photon-pair source and injected into the target waveguide. At the other end of the sample, a polarization beamsplitter (PBS) served to separate the photons remaining in the target waveguide according to their polarization, and route them to dedicated single-photon counting modules (SPCMs) to register coincidences between the two output ports. Crucially, a half-wave plate (HWP) placed in front of the PBS allowed us to freely choose the orientation θ of the observation polarization between the horizontal/vertical (H/V, θ = 0°) and diagonal/anti-diagonal (D/A, 45°) cases by an appropriate orientation θ/2 of its fast axis (Fig. 4a). In turn, the characteristic HOM patterns were recorded by varying the photons’ relative delay τ at the injection facet between τ = 0 for the indistinguishable configuration and $${\left| \tau \right| > {1}\,{{{\mathrm{ps}}}}}$$ for the fully distinguishable case.

In a first set of experiments we set the birefringence of the target waveguide such that the cumulative phase difference along the sample yielded $${{{{\mathrm{{\Delta}}}}}{\beta } {z}} = {0}$$ modulo π, meaning that any observed change to the photons’ polarization state at the output is directly associated with the respective loss rates ɣH/V instantiated by the ancillary arrays. Notably, this configuration removes any impact of the polarization coupling dynamics in the target waveguide on the HOM interference, allowing us to directly characterize the influence of observation basis choice. As shown in Fig. 4b for six representative orientation angles θ of the HWP, the obtained HOM pattern can be continually tuned between the conventional HOM dip regime in the H/V basis (θ = 0°) with a visibility of vHV = −94.6 ± 1.0%, via the case of HOM suppression (v = 2.9 ± 0.9% at θ = 18°), to a pronounced HOM peak displaying a visibility of vAD = +54.1 ± 0.9% in the A/D basis (θ = 45°). Note that an ideal HOM peak visibility of vAD = 1 would be reached in the limiting case of zero transmission in one of the polarizations H or V (Extended Data Fig. 2 presents an equivalent realization of this configuration with a high-contrast polarizer).

To gain a more detailed understanding of the interplay between the choice of basis and the polarization coupling dynamics, we implemented three additional settings: a system with identical losses ($${\bar \gamma }= {0.0639}\,{{{\mathrm{cm}}}}^{ - 1}$$, $${\Delta \gamma }= {0.0396}\,{{{\mathrm{cm}}}}^{ - 1}$$) but a non-trivial cumulative phase of $${{{{\mathrm{{\Delta}}}}}{\beta }{z}} = {\uppi }/{4}$$ modulo π/2, representing a balanced A/D polarization coupler, and the ‘lossless’ counterparts ($${\bar \gamma }= {0}$$; that is, in the absence of ancillary arrays) for the cases of $${{{{\mathrm{{\Delta}}}}}{\beta } {z}} = {0}$$ and $${{{{\mathrm{{\Delta}}}}}{\beta } {z}} = {\uppi }/{4}$$. Figure 4c summarizes the dependence of the visibility on basis orientation and polarization coupling (Extended Data Fig. 3 presents individual basis-dependent HOM traces). Although the A/D basis clearly allows for both HOM dips and HOM peaks to be observed depending on the specific combination of cumulative phase and basis orientation, the H/V basis universally enforces a suppression of the two-photon coincidences (vHV ≈ −1). This prevalence of the conventional HOM dip is inextricably linked to the structure of the two-photon state

$${\varPsi}_{{{{\mathrm{ind}}}}}{\left( {z} \right)} = {\rm{e}}^{ - 2{{{\mathrm{i}}}}{\bar \beta }{z}}{\left( {\frac{{{\rm{e}}^{ - {2}{{{\mathrm{i}}}}{\Delta}\beta z}{\rm{e}}^{ - {2}{\gamma }_{{{\mathrm{V}}}}z}}}{{\sqrt 2 }}{|{{{\mathrm{VV}}}}\rangle }- {\frac{{{\rm{e}}^{ - {2}{{{\mathrm{i}}}}{\Delta}\beta z}{\rm{e}}^{ - {2}{\gamma }_{{{\mathrm{H}}}}z}}}{{\sqrt 2 }}}{|{{{\mathrm{HH}}}}\rangle }} \right)}.$$
(2)

As it evolves from the injected state $${\varPsi}{\left( {0} \right)} = {\left( {{|{{{\mathrm{AD}}}}\rangle } + {|{{{\mathrm{DA}}}}\rangle }} \right)}/{\sqrt 2} = {\left( {{|{{{\mathrm{VV}}}}\rangle } - {|{{{\mathrm{HH}}}}\rangle }} \right)}/{\sqrt 2}$$ by propagating though the system, the bunching of indistinguishable photons in the H and V polarizations is retained, regardless of the propagation phasors $${\rm{e}}^{ - 2{{{\mathrm{i}}}}{\left( {{\bar \beta } \pm {{{\mathrm{{\Delta}}}}}\beta } \right)}z}$$ or the specific values of the loss rates $$\gamma_{\rm{H/V}}$$. It follows that the indistinguishable photons never coincide in the H/V basis (as indicated in Fig. 1b). By contrast, the two-photon wavefunction of distinguishable particles injected as $${|{{{\mathrm{A}}}},{{{\mathrm{D}}}}\rangle}$$ evolves as

$$\begin{array}{ll}{{{\varPsi}_{{\rm{dis}}}}\left( z \right)}=&{ {{\rm{e}}^{ - 2{\rm{i}}\bar \beta z}}\left( {\frac{{{{\rm{e}}^{ - 2{\rm{i}}\Delta \beta z}}{{\rm{e}}^{ - 2{\gamma _{\rm{V}}}z}}}}{2}|{\rm{V}},{\rm{V}}\rangle + \frac{{{{\rm{e}}^{ - \left( {{\gamma _{\rm{V}}} + {\gamma _{\rm{H}}}} \right)z}}}}{2}\left( {|{\rm{V}},{\rm{H}}\rangle } \right.} \right.}\\{}&\quad \quad \quad \quad {\left. {\left. { - |{\rm{H}},{\rm{V}}\rangle } \right) - \frac{{{{\rm{e}}^{2{\rm{i}}\Delta \beta z}}{{\rm{e}}^{ - 2{\gamma _{\rm{H}}}z}}}}{2}|{\rm{H}},{\rm{H}}\rangle } \right)}\end{array}$$
(3)

and, therefore, yields detectable photon pairs in the states $${|{{{\mathrm{V}}}},{{{\mathrm{H}}}}\rangle}$$ and $${|{{{\mathrm{H}}}},{{{\mathrm{V}}}}\rangle}$$ for any finite z. In this vein, the impact of non-Hermiticity on the HOM trace is most apparent in the absence of interference terms ($${{{{\mathrm{{\Delta}}}}}{\beta } {z}} = {0}$$ modulo π). If, in addition, the losses are polarization-independent, that is for a trivial loss mismatch $$\left(\Delta \gamma = 0\right)$$, the coincidence rates of distinguishable and indistinguishable particles necessarily coincide and the HOM visibility is suppressed (vAD = 0). By contrast, in the limiting case of $${{\gamma }_{\rm{H}} \to \infty}$$, only the first terms of equations (2) and (3) remain: both photons are then V-polarized and register as pairs in the AD basis with a probability of 1/4. Comparing the transmitted amplitudes of the $${|{{{\mathrm{VV}}}}\rangle}$$ and $${|{{{\mathrm{V}}}},{{{\mathrm{V}}}}\rangle}$$ terms, respectively, one notices that they differ by a factor of $${\sqrt 2}$$: compared to the baseline of the fully distinguishable case, twice as many indistinguishable photon pairs can, in principle, be observed, corresponding to a HOM peak with ideal maximal visibility of vAD = 1, whereas finite loss contrasts yield maximal visibilities in the range of 0 < vAD < 1 (cf. Fig. 4b, where a maximal visibility of vAD = 54.1 ± 0.9% was observed for a loss contrast of $$\Delta \gamma = 0.0396\, {\rm{cm}}^{-1}$$).

As shown in Fig. 4c, the loss contrast is the crucial variable in determining the specific impact that the HOM interference within the sample has on vAD: the polar plot representation of the visibility landscape depending on θ and $${{{{\mathrm{{\Delta}}}}}{\beta }{z}}$$ illustrates that the case of vanishing loss contrast ($$\Delta \gamma = 0$$, left box) only allows for deviations (v ≈ 0) from strong suppression (v ≈ −1) in a narrow region surrounding the AD basis (θ = 45°) for trivial cumulative phases ($${{\Delta}{\beta } {z}} = {0}$$). On the other hand, two-photon interference for a balanced splitting ratio $${({{\Delta}{\beta } {z}} = {\uppi }/{4})}$$ exhibits basis-independent visibilities v ≈ −1, which also holds for fermionic two-photon states (as calculated in Supplementary Section 3) with visibility v ≈ 1. By contrast, a non-trivial loss mismatch ($$\Delta \gamma = 0.0396\, {\rm{cm}}^{-1}$$, right box) leads to nearly $${{{{\mathrm{{\Delta}}}}}{\beta } {z}}$$-independent visibilities for any given basis orientation θ. This calculated behaviour is confirmed by the observed visibilities. These measurements clearly show that the requirement of $${{{{\mathrm{{\Delta}}}}}{\beta } {z} \approx {\uppi }/{4}}$$ for quantum interference in the lossless case can be overcome by introducing non-trivial loss contrasts. This confirms that a new mechanism for tailoring two-photon coincidences has indeed been identified.

In conclusion, our findings outline a new approach to manipulate the coincidence dynamics of biphotons through the interplay of non-Hermitian systems and multiple-particle quantum mechanics. To this end, the polarization degree of freedom allows both for a seamless adjustment of the observation basis and for the implementation of complex loss profiles, facilitating a clear distinction between non-Hermitian photon dynamics and the basis-independent HOM visibilities for Hermitian quantum interference. Notably, the techniques described here can be readily combined with the versatility afforded by the three-dimensional spatial arrangement of birefringent waveguides25, specific exchange symmetries of the input states31 and waveguide structures in higher synthetic dimensions6, and open up a number of fascinating opportunities. For example, probabilistic quantum gates clearly stand to benefit greatly from the capability to introduce sophisticated loss realization into integrated optical platforms. Arbitrary quantum-optical transformations such as the singular value decomposition32 can be emulated in photonic systems, and even experiments on the evolution of multiple-particle quantum states in complex non-Hermitian systems become experimentally accessible. Taking a broader perspective, the reported basis-sensitive non-classical interference during photon evolution in open systems provides a first glance at previously unexplored physics behind the HOM dip. Along these lines, it will be of particular interest to leverage non-Hermitian HOM interferometry to explore photon-pair evolution in topological edge states33 and PT-symmetric quantum systems23, as well as quantum interference phenomena that arise from the Harper–Hofstadter model34, all of which are brought into experimental reach by our platform6.

Methods

Waveguide fabrication and birefringence characterization

We used the femtosecond-laser direct-writing technique25 to inscribe systems of evanescently coupled single-mode waveguides. To this end, ultrashort laser pulses with 270-fs duration at a carrier wavelength of λ = 517 nm and a repetition rate of 333 kHz from a fibre amplifier system (Coherent Monaco) were focused into the bulk of 150-mm-long fused-silica samples (Corning 7980) through a ×50 objective (NA = 0.60), forming waveguides along desired trajectories by translating the sample with a precision positioning system (Aerotech ALS180).

The combination of elliptical material modifications with residual stress fields from the rapid quenching immediately following the inscription pass imbues the laser-written waveguides with an inherent birefringence, the strength of which can be tuned via the writing speed35. We characterized the waveguides using classical light and crossed polarizers36. Having placed the sample between two PBSs and simultaneously rotating them by an angle φ (while maintaining their respective crossed orientation), the ratio of the transmitted (IT) and total intensities (Itotal) was measured with two photodiodes and evaluated according to the theoretical model

$${\frac{{I_{{{\mathrm{T}}}}}}{{I_{{{{\mathrm{total}}}}}}}} = {\sin ^2}{\left( {2\varphi } \right)} {\sin ^2}{\left( {{{{\mathrm{{\Delta}}}}}{\beta } {z}} \right)},$$

allowing for the relevant values of 0 and π/4 of the cumulative phase difference $${{{{\mathrm{{\Delta}}}}}{\beta } {z}}$$ to be identified by max(IT/Itotal) = 0.5 and max(IT/Itotal) = 0 up to multiples of π/2 and π, respectively.

For our experiments we chose a separation of 27.5 μm between the target waveguide and the ancillary arrays, corresponding to coupling strengths of CH = 0.154 cm−1 and CV = 0.065 cm−1 of the polarization eigenstates to their respective ‘sinks’. To ensure that photons, once extracted from the target guide, were swiftly transported away, an array pitch of 20 μm was used to set the intra-array coupling strengths to 0.551 cm−1 and 0.335 cm−1 for the horizontal and vertical polarizations, respectively (see Extended Data Fig. 1 for more details on the dependence of the coupling strength on polarization and waveguide separation).

Biphoton creation and detection

For the generation of photon pairs, we used a type I SPDC source (Fig. 4a). A bismuth borate (BiBO) crystal was pumped by a 100-mW continuous-wave laser diode (Coherent OBIS) at λ = 407 nm, producing wavelength-degenerate horizontally polarized photon pairs at λ = 814 nm, which were subsequently collected by polarizing fibres and routed to the sample via a polarization combiner with adjustable principal axes of the output fibre. A variable delay τ between the arrival times of the two photons at the sample was implemented by means of a motorized translation stage (PI). The degree of indistinguishability of the photons was characterized by recording the HOM dip with a fibre-based beamsplitter, yielding a visibility of up to v = −97.2 ± 1.1%. To record coincidences, the single photons at the output of the PBS were detected with SPCMs (Excelitas Technologies, detection efficiency of >50%, dark counts of <50 s−1, dead time of 20 ns), the signals of which were processed with a correlation card (Becker & Hickl).

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# How to calculate the mass of air that fits into a balloon of a certain size?

A hot air balloon has a $\pu{500 m^3}$ volume. How much mass of normal air fits into the balloon, if the relative molar mass is $\pu{28.8 g/mol}$?

I am not sure if I should simply use the $PV=nRT$ formula and therefore
$$1\times500= m/28.8 \times 273 \times 8.31 .$$

I feel that there is something wrong, can anyone enlighten me with the right formula or the basic concept of solving this kind of questions?

$$m = \frac{\pu{101325 Pa} \times \pu{500 m3} \times \pu{28.8 g mol-1}}{\pu{8.31 J K-1 mol-1} \times \pu{290 K}^{[1]}}$$


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BREAKING NEWS
Non-bonding orbital

## Summary

A non-bonding orbital, also known as non-bonding molecular orbital (NBMO), is a molecular orbital whose occupation by electrons neither increases nor decreases the bond order between the involved atoms. Non-bonding orbitals are often designated by the letter n in molecular orbital diagrams and electron transition notations. Non-bonding orbitals are the equivalent in molecular orbital theory of the lone pairs in Lewis structures. The energy level of a non-bonding orbital is typically in between the lower energy of a valence shell bonding orbital and the higher energy of a corresponding antibonding orbital. As such, a non-bonding orbital with electrons would commonly be a HOMO (highest occupied molecular orbital).

According to molecular orbital theory, molecular orbitals are often modeled by the linear combination of atomic orbitals. In a simple diatomic molecule such as hydrogen fluoride (chemical formula: ${\displaystyle {\ce {HF}}}$), one atom may have many more electrons than the other. A sigma bonding orbital is created between the atomic orbitals with like symmetry. Some orbitals (e.g. px and py orbitals from the fluorine in ${\displaystyle {\ce {HF}}}$) may not have any other orbitals to combine with and become non-bonding molecular orbitals. In the ${\displaystyle {\ce {HF}}}$ example, the px and py orbitals remain px and py orbitals in shape but when viewed as molecular orbitals are thought of as non-bonding. The energy of the orbital does not depend on the length of any bond within the molecule. Its occupation neither increases nor decreases the stability of the molecule, relative to the atoms, since its energy is the same in the molecule as in one of the atoms. For example, there are two rigorously non-bonding orbitals that are occupied in the ground state of the hydrogen fluoride diatomic molecule; these molecular orbitals are localized on the fluorine atom and are composed of p-type atomic orbitals whose orientation is perpendicular to the internuclear axis. They are therefore unable to overlap and interact with the s-type valence orbital on the hydrogen atom.

Although non-bonding orbitals are often similar to the atomic orbitals of their constituent atom, they do not need to be similar. An example of a non-similar one is the non-bonding orbital of the allyl anion, whose electron density is concentrated on the first and third carbon atoms.[1]

In fully delocalized canonical molecular orbital theory, it is often the case that none of the molecular orbitals of a molecular are strictly non-bonding in nature. However, in the context of localized molecular orbitals, the concept of a filled, non-bonding orbital tends to correspond to electrons described in Lewis structure terms as "lone pairs."

There are several symbols used to represent unoccupied non-bonding orbitals. Occasionally, n* is used, in analogy to σ* and π*, but this usage is rare. Often, the atomic orbital symbol is used, most often p for p orbital; others have used the letter a for a generic atomic orbital. (By Bent's rule, unoccupied orbitals for a main-group element are almost always of p character, since s character is stabilizing and will be used for bonding orbitals. As an exception, the LUMO of phenyl cation is an spx (x ≈ 2) atomic orbital, due to the geometric constraint of the benzene ring.) Finally, Woodward and Hoffmann used the letter ω for non-bonding orbitals (occupied or unoccupied) in their monograph Conservation of Orbital Symmetry.

## Electron transitions

Electrons in molecular non-bonding orbitals can undergo electron transitions such as n→σ* or n→π* transitions. For example, n→π* transitions can be seen in ultraviolet-visible spectroscopy of compounds with carbonyl groups, although absorbance is fairly weak.[2]

## References

1. ^ Anslyn, Eric V.; Dougherty, Dennis A. (2006). Modern Physical Organic Chemistry. University Science Books. pp. 841–842. ISBN 978-1-891389-31-3.
2. ^ Alfred D. Bacher. "Theory of Ultraviolet-Visible (UV-Vis) Spectroscopy". UCLA Chemistry Department. Retrieved 1 February 2012.


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# Find Distance Function from Acceleration Function

The (non-constant) acceleration as a function of time, $a(t)$, is defined and known over $[t_0, t_2]$. It is also known that $a(t)$ is integrable. Also, $a(t)=\frac{dv(t)}{dt}$ and $v(t)=\frac{dx(t)}{dt}$, where $v(t)$ is the velocity function and $x(t)$ is the distance function. $t_1$ is a known time within $[t_0, t_2]$. Given $v(t_1)$ and $x(t_1)$, is it possible to find $x(t)$ over the entire interval? If so, how can this be done? Rigor would be appreciated.

#### Solutions Collecting From Web of "Find Distance Function from Acceleration Function"

By the Fundamental Theorem of Calculus,
$$v(t)-v(t_1)=\int_{t_1}^{t}a(s)\ ds.$$
Similarly,
\begin{align*} x(t)-x(t_1)&=\int_{t_1}^t v(s)\ ds\\ &=\int_{t_1}^t\left(v(t_1)+\int_{t_1}^{s}a(u)\ du\right)\ ds\\ &= (t-t_1)v(t_1)+\int_{t_1}^t\int_{t_1}^{s}a(u)\ du\ ds. \end{align*}
Rearranging yields the solution
$$x(t)=x(t_1)+(t-t_1)v(t_1)+\int_{t_1}^t\int_{t_1}^{s}a(u)\ du\ ds,\qquad t\in [t_0,t_2]$$


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# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Sharkovskiĭ's theorem holds for some discontinuous functions

### Tom 179 / 2003

Fundamenta Mathematicae 179 (2003), 27-41 MSC: Primary 26A15; Secondary 26A18, 26A21, 54C30, 37E99. DOI: 10.4064/fm179-1-3

#### Streszczenie

We show that the Sharkovski{ĭ} ordering of periods of a continuous real function is also valid for every function with connected $G_\delta$ graph. In particular, it is valid for every ${\rm DB}_1$ function and therefore for every derivative. As a tool we apply an Itinerary Lemma for functions with connected $G_\delta$ graph.


text:
## Projected likelihood contrasts for testing homogeneity in finite mixture models with nuisance parameters

##### Authors
Sengupta, Debapriya
Mazumder, Rahul
##### Description
This paper develops a test for homogeneity in finite mixture models where the mixing proportions are known a priori (taken to be 0.5) and a common nuisance parameter is present. Statistical tests based on the notion of Projected Likelihood Contrasts (PLC) are considered. The PLC is a slight modification of the usual likelihood ratio statistic or the Wilk's $\Lambda$ and is similar in spirit to the Rao's score test. Theoretical investigations have been carried out to understand the large sample statistical properties of these tests. Simulation studies have been carried out to understand the behavior of the null distribution of the PLC statistic in the case of Gaussian mixtures with unknown means (common variance as nuisance parameter) and unknown variances (common mean as nuisance parameter). The results are in conformity with the theoretical results obtained. Power functions of these tests have been evaluated based on simulations from Gaussian mixtures.
Comment: Published in at http://dx.doi.org/10.1214/193940307000000194 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)
##### Keywords
Mathematics - Statistics Theory, Statistics - Methodology, 62G08, 60G35 (Primary) 60J55 (Secondary)


text:
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.

QUESTION

# What is the density of CO gas if 0.196 g occupies a volume of 100 mL?

The of CO gas if 0.196g occupies a volume of 100mL is 0.00196g/(mL) or 0.00196 grams per milliliter.

Using the density formula, D=m/V where d is density, m is mass, and v is volume, we will simply plug in the values in the question.

m=0.196g and v=100mL therefore D=(0.196g)/(100"mL").

After doing the division the result is D=0.00196g/(mL).

For future reference, this equation can be used to solve for mass or volume too, if you manipulate it.

For mass, it's m=DV.

For volume, it's V=m/D.


text:
# Tag Info

18

Your code is quite nice, and can't really be improved. However, let's put the “generator” back into the code. Go has channels which can be used to write elegant generators/iterators. We spawn of a goroutine that fills the channel with the fibonacci sequence. The main thread then takes as many fibonacci numbers as it needs. So let's write a function ...

16

Note that your code classifies age = 0 as "Adult" which you probably want to be "Infant". (This is because age = 0 is not included in any of your specified ranges and so the last else branch will be executed.) Two completely different solutions will be presented: The clearest and most obvious using switch. An elegant, very compact and faster way using ...

14

I don’t know Go but here are a few general things that I noticed: func between(start, end, value byte) bool { if value > end { return false } else if value < start { return false } return true } As a general pattern, don’t write if condition return true else return false or any permutation thereof. Instead, return the ...

12

encoding/binary package may have what you need. Check this: http://golang.org/pkg/encoding/binary/#example_Read Your code with modified read_int32 function could be: package main import ( "bytes" "encoding/binary" "fmt" ) func read_int32(data []byte) (ret int32) { buf := bytes.NewBuffer(data) binary.Read(buf, binary.LittleEndian, &...

12

Chan sounds like the abbreviated name for Channel and apparently it's a channel. So I'd use Channel or maybe even GoChannel. The most commonly used naming convention I have seen for private members is to prefix them with an underscore. This way you can see at the first glance whether it's a local variable or a class member. This also means you can get rid of ...

12

Rather than tie yourself to only one type (string), you could use the reflect package as well as interfaces to make it somewhat type indifferent. The following is my reworking of your code: package main import "fmt" import "reflect" func in_array(val interface{}, array interface{}) (exists bool, index int) { exists = false index = -1 switch ...

12

This entire answer refers to my rewrite of the asker's code on the Go Playground and is mostly a summary of our discussion on the chat.so Go/Golang room. For reference, I'm reproducing the code below: package main import ( "bufio" "io" "os" "strconv" ) // charSet is a limited bitset to contain all lowercase latin characters (a-z). type ...

11

Without the select operator, you haven't got Go channels - you've just got buffered queues, which are much easier to implement but much less useful. Also, it's important to allow channels with a zero size buffer - in that case the sender should synchronise with the receiver.

11

Review I - package go-fuzzywuzzy After a quick read of your Go code, the Go code for package go-fuzzywuzzy, and the Go documentation, it is reasonable to hope for at least a 60% to 95% improvement in Go performance. For example, $go build fuzzy.go && time ./fuzzy real 0m55.183s user 0m58.858s sys 0m0.944s$ After moving one line in ...

10

Style Use CamelCase only. last_status -> lastStatus, site_json -> siteJSON, etc. Even in constants. UP -> Up or StatusUp, etc. Imports are usually split into groups by source, so separate memcache from stdlib with a newline. Stuttering. E.g. configuration := Configuration{} could be c := Configuration{}. People would search for main() in main.go. So I would ...

10

Instead of checking with an if statement if errStat is null and then returning false: if errStat != nil { return false } return true you can return a Boolean expression: return errStat == nil

9

First let's mention a bug / issue: Your passwords variable is accessed (read and modified) from multiple goroutines without any syncronization: this is a race condition! Primary problem Your memory problem arises from launching a tremendous amount of goroutines. For example when you call compute() with n = 6 (to try passwords with a length of 6), it will ...

9

A vs. B type questions are seldom good questions for a code review. Often the answer is "C". That's true this time too. You don't need a go routine for the "ticker". Your second implementation only waits for 1 second, instead of 2. your "quit" channels don't need to be buffered Why don't you just do: func main() { for t := range time.NewTicker(2 * ...

9

This covers an interesting topic. Great work! Because I am unfamiliar with this area, I utilized your unit testing to ensure changes I make did not break functionality. If they do, I apologize and please let me know. Utilize implicit repetition and iota Rather than manually defining the type and value of nCode, aCode, etc. we an implicitly get the value ...

8

Disclaimer: I don't know Go. Error handling The sendRequest method may fail, returning nil, but the caller in main continues happily as if nothing happened. In fact, looking at the context of the call in main, my first impression was that sendRequest magically always succeeds, which is not the case, and so a bit misleading. Likewise, parseJson may also ...

8

import sanitize "github.com/kennygrant/sanitize" is better written as just plain ol': import "github.com/kennygrant/sanitize" since the rename is redundant. As already mentioned, it's more common to be a bit more explicit with error handling in Go and let utility functions (such as your sendRequest) return any errors for handling "higher up". In ...

8

Bug Your code clearly states: Src: 0, 0 Dst: 0, 5 Excpeted answer: {(0,0), (1,1), (2,2), (2,3), (2,4), (1,5), (0,5)} But, when I run your code, I get: [{0 5} {1 5} {2 4} {2 3} {2 2} {1 1} {0 0}] Everything in reverse order. It is clear that your back-track to reverse the path taken is appending each point to the ans slice, instead of inserting each ...

8

Nice question. Consistent error handling. Regardless of the language, the key to a good program is consistency. Your code combines a combination of panic and error handling mechanisms. You should only be using error values - panicing is an over-the-top reaction and is "not cool". Additionally, you have a copy/paste and error handling problem here too: // ...

8

let me know what I can improve in this code. For a real-world code review, code should be correct, maintainable, reasonably efficient, and, most importantly, readable. Writing code is a process of stepwise refinement. Here's your code. Consider it as a first draft. func BubbleSort(arr []int) []int { keepRunning := true for keepRunning { ...

7

The only weak point is that the implementation is not copy safe nor there exist mechanism for ensuring copy protection. I would hide its underlying type and return as sync.Locker, so it can't be mis-used: type spinLock uint32 func (sl *spinLock) Lock() { for !atomic.CompareAndSwapUint32((*uint32)(sl), 0, 1) { runtime.Gosched() //without this it ...

7

There's not really a one "right" way of doing things, and all of the others are wrong. However, there's a number of things that are considered good practice, and WRT to those, your code can do with a bit of TLC. To that end, I'll just run through your code line by line, leaving comments on both style, and things I deem to be missing, hopefully ending up ...

6

This improves slightly on Keeth's answer by making the source of the magic numbers more clear. func (r *rot13Reader) Read(p []byte) (n int, err error) { n, err = r.r.Read(p) for i := 0; i < n; i++ { c := p[i] switch { case 'a' <= c && c <= 'z': p[i] = (c-'a'+13)%26 + 'a' case 'A' <= c &...

6

First, let's rewrite your code in idiomatic Go. // For loop using a for clause func NewRectangleF(height, width int, value byte) [][]byte { r := make([][]byte, 0, height) for i := 0; i < height; i++ { r = append(r, bytes.Repeat([]byte{value}, width)) } return r } // For loop using a range clause func NewRectangleR(height, width ...

6

You have correctly noticed that this isn't extremely idiomatic. However, this code also has some issues that are completely language independent: You have a large amount of magic numbers. Either explain them with a short comment, or (preferably) calculate them. Let the compiler do constant folding; don't do it yourself. Your code does not have a single ...

6

Inside SendRequst you are calling SendRequest again, which will make your stack full of recursive calls.

6

This seems like a good solution except for one thing: func power(a, b int) int { return a ^ b } The ^ operator doesn't do what you think it does. It's a bitwise xor operator. There is no power operator in Go, there's only math.Pow for float64s. On a side note, I personally would create a type for operators: type Op func(int, int) int It's easier to ...

6

There are a few things I would point out as being poor go style. The two different nesting loops is where I would start: for i := range A { for j := i + 1; j < len(A); j++ { The outer loop does a index-only range on the A slice, which in itself is not a problem, but logically it is for i := 0; i < len(A); i++ {. Again, this is not really a ...

6

Your code is neat, and structured just like I would expect to see Java ;-) - separate files for each "class". In your case, though, the separation in to files is probably over the top, and if you are talking idiomatic go, I would not expect to see separate files for the NotGate and the XorGate. There is a small bug in your code, you use: fmt.Print("%d: {%t,...

6

Combine the mutex with the thing it is locking into a single struct. Often mutex elements of a struct are unnamed, since you only need to lock and unlock the whole thing type SafeMap struct { sync.Mutex URLs map[string]bool } which you can then use like so: url := "http://some_example.com" fetched := SafeMap{URLs:map[string]bool{}} fetched.Lock() ...

6

Bugs Your code does have a problem with the concurrency, and it's a big one.... your buffered channels are not going to be able to complete the system. The issue is the unbuffered resc and errc channels: resc, errc := make(chan string), make(chan error) Your code attempts to loop through a million input records, but your concurrency is set relatively low ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# A person invests $5000 in treasury notes and bonds. The notes pay 8% annual interest and the bonds pay 10% annual interest. If the annual in- come is$480, how much is invested in treasury notes?

Investment in treasury notes is $1000. #### Explanation: Let the money invested in treasury notes be $x. Annual income on it at 8% per annum for a year will be $\frac{8}{100} \times x$.
Then money invested in bonds at 10% would be $(5000-x) and annual income on it for a year will be $\frac{10}{100} \times \left(5000 - x\right)$. As total yield is $480$, we have $\frac{8 x}{100} + \frac{10 \left(5000 - x\right)}{100} = 480$- multiplying both sides by $100$or $8 x + 10 \left(5000 - x\right) = 48000$or $8 x + 50000 - 10 x = 48000$or $- 2 x = 48000 - 50000 = - 2000$i.e. $x = - \frac{2000}{-} 2 = 1000$Hence, investment in treasury notes is $1000.


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# Adadelta optimize weights with 0 grad?

Suppose the following very trivial model:

``````class Net(torch.nn.Module):
def __init__(self, img_size):
super(Net, self).__init__()
self.l1 = torch.nn.Linear(1, 1)
self.optimizer = torch.optim.Adadelta(self.parameters())
self.loss_function = torch.nn.MSELoss()

def forward(self):
return self.l1(torch.tensor([1], dtype=torch.float))

def backward(self, loss):
self.optimizer.zero_grad()
loss.backward()
if not self.l1.weight.grad is None:
#(*) changing the grad to 0
self.l1.weight.grad[0] = torch.tensor([0], dtype=torch.float)
self.optimizer.step()
``````

In the backward method (*) we are changing the value of the grad to 0.

My understanding was that this mean that the weight is not contributing to the loss, so, the optimizer should not optimize the weight, but it happens anyway.

Why the network keeps learning even when the grad has been set to zero?


text:
• ### Jet Substructure Studies with CMS Open Data(1704.05842)

Sept. 28, 2017 hep-ph, hep-ex
We use public data from the CMS experiment to study the 2-prong substructure of jets. The CMS Open Data is based on 31.8/pb of 7 TeV proton-proton collisions recorded at the Large Hadron Collider in 2010, yielding a sample of 768,687 events containing a high-quality central jet with transverse momentum larger than 85 GeV. Using CMS's particle flow reconstruction algorithm to obtain jet constituents, we extract the 2-prong substructure of the leading jet using soft drop declustering. We find good agreement between results obtained from the CMS Open Data and those obtained from parton shower generators, and we also compare to analytic jet substructure calculations performed to modified leading-logarithmic accuracy. Although the 2010 CMS Open Data does not include simulated data to help estimate systematic uncertainties, we use track-only observables to validate these substructure studies.
• ### Exposing the QCD Splitting Function with CMS Open Data(1704.05066)

Sept. 25, 2017 hep-ph, hep-ex
The splitting function is a universal property of quantum chromodynamics (QCD) which describes how energy is shared between partons. Despite its ubiquitous appearance in many QCD calculations, the splitting function cannot be measured directly since it always appears multiplied by a collinear singularity factor. Recently, however, a new jet substructure observable was introduced which asymptotes to the splitting function for sufficiently high jet energies. This provides a way to expose the splitting function through jet substructure measurements at the Large Hadron Collider. In this letter, we use public data released by the CMS experiment to study the 2-prong substructure of jets and test the 1 -> 2 splitting function of QCD. To our knowledge, this is the first ever physics analysis based on the CMS Open Data.
• ### Systematics of quark/gluon tagging(1704.03878)

Aug. 8, 2017 hep-ph, hep-ex
By measuring the substructure of a jet, one can assign it a "quark" or "gluon" tag. In the eikonal (double-logarithmic) limit, quark/gluon discrimination is determined solely by the color factor of the initiating parton (C_F versus C_A). In this paper, we confront the challenges faced when going beyond this leading-order understanding, using both parton-shower generators and first-principles calculations to assess the impact of higher-order perturbative and nonperturbative physics. Working in the idealized context of electron-positron collisions, where one can define a proxy for quark and gluon jets based on the Lorentz structure of the production vertex, we find a fascinating interplay between perturbative shower effects and nonperturbative hadronization effects. Turning to proton-proton collisions, we highlight a core set of measurements that would constrain current uncertainties in quark/gluon tagging and improve the overall modeling of jets at the Large Hadron Collider.
• ### How Much Information is in a Jet?(1704.08249)

June 7, 2017 hep-ph, hep-ex
Machine learning techniques are increasingly being applied toward data analyses at the Large Hadron Collider, especially with applications for discrimination of jets with different originating particles. Previous studies of the power of machine learning to jet physics has typically employed image recognition, natural language processing, or other algorithms that have been extensively developed in computer science. While these studies have demonstrated impressive discrimination power, often exceeding that of widely-used observables, they have been formulated in a non-constructive manner and it is not clear what additional information the machines are learning. In this paper, we study machine learning for jet physics constructively, expressing all of the information in a jet onto sets of observables that completely and minimally span N-body phase space. For concreteness, we study the application of machine learning for discrimination of boosted, hadronic decays of Z bosons from jets initiated by QCD processes. Our results demonstrate that the information in a jet that is useful for discrimination power of QCD jets from Z bosons is saturated by only considering observables that are sensitive to 4-body (8 dimensional) phase space.
• ### Holography with Schroedinger Potentials(hep-th/0610218)

Oct. 19, 2006 hep-th, hep-ph
We examine the analogue one-dimensional quantum mechanics problem associated with bulk scalars and fermions in a slice of AdS_5. The Schroedinger'' potential can take on different qualitative shapes depending on the values of the mass parameters in the bulk theory. Several interesting correlations between the shape of the Schroedinger potential and the holographic theory exist. We show that the quantum mechanical picture is a useful guide to the holographic theory by examining applications from phenomenology.


text:
## PREP 2015 Question Authoring - Archived

### Scientific Notation Error in OPL problem

by tim Payer -
Number of replies: 2
Hello,

I am writing solutions to a number of selected OPL problems and came across a possible error in a problem that had the original file path of:

Library/Rochester/setDerivatives2Formulas/c2s5p3

The (algorithmic) problem answer has a number of zeros after the decimal before significant figures are encountered. As a hint to use scientific notation the student is given a prompt  in the wording of the problem that states:

"Tip: You can enter an answer such as 3.14e-1 for 0.314."

The problem is that when entered in this form the numerical value of e = 2.718241824.. is used in the calculation of the students answer.

I checked the course configuration and did not see a feature where scientific notation could be enabled for student answers in this way. Is there something else I should have checked to enable scientific notation?

Thanks,  tim

Below is the altered code for the problem with solutions:

## DESCRIPTION
##   The Chain Rule
## ENDDESCRIPTION

## Tagged by nhamblet

## DBsubject(Calculus - single variable)
## DBchapter(Differentiation)
## DBsection(Chain rule (without trigonometric functions))
## Institution(Rochester)
## Level(2)
## MO(1)
## KEYWORDS('Derivative', 'Polynomial', 'Chain')
## Library/Rochester/setDerivatives2Formulas/c2s5p3

DOCUMENT();        # This should be the first executable line in the problem.

"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGauxiliaryFunctions.pl",
"MathObjects.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1; Context("Numeric");$denom= 0;
while ($denom <= 0 ) {$t1 = random(-4,4,1);
$a= non_zero_random(-4,4,1);$a2 = $a*2;$b=   non_zero_random(-4,4,1);
$gcd = ($a2,$b);$a3 = $a2/$gcd;
$b3 =$b/$gcd;$gc8 = $gcd*(-8);$at3 = $a3*$t1;
$tsq =$t1*$t1;$bt1 = $b*$t1;
$abt3 =$at3+$b3;$atsq = $a*$tsq;
$c= non_zero_random(-4,4,1);$btc = $bt1 +$c;
$num =$gc8*$abt3;$den1 = $atsq+$btc;
$den2 =$den1**9;
$gcd = gcd($num,$den2);$num3 =$num/$gcd;
$den3 =$den2/$gcd;$ans = $num/$den1**9;
$d = random(-4,4,1);$denom = $a*$t1**2 +$b*$t1 +$c; }$fp= Compute("-8*$denom**(-9)*(2*$a*$t1 +$b)");

# Present the text.
TEXT(EV2(<<EOT));
Calculate $$f'(t1)$$ to 3 significant figures where
$f(t) = (a t^2 + b t + c )^{-8}$

$$f'( t1 ) =$$\{ ans_rule(20) \}

$PAR Tip: You can enter an answer such as 3.14e-1 for 0.314.$PAR
EOT

ANS($fp->cmp); SOLUTION(EV3(<<'END_SOLUTION'));$PAR SOLUTION $PAR Use the chain rule to find the derivative of composed functions.$BR
The function of  $$f(t) = (a t^2 + b t + c )^{-8}$$  is a composed function because the outer function of the negative exponent is "holding" the inner function of a quadratic equation. The chain rule applies the derivative on the outermost function first and then works inward applying the derivative to each successive inner function in a series of products. With each application of the derivative on the outer function the inner contents are left alone. In this way a "chain" of derivative products are formed. The chain rule can be expressed with notation as shown below: $BR$BR
Given that p(x) = f(g(x)), $BR$BR
The derivative is applied by the chain rule: $BR$BR
$$p'(x) = f'(g(x)) \cdot g'(x)$$   $BR$BR
Apply the derivative to f(t): $BR$BR
$$f(t) = (a t^2 + b t + c )^{-8}$$ $BR$BR
Apply the prime tics for the chain rule:$BR$BR
$$f'(t) = \left((a t^2 + b t + c )^{-8}\right)' \left(a t^2 + b t + c \right)'$$ $BR$BR
Apply the derivative: $BR$BR
$$f'(t) = -8(a t^2 + b t + c )^{-9} \left(a2 t + b \right)$$ $BR$BR
Push the quantity of the negative exponent down to the denominator: $BR$BR
$$\displaystyle{f'(t) = \frac{-8(a2 t + b) }{(a t^2 + b t + c )^{9}} }$$ $BR$BR
Pull any common factors from the numerator: $BR$BR
$$\displaystyle{f'(t) = \frac{-8 \cdot gcd(a3 t + b3) }{(a t^2 + b t + c )^{9}} }$$ $BR$BR
Combine factors for the reduced derivative: $BR$BR
$$\displaystyle{f'(t) = \frac{gc8(a3 t + b3) }{(a t^2 + b t + c )^{9}}}$$ $BR$BR
Evaluate the derivative at t = $t1:$BR
$BR $$\displaystyle{f'(t1) = \frac{gc8(a3 \cdot (t1) + b3) }{(a \cdot (t1)^2 + b \cdot (t1) + c )^{9}}}$$$BR
$BR Combine factors and squared values:$BR
$BR $$\displaystyle{f'(t1) = \frac{gc8(at3 + b3) }{(a \cdot tsq + bt1 + c )^{9}}}$$$BR
$BR Combine factors and terms:$BR
$BR $$\displaystyle{f'(t1) = \frac{gc8(abt3) }{(atsq + btc )^{9}}}$$$BR
$BR Combine factors and terms:$BR
$BR $$\displaystyle{f'(t1) = \frac{num }{(den1 )^{9}} =\frac{num }{den2} }$$$BR
$BR An exact value:$BR
$$\displaystyle{f'(t1) = \frac{num3 }{den3} }$$ $BR$BR
A decimal approximation:$BR$BR
$$\displaystyle{f'(t1) = ans}$$ $BR$BR

END_SOLUTION

ENDDOCUMENT();        # This should be the last executable line in the problem.

### Re: Scientific Notation Error in OPL problem

by Arnold Pizer -
Hi Tim,

The hint should read "Tip: You can enter an answer such as 3.14E-1 for 0.314." See http://webwork.maa.org/wiki/Available_Functions

I don't remember but ages ago before we had the constants pi and e and/or before the current parser, maybe 3.14e-1 did work for scientific notation or it could be a typo in the problem.

Arnie


text:
Article | Open | Published:

# Akinetic swept-source optical coherence tomography based on a pulse-modulated active mode locking fiber laser for human retinal imaging

## Abstract

Optical coherence tomography (OCT) is a noninvasive imaging modality that can provide high-resolution, cross-sectional images of tissues. Especially in retinal imaging, OCT has become one of the most valuable imaging tools for diagnosing eye diseases. Considering the scattering and absorption properties of the eye, the 1000-nm OCT system is preferred for retinal imaging. In this study, we describe the use of an akinetic swept-source OCT system based on a pulse-modulated active mode locking (AML) fiber laser at a 1080-nm wavelength for in-vivo human retinal imaging. The akinetic AML wavelength-swept fiber laser was constructed with polarization-maintaining fiber that has an average linewidth of 0.625 nm, a spectral bandwidth of 81.15 nm, and duty ratio of 90% without the buffering method. We successfully obtained in-vivo human retinal images using the proposed OCT system without the additional k-clock and the frequency shifter that provides a wide field of view of 43.1°. The main retina layers, such as the retinal pigment epithelium, can be distinguished from the OCT image with an axial resolution of 6.3 μm with this OCT system.

## Introduction

Optical coherence tomography (OCT) is a noninvasive imaging modality that can provide high-resolution, cross-sectional images of tissues1,2,3. OCT has evolved over the past decades as one of the most important clinically applicable tests in ophthalmology, cardiology, and dermatology1,4,5,6. Especially in ophthalmology, OCT has become one of the most valuable imaging tools for diagnosing eye diseases7,8,9. In ophthalmic OCT application, wavelengths of 800 nm, 1000 nm, and 1300 nm are mostly used as light sources3,7,10,11. Although 1300-nm OCT is used for dermatological, cardiological, and corneal imaging, the high water absorption feature of the vitreous body of the human eye limits the depth of retinal OCT imaging. The 800-nm OCT system provides more detailed morphological information, microvascular mapping of the retina with a relatively high axial resolution, and it has been successfully popularized in the clinic setting. However, it is difficult to image deeper posterior areas of the eye, such as the choroid region, because of the strong absorption and reflection of the retinal pigment epithelium (RPE) and highly pigmented choroidal melanocytes. Considering the scattering and absorption properties of the eye, the 1000-nm OCT system is preferred for retinal imaging10.

In order to successfully apply a biomedical imaging modality to the clinical setting, it must be non-invasive, non-labeling, real-time, in-vivo imaging with high-resolution, a sufficient imaging range, and short acquisition time. Therefore, the 1000-nm OCT system needs to have adequate imaging speed (more than 100 kHz) and imaging depth range to capture the deeper choroid layer, including the microvascular structure. One of the high-speed OCT systems is the spectral domain OCT (SD-OCT) system, which uses a broadband light source, such as a superluminescent diode, and a spectrometer to measure the optical interference signals. However, with the SD-OCT system, the imaging speed and imaging depth range are limited by the characteristics of the spectrometer12. Silicon-based spectrometers are widely used for the SD-OCT system; however, this device has a low sensitivity at a 1000-nm wavelength than at an 800-nm wavelength. Therefore, an InGaAs CCD camera can be used for a 1000-nm SD-OCT system; however, it is not cost effective, and it has a low pixel number, making it difficult to obtain an OCT image of the deeper retinal layer.

Another high-speed OCT system is swept-source OCT (SS-OCT), which uses a wavelength-swept laser as the light source. SS-OCT systems have advantages of a high imaging speed and longer imaging depth range with a simple and cost-effective detecting scheme (single photon detector) compared with other OCT systems, such as time domain-OCT and SD-OCT7,11,13,14,15,16,17. The characteristics of SS-OCT systems mainly depend on the performances of the wavelength-swept source. Many kinds of wavelength-swept sources have been reported, and they tune a lasing wavelength based on mechanical movements of a wavelength-tunable component in the laser cavity. The sweeping behaviors are determined by the mechanical movements of the wavelength-tunable components, such as a the Fabry-Perot tunable filter18,19, a polygon mirror scanner20, tunable microelectromechanical system filter with a semiconductor optical amplifier (SOA)21, or vertical cavity surface-emitting laser7,16.

This mechanical movement requires depletion and accumulation of momentum for its completion, which can be affected to hysteresis or experience undesirable drifts of operational scheme, such as thermal drift because of friction. Unstable or drifting mechanical movements can lead to overall degradation of source performance11,14,15,22.

Recently, Insight Photonic Solutions, Inc. (Lafayette, CO, USA) reported an akinetic wavelength-swept laser with an integrated semiconductor opto-electronic design without the need for external cavity coupling at the center wavelength of 1060 nm, 1310 nm, and 1550 nm14,15,22. The company successfully showed an SS-OCT image of the human retina, skin, and tooth and an OCT angiogram of human skin owing to the phase-stable swept source. However, the swept source contains valid and invalid points that can limit available sampling points and cause data acquisition to be complex. Therefore, a k-clock has to be used to synchronize acquisition of the DAQ card with a delay on the start of the sweep in order to remove invalid data points15, and only 780 sample numbers are used for OCT images, which can affect the image quality.

Another type of akinetic wavelength-swept laser is based on the active mode locking (AML) method with dispersive fiber cavity11,23,24,25,26,27,28,29,30,31,32,33. Its feasibility for use with fiber Bragg grating sensor systems25 and SS-OCT systems11,23,26,27,28,29,30,31,32,33 with AML wavelength-swept sources has been reported. The SS-OCT images showed various samples, such as cover glasses, adhesive tape, and an in-vivo human finger, at a sweep rate of 1 kHz. In addition, the in-vivo OCT image of a human cornea was reported at a sweep rate of 100 kHz by directly modulating the gain medium with the sinusoidal signal11. Although the mechanical tunable filters are successfully eliminated with AML methods, the main limitation of in-vivo OCT imaging remains: a short image depth range (coherence length of the laser). Therefore, in a previous study, a frequency shifter was inserted into the interferometer to extend the imaging depth range, but the whole corneal structure still could not be shown simultaneously11. In addition, the center wavelengths of all studies were 1300 nm and 1550 nm, which are unsuitable for in-vivo human retinal OCT imaging11,23,26.

In the present study, we describe the use of a pulse-modulated AML wavelength-swept fiber laser with a 1080-nm wavelength for in-vivo human retinal OCT imaging. An electro-optic intensity modulation was inserted to the laser cavity which has 3 dB bandwidth of 20 GHz for the high speed pulse modulation of 50 ps duration. A ring laser cavity, with a length of ~1.542 m (in air), was constructed with all polarization maintain fiber to enhance environmental resistance. The average linewidth of 0.0625 nm and the spectral bandwidth of 81.85 nm were obtained in the static AML condition. The sweeping behavior was determined by a fully electrical signal without a mechanical tuning mechanism. Therefore, the swept laser was driven with a uni-directional linear sweep mechanism with a duty ratio of ~90%. Therefore, we extracted a fully synchronized sweep spectrum without observing a difference between forward and backward sweeping. The -6 dB roll-off of 2.5 mm and -20 dB roll-off of 9 mm were obtained at swept rate of 100 kHz for the SS-OCT system. We successfully obtained in-vivo retina images of a human using the OCT system without the additional k-clock and the frequency shifter providing wide field of view (FOV) of 43.1°. The main retina layers such as internal limiting membrane, retinal nerve fiber layer, photoreceptor layers, RPE, choroidal stroma, were clearly distinguished through the OCT image with axial resolution of 6.3 μm.

## Results

### Pulse-modulated active mode locking wavelength-swept laser

Figure 1a shows the experimental setup of a proposed pulse-modulated AML wavelength-swept fiber laser at a 1080-nm wavelength range for in-vivo human retinal OCT imaging. The laser cavity consisted of an SOA, optical circulator, and chirped fiber Bragg grating (CFBG) and electro-optic intensity modulators. The electro-optic intensity modulator was inserted for high-speed pulse modulation because the commercial SOA has a switching time larger than 500 ps. All components were made of a polarization-maintaining fiber to protect polarization-dependent properties and ensure environmental resistance. A radiofrequency (RF) signal from an RF signal generator was connected to the pulse generator. The pulse signal with 50 ps pulse duration was used to modulate the gain and loss of the laser cavity in order to achieve the AML condition synchronized with RF signal generator. An arbitrary waveform from an arbitrary function generator (AFG) was used to electrically vary the center frequency of the RF signal generator by using the frequency modulation (FM) function. The optical components were spliced as short as possible to provide a better sweeping property. The total cavity length was 1.542 m (in air), which represents a free spectral range (FSR) of 132.89 MHz. The FSR of the laser cavity is expressed as follows:

$$FSR=\frac{c}{{n}_{eff}L}$$
(1)

where c is the speed of light in the vacuum, neff is the effective refractive index, and L is the cavity length.

In a harmonic mode-locking condition in a fiber laser cavity, the lasing wavelength, λm, is determined by the modulation frequency fm of the applied voltage into the electro-optic intensity modulator with an RF pulse, which is an integer (N, mode-locking order) multiplied by the FSR, as follows.

$${f}_{m}=N\cdot FSR$$
(2)

The zero-order FSR, FSR0, is given by the cavity length and speed of light in the fiber cavity: FSR0 = c0/nL. However, in a chromatic dispersive medium, the FSR becomes a function of the wavelength (or wavenumber). The relationship between the lasing wavelength and modulation frequency can be expressed as follows11:

$${\rm{\Delta }}{\lambda }_{m}=-\,S{\rm{\Delta }}{f}_{m}$$
(3)

where Δλm is the deviation in the lasing wavelength, S is the tuning sensitivity, and Δfm is the deviation of the modulation frequency. The sensitivity is defined as follows:

$$S=\frac{{n}_{eff}L}{c{D}_{total}{f}_{m0}}$$
(4)

where neff is the effective refractive index, Dtotal is the dispersion parameter, and fm0 is the modulation frequency11.

Figure 1b shows the static output spectra of the pulse-modulated AML wavelength-swept fiber laser with discrete variation of the modulation frequency in the whole wavelength tuning range. As modulation frequency fm varied from 758.3 MHz to 835.3 MHz (N = 6), the output wavelength λm changed from 1120.15 nm to 1038.3 nm. Figure 1c shows the relationship between RF modulation frequency (fm) and output wavelength (λm) of the pulse-modulated AML wavelength-swept fiber laser. The red line in Fig. 2a is the first-order fitting of the measured data. The measured R-square values of the first-order fitting and residual sum of square were 0.99949 and 2.95549, respectively, which show a high linear relationship according to equations (Eqs 3 and 4). The experimentally measured sensitivity parameter S was 1.064 nm/MHz. The theoretical value of the sensitivity was 1.056 nm/MHz according to Eq. 4.

A total tuning range of 81.85 nm was obtained with the center wavelength of 1079.23 nm. The theoretical maximum tuning range can be expressed as follows:

$${\rm{\Delta }}{\lambda }_{{\max }}=\frac{1}{|{D}_{total}|{f}_{m0}}$$
(5)

It was determined as 139.4 nm based on this equation. In this study, the tuning range of the AML wavelength-swept fiber laser was limited by the gain bandwidth of the SOA and CFBG in the cavity. Figure 2a shows the reflection property of the CFBG. The tuning bandwidth of the laser was primarily limited by the 3-dB reflection bandwidth of the CFBG and optical circulator. The tuning range can be improved by using the wider band SOA, optical circulator, and CFBG.

With the pulse-modulated AML wavelength-swept fiber laser, the relatively wide tuning range without linewidth broadening could be obtained in the whole wavelength range, unlike with direct sinusoidal modulation. Since the wavelength components are almost continuously existed in different time in the laser cavity according to the dispersion, the RF signal should select wavelength component in the short time for narrow linewidth. In addition, with the direct sinusoidal modulated AML laser, a high DC current above the gain threshold should be applied to the SOA before RF modulation to achieve a wide tuning range. This high DC current caused a relatively high-amplified spontaneous emission (ASE) signal, which can also affect the linewidth broadening at the wavelength of the high-gain region. Therefore, the DC current should be controlled to preserve a narrow linewidth, which limits wider tuning bandwidth11. Figure 1d shows the static output spectrum with discrete variation of the modulation frequency (786.5–794.5 MHz) at the narrow wavelength tuning range (1084.39–1075.88 nm) in the linear scale. As shown in Fig. 1d, the spectral shapes of the output laser showed a stable and similar tendency, indicating stable lasing through all wavelengths, unlike with direct sinusoidal modulation (Supplementary Figure S1). The measured 3-dB linewidth of the lasing peak was 0.053 nm at the wavelength of 1079.90 nm, which is lower than that of the direct sinusoidal modulated AML wavelength-swept laser. The average 3-dB linewidth of the sinusoidal-modulated (Supplementary Figure S1) and pulse-modulated AML laser (Fig. 1e) was measured as 0.29 nm and 0.0625 nm, respectively. The measured output power of pulse-modulated AML fiber laser was 2.1 mW at 1079.90 nm.

### Sweeping properties of the pulse-modulated AML wavelength-swept laser

With the wavelength-swept laser, a lasing property could be preserved until the sweep rate reached a single round-trip frequency. The single round-trip frequency was defined as follows19:

$${f}_{single \mbox{-} roundtrip}=\frac{\delta \lambda \cdot c}{{\rm{\Delta }}\lambda \cdot {\rm{L}}\cdot {\rm{n}}}$$
(6)

In this experiment, the sweep rate fsingle-roundtrip for a single round trip was about 102.22 kHz, where δλ = 0.0625 nm, c = 3 × 108, Δλ = 81.25 nm, L = 1.542 m, and n = 1.464. Although wavelength sweeping could be operated at more than 1 MHz, the sweep rate had to be below this frequency to obtain sufficient gain for the lasing. Figure 2b and c show the sweep spectra of the pulse-modulated AML wavelength-swept fiber laser with various sweep rates, fs. The spectra were measured by a peak-hold mode of an optical spectrum analyzer. During wavelength sweeping, the ramp (uni-directional) waveform of the AFG was applied to the RF signal generator with FM function of the RF generator (Fig. 1a). As shown in Fig. 2b, the optical output power was decreased at a sweep rate of 200 kHz because the sweep rate was higher than fsingle-roundtrip. To preserve output power at a sweep rate of 200 kHz, the sweep bandwidth had to be half of 100 kHz. In addition, the output power and other laser properties simultaneously degraded, such as the optical linewidth. Figure 3 shows the interferogram of the AML wavelength-swept laser at different conditions when the optical path differences were 1 mm (3a, b, c) and 2 mm (3d, e, f). Compared with the 100-kHz sweep rate, the coherence property was highly degraded at the sweep rate of 200 kHz with the same spectral bandwidth of 81.15 nm, as shown in Fig. 3b and d. For the same laser cavity configurations such as the cavity length, linewidth, gain, the operating spectral bandwidth should be narrower in the case of a 200-kHz sweep rate. Figure 3c and f show the interferogram of the AML wavelength-swept laser at the sweep rate of 200 kHz and operating spectral bandwidth of 40.07 nm (Fig. 2c). By dividing the optical spectral bandwidth in half (100 kHz), the other sweep properties were well preserved at the sweep rate of 200 kHz. In order to use the wide spectral bandwidth at a higher sweep rate, the cavity length should be as short as possible. A wider spectral bandwidth at a higher sweep rate with sufficient lasing properties are expected to be available when the length of the laser cavity is decreased further by using improved fiber splicing techniques.

Here, we can confirm that the duty ratio of the laser is almost 90% because we applied the single-directional drive function thanks to removal of the mechanical tunable filter, which can remove non-linear properties of bi-directional sweeping. A sampling rate of the SS-OCT system, fsampling, can be defined as follows:

$${f}_{sampling}=\frac{{f}_{s}\cdot {N}_{sample}}{{R}_{duty}\,}$$
(7)

where fs is the sweep rate, Nsample is the sample number for every A-scan and Rduty is the duty ratio. The higher sample number can be achieved at the higher high duty ratio, which makes it possible to obtain better imaging qualities. A conventional mechanical swept laser has an effective duty ratio less than 50% in order to achieve single-directional sweep modes, which require a higher speed digitizer or lower sample number. Therefore, a buffering method is conventionally used for common SS-OCT systems to achieve the duty ratio of about 90%20,21,34,35. An advantage of our system is that the entire spectrum is single-directionally swept without the use of a buffering method.

### Characteristics of the swept-source optical coherence tomography system

Figure 4a show the experimental setup of the SS-OCT system based on the pulse-modulated AML wavelength-swept fiber laser. For the best OCT imaging with stable laser conditions and data processing capacity, we selected an optimal sweep rate of 100 kHz for the pulse-modulated AML wavelength-swept fiber laser, with an output power of 18.2 mW after the booster optical amplifier. The measured spectral bandwidth of the laser was 81.15 nm. Therefore, according to Eq. (8), the related OCT resolution was theoretically 6.3 μm in air36.

$${\rm{\delta }}z=\frac{{\rm{2ln2}}}{\pi }\frac{{\lambda }_{0}^{2}}{n\Delta \lambda }$$
(8)

The SS-OCT system was configured with a Michelson interferometer. To analyze the OCT performance of the pulse-modulated wavelength-swept AML fiber laser, we measured the point-spread function (PSF) roll-off over the imaging depth in OCT application. For better coherence properties during the sweeping, the laser should be swept from a short to a long wavelength region (forward sweeping). When the optical pulse passes through the gain medium (e.g. SOA) of laser cavity, the optical redshift is induced by the self-phase modulation. This red shifting from modulation of the gain medium allows stable pulsation for short to long wavelength tuning37. In contrast, the coherence properties were degraded for backward sweeping (long to short wavelength region). To examine this experimentally, two types of waveforms from an AFG were applied to the laser. One was a forward sweeping, and the other was a backward sweeping with single-directional linear drive function. The characteristics of these waveforms were analyzed and compared. Figure 4b and c show the PSF roll-off of the AML wavelength-swept fiber laser for forward sweeping and backward sweeping, respectively. As expected, the PSF signals were significantly decreased in backward sweeping which showed almost meaningless results at >1 mm (Supplementary Figure S2). In forward sweeping, the total 20-dB roll-off imaging range of 9 mm was obtained which is almost 5 times higher than that of sinusoidal modulation. The measured sensitivity of the system was 119.8 dB. The proposed laser is basically a pulse laser, therefore, the pulse duration can affect a fringe visibility as well as roll-off of the OCT system. In term of the coherence behavior, the pulse width is one of the factors which could affect the fringe visibilities because of the time mismatch of the time-delayed pulses pair from reference and sample arms. However, the 20-dB roll-off of 9 mm is enough to image the human retina since the structural thickness and image range of a posterior eye is around 3 mm. In our previous study, we inserted a frequency shifter to enhance the imaging depth range because of the low coherence length of the sweeping source11. However, in this study, we obtained enough imaging range for in-vivo retina OCT imaging thanks to the linewidth enhancement of the pulse modulation. Additionally, the OCT system could be operated with a sweep rate of 200 kHz when the spectral bandwidth was decreased to 40.07 nm (Supplementary Figure S3).

### In-vivo optical coherence tomography image based on the pulse-modulated AML wavelength-swept laser

Using the proposed SS-OCT system, we firstly obtained in-vivo OCT images of a human finger. The measured transverse spot size on the sample was 20 µm, and the measured axial resolution was 6.3 μm. The average incident power on the finger surface was 1.8 mW, which is below the American National Standards Institute (ANSI) limit of 1.925 mW at a wavelength of 1060 nm38. Figure 5a and b show the 2-dimensional (2D) cross-sectional image of a human finger and nail/finger boundary, and Fig. 5c shows the 3-dimensional (3D) OCT image of the human finger (Supplementary Video S1). Because of the longer imaging range and high axial resolution of the system, the high contrast OCT image could be obtained with the pulse-modulated AML wavelength-swept laser compared with the sinusoidal-modulated AML wavelength-swept laser.

For retinal OCT imaging, the optical lens at the sample arm was removed, and the light was focused by the eye of a patient. The average incident power on the eye surface was 1.3 mW, which is less than the ANSI limit of 1.925 mW at a wavelength of 1060 nm38. Figure 6a and b show the cross-sectional OCT image of the human retinas of the left and right eyes, respectively. The main layers of the retina, such as the internal limiting membrane, retinal nerve fiber layer, photoreceptor layers, RPE, choroidal stroma, could be distinguished because of the high axial resolution of the proposed sweep source and the OCT system. The 3D OCT images were created from a Fourier transform of 500 × 500 × 4096 voxels per volume data set. The acquisition times were 5 milliseconds and 2.5 seconds for 2D and 3D OCT images, respectively. Figure 6c shows the enface image that was obtained from the 3D OCT image. Compared with the commercial fundus image at 50° (Supplementary Figure S4), we confirmed that the image range of the retinal OCT system was 43.1°, which can be used to simultaneously observe macular and nerve bundle regions. The 3D OCT images of human retina were obtained after motion compensation38 (Fig. 7, Supplementary Videos S2 and S3).

## Discussion

Akinetic wavelength-swept sources have many advantages, such as phase stability, a high duty ratio without a buffering method, linear sweeping, and a high repetition rate. Many studies have reported on the akinetic AML wavelength-swept laser; however, the sweeping performances were insufficient to obtain high-speed in-vivo OCT imaging. In addition, the center wavelength range was unsuitable for in-vivo retinal OCT imaging because of the high water content of the human eye. We successfully obtained in-vivo OCT images of the human retina using a pulse-modulated akinetic AML wavelength-swept fiber laser with the center wavelength of 1080 nm. To enhance environmental resistance, the entire fiber-optic ring laser cavity was constructed with polarization-maintaining fiber instead of single-mode fiber. The pulse modulation suppresses the DC current, which affects the linewidth broadening; therefore, the average linewidth of 0.0625 nm and spectral bandwidth of 81.85 nm were obtained in the static AML condition, showing stable spectral shapes over the tuning range. The proposed wavelength-swept laser has the unidirectional sweeping characteristic of about a 90% duty ratio, which can reduce the sampling rate compared with the commercialized bi-directional swept source under the 60% duty ratio. We successfully obtained in-vivo 3D OCT images of human retinas at a sweep rate of 100 kHz without an optical frequency shifter. Further study of the following is needed to improve the system: (1) shortening of the laser cavity length to increase swept performance at the high sweep rate, (2) using high-speed and more stable RF electronics for phase stability (3) optimizing the pulse duration of the laser to enhance the fringe visibility (4) enhancement of the spectral bandwidth using a broadband CFBG and gain medium, (5) applying wider-angle retinal OCT angiography using a laser with a higher sweep rate.

## Methods

### In-vivo optical coherence tomography imaging and 3-dimensional image reconstruction

This study was approved by the institutional review board of the Gwangju Institute of Science and Technology (approval number 20180629-HR-36-02-02), and written informed consent was obtained from 2 healthy subjects before conducting the imaging procedure. All experiments were performed in accordance with relevant guidelines and regulations. To obtain the 3D image of the human retina, a subject was positioned on the commercial OCT headrest. The averaged laser power energy on the eye surface was 1.3 mW, which is less than the ANSI limitation of 1.925 mW for a wavelength of 1060 nm37. All volumetric data (500 × 500 × 4096 pixels) were rendered using the commercial software Amira (FEI Visualization Sciences Group) after motion compensation39.

### Pulse-modulated active mode locking wavelength-swept laser

The SOA (SOA-1060-90-PM-30dB, Innolume Inc.) has a 3-dB bandwidth of 90 nm and center wavelength of 1070 nm when a current of 400 mA is applied. The optical circulator was inserted for ring cavity configuration and used for unidirectional lasing by protecting bi-directional emission of the SOA and optical reflection between the fiber components. The optical circulator (Oz Optics Inc.) has a 3-dB operating bandwidth of 100 nm with a center wavelength of 1060 nm and an isolation more than 20 dB. The CFBG with a dispersion parameter of 9 ps/nm at λ = 1060 nm was inserted into the laser cavity to provide chromatic dispersion in the wavelength-dependent AML condition. We used anomalous dispersion for the fiber laser cavity because it can suppress SPM-induced spectral broadening and create a narrower linewidth of the laser29. Additionally, the CFBG acts as an optical output coupler in the fiber laser cavity, which has a reflectivity of 60 ± 3%, and the operating spectral bandwidth of the CFBG is about 120 nm. An electro-optic intensity modulator (NIR-MX-LN-20, iXblue Inc.) was inserted to the laser cavity which has 3 dB bandwidth of 20 GHz for the high speed pulse modulation of 50 ps duration. A pulse generator (EPG-210, Alnair Lab) was used to modulate electro-optic intensity modulator with electrical pulse of 50 ps duration which is synchronized with frequency of the RF signal generator.

### Interferogram and PSF measurement

The interferogram and PSF measurement system was constructed based on a Michelson interferometer, which consists of four broadband 50/50 couplers (Supplementary Figure S5). The optical path difference was determined by translating the pair of optical lens (AC254-035-C-ML, Thorlabs) and mirror of one of the interferometer arms. The interference signal was detected by a balanced photodetector (PDB460C, Thorlabs), and 4096 samples were collected by every A-line with a 12-bit, 500 MS/s digitizer (ATS9360, AlazarTech, Inc.). Although the AML wavelength-swept fiber laser can be driven with linear sweeping signal, the system should be calibrated for high wavenumber linearity. The linearization in wavenumber domain was performed with a phase oriented fringe analyzing technique40,41. The interference signals were collected for every 0.25 mm optical path difference between the sample and reference arms. The optical phase was extracted from an arbitrary interference signal, and the calibration table was written as look-up table (LUT) to perform the wavenumber linearization in real-time. The maintenances of the PSF correspond to the OPD variation was confirmed. The PSF signals were collected for every 500-μm path difference after calibration.

### Optical coherence tomography system

The swept-source OCT system was constructed based on a Michelson interferometer which consisted of a broadband 70/30 coupler, couple of 50/50 couplers. The average power illuminated on the sample was low as 1.5 mW. The OCT interference signal was detected by a balanced photodetector (PDB460C, Thorlabs) and 4096 samples were collected via every A-line with a 12-bit, 500 MS/s digitizer (ATS9360, Alazartech Inc.).). The interfered signals were linearly sampled without k-clock and the spectrum interpolation for calibration to the k-space was performed on all A-line data prior to the fast Fourier transform (FFT) process. A tube lens is used to control diopter for each patient and the length of reference arm is automatically controlled with range of ±2  mm. An automatically controlled adjustable polarization controller is inserted in the reference arm to enhance image contrast and suppress the polarization dependent noise. The scan range was set as wide as 12 × 12 mm2 with an interval of 24 μm for both B- and C-scans such that one 3D OCT image was composed of equally spaced 500 × 500 × 4096 voxels. Considering the sweep rate of the 100 kHz laser, the frame rate of 2D image was 200 fps, and one 3D volume image was acquired within 2.5 s, which is faster than the normal human tear breakup time (TBUT). The axial resolution of the system was measured to be 6.3 μm in air.

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## Acknowledgements

This work was supported in part by “Research on Advanced Optical Science and Technology” grant funded by the GIST in 2018, the Industrial Technology Innovation Program, (N0002310, 10063062, 10063364) of the Ministry of Trade, Industry and Energy of Korea, and grant of the Korea Health Technology R&D Project through the Korea Health Industry Development Institute, funded by the Ministry of Health & Welfare (HI18C0858), the National Research Foundation of Korea (NRF) grant funded by Ministry of Science and ICT, Republic of Korea (No. NRF-2017R1A4A1015627).

## Author information

### Affiliations

1. #### Advanced Photonics Research Institute, Gwangju Institute of Science and Technology, Gwangju, 61005, South Korea

• Hwi Don Lee
• , Jun Geun Shin
•  & Tae Joong Eom
2. #### Department of Cogno-Mechatronics Engineering, Pusan National University, Busan, 46241, South Korea

• Gyeong Hun Kim
•  & Chang-Seok Kim

• Boram Lee

### Contributions

H.D.L. and G.H.K. equally contributed. H.D.L., G.H.K., B.L. and J.G.S. conceived and designed the study. All authors contributed to writing the manuscript. C.-S.K. and T.J.E. initiated and supervised the project. C.-.S. Kim. and T.J.E. are equally contributed.

### Competing Interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to Chang-Seok Kim or Tae Joong Eom.