Discrete optics in femtosecond-laser-written photonic structures

For α > 0, G(β_x d) moves with constant velocity α/λ towards +β_x d during propagation.

“Hence, G(β x d) moves for α > 0 with a constant velocity towards +β x d during the propagation, where the velocity is given by α/ - λ [35].”

• cites [35] Longhi S, 2006, Europhys. Lett. sourcePartly supported

  The passage on spectral coefficients (section labeled Abstract; page unavailable) gives c₁,₂(z, κ) = f₁,₂(Fz/h)g(κ − Fz/h), supporting a constant shift toward positive κ during propagation. The paper uses F and κ, however, and does not establish that the cited sentence’s G(βₓd) moves at the stated velocity α/λ. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

After every full period, an effective coupling constant c_eff = c J0(2πωA0/z0) can be defined.

“One can formally show [25,33,47] that after every full period z → z + z 0 , an effective coupling constant”

• cites [25] sourceNot checked — no full text of the source
• cites [33] sourceNot checked — no full text of the source
• cites [47] sourceSupported

Fs-laser-written waveguide arrays are today the only systems in which complex two-dimensional interfaces can be realized.

“This allows analysis of the influence of boundary and interfacing effects, making fs-laser-written waveguide arrays today the only systems where complex two-dimensional interfaces can be realized [93].”

• cites [93] sourcePartly supported

  The paper (p. 3) describes a femtosecond-laser-written array containing five types of wedge-like interface in one structure, supporting the claim that this method can realize complex two-dimensional interfaces. It does not say these arrays are the only systems capable of doing so. (LLM, conf=0.87)

  Experimental — accuracy not yet measured

The greater power needed for surface soliton formation relative to the bulk arises from the repulsive nature of the array surface, which the nonlinearity must additionally overcome.

“This is due to the repulsive character of the array surface [90], which the nonlinearity has to overcome in addition.”

• cites [90] sourceSupported

Arrays of evanescently coupled waveguides allow the dispersion and diffraction of propagating light to be tuned.

“arrays of evanescently coupled waveguides have been brought into focus as a particular representation of functionalized optical materials, in which the dispersion and diffraction of propagating light can be specifically tuned.”

Waveguide arrays exhibit a band structure with allowed propagation bands separated by photonic bandgaps.

“the reduced band structure with the allowed propagation bands, separated by photonic bandgaps, is shown.”

In 2004, waveguide arrays fabricated by femtosecond direct writing were demonstrated for the first time.

“In 2004, waveguide arrays fabricated by the femtosecond (fs) direct-writing method were demonstrated for the very first time [6,7].”

• cites [6] sourceNot checked — no full text of the source
• cites [7] Itoh K, 2006, MRS Bull. sourceNot checked — no full text of the source

The centre of the beams experiences a shift of about 30 waveguides, according to the coupling strength at a lattice site spacing of d=11 μm.

“The centre of the beams experiences a shift of about 30 waveguides, according to the coupling strength at the lattice site spacing of d = 11 μm.”

• cites [40] sourceNot checked — no full text of the source

For a narrow excitation of a single waveguide, light cannot distinguish left and right because the complex-conjugated wave packet is also a solution.

“In this case light can no longer distinguish between left and right, since also the complex-conjugated wave packet ϕ * m is a solution of equation (17).”

By that point, the beam has experienced a transverse shift proportional to the coupling coefficient c.

“the beam experienced until now a transverse shift (which is proportional to the coupling coefficient c).”

The guiding properties of individual waveguides strongly depend on the writing parameters.

“The guiding properties of the individual waveguides are strongly dependent on the writing parameters.”

At β_x d=π/2, transverse group velocity reaches its maximum due to the maximal absolute first derivative of the band structure.

“At this point, the transverse group velocity reaches its maximum due to the maximal absolute value of the first derivative ∂β z /∂β x of the band structure”

Due to band structure symmetry, propagation from π/α to 2π/α is similar to the first half period but with opposite sign of transverse group velocity.

“Due to the symmetry of the band structure the propagation from z = π/α to z = 2π/α is similar to the first half period but with opposite sign of the transverse group velocity”

Propagating to z=π/α, G reaches β_x d=π, the edge of the Brillouin zone, where transverse group velocity vanishes.

“Propagating further to z = π/α, G(β x d) reaches b x d = π which is the edge of the Brillouin zone, where the transverse group velocity vanishes”

At the beginning of the Bloch cycle, transverse group velocity vanishes because the band structure derivative is zero at β_x d = 0.

“At the beginning of the Bloch cycle one obtains a vanishing transverse group velocity, which arises from the vanishing derivative of the band structure at β x d = 0.”

With pulse duration and energy fixed, changing the writing velocity tunes the refractive index modulation.

“Keeping these parameters fixed, by only changing the writing velocity v a specific tuning of the refractive index modulation is achieved.”

Propagating from z=0 to z=π/(2α) moves G from β_x d=0 to β_x d=π/2.

“Following the propagation from z = 0 to z = π/(2α) corresponds to a motion in reciprocal space of G(β x d) from β x d = 0 to β x d = π/2”

The refractive index modulation decreases for increasing writing velocities because successive pulses overlap less.

“the refractive index modulation decreases for increasing writing velocities due to the smaller overlap between successive pulses, as shown in figure 7(b).”

Bragg reflection at the periodic structure changes transverse momentum, transferring G to β_x d=-π.

“The occurring Bragg reflection on the periodic structure changes the transverse momentum, which corresponds to a transfer of G(β x d) to the other side of the Brillouin zone β x d = -π”

For a broad excitation, a radius of curvature of R=1.9 m yields an oscillation period of 75 mm.

“A radius of curvature of R = 1.9 m yields to an oscillation period of 75 mm.”

• cites [40] sourceNot checked — no full text of the source

For writing velocities between about 500 and 1500 μm, waveguide losses measured by a cut-back method are approximately constant.

“for writing velocities of about 500 μm < v writing < 1500 μm the waveguide losses, measured by a cut-back method, are approximately constant (see figure 7(a)).”

Dynamic localization was experimentally realized in fs-laser-written waveguide arrays.

“The experimental realization of DL in fs-laser-written waveguide arrays is presented in figure 15.”

The lack of perfect refocusing is more pronounced after two periods, when imperfections and coupling effects have accumulated.

“This is even more pronounced after two periods, where imperfections and coupling effects have accumulated.”

Further propagation refocuses the light field at the edges of the Brillouin zone.

“Further propagation results in refocusing of the light field at the edges of the Brillouin zone.”

The dispersion relation of the first band in the binary lattice is β±z = ±√(σ2+4c2cos2β_x d).

“The corresponding dispersion relation of the first band can be calculated as”

Propagating further moves G from β_x d=-π/2 to the opposite band edge at β_x d=+π/2 in both minibands.

“Propagating further results in a motion of G(β x d) from β x d = -π/2 to the opposite band edge at β x d = +π/2 in both minibands”

Increasing the gap to σ = 0.41 c by reducing the writing velocity to 88 mm min−1 in every second waveguide decreases the tunnelling rate to about 25%.

“Increasing the gap between the bands further to σ = 0.41 c by reducing the writing velocity to 88 mm min -1 in every second waveguide decreases the tunnelling rate to ∼25%.”

The paper presents a generalized DL that softens the stringent resonance condition to achieve approximate DL over an extremely broad spectral region.

“Here, we present a generalized DL, in which the stringent resonant condition ( 23) is softened in order to achieve an approximated DL over an extremely broad spectral region.”

For σ=c/2, the band splits into two minibands separated by a gap of size ∼σ.

“in figure 12(d) the detuning is set to σ = c/2.As predicted, the band splits into two minibands separated by a gap with a size ∼σ .”

In comparison, a straight lattice exhibits strong diffraction for all spectral components.

“in figure 18(f) the spectrally resolved output pattern for the straight lattice is shown, exhibiting strong diffraction for all spectral components.”

Detuning the array with σ = 0.22 c corresponds to a Zener tunnelling rate of 65%.

“the array is detuned with σ = 0.22 c, corresponding to a tunnelling rate of 65%”

The experiments demonstrate that this device achieves DL over the entire visible spectral region from 450 nm to 700 nm.

“The experiments demonstrate that with such a device DL is achieved over the entire visible spectral region from 450 nm to 700 nm.”

Because the investigated systems have about 3% higher-order coupling and small imperfections, perfect refocusing is not obtained.

“Since the investigated systems feature slight higher-order coupling to sites beyond direct neighbours with a relative strength of 3% and also contain small imperfections inside the array, no perfect refocusing of the excitation is obtained.”

When a binary lattice is excited with a broad Gaussian beam with flat phase, the initial narrow spectral distribution is centred at β_x d=0.

“When exciting the lattice with a broad Gaussian beam with a flat phase, the resulting initial narrow spectral distribution G(β x d) is centred around β x d = 0”

A sample with two bending segments, meeting first-order and second-order DL conditions, was fabricated.

“Therefore, a sample was fabricated which exhibits a bending consisting of two segments where the first segment meets condition (23) for η 1 = 2.405 (first-order DL) and the second one for η 2 = 5.520 (second-order DL).”

Due to the acceleration theorem, G moves toward the band edge β_x d=π/2 during propagation.

“Due to the acceleration theorem, G(β x d) moves towards the band edge β x d = π/2 during the propagation.”

Because the lower miniband is occupied, the Bragg reflected part in the upper band interferes with the fraction that tunnels from the lower band to the upper band.

“Since the lower miniband is now occupied, the Bragg reflected part of the wave packet in the upper band (from β x d = π/2 to β x d = -π/2) interferes with the fraction that tunnels from the lower band (β x d = π/2) to the upper band (β x d = -π/2)”

In the 2D unsegmented array, light spreads across a significant part of the array for two excitation sites.

“The observed diffraction patterns in the unsegmented array for two different excitation sites are shown in figures 23(a) and (b).In both cases, light spreads across a significant part of the array.”

Increasing input peak power changes the phase relation between adjacent guides due to nonlinear refractive index changes, causing delocalization even though the nonlinearity is focusing.

“increasing the input peak power affects the sensitive phase relation between the adjacent guides due to the induced nonlinear change of the refractive index in the guides, resulting in delocalization of the output pattern, although the nonlinearity is focusing.”

In an unsegmented 1D array, light at 633 nm experiences strong diffraction and spreads across the array for central and edge excitations.

“In the unsegmented array, light at a wavelength of λ = 633 nm experiences strong diffraction and spreads across the array, as shown in the fluorescence images (figures 22(a) and (b)) for central and edge waveguide excitations.”

The power required for soliton formation in a 2D lattice is usually considerably higher than in the 1D case for the same fabrication parameters.

“the power required for soliton formation in a 2D lattice is usually considerably higher compared to the 1D case (for the same fabrication parameters).”

At P_peak = 1400 kW the linear coupling is reduced.

“At P peak = 1400 kW (see figure 29(b)) the linear coupling is reduced”

At P_peak = 500 kW, some light is trapped in the excited guide.

“at P peak = 500 kW input peak power the pattern already shows some light which is trapped in the excited guide.”

Intrinsic waveguide losses in the experiments are approximately 0.4 dB/cm.

“the intrinsic waveguide losses (approximately 0.4 dB cm -1 in our experiments)”

Preliminary numerical simulations indicate that the imaging method is applicable to large bandwidth signals and short pulses.

“preliminary numerical simulations have revealed that this imaging method is also applicable to large bandwidth signals and accordingly for short pulses as well.”

Localization is never perfect because of pulsed excitation, leaving some linear background in non-excited guides.

“there is always some linear background in the non-excited guides and localization will never be perfect due to the excitation with pulsed light.”

The segmentation-induced restoration is independent of array boundaries, as observed for central and edge excitation.

“This effect is independent of the presence of array boundaries, as can be observed for central as well as edge excitation”

Under segmentation, the light returns to the respective excited waveguides at the end facet.

“Under the influence of segmentation, the light has returned to the respective excited waveguides when it reaches the end facet, as shown in figures 23(e) and (f).”

At lower peak power almost all guided energy is coupled to adjacent waveguides due to linear coupling.

“in the lower peak power range almost all of the guided energy is coupled to the adjacent waveguides due to linear coupling”

For a 105 mm sample with d = 34 μm and z0 = 105 mm, at P_peak = 70 kW light dynamically localizes in the excited channel at the output facet.

“For P peak = 70 kW launched into a single waveguide light dynamically localizes in the excited channel at the output facet.”

The nonlinear refractive index is a function of writing velocity because writing modifies the molecular structure of the bulk material.

“since during the writing process the molecular structure of the bulk material is modified, also the nonlinear refractive index is a function of the writing velocity, as summarized in figure 24(b).”

At P_peak = 1030 kW, the light spreads significantly because the phase relation between adjacent guides is changed so DL is no longer observed.

“When the peak power is increased to P peak = 1030 kW, the light spreads significantly since the phase relation between adjacent guides is changed so that DL is no longer observed.”

In a periodically curved waveguide array, the suppression of linear diffraction due to DL is overcome by nonlinearity, so the beam spreads.

“The suppression of the linear diffraction due to DL is overcome by the nonlinearity, so the beam spreads.”

The threshold for soliton formation in a straight lattice is well below the threshold in the curved lattice.

“The threshold for soliton formation in this lattice is well below the threshold in the curved lattice.”

At P_peak = 1000 kW, the output intensity pattern is almost completely localized.

“at a peak power of P peak = 1000 kW the output intensity pattern is almost completely localized [73].”

• cites [73] sourceNot checked — no full text of the source

The power needed to form a soliton at the surface is considerably greater than that needed in the array bulk.

“The required power for soliton formation at the surface is considerably higher than in the bulk of the array.”

As power increases, light begins to penetrate into the hexagonal region at Ppeak = 1400 kW and Ppeak = 2000 kW.

“For increasing power the light starts to penetrate into the hexagonal region (figures 32(b) and (c), for P peak = 1400 kW and P peak = 2000 kW, respectively)”

The generalized dispersion relations for the square lattice and the hexagonal lattice differ.

“Importantly, the generalized dispersion relations (compare equation ( 7)) for the square lattice (figure 31(b)) and the hexagonal lattice (figure 31(c)) are different.”

The writing process is highly accurate, producing waveguide arrays with sharply defined boundaries.

“The high accuracy of the writing process results in waveguide arrays which exhibit sharply defined boundaries.”

When a waveguide in the first (interface) row of the square array is excited at low peak power Ppeak = 40 kW, the light is almost confined to the square region because it is reflected at the interface.

“When a waveguide in the first (interface) row of the square array is excited, for low peak power P peak = 40 kW the light is almost confined to the square region since it is reflected at the interface (figure 32(a)).”

Lattices of evanescently coupled waveguides are a unique discrete physical system, not only a model system for comparative investigation of discreteness.

“lattices of evanescently coupled waveguides are not just a simple model system for a comparative investigation of discreteness, but constitute a unique discrete physical system on their own as well [2].”

• cites [2] sourceNot checked — no full text of the source

In a continuous medium, a light beam undergoes normal diffraction and spreads during propagation.

“In a continuous medium, a light beam experiences normal diffraction and always spreads during propagation (figure 2(a)).”

In most relevant cases, the coupled mode approach is fully adequate and describes all observed propagation effects.

“in most relevant cases the coupled mode approach is fully adequate, describing all of the observed propagation effects.”

When ultrashort laser pulses are tightly focused into bulk material, nonlinear absorption causes optical breakdown and micro plasma formation, inducing a permanent change in molecular structure.

“When ultrashort laser pulses are tightly focused into the bulk material, nonlinear absorption takes place leading to optical breakdown and the formation of a micro plasma, which induces a permanent change in the material's molecular structure.”

Before femtosecond writing, available techniques had strong limitations for fabricating waveguide arrays with tailored shapes and functionalities in one and two dimensions.

“revealed strong limitations of the available techniques with respect to the fabrication of waveguide arrays with specifically tailored shapes and functionalities in one and two dimensions.”

• cites [5] sourceNot checked — no full text of the source

Femtosecond direct writing offers a versatile method for fabricating waveguide arrays with distinct properties and features.

“This technique offers a versatile method for the fabrication of waveguide arrays with distinct properties and features.”

• cites [6] sourceNot checked — no full text of the source
• cites [7] Itoh K, 2006, MRS Bull. sourceNot checked — no full text of the source

Weakly guiding evanescently coupled photonic waveguide structures provide an excellent model system for analyzing discrete systems.

“weakly guiding and evanescently coupled photonic waveguide structures provide an excellent model system for the analysis of the characteristics of discrete systems [1]”

• cites [1] sourceNot checked — no full text of the source

Experimental demonstration of discrete phenomena in waveguide arrays reveals fundamental properties of discreteness in general.

“Hence, the experimental demonstration of discrete phenomena in waveguide arrays reveals the fundamental properties of discreteness in general.”

• cites [1] sourceNot checked — no full text of the source

The Schrödinger equation for quantum wavefunction dynamics and the optical paraxial Helmholtz equation for field envelope evolution are equivalent under correspondences between time and distance, Planck constant and reduced wavelength, mass and bulk refractive index, and potential and refractive index change.

“The equivalence is evident when the correspondence between the characteristic values is borne in mind: time t and distance z, Planck's constant h = h/2π and reduced wavelength λ = λ/2π , mass m and refractive index of the bulk n 0 , and potential V and refractive index change n = n 0n(x, y, z).”

Structural changes from femtosecond writing are permanent and stable after fabrication.

“all structural changes are permanent and stable after fabrication.”

In a straight array, a plane wave does not change during propagation.

“In a straight array the plane wave does not change during propagation (figure 9(a)).”

During femtosecond writing, non-bridging oxygen hole centres are generated by breaking molecular bonds.

“During the writing process non-bridging oxygen hole centres (NBOHCs) are generated due to breaking of the molecular bonds.”

Curvature of individual waveguides can impose transverse gradients on the waveguide array.

“there is a much more sophisticated way to impose transverse gradients on the waveguide array, employing a curvature of the individual waveguides [32].”

• cites [32] sourceNot checked — no full text of the source

Because the Schrödinger equation and the paraxial Helmholtz equation are similar, waveguide arrays provide an exceptional model system in which wave dynamics are the same.

“Due to the similarity of the Schrödinger equation ( 1) and the paraxial Helmholtz equation (2), waveguide arrays provide an exceptional model system since the wave dynamics are the same in both systems.”

Employing curvature allows optical simulation of a constant transverse electrical field.

“Therefore, employing a curvature allows for the optical simulation of a constant transverse electrical field (z).”

As G moves through the band structure without changing shape, the spatial transverse profile of the beam remains unchanged.

“As G(β x d) moves through the band structure without changing its shape, the spatial transverse profile of the beam remains in its shape as well.”

Fluorescence microscopy allows direct monitoring of the light intensity distribution inside the samples.

“the light intensity distribution can be directly monitored inside the samples using a fluorescence microscopy technique [24,25].”

• cites [24] sourceNot checked — no full text of the source
• cites [25] sourceNot checked — no full text of the source

A transverse force q(z) in the Schrödinger equation is equivalent to a detuning of the local refractive index of the waveguides.

“In the optical setting, the transverse force q (z) is equivalent to a detuning of the local refractive index of the waveguides.”

The arc length passed by a plane wave in a curved array depends on the position of the individual guides.

“the arc length which is passed by the plane wave is now a function of the position of the individual guides.”

The challenge for observing Bloch oscillations in atomic lattices was that electron coherence lifetime is shorter than the oscillation time scale of 40 ps for an atomic lattice period of 10 Å and an electric field strength of 10^6 V m^-1.

“The challenge was to overcome the limit of a short electron coherence lifetime, which is lower than the oscillation time scale [1] of 40 ps for an atomic lattice period of 10 Å and an electric field strength of 10 6 V m -1 [31].”

• cites [1] sourceNot checked — no full text of the source
• cites [31] sourceNot checked — no full text of the source

Waveguides can be written along arbitrary paths because the only limiting factor in focus placement is the focal length of the writing objective.

“Such a guide can be written along arbitrary paths since the only limiting factor in the placement of the focus is the focal length of the writing objective [17].”

• cites [17] sourceNot checked — no full text of the source

In a slightly curved array, a plane wave propagates almost straight and is almost unaffected by the bending.

“If the waveguides are slightly curved, the plane wave will still propagate straight, being almost not affected by the bending (figure 9(b)).”

The transverse group velocity given by the derivative of the band determines the path of beam propagation.

“The transverse group velocity given by the derivative of the band determines the path of the beam propagation, which follows the band structure.”

Low focusing ensures that the created refractive index change is almost independent of writing depth, so large waveguide structures remain homogeneous in the vertical dimension.

“This relatively low focusing ensures that the created refractive index change is almost independent of the writing depth inside the material, so even large waveguide structures are still homogeneous in the vertical dimension.”

Bloch oscillations are accompanied by tunnelling to higher order bands, known as Zener tunnelling.

“Bloch oscillations come along with tunnelling to higher order bands [41], which is known as the Zener tunnelling [42].”

• cites [41] sourceNot checked — no full text of the source
• cites [42] sourceNot checked — no full text of the source

Optical Bloch oscillations in a waveguide array require a parabolic curvature x0 ∼ z^2 such that ẍ0 = const.

“The optical analogue in a waveguide array would then require a parabolic curvature x 0 ∼ z 2 of the array such that ẍ0 = const.”

The basic idea relies on the fact that the resonance condition has infinite solutions.

“The basic idea relies on the fact that equation ( 23) has infinite solutions as sketched in figure 18(a) [56].”

• cites [56] sourceNot checked — no full text of the source

Combining both resonances yields a strongly reduced effective coupling that effectively vanishes over a broad spectral region.

“A combination of both resonances would therefore yield a strongly reduced effective coupling, which effectively vanishes over a broad spectral region”

Truncated array structures cause considerable distortions of the self-imaging effect.

“truncated array structures lead to considerable distortions of the self-imaging effect.”

Observing dynamic localization is difficult in most quantum systems because the required field strengths are too high.

“the observation is difficult in most quantum systems as the required field strengths are too high.”

The main drawback of optical Bloch oscillations and dynamic localization for self-imaging is their small bandwidth of operation.

“The main drawback of these mechanisms is the rather small bandwidth of operation.”

A sufficiently broad beam experiences rapid tunnelling because the size of the tunnelling regions is set by the width of G(β x d).

“As the size of the tunnelling regions is given by the width of G(β x d), a sufficiently broad beam experiences rapid tunnelling.”

In the optical analogue, a transverse ac field can be mimicked by periodic bending of the waveguides.

“In the optical analogue, the transverse ac field can be mimicked by a periodic bending of the waveguides”

Near the resonances at η_j, the sign of the effective coupling alternates.

“in the vicinity of the resonances at η j the sign of the effective coupling alternates.”

Dynamic localization is exact only when coupling is restricted to immediate neighbours.

“DL is only exact for coupling restricted to the immediate neighbours.”

For a periodic ac field, the entire band collapses, preventing the wave packet from spreading over the lattice.

“for a periodic ac field, the entire band collapses, preventing the wave packet from spreading over the lattice.”

Dynamic localization is based on fundamentally different mechanisms from Bloch oscillations.

“DL is based on fundamentally different mechanisms.”

In a finite 1D array, the propagation equation is fulfilled by a set of M propagation-invariant orthonormal supermodes with the given shape and propagation constant.

“One can show [55] that this equation is fulfilled by a set of M propagation invariant orthonormal supermodes”

• cites [55] sourceNot checked — no full text of the source

At z0 the effective coupling vanishes for the resonance condition ωA0 z0 = η_j, where η_j is a root of J0.

“at z 0 the effective coupling vanishes for the resonance condition”

This is in strong contrast to Bloch oscillations, which occur for every transverse constant field.

“This is in strong contrast to Bloch oscillations, which are obtained for every transverse constant field.”

2D discrete solitons exhibit unusual features with no counterpart in the continuous regime.

“Hence, 2D discrete solitons exhibit unusual features, with no counterpart in the continuous regime [5].”

• cites [5] sourceNot checked — no full text of the source

The phase-shift-induced image reconstruction can be extended to a 2D square lattice as long as diagonal coupling is negligible.

“The concept of phase shift-induced image reconstruction can be extended to the 2D square lattice, since it can be described by the superposition of horizontal and vertical 1D lattices as long as the impact of coupling between waveguides along the diagonal is negligible.”

• AssumptionRests on an assumption: A two-dimensional square waveguide lattice can be described as a superposition of horizontal and vertical one-dimensional lattices.
• AssumptionThe authors state this as an assumption: Coupling between waveguides along the diagonal in the two-dimensional square lattice is negligible.

In the curved lattice, solitons form only at much higher peak power than in straight lattices because bending alters the phase relation between adjacent guides.

“solitons form only for much higher peak power compared to their counterparts in straight lattices due to the altered phase relation between adjacent guides which is caused by the bending.”

The additional phase shift from the nonlinear term moves the propagation constant of the excited guide into the semi-infinite gap above the first band, isolating it from the rest of the array.

“The additional phase shift induced by the nonlinear term γ |ϕ m | 2 ϕ m in equation ( 27) moves the propagation constant of the excited guide into the semi-infinite gap above the first band as shown in figure 25(a).Hence, this guide cannot 'communicate' with the rest of the array, so it forms an isolated state and the light remains 'trapped'.”

Experimental observation of surface waves at natural material interfaces was severely limited by unrealistically high power requirements.

“the progress in their experimental observation was severely limited because of unrealistically high power levels required for a surface wave excitation at the interfaces of natural materials.”

Shallow refractive index variations of a waveguide array support the stability of 2D solitons.

“one can show that the shallow refractive index variations of a waveguide array support the stability of 2D solitons [3].”

• cites [3] sourceNot checked — no full text of the source

In some materials with negative γ, the propagation constant can decrease into the finite gap between first and second bands, forming discrete gap solitons.

“In some materials, under particular conditions, γ may be negative, so the propagation constant can be decreased until it reaches the finite gap between the first and the second band, resulting in the formation of discrete gap solitons [72].”

• cites [72] sourceNot checked — no full text of the source

The effective nonlinearity can be tuned to tailor nonlinear propagation properties.

“Thus, the effective nonlinearity can be specifically tuned in order to tailor the nonlinear propagation properties.”

The formation of discrete surface solitons is subject to different physical factors than discrete solitons without boundary interaction.

“The formation of discrete surface solitons is subject to somewhat different physical factors than for discrete solitons without any boundary interaction.”

The segmented region is sufficiently small compared to the coupling length (~15 mm) to satisfy the abrupt phase shift condition with reasonable accuracy, as confirmed by simulations.

“Obviously, the segmented region is sufficiently small compared to the coupling length l c = π/2c ≈ 15 mm to satisfy the condition of an abrupt phase shift with reasonable accuracy, as is confirmed by the agreement with the respective simulations”

The image resolution is limited only by the number of waveguides in the array.

“The resolution of the image is only restricted by the number of waveguides in the array.”

Assuming a sample of length L and introducing the phase shift at L/2, diffraction in the first half is reversed in the second half, so the initial intensity distribution is imaged on the output facet.

“assuming a sample of length L and introducing the phase shift at position L/2, the diffraction occurring in the region z < L/2 is reversed in the region z > L/2, so the initial intensity distribution |ϕ(0)| 2 is imaged on the output facet”

The critical power for soliton formation can be roughly estimated as |φ_m|^2_crit = 2c/γ.

“the critical power for soliton formation can be roughly estimated by |ϕ m | 2 crit = 2c/γ .”

Experimental data and simulation show excellent agreement in the dependence of amplitudes in individual waveguides.

“A comparison of the experimental data with simulation is shown in figure 27, which still exhibits an excellent agreement in the dependence of the amplitudes in the individual waveguides.”

The shallow refractive index modulation in waveguide arrays facilitates nonlinear optical surface waves, allowing observation at moderate power levels at the edge of semi-infinite arrays in 1D and 2D.

“the relatively shallow refractive index modulation in waveguide arrays facilitates the formation of nonlinear optical surface waves significantly and they can be observed already at moderate power levels at the edge of semi-infinite arrays in the 1D [85,86] as well as in the twodimensional case [87][88][89].”

• cites [85] sourceNot checked — no full text of the source
• cites [86] sourceNot checked — no full text of the source
• cites [87][88][89]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In a finite lattice, the required periodicity is approximately satisfied when boundary influence on mode propagation can be neglected.

“The required periodicity is also approximately satisfied in a finite lattice, when the influence of the boundary on the propagation of the mode can be neglected.”

• AssumptionThe authors state this as an assumption: In a finite waveguide lattice, the influence of the boundary on the propagation of the mode can be neglected.

Introducing a π phase shift in every second waveguide causes pairwise conversion of individual supermodes.

“introducing a phase shift of π in every second waveguide results in a pairwise conversion of the individual supermodes”

The femtosecond laser writing technique is ideally suited for investigating surface phenomena.

“and proves that the fs laser writing technique is ideally suited for studying surface phenomena.”

The fs writing technique allows processing a variety of materials that exhibit other kinds of nonlinearity, in particular a χ(2) nonlinearity, as found for example in LiNbO3.

“Although all examples in this tutorial have been restricted to fused silica, the fs writing technique allows us to process a variety of materials [94][95][96][97], which exhibit other kinds of nonlinearity, in particular a χ (2) nonlinearity, as can be found e.g. in LiNbO 3 [98][99][100].”

• cites [94][95][96][97]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [98][99][100]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Writing waveguides along arbitrary paths with high precision enables the implementation of external potentials by curving the guides.

“The additional possibility of writing the waveguides along arbitrary paths with high precision allows the implementation of external potentials by curving the guides.”

The sharply defined boundaries guarantee that the output pattern lacks scattering or noticeable statistical distortions from small waveguide displacements or inhomogeneities.

“This ensures the absence of scattering or noticeable statistical distortions of the output pattern caused by small waveguide displacements or inhomogeneities”