Patent Description:
<CIT>, <CIT> and <CIT> describe optical coupling apparatus comprising a first waveguide coupled to a second waveguide, so as to transfer a photon received in the first waveguide to the second waveguide. <CIT> describes a coupled system configured to adiabatically couple light between a first waveguide and a second waveguide. <CIT> describes a photonic integrated circuit comprising first and second waveguides. <CIT> describes a system for generating clock signals for a photonic quantum computing system. <CIT> describes methods and systems for synchronizing a first clock with a second clock.

The invention concerns an integrated photonics vertical coupler according to claim <NUM> and a method according to claim <NUM>.

Further, optional features are defined by the dependent claims appended thereto. Systems and methods for an integrated photonics vertical coupler are provided herein. In certain embodiments, a device includes a first waveguide having a first photon and a second photon propagating therein, wherein the first photon and the second photon are propagating in orthogonal modes. Further, the device includes a second waveguide having a second coupling portion in close proximity with a first coupling portion of the first waveguide, wherein a physical relationship between the first waveguide and the second waveguide along the length of the second coupling portion causes an adiabatic transfer of the first photon and the second photon into distinct orthogonal modes of the second waveguide at different locations in the second coupling portion.

Understanding that the drawings depict only some embodiments and are not therefore to be considered limiting in scope, the exemplary embodiments will be described with additional specificity and detail using the accompanying drawings, in which:.

In accordance with common practice, the various described features are not drawn to scale but are drawn to emphasize specific features relevant to the example embodiments.

In the following detailed description, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration specific illustrative embodiments. However, it is to be understood that other embodiments may be utilized and that logical, mechanical, and electrical changes may be made, whilst remaining within the scope of the invention as defined by the claims.

Systems and methods for an integrated photonics source and detector of entangled photons are provided herein. In certain embodiments, hardware is described herein that enables methods for precise and secure synchronization of optical atomic clocks using the quantum interference of time-entangled photons. For example, the optical atomic clocks on orbiting satellites may be precisely and securely synchronized. Deployed across a swarm of LEO/MEO satellites, embodiments described herein may enable improved modalities of signal intelligence based on the coherent combination of distributed radio or optical apertures, including real-time computational interferometry for increased sensitivity to weak signals, and active beam forming radar/imaging for increased covertness by reducing both signal spillover and time-on-target.

Additionally, clock synchronization schemes, described herein, may use a chip-scale, ultra-high-flux source and interferometer for time-energy entangled bi-photons, with a reduced size, weight, and power, high pair production rate, and high flux-to-background ratio for entangled photon pairs. Also, for increased size, weight, and power reduction and improved deployability in small satellite platforms, devices described herein may be integrated onto a chip. In particular, both a photon source and interferometric detector may be integrated onto a chip.

In certain embodiments, entangled photons may be generated through a spontaneous parametric degenerate down-conversion of pump photons, also known as degenerate difference frequency generation. Typically, the above method for photon generation may yield entangled photons that have orthogonal polarizations to one another. Typically, free-space optics are used to separate the entangled photons and convert them into the same polarization state for use within a clock synchronization scheme. Embodiments described herein provide a chip-scale photonic integrated circuit having on-chip guided wave photonics for separating the entangled photons and converting the separated photons into the same polarization state.

In some embodiments, a chip-scale photonic integrated circuit may produce and interfere time-entangled photons. The chip-scale photonic integrated circuit may realize the optical functions for producing and interfering the photons on a hybrid optical waveguide platform which combines the nonlinear properties of periodically poled potassium titanyl phosphate (ppKTP) waveguides or waveguides made from similar material to ppKTP with the low transmission loss, high confinement, and filtering capabilities of silicon nitride waveguides or other waveguides made from similar material to silicon nitride. The chip-scale approach using the combination of waveguides made from different materials enable improvements over previous types of sources based in fiber and free-space optics.

In some embodiments, materials that have both nonlinear properties and low transmission loss, high confinement, and filtering capabilities could be used to implement similar optical functions for producing and interfering the photons in an optical waveguide platform based on a single material system, such as lithium niobate.

In certain embodiments, the optical functionality for producing and receiving entangled photons is implemented on a single, integrated platform, yielding reduced optical losses, enhanced mode overlap, efficient filtering of photons, increased interferometer contrast, and improved mechanical robustness, all while reducing size, weight, and power when compared to fiber or free space based systems. Additionally, embodiments described herein permit higher precision time synchronization when used in a system while enabling usage on smaller satellite platforms, such as microsats.

<FIG> is a diagram illustrating a system <NUM> for a Hong-Ou-Mandel (HOM) interferometer. As used herein, the chip scale integrated circuit may be used within a HOM interferometer. As used herein, a HOM interferometer is a device that may produce a pump photon <NUM>. The system <NUM> may split the pump photon into two daughter photons <NUM> (referred to separately herein as photons <NUM>-A and <NUM>-B). For example, the pump photon <NUM> may be produced by a laser source that produces a laser having a wavelength of <NUM> or other desired wavelength.

In certain embodiments, the pump photon <NUM> is split into daughter photons <NUM> that are guided through optical structures for recombination. For example, the pump photon <NUM> is split by optical structure <NUM> into daughter photons <NUM>-a and <NUM>-b. The daughter photons <NUM> may each have a wavelength that is twice the wavelength of the pump photon <NUM> (i.e., where the pump photon <NUM> could have a wavelength of <NUM>, the daughter photons <NUM> may each have a wavelength of <NUM>). Additionally, the system <NUM> may include guiding optics <NUM> that guide the daughter photons <NUM> to beamsplitter <NUM>, upon which the daughter photons <NUM> are combined, such that quantum superpositions <NUM>-c and <NUM>-d of the daughter photons impinge on detectors <NUM> for reception. For example, a detector <NUM>-a may receive and detect the daughter photon <NUM>-a and the detector <NUM>-b may receive and detect the daughter photon <NUM>-b; or detector <NUM>-a may receive and detect the daughter photon <NUM>-b and the detector <NUM>-b may receive and detect the daughter photon <NUM>-a or detector <NUM>-a may receive and detect both daughter photons <NUM>-a and <NUM>-b; or detector <NUM>-b may receive and detect both daughter photons <NUM>-a and <NUM>-b, in the manner of a HOM interferometer.

In some embodiments, when the detectors <NUM> receive the associated daughter photons <NUM>, the detectors <NUM> may provide the signals to an electronic correlator device <NUM>, where the electronic correlator device <NUM> combines the electrical signals of the two detectors <NUM> for the performance of HOM interferometry. The electronic correlator device <NUM> quantitively determines the degree of temporal correlation of the signals produced by the detectors <NUM>. For example, the electronic correlator <NUM> may show that the coincidence rate of the signals provided by the photodetectors <NUM> may drop towards zero when the daughter photons <NUM> overlap substantially perfectly in time. This drop towards a zero rate of coincident detections is known as the HOM dip illustrated in the trace graph <NUM>. The dip occurs when the two daughter photons <NUM> are substantially identical in all properties. When the photons <NUM> become distinguishable, including and especially in regards to the equality of their times-of-flight between the source region <NUM> and the beam splitter <NUM>, the HOM dip disappears. In this way the system <NUM> is sensitive to the quality of the times-of-flight of the daughter photons 103between the source region <NUM> and the beam splitter <NUM> being substantially perfectly equal.

<FIG> illustrates different optical paths <NUM> and <NUM> on a chip-scale device <NUM> that is both capable of generating a photon, splitting the photon into daughter photons, providing the daughter photons as outputs (such as into free space or optical fibers), receiving the daughter photons which may have been reflected from remote mirrors or optical systems, and providing the received photons to an interferometer for performing HOM interferometry. As shown, <FIG> illustrates a source path <NUM> and an interferometer path <NUM>. In the source path <NUM>, an incoming pump photon is split into daughter photons which may be separated and directed to different remote platforms,. In the interferometer path <NUM>, the daughter photons reflected by the remote platforms are received and interfered in the manner of HOM interferometry.

In certain embodiments, the chip-scale device <NUM> utilizes the nonlinear optical effect of degenerate spontaneous parametric down conversion (dSPDC), in which a pump photon <NUM> splits into two "twin" daughter photons <NUM> and <NUM> that are "born" at nearly the same instant (e.g., within <<NUM> femtoseconds of one another). This simultaneity, enforced by quantum mechanics, may be exploited for synchronizing separated atomic clocks. To synchronize the separated atomic clocks, (i.e., when the different atomic clocks are located on different satellites) the synchronization is achieved by projecting daughter photons <NUM> and <NUM> from the chip-scale device <NUM>, reflecting some of the photons <NUM> and <NUM> from each of the satellites, and providing them for recombining in a Hong-Ou-Mandel (HOM) interferometer <NUM>, in which a purely quantum mechanical interference "dip" in the coincidence rate is observed only when the paths are substantially exactly equal as described above with respect to <FIG>. The arrival times of some of the entangled photons from each satellite may be compared over a classical channel, enabling controllers to synchronize the clocks with great precision (i.e., potentially with femtosecond precision).

In some embodiments, the chip scale device <NUM> is a chip-scale photonic integrated circuit that produces and interferes time-entangled photons. The chip-scale device <NUM> may include optical functions and components on a hybrid optical waveguide platform which combines the nonlinear properties of ppKTP waveguides (or other waveguides made from materials having similar properties) with the high confinement and filtering capabilities of silicon nitride waveguides. This combination permits miniaturization, efficiency, robustness, while increasing the useable flux of twin-photons <NUM> and <NUM>.

In some embodiments, the chip-scale device <NUM> may include optical functions and components on a single optical waveguide material platform that has both nonlinear properties and low transmission loss, high confinement, and filtering capabilities, such as lithium niobate.

In certain embodiments, the chip scale device may generate a pump photon <NUM>, and from the pump photon <NUM> in the source path <NUM>, and may generate, by dSPDC, daughter photons 206a and 206b in the photon producing waveguide. Each of the twin photons 206a and 206b may occupy a different waveguide mode, either Transverse Electric (TE), or Transverse Magnetic (TM). A Vertical Coupler (VC) region may adiabatically draw the daughter photons 206a and 206b out of the photon producing waveguide and into a photon conditioning waveguide patterned on top of the photon producing waveguide. Additionally, the TM and TE photons may be separated by two diffractive waveguide mode splitters (MS). The TE photon may then pass through a bandpass filter (BPF) to reject background photons, through a second MS, then may leave the chip <NUM> as emitted photon <NUM>. Meanwhile, the original TM photon may be converted into a TE mode by a diffractive mode converter (MC), which may also reverse the direction of propagation of the photon. This (now TE polarized) photon may pass through its own bandpass filter and leave the chip <NUM> as emitted photon <NUM>. The various functions performed on the chip may be performed by a photon conditioning waveguide (in some embodiments, made from silicon nitride or other similar material), where waveguide structures are patterned in a film deposited on top of the substrate containing the photon producing waveguide.

In additional embodiments, the interferometer path <NUM>. the twin-photons <NUM> and <NUM> may be reflected or sent back from remote satellites or other remote systems and are recoupled into the photonics component waveguides on the chip-scale device <NUM> to complete an HOM interferometer <NUM>. (In some implementations, the photons may also have their polarizations rotated by <NUM> degrees by conventional waveplates). Although the twin photons <NUM> and <NUM> may re-enter the same waveguides from which they were earlier emitted, because of their now rotated polarizations, they may couple into the orthogonal waveguide mode (i.e., TM). Each photon then may interact with a diffractive mode splitter (MS) that may reverse the direction of propagation in the waveguide, sending the photons <NUM> and <NUM> to the <NUM>/<NUM> waveguide coupler. The output ports of the interferometer may be directed onto photon detectors <NUM>, such as single-photon avalanche photodetectors (SP-APDs), where the photons <NUM> and <NUM> may be detected. The detected signal outputs of the photon detectors <NUM> may be directed to an electronic correlator <NUM>, which may determine the degree of coincidence of the arrive times of the signals, thus completing an HOM interferometer <NUM>.

<FIG> illustrate the propagation of the two photons produced by the photon producing waveguide, into a photon vertical coupling waveguide, and through the photon conditioning waveguide network. As discussed above, the photon producing waveguide produces two photons having orthogonal waveguide modes: one mode propagating in the TM mode and the other propagating in the TE mode. Depending on the mode of the photon, the photons propagate along different paths through the waveguide network, such that the waveguide network provides two photons propagating in the TE mode off of the chip and receives two photons back onto the chip, propagating in the TM mode. <FIG> illustrates the path of the photon originally in the TE mode of the photon vertical coupling waveguide <NUM>. <FIG> illustrates the path of the photon originally in the TM mode of the photon vertical coupling waveguide <NUM>. <FIG> illustrates the path of the photons through the photon conditioning waveguide network <NUM> that are received from external devices.

In certain embodiments illustrated in <FIG>, the photon in the TE mode of the photon vertical coupling waveguide <NUM>, passing into the photon conditioning waveguide network <NUM>, passes through a mode splitter <NUM> without diffraction. Then the photon passes through a bandpass filter <NUM>, which filters out fluorescence, as well as stray pump light coupled from the photon producing waveguide <NUM>. The photon then passes through the mode splitter <NUM> without diffraction and is emitted through the output port <NUM>.

In certain embodiments illustrated in <FIG>, the photon in the TM mode of the photon vertical coupling waveguide <NUM>, passing into the photon conditioning waveguide network <NUM>, is diffracted by the mode splitter <NUM>. The photon is further diffracted by the mode splitter <NUM>, whereupon the photon enters the mode converter <NUM>. The mode converter <NUM> again diffracts the photon but converts the photon from the TM mode into the TE mode. As the photon is now in the TE mode, the photon is not diffracted by the mode splitter <NUM>. The photon then passes through the bandpass filter <NUM>, which filters out fluorescence, as well as stray pump light coupled from the photon producing waveguide. The photon then passes through the mode splitter <NUM> without diffraction and is emitted through the output port <NUM>.

In additional embodiments illustrated in <FIG>, the two daughter photons emitted from the photon conditioning waveguide network <NUM> may be sent back from another optical device such that they are recoupled into the photon conditioning network <NUM> in TM modes at the waveguides <NUM> and <NUM>. The two received photons in the TM modes may propagate into the waveguides to the mode splitters <NUM> and <NUM> respectively. Both mode splitters <NUM> and <NUM> diffract the received photons. The photons then are interfered with one another via a <NUM>/<NUM> coupler <NUM> before being output on ports <NUM> and <NUM> for subsequent detection by photon detectors. In embodiments discussed above, the TM mode from the photon vertical coupling waveguide <NUM> is converted by the photon conditioning waveguide network <NUM> into a TE mode for transmission out of the chip-scale device, while the light received back into the device for subsequent interferometric detection is in the TM mode. However, in another embodiment, the TE mode from the photon vertical coupling waveguide is converted by the photon conditioning waveguide network <NUM> into a TM mode for transmission out of the chip-scale device, while the light received back into the device for subsequent interferometric detection is in the TE mode.

<FIG> illustrates the different photonics components within the chip-scale device <NUM>. For example, the chip scale device <NUM> includes a photon producing waveguide <NUM>, a photon vertical coupling waveguide <NUM>; and a photon conditioning waveguide network (similar to the photon conditioning waveguide network <NUM> in <FIG>) comprising mode splitters, <NUM>, <NUM>, <NUM>, and <NUM>; mode converter <NUM>; bandpass filters <NUM> and <NUM>; input/output waveguides <NUM>, <NUM>, <NUM>, and <NUM>; and <NUM>/<NUM> coupler <NUM>. Possible embodiments for the vertical coupler <NUM>; mode splitters <NUM>, <NUM>, <NUM>, and <NUM>; mode converter <NUM>, and bandpass filters are described in greater detail below.

<FIG> is a side view diagram illustrating the operation of a vertical coupler. To efficiently couple photons out of photon producing waveguide <NUM> into the photon vertical coupling waveguide <NUM>, a stacked waveguide is formed. Further, a relatively thin photon vertical coupling waveguide <NUM> in relation to the width of the photon producing waveguide <NUM> has little perturbation on the shape of the weakly confined modes in the photon producing waveguide <NUM>. As discussed herein, the photon vertical coupling waveguide <NUM> is gradually widened throughout the overlapping portions of the stacked waveguide. For example, the photon vertical coupling waveguide <NUM> may widen from <NUM> to <NUM> over a distance of ~<NUM> microns. The gradual widening of the photon vertical coupling waveguide <NUM> adiabatically draws the photons from the photon producing waveguide <NUM> into a much more tightly confined waveguide mode. Additionally, the transfer preserves the polarization modes of the propagating photons (i.e., TE→ TE, and TM→TM), with essentially zero mode cross-coupling. As such photons propagating in different modes may be coupled out of the photon producing waveguide <NUM> at different locations. Accordingly, as different modes may be coupled from out of the photon producing waveguide <NUM> at different locations, the vertical coupler may be implemented in other applications such as in a mode splitter and the like.

In further embodiments, the material used to produce the photon producing waveguide and the material used to produce the photon vertical coupling waveguide may have a large difference between their respective indexes of refraction. For example, where KTP is used for the photon producing waveguide, the photon vertical coupling waveguide may be made using silicon enriched nitride films.

<FIG> is a diagram illustrating certain aspects of a mode splitter as found in the chip-scale device <NUM>. In particular, <FIG> shows an isometric view <NUM> of a mode splitter, a detailed isometric view <NUM> of a portion of the mode splitter, and a frequency response graph <NUM> of the coupling of the different modes within the mode splitter.

In certain embodiments, as shown in the isometric view <NUM>, the mode splitter may include a single input port <NUM>. Through the input port the mode splitter may receive two photons as an input <NUM> that are propagating in different orthogonal modes within the waveguide. For example, one photon may be propagating in the TE mode and another photon may be propagating in the TM mode. The mode splitter may pass one of the received photons at the input port <NUM> through to the output port <NUM> as an output photon <NUM>. For example, the mode splitter may pass the TE mode photon received at the input port <NUM> directly through to the output port <NUM>. Additionally, the mode splitter may diffract one of the propagating photons so that one of the propagating photons is coupled into a contra-directional waveguide and passed through to the output port <NUM> as output <NUM>. For example, the TM mode may be diffracted by a coupling portion <NUM> of the mode splitter and passed to the output port <NUM>.

In some embodiments, as shown in the detailed isometric view <NUM> of the coupling portion <NUM> of the mode splitter, to split the two orthogonally polarized photons into different paths, the mode splitter may include a chirped-grating-assisted contra-directional mode coupler. As shown, view <NUM> depicts the waveguide structure and graph <NUM> shows the results of a calculation of its spectral response. As shown, the coupling portion consists of two closely spaced waveguides <NUM> and <NUM>. The waveguide <NUM> may further be patterned with a modulated sidewall <NUM>, thus, creating an in-waveguide diffraction grating which has a large overlap integral for the TM-to-TM transition from one waveguide to the other. The effect of the modulation is to couple the TM mode from the forward direction in the waveguide <NUM> to the backward direction in the waveguide <NUM>; whereas the TE mode passes through the mode splitter in the forward direction, remaining in waveguide <NUM>. Additionally, the frequency of the modulated sidewall <NUM> may change along the length of the mode splitter, to allow for a desired frequency response for the mode splitter.

<FIG> is a diagram illustrating certain aspects of a mode converter as found in the chip-scale device <NUM>. In particular, <FIG> shows an isometric view <NUM> of a mode splitter, a detailed isometric view <NUM> of a converting portion of the mode splitter, and a frequency response graph <NUM> of the conversion of the modes within the mode converter.

In certain embodiments, as shown in the isometric view <NUM>, the mode converter may include a single port <NUM>. Through the port <NUM> the mode converter may receive a photon as an input <NUM> that is propagating in a particular mode within the waveguide. For example, the photon received through the port <NUM> may be propagating in the TM mode. The mode converter may convert the mode from one mode into an orthogonal mode within a converting portion <NUM>, where the mode converter converts the photon into an orthogonally propagating mode to be output through the port <NUM> as an output. For example, when the photon received on the port <NUM> is in the TM mode, the photon output through the port <NUM> may be in the TE mode.

In some embodiments, as shown in the detailed isometric view <NUM> of the converting portion <NUM> of the mode converter, to make all the waveguide paths as similar as possible for the two photons, the chip-scale device may flip the in-waveguide polarization of the TM photon using a single waveguide grating structure designed with asymmetrically modulated sidewalls <NUM> and <NUM>. For example, the modulation of the sidewalls may be out of phase with each other such that the transverse cross-section of the waveguide along the length of the modulation is constant. This asymmetric modulation creates a cross coupling between the TM mode in the forward direction and the TE mode in the backward direction. As shown in the graph <NUM>,Error! Reference source not found. mode conversion only occurs within the stopband of the grating. To control the stopbands of the grating, the length of the converting portion <NUM> may be changed along with the modulation frequency of the modulated sidewalls <NUM> and <NUM>. For example, the frequency of the modulated sidewalls may either decrease or increase along the length of the converting portion of the mode converter.

<FIG> is a diagram illustrating certain aspects of a bandpass filter as found in the chip-scale device <NUM>. In particular, <FIG> shows an isometric view <NUM> of a bandpass filter, a detailed isometric view <NUM> of a filtering portion of the bandpass filter, and a frequency response graph <NUM> of the filtering of photons by the bandpass filter.

In certain embodiments, as shown in the isometric view <NUM>, the bandpass filter may include a single port <NUM>. Through the port <NUM>, the bandpass filter may receive a photon as an input <NUM> that is propagating in a particular mode within the waveguide. For example, the photon received through the input port <NUM> may be propagating in the TE mode. The bandpass filter may filter photons having unwanted wavelengths in a filtering portion <NUM> and provide the filtered photons as output <NUM> through the output port <NUM>.

In some embodiments, as shown in the detailed isometric view <NUM> of the filtering portion <NUM> of the bandpass filter, to reject any background fluorescence photons that may be propagating in the waveguides, as well as to reject any residual pump photons, a waveguide bandpass filter is implemented. As shown, the filter is made from two high reflectivity waveguide gratings <NUM> and <NUM> that manifest by a chirp in the modulation period along the length of the waveguide, in other words, the modulation of the waveguide gratings symmetrically, longitudinally varies along the length of the sidewalls of the filters. Light just outside of the passband is diffracted back down the waveguide, while light at the pump wavelength is scattered out of the waveguide entirely. In some embodiments the spectral location of the waveguide gratings <NUM> and <NUM> may change along the length of the filtering portion <NUM> of the passband.

<FIG> is a method <NUM> of using a chip-scale device to produce and interfere pairs of correlated photons, as described above. The method <NUM> proceeds at <NUM>, where a pair of photons are generated in a photon producing waveguide. Additionally, the method <NUM> proceeds at <NUM>, where the pair of photons is coupled into a photon vertical coupling waveguide. Further, the method <NUM> proceeds at <NUM>, where one of the photons in the pair of photons is converted in a photon conditioning waveguide network so that photons are propagating in identical modes in two different waveguides. In certain embodiments, the method <NUM> proceeds at <NUM>, where the photons are provided to one or more external devices. Further, the method <NUM> proceeds at <NUM>, where the photons are received from the one or more external devices. Additionally, the method <NUM> proceeds at <NUM>, where interferometry is performed on the received photons.

<FIG> is a method <NUM> for vertically coupling two photons from a first waveguide into a second waveguide. The method <NUM> proceeds at <NUM>, where a first photon and a second photon are generated in a first waveguide in a first waveguide layer. Further, the first photon and the second photon may be in different modes that are orthogonal to one another. For example, the first photon may be propagating in the TE mode and the second photon may be propagating in the TM mode. Additionally, the method <NUM> proceeds at <NUM>, where the first photon is coupled from the first waveguide into a second waveguide at a first location within a coupling portion of the second waveguide. Moreover, the method <NUM> proceeds at <NUM>, where the second photon is coupled from the first waveguide into the second waveguide at a second location distinct from the first location within the coupling portion. For example, the first photon and the second photon are coupled into one of the first location and the second location based on the mode of propagation within the first waveguide.

Claim 1:
An integrated photonics vertical coupler comprising:
a first waveguide (<NUM>) formed in a first waveguide layer, configured to receive a pump photon (<NUM>) and to generate a first photon (206a) and a second photon (206b) within the first waveguide (<NUM>) by spontaneous parametric degenerate down-conversion of the pump photon, wherein the first photon and the second photon will propagate in orthogonal modes; and
a second waveguide (<NUM>) formed in a second waveguide layer patterned on the first waveguide layer, the second waveguide having a second coupling portion in close proximity with a first coupling portion of the first waveguide (<NUM>), wherein a physical relationship between the first waveguide (<NUM>) and the second waveguide (<NUM>) along the length of the second coupling portion is suitable to cause an adiabatic transfer of the first photon and the second photon into distinct orthogonal modes of the second waveguide (<NUM>) at different locations in the second coupling portion;
wherein the physical relationship comprises changing a width of the second waveguide (<NUM>) along the length of the second coupling portion.