Exotic light dynamics around a fourth order exceptional point
aa r X i v : . [ phy s i c s . op ti c s ] A ug Exotic light dynamics around a fourth order exceptional point
Sibnath Dey, Arnab Laha and Somnath Ghosh ∗ Unconventional Photonics Laboratory, Department of Physics,Indian Institute of Technology Jodhpur, Rajasthan-342037, India
The physics of exceptional point (EP) singularities, has been a key to a wide range of unique phys-ical applications in open systems. In this context, the mutual interactions among four coupled statesaround a fourth-order EP (EP4) in a physical system is yet to be explored. Here, we investigatethe unique features of an EP4 in a fabrication feasible planar optical waveguide with a multilayergain-loss profile based on only two tunable parameters. A unique ‘fourth-order β -switching’ phe-nomenon due to quasi-static gain-loss variation around EP4 has been explored. An exclusive chirallight dynamics following the dynamical variation of the gain-loss profile has been reported for thefirst time, which enables a special type of asymmetric higher-order mode conversion scheme. Here,all the coupled modes associated with an EP4 are fully converted into different specific higher-ordermodes based on the choice of encirclement directions. The proposed scheme would present EP4 asa new light manipulation tool for integrated photonic devices. ∗ somiit@rediffmail.com I. INTRODUCTION
Nonconservative or open systems behold productive physical significance as they interact with the environmentand hence are dissipative or active. Non-Hermitian quantum mechanics describes such an open physical system byconverting it into an effective Hamiltonian [1, 2]. In a particular system, the non-Hermiticity can be introducedby integrating gain-loss. Recently, the attractive features of the non-Hermitian quantum analogy have fertilized theplatform for the topological study of various quantum-inspired photonic devices. One of the astonishing topologicalfeatures of such type of photonic systems is the appearance of branch point singularities namely, exceptional points(EPs) [3, 4] in the system parameter space. An EP of order n can be defined as a particular branch point singularity insystem’s parameter space, where n number of eigenvalues and the corresponding eigenstates simultaneously coalesceand hence the effective Hamiltonian of the underlying system becomes defective [1–4]. However, it has been establishedthat the functionality of an EP of order n can be realized with simultaneous presence ( n −
1) second-order EPs [5–7].In this paper, we conventionally use the abbreviation ‘EP’ to define a second-order exceptional point, whereas thehigher-order exceptional points are abbreviated as ‘EP n ’ with n = 3 , ± π phase at the end of the encirclement process [8, 41]. However, when weconsider time or analogous length scale dependent (dynamical) parametric variation around an EP, the adiabaticityof the system’s dynamics breaks down. Here the competition of the dynamical EP-encirclement process with theadiabatic theorem leads to an asymmetric mode conversion phenomenon where only one eigenstate follows the adia-batic expectation [42]. An asymmetric mode conversion enabled by a dynamical EP encirclement process refers theconversion of light into a specific dominating mode, regardless of the choice of inputs. Here, the clockwise (CW)and counter-clockwise (CCW) dynamical parametric variation around an EP leads to different dominating modes,beyond the adiabatic restriction [42]. The effect of dynamical parametric encirclement process around a single EP[9, 10, 14, 26, 40]. or two connecting EPs (or an analogous EP3) [13, 43] have widely been investigated using variousplanar or coupled guide-wave systems.In this context, the presence of fourth-order EPs (EP4s) deals with more complex topology of the underlying system.It should be quite interesting, if one can manipulate the simultaneous interaction among the four states in the vicinityof an EP4. In general, dynamical encirclements of second- or third- order EPs have attracted enormous attentionto develop mode-conversion devices [9, 10, 13, 14, 26, 40, 43]. Thus, a general question comes out in the context ofEP4, i.e., what is the outcome of dynamical encirclement around an EP4? However, with proper parametric controlin a planar geometry, the simultaneous interaction among four coupled states and successive state exchange amongthem around an EP4 has never been explored. Moreover, the device implementation of higher-order mode conversionscheme, using higher-order EPs is still a challenge.In this paper, to address the issues mentioned above, we investigate a specially designed gain-loss assisted multi-mode supported planar optical waveguide to host an EP4. Here, non-Hermiticity has been introduced in termsof a multilayer gain-loss profile based on only two control parameters. This is beyond the general prediction forencountering a higher-order EP [6], where it was shown that ( n + n − / n . With proper variation of two control parameters, an EP4 has been embedded by encounteringthree connecting EPs among four coupled modes. We present an unique ‘fourth-order β -switching’ phenomenonby considering a quasi-static parametric variation around the identified EP4. For the first time, we investigatethe unconventional propagation characteristics of the coupled modes following a dynamical parametric encirclementprocess, enclosing the identified EP4. Here, we explore a special type of chiral dynamics of the coupled modesaround the EP4, where four coupled modes (associated with the respective EP4) are converted into a particulardominating higher-order mode, irrespective of the choice of inputs, however, depending on the encirclement direction.An analytical treatment has also been presented to establish the unique state-dynamics around an EP4. The proposedscheme opens up a potential platform to develop higher-order optical mode converters for the integrated or on-chipphotonic circuits. II. ANALYTICAL STRUCTURE OF AN EP4
To develop an analogous analytical model to study the appearance of an EP4, we can consider a simple 4 × H having the form H + λH q H = ǫ ǫ ǫ
00 0 0 ǫ + λ ω i ω j ω i ω k ω k ω l ω j ω l . (1)Here, H corresponds to a passive Hamiltonian, which is subjected to a parameter-dependent complex perturbation H q . λ represents complex parameter. Here, H consist of four passive eigenstates ǫ l ( l = 1 , , , H q consistof four interconnected perturbation parameters w i , w j , w k and w l . Now, four eigenvalues E l ( l = 1 , , ,
4) of H arecalculated by solving the eigen-value equation |H − EI | = 0 ( I → × E + s E + s E + s E + s = 0 , (2)with s = − ( ǫ + ǫ + ǫ + ǫ ) , (3a) s = ǫ ǫ + ǫ ǫ + ǫ ǫ + ǫ ǫ + ǫ ǫ + ǫ ǫ − λ (cid:0) ω i + ω j + ω k + ω l (cid:1) , (3b) s = − ( ǫ ǫ ǫ + ǫ ǫ ǫ + ǫ ǫ ǫ + ǫ ǫ ǫ ) + λ (cid:8) ( ǫ + ǫ ) ω l + ( ǫ + ǫ ) ω j + ( ǫ + ǫ ) ω i + ( ǫ + ǫ ) ω k (cid:9) , (3c) s = ǫ ǫ ǫ ǫ − λ (cid:0) ǫ ǫ ω l + ǫ ǫ ω j + ǫ ǫ ω i + ǫ ǫ ω k (cid:1) − λ ( ω i ω l + ω j ω k ) . (3d)Using Ferrari’s method [44], the roots of the Eq. 2 can be written as E , = − s − η ± r − η − m + m η , (4a) E , = − s η ± r − η − m − m η ; (4b)where, η = 12 r − m (cid:16) k + m k (cid:17) , (5a) k = (cid:18) m q m − m (cid:19) / . (5b)Here, m = − s s , (6a) m = s − s s s , (6b) m = s − s s + 4 s ) , (6c) m = 2 s − s ( s s + 8 s ) + 27( s s + s ) . (6d)Eq. 4 represents the eigenvalues of H . Now, among the four coupled eigenvalues E l ( l = 1 , , , { E , E } , { E , E } and { E , E } . Here, threedifferent EPs are identified by the individual coalescence of E with E or E , and E with E , by specific settingsof perturbation. Under these specific conditions, the system hosts three different EPs through which the eigenvalues(given by Eq. 4) are analytically connected. As it has already been established that the functionality of an EP oforder N can be achieved by ( N −
1) connecting EPs of the order 2 [5], the identified three connecting EPs for thechosen pairs { E , E } , { E , E } and { E , E } give an analogous effect of an EP4 between E l ( l = 1 , , , k + m k = 0 (7)In the following section, we have implemented the above analytical treatment using the framework of a prototypeof a few-mode supported planar optical waveguide with a multilayer gain-loss profile. Here, four chosen modes toencounter an EP4 in the presence of three connecting EPs are analogous to the eigenvalues of H , where the gain-lossprofile can be considered as perturbation which controls the coupling among the chosen modes. III. DESIGN OF A GAIN-LOSS ASSISTED PLANAR OPTICAL WAVEGUIDE AND ENCOUNTER OFMULTIPLE EPS
We design a special type of planar optical waveguide, schematically shown in Fig. 1(a), with x -axis as the transversedirection and z -axis as the propagation direction. The designed waveguide of width W covers the region − W/ ≤ x ≤ W/
2, which consists of a core and a cladding, having the refractive indices n h and n l ( n h > n l ), respectively.During the operation, we normalize the operating frequency ω = 1 and set the total width W = 81 λ/π = 162 (with λ as a free space wavelength) and length L = 15 × in a dimensionless unit. But conventionally, we choose themicrometer unit for a real prototype. The passive refractive indices for core and cladding are chosen as n h = 1 . n l = 1 .
46, respectively. Such a prototype can suitably be realized by thin film deposition technique of glass materialsover a thick silica glass substrate.Here, the non-Hermiticity has been introduced by a specific transverse distribution of a multilayer gain-loss profile.Inhomogeneity in the gain-loss profile has been controlled by considering two tunable parameters such as γ to representthe gain coefficient, and τ to maintain a fixed loss-to-gain ratio along the transverse direction. The spatially distributedimaginary part of the refractive index profile represents such gain-loss distribution. The multilayer gain-loss profilehas been chosen in such a way that the transverse distribution of the complex refractive index profile n ( x ) can bewritten as follows: n ( x ) = n l + iγ, for W/ ≤ | x | ≤ W/ n h − iγ, for − W/ ≤ x ≤ − W/ − W/ ≤ x ≤ W/ ≤ x ≤ W/ n h + iτ γ, for − W/ ≤ x ≤ − W/ ≤ x ≤ W/ W/ ≤ x ≤ W/ − W/ ≤ x ≤ W/ W/ ≤ | x | ≤ W/
2) consists of only loss.The overall transverse distribution of n ( x ) has been shown in Fig. 1(b), where solid black line (corresponding toleft y -axis) and dotted red line (corresponding to right y -axis) represents the profiles of ℜ ( n ) and ℑ ( n ) (for a specific γ = 0 .
007 and τ = 3 . { γ, τ } , however, the coupling parameters { γ, τ } can be variedalong the z -direction to modulate the gain-loss profile for controlling the interactions. Here, the causality conditionis satisfied, i.e., the independent tunability of ℑ ( n ) along the propagation axis irrespective of the choice of ℜ ( n ) isrealized only at the single operating frequency, as per Kramers-Kronig relation [13, 45]. We have chosen operatingparameter in such a way that the waveguide support six scalar modes (linearly polarized) ψ j ( j = 1 , , , , ,
6) inincreasing order. We compute the propagation constants of the respective modes β j ( j = 1—6) by solving the scalarmode equation. According to instantaneous mode profile ψ ( x ), the scalar mode equation is given by[ ∂ x + n ( x ) ω − β ] ψ ( x ) = 0 (9)We choose the small index difference between core and cladding to consider scalar wave approximation, and derive Eq.9 from the Maxwell’s equations. Even though the waveguide supports six quasi-guided modes, to encounter an EP4 in -81 0 81 W ( n ) -0.0400.1 ( n ) (b) I n t e n s it y ( a . u ) -81 0 81 W .... .... (c.1) I n t e n s it y ( a . u ) -81 0 81 W .... .... (c.2)(c)(a) FIG. 1. (a)
Schematic diagram of the proposed waveguide, having width W (along the transverse axis x ) and length L (alongthe propagation axis z ). n h and n l are the passive refractive indices of core and cladding, respectively. (b) The profiles of ℜ ( n ) (solid black line; corresponding to left vertical axis) and ℑ ( n ) (dotted red line; corresponding to right vertical axis) fora specific γ = 0 .
007 and τ = 3 . (c) Field intensity profiles (normalized) of four chosen modes ψ i ( i = 2 , , , ψ and ψ and (c.2) shows ψ and ψ . the presence of three connecting EPs, we have chosen only four modes ψ i ( i = 2 , , , ψ i ( i = 2 , , ,
5) have been shown in Fig. 1(c). For proper visualization they have been shown intwo different plots, where Fig. 1(c.1) shows ψ and ψ , and in Fig. 1(c.2) displays ψ and ψ .Here, an EP4 has been embedded by encountering three connecting EPs among four chosen quasi-guided modes. Toencounter an EP between two quasi-guided modes, we have identified the abrupt transition between two topologicallydissimilar avoided crossings between the corresponding β -values with crossing/ anticrossing of their real and imaginaryprats, i.e., ℜ [ β ] and ℑ [ β ] for two different τ -values, while varying γ within a specified limit. Here, we have to identifytwo such nearby τ , for which the coupled β -values exhibit a crossing in ℜ [ β ] and a simultaneous anticrossing in ℑ [ β ]for particular τ , and the vice-versa, i.e., an anticrossing in ℜ [ β ] and a simultaneous crossing in ℑ [ β ] for another τ .This is the standard technique to identify an EP [3, 9, 14, 40], where for an intermediate τ , two coupled β -valuescoalesce.Using this standard method, we have identified three connecting EPs among ψ i ( i = 2 , , , ℜ [ β ] and ℑ [ β ] for two distinct nearby τ -values, we have identified an intermediate τ = 3 .
295 for ψ and ψ , where we observed that the corresponding β -values coalesce at γ = 0 . ∼ (0 . , . (1) ) in the ( γ, τ )-plane. In a similar way, we find two other EPsin the ( γ, τ )-plane, such as EP (2) at ∼ (0 . , . ψ and ψ , and EP (3) at ∼ (0 . , . ψ and ψ . IV. MODAL PROPAGATION CHARACTERISTICS THROUGH THE WAVEGUIDE IN THEPRESENCE OF MULTIPLE EPSA. Adiabatic Modal Dynamics: β -switching Here, we study the trajectories of the β -values of the chosen modes in the complex β -plane due to the effect ofvarious quasi-static parametric encirclement processes around the identified EPs in the ( γ, τ )-plane. The shape ofthe parameter space has been chosen in such a way, so that we consider γ = 0 at both beginning and the end ofthe encirclement process, which is a crucial requirement for device implementations [9, 40, 42]. To ensure such acondition, we choose a specific contour to enclose single or multiple EPs as γ ( φ ) = γ sin ( φ/
2) ; τ ( φ ) = τ + r sin ( φ ) . (10)where γ , τ and r are three characteristics parameters to enable the stroboscopic variation of γ and τ over thetunable angle φ (0 ≤ φ ≤ π ). To encircle a particular EP, we have to chose a γ which must be greater than the γ -value associated with the respective EP. This type of encirclement method is undoubtedly necessary to check the -3 -3 -3 -3 Loop-2 (a) ...... -2 -2 ...... ...... ...... Loop-3 ...... Loop-1 ...... ...... ...... ...... -3 (c) (d)(b) ...... ... ...... ( ) ( )( ) () () () FIG. 2. (a)
Three chosen parametric loops in ( γ, τ )-plane to encircle three identified EPs, individually, where Loop-1, Loop-2,and Loop-3 encircle EP (1) , EP (2) and EP (3) , respectively. Trajectories of complex β i ( i = 2 , , ,
5) (represented by dotted blue,pink, green and black curves) in the complex β -plane following the quasi-static variation of γ and τ along (b) Loop-1, showingthe adiabatic switching between β and β , unaffecting β and β (shown in the insets); (c) Loop-2, showing the adiabaticswitching between β and β , unaffecting β and β (shown in the insets); and (d) Loop-3, showing the adiabatic switchingbetween β and β , unaffecting β and β (shown in the insets). The circular markers of respective colors represent their initialpositions (i.e., for φ = 0). Arrows of respective colors indicate their direction of progressions. second-order branch point behavior of the identified EPs by scanning the alongside regions around them. The chosenparameter spaces in ( γ, τ )-plane for encircling each of the identified EPs individually has been shown in Fig. 2(a).Here, Loop-1 (with γ = 0 . τ = 3 .
3, and r = 1; shown by green contour), Loop-2 ( γ = 0 . τ = 5, and r = 1 .
5; shown by red contour), and Loop-3 ( γ = 0 . τ = 2 .
9, and r = 3; shown by black contour) individuallyenclose EP (1) , EP (2) and EP (3) , respectively.In Figs. 2(b–d), we have shown the trajectories of β i ( i = 2 , , ,
5) following a sufficiently slow variation of γ and τ along the Loop-1. The trajectories of β i ( i = 2 , , ,
5) are represented by dotted blue, pink, green and black curves,respectively, where the circular markers of respective colors represent their initial positions at the beginning of theencirclement process (i.e., for φ = 0). Each point on the trajectories of β i ( i = 2 , , ,
5) are generated by each pointevolution of γ and τ along the operating loop. Arrows of respective colors indicate their direction of progressions.Now, for a total 2 π complete rotation along Loop-1 (that encloses only EP (1) ), β and β adiabatically exchangetheir initial positions and form a complete loop in complex β -plane, as shown in Fig. 2(b), as only β and β are analytically connected through EP (1) . Such a permutation between two connected β -values can be called as a β -switching (analogous to EP-aided flip-of-sates phenomenon) phenomenon. It can be also observed that, β and β regain their initial positions for a further one-round encirclement along the same contour. Such a β -switchingphenomenon reveals the second-order branch point behavior of EP (1) . Now, the encirclement process following Loop-1 does not affect the trajectories of β and β and hence they remains in the same states making individual loops inthe complex β -plane, as can be seen in two insets of Fig. 2(b). Similar trajectories of β i ( i = 2 , , ,
5) have been shownin Fig. 2(c), while varying the γ and τ along Loop-2, which encloses only EP (2) . Here, the adiabatic β -switching hasbeen observed between β and β at the end of the encirclement process, as they are analytically connected throughEP (2) . However, in this case, β and β remains in the same states, which have been shown in the insets of the Fig.2(c). A similar adiabatic β -switching between β and β (two connecting β ’s through EP (3) ), unaffecting β and β (as shown in the insets), has been exhibited in Fig. 2(d), when we considerthe stroboscopic variation of γ and τ along Loop-3 (that encloses only EP (3) ). In Figs. 2(b–d), the trajectories ofunaffecting states for the respective cases have been shown in insets for clear visualization of the trajectories.Now, we consider a new parametric loop (say, Loop-4) to encircle all the identified EPs, as shown in Fig. 3(a), forwhich we have chosen γ = 0 . τ = 4, and r = 3. The associated trajectories of β i ( i = 2 , , ,
5) has been shown inFig. 3(b). Here, with the quasi-static variation of γ and τ along Loop-4, all four propagation constants β , β , β , and β exchange their own identities and switch successively following the manner β → β → β → β → β to make acomplete loop in the complex β -plane. Thus, interestingly the system shows a fourth-order branch point behavior forthe corresponding eigenvalues if all three connecting EPs are quasi-statically encircled in the parameter space. Thisunique type of successive β -switching phenomenon around three connecting EPs indeed confirms the appearance of anEP4 in system parameter space, where all four chosen modes are analytically connected. Specifically, the successiveswitching among four β -values, enabling by an EP4 with the simultaneous presence of three EPs, can be refereed as‘fourth-order β -switching’. Here, we have also observed that if we reverse the encirclement direction (i.e., CCW), thenthe chosen complex β -values switch successively following the manner β → β → β → β → β (exactly the reverseprogressions from the previous observation) in the complex β -plane. Such type of adiabatic fourth-order β -switchingphenomenon has been reported for the first time in a guided wave system. ( ) ( ) ...... ...... ...... ...... Loop-4 (b) EP (1) EP (2) EP (3) (a) (c) Loop-4
FIG. 3. (a)
A parametric loop in ( γ, τ )-plane to encircle three identified EPs, simultaneously. (b)
Corresponding trajectories β i ( i = 2 , , , β -switching phenomenon following the manner β → β → β → β → β . The colorsand markers carry the same meaning as described in the caption of Fig. 2. (c) Topological structure of the Riemann surfacesassociated with real parts of β i ( i = 2 , , ,
5) as a function of γ and τ within specified ranges. The curves on the surfacerepresent the successive switching of β -values from their respective surfaces. In the ground surface, the chosen parameter spacehas been shown separately for proper understanding. In Fig. 3(c), we have demonstrated the formation of the Riemann sheets associated with β i ( i = 2 , , ,
5) by choosinga specific range for both γ and τ (which is sufficient to accommodate Loop-4, and to consider the interaction regime).Here, the overall distribution of ℜ ( β ) as a function of γ and τ , reveals the simultaneous interaction of β i ( i = 2 , , , ℜ ( β ) associated with the fourth-order β -switching phenomenon, asshown in Fig. 3(b), has been shown (by the dotted blue, pink, green and black curves) on the associated distributionof the Riemann surfaces, where it is evident that the real parts of β i ( i = 2 , , ,
5) switch successively following themanner β → β → β → β → β from their respective surfaces. For a clear understanding, the parameter space(including the location of three EPs) has been been shown separately in the ground surface of Fig. 3(c) [exactly sameloop shown in Fig. 3(a)]. B. Dynamical parametric encirclement: beam propagation dynamics
The β -switching phenomenon due to quasi-static encirclement of the coupling parameters around single or multipleEPs, as described in the preceding section, follow the adiabatic theorem. However, if we consider the time dependence(dynamical) in the parametric variation, then the breakdown of inversion symmetry in the overall gain-loss variationalong the time scale compete with adiabatic theorem [42]. The breakdown in the adiabatic theorem has been studiedaround second- and third-order EPs, which enables chiral or nonchiral dynamics [9, 10, 13, 40, 43]. However, the chiralaspect due to breakdown in system’s adiabaticity following a dynamically encircled EP4 has never been reported. Inthis section, we explore the effect of such dynamical encirclement around the embedded EP4 and study the chiralaspect of the proposed waveguide device for the first time.Now, to study the actual propagation of modes through the waveguide, we have to consider a length dependentgain-loss distribution to encircle single or multiple EPs dynamically. Such dynamical encirclement can be achieved bymapping the overall ℑ ( n ) associated with a EP-encircling parameter space throughout the entire length of the waveg-uide. Here, the propagation of the eigenmodes are governed by the time-dependent Schrodinger equation(TDSE).The parameter space mapping demands the simultaneous closed variation of γ and τ along the z -axis throughout
15 50-500 0310 -2 ( n ) (a) (b) Loop-4Loop-1
FIG. 4. Two chosen parameter spaces in the ( γ, τ )-plane; where Loop-1 encircles only EP (1) , and Loop-4 encircles all theidentified EPs, i.e., EP (1) , EP (2) , and EP (3) , simultaneously. (b)
Length dependent distribution ℑ [ n ( x, z )] after mappingLoop-4. the total waveguide length. However, for each transverse cross section (i.e., for a fixed z ), the refractive index profilecorresponds to a specific set of { γ, τ } . Such a parameter space mapping has been achieved by the consideration of φ = 2 πz/L in Eq. 10 as γ ( z ) = γ sin h πzL i ; τ ( z ) = τ + r sin (cid:20) πzL (cid:21) . (11)Here, Z = 0 and L are associated with φ = 0 and 2 π , respectively. Thus, one complete encirclement (0 ≤ φ ≤ π )following Eq. 10 is equivalent to one complete pass (0 ≤ z ≤ L ) of light following Eq. 11 along the propagationlength. Here CW encirclement (0 ≤ φ ≤ π ) has been realized by the propagation of light form z = 0 to z = L ,whereas the CCW encirclement (2 π ≤ φ ≤
0) has been realized by the propagation of light in the opposite direction,i.e., form z = L to Z = 0. As for both Z = 0 and L , γ = 0, we can excite and retrieve the passive modes at both theinput and output, which is not achievable using conventional circular parametric loop for EP encirclement [9, 22, 40].Now, we perform two different dynamical encirclement processes following Loop-1 and Loop-4, as shown in Fig.4(a) individually. Fig. 4(b) represents the overall variation ℑ [ n ( x, z )] after mapping Loop-4 (following Eq. 11) alongthe entire length of the waveguide. For sufficiently slow variation of ℑ ( n ) along the z direction, the dynamics of thequasi-guided modes are governed by the (1+1) D scalar beam propagation equation, that can be rewritten under theparaxial approximation as: 2 iω ∂ψ ( x, z ) ∂z = − (cid:20) ∂ ∂x + ∆ n ( x, z ) ω (cid:21) ψ ( x, z ) , (12)where ∆ n ( x, z ) ≡ n ( x, z ) − n l . We use split steps Fourier method to solve Eq. 12 [46].During the dynamical EP-encirclement process, it has been established that only the mode that evolves with thelower average loss follow the adiabatic expectations governed by the associated β -trajectories. Here the individualdecay rate of a particular mode during the evolution around an EP can be calculated by averaging the losses over theentire contour in the complex β -plane (governed by corresponding quasi-static EP-encirclement process) as [40] ζ average = 12 π Z π ℑ ( β ) dφ (13)In Fig. 5, we have shown the beam propagation simulation results for the dynamical encirclement process alongLoop-1, which encloses only EP (1) . Here, to consider the encirclement in the CW direction, we choose input at Z = 0 to launch the light. The associated beam propagation results of the chosen modes ψ i ( i = 2 , , ,
5) have beenshown in the upper panel of Fig. 5. Here, it is evident that, the modes ψ and ψ associated with EP (1) , whichare individually excited from Z = 0 are essentially converted to ψ at z = L , beyond the adiabatic expectationsfrom the corresponding β -trajectories shown in Fig. 2(b). Here, as β evolve with lower average loss (to move tothe location of β ) in comparison to β , as can be seen in Fig. 2(b) (also calculated by using Eq. 13), the mode ψ is adiabatically converted into ψ , whereas ψ shows a nonadiabatic transition (NAT) and is converted into itself.However, the dynamical encirclement along Loop-1 does not affect the propagation of ψ and ψ , and hence theyevolve without any conversion. Now for the dynamical encirclement along Loop-1 in the CCW direction (i.e., whenthe modes are excited from z = L ), ψ evolves adiabatically and is converted to ψ (as in this case, β evolves withthe lower average loss in comparison to β ), wheres ψ behaves nonadiabatically and is converted to ψ at z = 0, asshown in the lower panel of Fig. 5. Here, also ψ and ψ remains unaffected. Thus, depending on the direction ofpropagation, we get the conversions { ψ , ψ } → ψ (for z = 0 → L ; CW rotation) and { ψ , ψ } → ψ (for z = L → ψ and ψ (unaffecting ψ and ψ ) by considering the dynamical encirclement along Loop-2, and also between ψ and ψ (unaffecting ψ and ψ ) by considering the dynamical encirclement along Loop-3.Now, we have shown the effect of dynamical encirclement around the embedded EP4 with the simultaneous presenceof three connecting EPs by mapping Loop-4 [as already shown in Fig. 4(b)] along the entire length of the waveguide.The associated beam propagation results and the expected output intensity profiles have been shown in Fig. 6. Toconsider a CW dynamical encirclement process, the light has been launched at z = 0, where the propagations of thechosen modes ψ i ( i = 2 , , ,
5) has been shown in the upper panel of Fig. 6(a). Here, it has been shown that allthe chosen modes, that are individually excited from z = 0, are finally converted into dominating ψ at z = L , i.e.,we get the conversions { ψ , ψ , ψ , ψ } → ψ . The overall output field intensity (normalized) has been shown in theupper panel of Fig. 6(b). Here, only ψ behaves adiabatically and is converted to ψ and the other modes ψ , ψ ,and ψ follow the NATs and converted into ψ . Such an adiabatic conversion of ψ ( → ψ ) can also be verified fromthe associated adiabatic β -trajectories, shown in Fig. 3(b), where β evolves with lower average loss (calculated usingEq. 13) in comparison to the others. L -81 0 81 W -81 0 81 W L -81 0 81 W -81 0 81 W FIG. 5. Beam Propagation simulation results for the propagations of ψ i ( i = 2 , , ,
5) following the dynamical parametricencirclement along the Loop-1. The upper panel shows the asymmetric conversions ( ψ , ψ ) → ψ , unaffecting ψ and ψ ,for propagation from z = 0 to z = L (i.e., CW dynamical encirclement). The lower panel shows the asymmetric conversions( ψ , ψ ) → ψ , unaffecting ψ and ψ , for propagation from z = L to z = 0 (i.e., CCW dynamical encirclement). Here, Were-normalize the intensities at each z for visualizing the propagations clearly, and hence the overall effect of loss is not evident. (cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13) -(cid:14)(cid:15) W L (cid:18)(cid:19)(cid:20) (cid:21)(cid:22) W (cid:23)(cid:24)(cid:25) (cid:26)(cid:27) W (cid:28)(cid:29)(cid:30) (cid:31) W !" L &’ W ()* +, W
542 33 4 5 OutputOutput (a) (b) .... CCW CW N o r m a li z e d f i e l d I n t e n s it y ( I ) FIG. 6. (a)
Propagation of four chosen modes ψ i ( i = 2 , , ,
5) following dynamical EP4-encirclement process along Loop-4.The upper panel shows the asymmetric conversions { ψ , ψ , ψ , ψ } → ψ , for the encirclement in CW direction (i.e., duringpropagation from z = 0 to z = L ). Whereas, The lower panel shows the asymmetric conversions { ψ , ψ , ψ , ψ } → ψ , for theencirclement in CCW direction (i.e., during propagation from z = L to z = 0). (b) The upper panel shows the overall outputfield intensity at z = L associated with the conversions shown in the upper panel of (a). Whereas, the lower panel shows theoverall output field intensity at z = 0 associated with the conversions shown in the lower panel of (a). If we reverse the encirclement direction by exciting the inputs ψ i ( i = 2 , , ,
5) from z = L , then we get theconversions { ψ , ψ , ψ , ψ } → ψ after the completion of the propagation (i.e., at z = 0), beyond the adiabaticexpectations governed by the associated β -trajectories. In this case, we have calculated the average losses of individualmodes by using Eq. 13, and observed that β evolves with lower average loss in comparison to the others. Thecorresponding beam propagation results have been shown in the lower panel of Fig. 6(a). Here, only ψ behavesadiabatically and is converted to ψ and the other modes ψ , ψ , and ψ follow the NATs and converted into ψ . Theoverall output field intensity (normalized) for this anticlockwise encirclement process has been shown in the lower0panel of Fig. 6(b).We also calculate the mode conversion efficiencies in terms of overlap integrals between input and output fields as C R → S = (cid:12)(cid:12)R ψ R ψ S dx (cid:12)(cid:12) R | ψ R | dx R | ψ S | dx ; { R, S } ∈ i. (14)Here, C R → S defines the conversion efficiency for the conversion ψ R → ψ S . For the conversions { ψ , ψ , ψ , ψ } → ψ ,as shown in the upper panel of Fig. 6(a), we find the maximum conversion efficiency of 77.18%. On the other hand,for the conversions { ψ , ψ , ψ , ψ } → ψ , as shown in the lower panel of Fig. 6(a), we find the maximum conversionefficiency of 86.64%.We also examine the robustness of the such a unique EP4-aided higher-order asymmetric mode conversion processagainst the parametric tolerances during the fabrication using state-of-the-art techniques. Accordingly, we haveappended an average 5% random parametric fluctuations on the dynamical encirclement process along Loop-4 andobserved that the higher-order asymmetric mode conversion process, as can be seen in Fig. 6, is omnipresent even inthe presence of moderate parametric tolerances. V. ANALYTICAL TREATMENT FOR UNCONVENTIONAL HIGHER ORDER MODALPROPAGATION
Based on only two coupling parameters, the unconventional dynamics of four coupled modes around an EP4 can cananalytically be treated as follows. Let assumes that the corresponding four-level Hamiltonian depends on two time-dependent potential parameters Γ ( t ) and Γ ( t ) (analogous to γ and τ ). Within the adiabatic limit, the eigenfunctionsfollow the TDSE during evolutions. Now to estimate the nonadiabatic correction terms during the propagation of foureigenmodes around an EP4, we consider possible conversions among ψ adR (Γ , Γ ) and ψ adS (Γ , Γ ) (with correspondingeigenvalues β adR (Γ , Γ ) and β adS (Γ , Γ ), respectively), where { R, S } ∈ { , , , } . Here, the dynamical nonadiabaticcorrection terms can be written asΘ NA R → S = ϑ R → S exp ( + i I T ∆ β ad R,S [Γ ( t ) , Γ ( t )] dt ) , (15a)Θ NA S → R = ϑ S → R exp ( − i I T ∆ β ad R,S [Γ ( t ) , Γ ( t )] dt ) ; (15b)with the pre-exponet terms ϑ R → S = * ψ ad R [Γ , Γ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m =1 ˙Γ m ∂∂ Γ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ad S [Γ , Γ ] + , (16a) ϑ S → R = * ψ ad S [Γ , Γ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m =1 ˙Γ m ∂∂ Γ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ad R [Γ , Γ ] + , (16b)and ∆ β ad R,S [Γ , Γ ] = β ad R [Γ , Γ ] − β ad S [Γ , Γ ] ≡ Re[∆ β ad R,S [Γ , Γ ] − i ∆ ζ ad R,S [Γ , Γ ] . (16c)In Eq. 15, the suffixes R → S and S → R corresponds to the conversion ψ ad R → ψ ad S and vice versa, respectively. T is the duration of encirclement. In Eq. 16c, (cid:12)(cid:12) ∆ ζ ad R,S (cid:12)(cid:12) represents the ‘relative gain’ between the two correspondingmodes. As the pre-exponent terms of Eq. 15 contain the time derivative of two potential parameters ( ˙Γ m ; m = 1 , T of the exponents of Eq. 15 beats the T − suppression associated with the associatedpre-exponent terms. Thus, if we consider a situation ∆ ζ ad R,S >
0, then for T → ∞ , Θ NA R → S → ∞ and Θ NA S → R → NA S → R followthe adiabatic expectations, whereas the conversion associated with Θ NA R → S dose not follow the adiabatic expectations.In a similar way, we can establish the vice-versa condition for ∆ ζ ad R,S <
0. Now around an EP, if CW parametricvariation gives ∆ ζ ad R,S > ζ ad R,S >
0. Thus for the dynamical parametricencirclement in any of the directions, only one state evolves adiabatically.For our proposed waveguide, we can consider { R, S } ∈ { , , , } to calculate the possible nonadiabatic correctionterms for different encirclement direction. Now, following the dynamical encirclement along Loop-1 (around only1EP (1) ), we have obtained ∆ ζ , > ζ , < ψ → ψ (adiabatic) and ψ → ψ (nonadiabatic). On the other hand, the waveguide allows the adiabatic conversionof ψ → ψ (adiabatic) and ψ → ψ (nonadiabatic) for CCW dynamical encirclement process. As the other twoconnecting EPs, i.e., EP (2) and EP (3) are away from the chosen parametric loop, the other possible nonadiabaticcorrection terms can be neglected. The beam propagation results shown in Fig. 5 support these analytical predictions.Now, we consider the situation of the dynamical encirclement along Loop-4 around the embedded EP4 with thesimultaneous presence of three connecting EPs. Here, during the clockwise encirclement we have obtained ∆ ζ , > ψ → ψ and nonadiabatic conversion of ψ → ψ . However, unlike Loop-1,Loop-4 encircles all the identified EPs, and hence we have to consider the other possible nonadiabatic correction terms.Now, the validity of ∆ ζ , > ζ , > ζ , > ψ → ψ and ψ → ψ . Thus for encirclement in the clockwise direction, we get the overall conversions { ψ , ψ , ψ , ψ } → ψ . On the other hand, the CCW encirclement along Loop-4 yields the nonadiabatic correctionterms, ∆ ζ , <
0, in addition with ∆ ζ , > ζ , >
0. Thus, we get the conversions { ψ , ψ , ψ , ψ } → ψ ,where only ψ follows the adiabatic expectation. The beam propagation results shown in the lower panel of Fig. 6support these analytical predictions. Thus if we dynamically encircle the embedded EP4 in the simultaneous presenceof three connecting EPs, then the device enables a specific chiral light dynamics, where irrespective of the input modes,the coupled modes associated with the EP4 is converted into a specific dominating higher-order mode depending onthe direction of light propagation. Here, for the propagation in two different directions, the waveguide delivers twodifferent dominating higher-order mode. Thus the proposed waveguide can be used as an efficient higher-order modeconverter, where one can achieve different higher-order modes by changing the direction of light propagations. VI. CONCLUSION
In summary, we report an exclusive topologically robust, compact and fabrication feasible scheme for higher-ordermode conversion using the framework of a gain-loss assisted and multimode supported planar optical waveguide.Here, introducing a spatial distributing of a multilayer gain-loss profile (based on only two tunable parameters), theinteractions among four chosen modes have been manipulated with encounter of three connecting EPs. The fourth-rootbranch point behavior of the embedded EP4 has been revealed in terms of a successive β -switching phenomena amongfour chosen modes following a quasi-static parametric encirclement process around three connecting EPs. Withoutusing any complex topology, the effect of dynamical parametric encirclement around the embedded EP4 has beenrevealed by customizing only a 2D parameter space associated with the gain-loss profile, which is the first-ever reportin the context of a dynamical EP4-encirclement scheme. The waveguide, hosting such a dynamical EP4-encirclementscheme, enables a unique chiral light dynamics, where depending on the direction of light propagation, the associatedfour coupled modes get converted to a specific dominating higher-order mode, irrespective of the choice of inputs.Here, owing to the device chirality, while operating around an EP4, two different dominating modes survive forpropagation in two different direction. Thus, the proposed specially configured waveguide indeed has a vast potentialto present itself as a higher-order asymmetric mode converter. Such a customized device is able to excite a particularmode in a multi-modal configuration, that would be suitable for various pumping scheme in integrated (or chip-scale)device applications. The proposed scheme, hosting the rich physics of an EP4, would open up opportunities to boostthe device applications associated with next-generation all-optical communication, and computing. ACKNOWLEDGMENTS
S.D. acknowledges the support from the Ministry of Human Research and Development (MHRD), Government ofIndia. A.L. and S.G. acknowledge the financial support from the Science and Engineering Research Board (SERB)[Grant No. ECR/2017/000491], Department of Science and Technology, Government of India. [1] N. Moiseyev,
Non-Hermitian Quantum Mechanics (Cambridge University press, Cambridge, New York, 2011).[2] T. Kato,