Straightening of light in a one dimensional dilute photonic crystal
SStraightening of light in a one dimensional dilutephotonic crystal
Zhyrair Gevorkian , Vladimir Gasparian , and Emilio Cuevas Yerevan Physics Institute, Alikhanian Brothers St. 2, 0036 Yerevan, Armenia Institute of Radiophysics and Electronics, Ashtarak-2, 0203, Armenia California State University, Bakersfield, California 93311-1022, USA Departamento de F´ısica, Universidad de Murcia, E-30071 Murcia, Spain * [email protected] ABSTRACT
Light transport in a dilute photonic crystal is considered. The analytical expression for the transmission coefficient is derived.Straightening of light under certain conditions in a one-dimensional photonic crystal is predicted. Such behavior is causedby the formation of a localized state in transversal motion. The main contribution to the central diffracted wave transmissioncoefficient is due to states, that either close to the conductance band’s bottom or deeply localized in the forbidden gap. Boththese states suppress mobility in the transverse direction and force light to be straightened. Straightening of light in the opticalregion along with small reflection make these systems very promising for use in solar cells.
Recent developments in material science have made possible the fabrication of photonic crystals that allow the observationof many peculiar effects , including perfect reflector for all polarizations over a wide selectable spectrum ,optical Halleffect , unidirectional scattering with broken time-reversal symmetry and the propagation of optical beams without spatialspreading. The latter, known as supercollimation, has attracted a lot of attention in various fields of physics. Supercollimationeffect has been observed in mesoscopic ( ∼ µ m ) and macroscopic centimeter-scale photonic crystals . The latterresults indicate that supercollimation effect is very robust and insensitive to possible irregularities or short-scale disorder in thephotonic crystal structure. A standard mechanism that can account for the supercollimation effect of light in two-dimensionalphotonic crystals is the flat character of dispersion curve ω ( (cid:126) q ) in the transverse direction due to the interference of the differenttransverse components of the wavevectors (see, for example, Ref. ).However, as we will show below, in diluted photonic crystals (DPC) exists another physical mechanism, based on photon’stransversal motion restriction, that can lead similar to supercollimation effect . The basic idea of this model can be easilyunderstood by considering the following geometry shown in Fig.1. First, suppose that a photon falls on the plates normally on0 x direction. In case if the photon’s wavenumber k x lies in the photonic band gap region, the transmission coefficient must besuppressed (see, for example, Ref. ). Next, imagine that the same photon falls down onto the system obliquely, as shown inFig.1 and assume that the transversal to plates component k x still remains in the photonic band gap. Clearly, for small andmoderate scattered angles transversal motion is suppressed due to the appearance of localized or low energy states. Hence, as adirect consequence of the restriction in the transverse direction, photon’s propagation parallel to plates will be enhanced.Note that in most papers on 1d photonic crystal photon main motion is normal to the plates, see, for example, and forrecent reviews . In our manuscript we are considering a different geometry (see Fig.1) where photon mainly propagatesparallel to the plates. The novelty of our consideration is in different geometry and in corresponding theoretical approach. Ourapproach allows to obtain closed analytical expression for transmission coefficient.In this paper we aim to present a complete and quantitative theoretical description of the supercollimation and straighteningof light effects within simplified DPC model, taking into account the above-mentioned restriction in the transverse direction.We will see, that the simplified DPC models, developed in Refs. , can help to provide new insights into the properties of thementioned effects.The main simplification is related to the ignoring of backscattering in a DPC. Within this approach, we have investigatedthe transport of light through a one-dimensional metallic photonic crystal with transverse to incident direction inhomogeneity.Independence of transmission coefficient on the incident light wavelength was found . Beside that, we have predicted andexperimentally observed a capsize, a drastic change of polarization to the perpendicular direction in DPC. The present worktakes one step further in the study of the supercollimation effect. We will remove the limitations on the normal incident light, a r X i v : . [ phy s i c s . op ti c s ] O c t igure 1. Geometry of the problem.discussed in Refs. and consider general oblique incidence case. The latter, as we will see below, can lead to an interestingphenomenon of straightening of light into the normal direction on exit of the photonic crystal. In this sense, the DPC can beused in solar cells to increase their efficiency. The point is that at oblique incidence a large amount of light energy is lost due toreflection. Preliminary straightening would reduce this loss. For other applications of 1d photonic crystals see also .Note, that the main difference of DPC model from other photonic crystal systems where supercollimation effects whereobserved is that in the former reflection is negligible due to the fact that the fraction of one component is very small.
Consider a system with inhomogeneous dielectric permittivity ε ( x , y ) (see Fig.1) and suppose that a plane wave enters thesystem from the z < θ . Following procedures described in previous work Ref. and - ,scalar Helmholtz equation is reduced to the following time-dependent Schr¨odinger equation for a particle with mass k i d φ dz = ˆ H ( x , y ) φ , (1)where ˆ H ( x , y ) = − k ∇ t + k ( − ε ( x , y )) . (2)Note, that the parabolic approximation, that is | d φ / dz | << k | d φ / dz | justified, if the characteristic scale L z of φ ( x , y , z ) along z is much longer than a wavelength, that is k L z >> Φ ( x , y , z ) = e ik z ∑ n c n e − iE n z φ n ( x , y ) . (3) here ˆ H φ n ( x , y ) = E n φ n ( x , y ) . (4)It follows from Eq.(3) that the local transmission amplitude of a central diffracted wave can be defined as in Ref. t ( x , y ) = ∑ E n < k c n e − iE n L φ n ( x , y ) , (5)where L is the system size in the z direction.The central diffracted wave transmission coefficient that is measured in the experiment can be estimated by the followingexpression T = S (cid:90) dxdy (cid:12)(cid:12)(cid:12)(cid:12) t ( x , y ) (cid:12)(cid:12)(cid:12)(cid:12) , (6)where S is the area of the system. Substituting Eq.(5) into Eq.(6), one has T = S ∑ E n < k | c n | . (7)In order to find the coefficients c n let us consider the Eq.(3) for z = Φ ( x , y , z = ) = ∑ n c n φ n ( x , y ) . (8)To proceed, we assume that the wave intruding the system has an amplitude 1 (the region z < z = Φ ( x , y , z = ) = e k sin θ x . Here we ignore the reflected waves from the dilute system. Within this approach, multiplyingboth sides of Eq.(8) by φ ∗ n ( x , y ) and integrating over the surface, one has c n = (cid:90) dxdy φ ∗ n ( x , y ) e ik sin θ x . (9)Substituting Eq.(9) into Eq.(7), we arrive at the final result for the transmission coefficient T = S ∑ E n < k (cid:90) d (cid:126) ρ d (cid:126) ρ (cid:48) φ ∗ n ( (cid:126) ρ ) φ n ( (cid:126) ρ (cid:48) ) e ik sin θ ( x − x (cid:48) ) , (10)where (cid:126) ρ ≡ ( x , y ) is a two dimensional vector on the xy plane. The equation above is the generalization of the previous result,obtained in Ref. , in case of oblique incidence of light at an arbitrary angle θ . In the succeeding subsections we examine thelimitations of the found equation for different models. Note that when k < E b ( E b is the bottom value of first energy band), the transmission coefficent (Eq.(10)) is equal to zero.When k is inside the allowed band, transmission coefficient can be represented by quasi-momentum (cid:126) q in the following way T = S ∑ n (cid:90) E n ( (cid:126) q ) < k d (cid:126) q π (cid:90) d (cid:126) ρ d (cid:126) ρ (cid:48) φ ∗ n (cid:126) q ( (cid:126) ρ ) φ n (cid:126) q ( (cid:126) ρ (cid:48) ) e ik sin θ ( x − x (cid:48) ) , (11)where (cid:126) q is integrated over the first Brilloin zone. For sake of simplicity and demonstration of the results, we will carry outfurther consideration in one-dimensional case. Assuming that plates are positioned periodically along the x axis (see Fig.1), we can represent the cross section of the potentialas a multitude of square potential wells. Metal layers of PC imitate the potential wells with depth V d = k ( ε − ) / b . The vacuum layer is charachterized by potential energy V = ε =
1. So the problem described by Eq.(2) is brought tothe Kronig-Penney model . The transmission coefficient T for one-dimensional configuration can be written as T = L x ∑ n (cid:90) E n ( q ) < k dq π (cid:90) dxdx (cid:48) φ ∗ nq ( x ) φ nq ( x (cid:48) ) e ik sin θ ( x − x (cid:48) ) . (12) , 00 , 20 , 40 , 60 , 81 , 0 b = 0 b = 0 . 0 6 m m (cid:3) p (cid:2)(cid:5) (cid:1) (cid:1) (cid:1) ( r a d ) p (cid:2)(cid:4) a = 0 . 6 m m k = 1 2 m m - 1 e = 4 Figure 2.
Transmission coefficient dependence on incident angle.Blue line is the theoretical plot with PC. Red line is thebackground theoretical plot without PC. In the inset the energy band scheme of transverse motion is shown.where L x is the system size in the x direction. Here sum is running over the bands with energy E n ( (cid:126) q ) < k . When k lies inthe energy gap the integration over (cid:126) q covers whole Brilloin zone − π / a , π / a . Using Bloch theorem φ nq ( x ) = e iqx u nq ( x ) , where u nq ( x ) is a periodical function, one obtains T = L x ∑ n (cid:90) E n ( q ) < k dq π ∑ lm (cid:90) la ( l − ) a dxe − i ( q − k sin θ ) x u ∗ nq ( x ) ×× (cid:90) ma ( m − ) a dxe i ( q − k sin θ ) x u nq ( x ) (13)Changing the variables one finds T = L x ∑ n (cid:90) E n ( q ) < k dq π ∑ l e − i ( q − k sin θ ) al × ∑ m e i ( q − k sin θ ) am × (14) × (cid:90) a dxe − iqx u ∗ nq ( x ) (cid:90) a dxe iqx u nq ( x ) , where wave functions in different layers of PC can be found from Eq.(18) (see Appendix). By substituting ∑ n e − inqa = πδ ( qa ) into Eq.(14), for transmission coefficient we will have T = a ∑ E n < k (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) a u n ( x ) e ik sin θ x dx (cid:12)(cid:12)(cid:12)(cid:12) (15)where u n ( x ) ≡ u nq = k sin θ ( x ) . The sum in Eq.(15) include states from different zones with both negative and positive energieswith the same quasimomentum k sin θ . The number of terms in the sum depends on parameters a , k , b and ε . Using formula from Supplement we numerically calculate the transmission coefficient of central diffracted wave for thefollowing parameters a = . µ m , b = . µ m , k = µ m − , ε =
4. The results are shown in Fig.2 . We compare with thevacuum case b ≡
0. In this case the sum Eq.(15) contains only one term with E = k sin θ /
2. Clear straightening effect isseen from Fig.2.Indeed, independent of incident angle, light leaves the system mainly on the 0 z direction because T is close to unity. It isclear, as mentioned above, that different energies give contribution to T . However, as indicate our numerical calculations, themain contribution to T comes from relatively small positive E n > E n < . As for the states with negative nergies, they form a very narrow band gap with almost localized states and with limited transport properties in transversedirection, see inset in Fig.2.From theoretical point of view, it is clear that under certain conditions when the incident photon wavenumber is trapped inthe gap of energy spectrum of transversal motion, in DPCs light can propagate without spreading out. These states also lead tothe suppression of mobility in transverse direction and force photons move only parallel to plates. Note, that depending onthe incident angle, the first or second type of states contribute differently to straightening effect, see Appendix. However, theresultant T , taking into account the contribution of all states (negative and positive) becomes almost incident angle independentand close to unity value. We have considered the transport of light through a 1 d dilute photonic crystal model at oblique incidence. Our theoreticalstudy, based on Maxwell’s equations with a spatially dependent inhomogeneous dielectric permittivity. For certain parameters’values of DPC the emerging light propagates in the normal direction despite the oblique incidence. This straightening effect isintimately connected with limited transport properties of photons in transverse direction.When considering the straightening effect we assume continuity of scalar wave field at the interface plane between twomedia (xy in Fig.1). Coming back to the em wave case this means that our consideration is correct for s-polarized waves,electric field vector of which is directed on 0y in geometry of Fig.1. As mentioned above the straightening effect in visiblerange could be utilized in solar cell elements to make their absorption efficiency higher. It could seem that the polarizationdependence of the effect will decrease the application efficiency because the natural light consists of both s- and p-polarizations.However it follows from Fresnel formulae that the p-polarized wave is essentially reflected only for very large incident angles θ > . In contrary a s-polarized wave is reflected even for moderate incident angles. Therefore using DPC to straightens-polarized light has sense.As was mentioned above, in the discussed model of dilute photonic crystal, we have ignored the back-scattered waves andreflection. In order to justify formally this approach, we estimate below the value of reflection amplitude r , using the standarddefinition r = √ ε e − √ ε e + ε e defined by ε e = ( − b / a ) + ε b / a . Taking ε = b / a ∼ .
1, onehas ε e ≈ . r ∼ .
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Acknowledgement
Zh.G. is grateful to Science Committee of Armenia for financial support (project 18T-1C082),A.Hakobian for help andA.Hakhoumian for useful discussions. E.C. and V.G. thank partial financial support by the Murcia Regional Agency of Scienceand Technology (project 19907/GERM/15). V.G. acknowledges the kind hospitality extended to him at Murcia Universityduring his sabbatical leave.
Author contributions statement
Zh.G. and V.G. develope the theory, E.C. carry out the numerical calculations. All the authors review the manuscript.
Additional information
Competing Interests:
The authors declare that they have no competing interests.
Bloch theorem states that the eigenstate φ (cid:126) q in a periodical potential can be represented in the form φ n (cid:126) q ( (cid:126) ρ ) = e i (cid:126) q (cid:126) ρ u n (cid:126) q ( (cid:126) ρ ) , (17) here u n (cid:126) q ( (cid:126) ρ ) is a periodical function satisfying the equation (cid:20) − k ( i (cid:126) q + (cid:126) ∇ ) + V ( (cid:126) ρ ) (cid:21) u n (cid:126) q ( (cid:126) ρ ) = E n ( (cid:126) q ) u n (cid:126) q ( (cid:126) ρ ) . (18)Below we present the solution of Scr ¨ o dinger equation Eq.(18) in a unit cell and using it calculate the transmission coefficient.First consider positive energies E > u q ( x ) = ( A cos β x + B sin β x ) e − iqx , < x < a − bu q ( x ) = ( C cos α x + D sin α x ) e − iqx , a − b < x < a (19)with β = √ k E and α = (cid:112) k ( V d + E ) . Remind that V d = k ( ε − ) / . . Here we omit index n for simplicity.The constants B , C , D can be expressed by A using boundary conditions. A itself can be found from the normalization condition (cid:82) a | u ( x ) | dx = ( k a sin θ ) = cos α b cos β ( a − b ) − α + β αβ sin α b sin β ( a − b ) (20)and following relations between coefficients C = A , D = β B / α and B = A cos α b − e ik sin θ cos β ( a − b ) e ik sin θ sin β ( a − b ) + βα sin α b (21)Substituting Eq.(19) into Eq.(15), using relations between coefficients and taking elementary integrals for a particular contribu-tion of E into transmission coefficient of central diffracted wave one has T = T × T (22)where T = a (cid:12)(cid:12)(cid:12)(cid:12) sin β ( a − b ) β + sin α b α + cos α b − e ik asin θ cos β ( a − b ) e ik asin θ sin β ( a − b ) + βα sin α b (cid:32) β ( a − b ) β − β sin α b α (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) (23)and T = (cid:20) a + sin 2 β ( a − b ) β + sin 2 α b α + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos α b − e ik asin θ cos β ( a − b ) e ik asin θ sin β ( a − b ) + βα sin α b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ×× (cid:18) a − b − sin 2 β ( a − b ) β − β sin 2 α b α + β b α (cid:19) ++ (cid:32) cos α b − e ik a sin θ cos β ( a − b ) e ik asin θ sin β ( a − b ) + βα sin α b + cos α b − e − ik asin θ cos β ( a − b ) e − ik asin θ sin β ( a − b ) + βα sin α b (cid:33) ×× (cid:18) sin β ( a − b ) β + β cos 2 α b α − β α (cid:19) (cid:21) − (24)Note that Eqs(22-24) determine the contribution of a particular and positive solution of the dispersion equation Eq.(20)(0 < E < k ) into transmission coefficient. In order to find the total transmission coefficient one must, for given parameters a , b , k and ε , find all positive solutions with E < k and sum up their contributions. Beside the mentioned positive solutions,one should take into account also the contribution of the negative solutions E <
0. The dispersion equation and transmissioncoefficient for this case can be found from Eqs.(20,23,24) by analytical continuation. The dispersion equation in this caseacquires the formcos ( k sin θ a ) = cos α b cosh β ( a − b ) − α − β αβ sin α b sinh β ( a − b ) (25)where β = (cid:112) k | E | , α = (cid:112) k ( V d − | E | ) . Corresponding transmission coefficient has the form T = T × T (26) here T = a (cid:12)(cid:12)(cid:12)(cid:12) sinh β ( a − b ) β + sin α b α + cos α b − e ik asin θ cosh β ( a − b ) e ik asin θ sinh β ( a − b ) + βα sin α b (cid:32) β ( a − b ) β − β sin α b α (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) (27)and T = (cid:20) a + sinh 2 β ( a − b ) β + sin 2 α b α − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos α b − e ik asin θ cosh β ( a − b ) e ik asin θ sinh β ( a − b ) + βα sin α b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ×× (cid:18) a − b − sinh 2 β ( a − b ) β + β sin 2 α b α − β b α (cid:19) ++ (cid:32) cos α b − e ik a sin θ cosh β ( a − b ) e ik asin θ sinh β ( a − b ) + βα sin α b + cos α b − e − ik asin θ cosh β ( a − b ) e − ik asin θ sinh β ( a − b ) + βα sin α b (cid:33) ×× (cid:18) sinh β ( a − b ) β + β cos 2 α b α − β α (cid:19) (cid:21) − (28)Now using the above mentioned expressions one can calculate transmission coefficient of central diffracted wave. Taking a = . µ m , b = . µ m , ε = k = µ m − and numerically calculating we get for θ = E s = . T s = . E m = . T m = | E b | = . T b = . E s , E m <
12 are the positive solution of dispersion equation Eq.(20), T s , m are the corresponding partial transmission coefficients calculated using Eq.(24). Correspondingly E b , T b are contributionsfrom negative solution calculated using Eqs.(25) and (28). The resulting transmission coefficient for incident angle θ = T = T s + T b + T m = . θ = π / E s = . T s = . E m = . T m = . | E b | = . T b = . T = T s + T b + T m = . θ = π / E s = . T s = . E m = . T m = . | E b | = . T b = . T = T s + T m + T b = . s , b modes. The contribution of the m mode with not small positive energy is negligible and notshown in band scheme Fig.3.The angle dependence of transmission coefficient is presented in Fig.2 of main text. Note that inthe vacuum case b = E m = k sin θ /
2. Weuse this value when calculating transmission coefficient in vacuum case, see Fig.2.2. Weuse this value when calculating transmission coefficient in vacuum case, see Fig.2.