SSpace-Time Metamaterials.
Andrei Rogov and Evgenii Narimanov
School of Electrical and Computer Engineering, and Birck Nanotechnology Center, Purdue University, West Lafayette IN 47907 (Dated: January 17, 2018)Despite more than a decade of active research, the fundamental problem of material loss remains a majorobstacle in fulfilling the promise of the recently emerged fields of metamaterials and plasmonics to bring inrevolutionary practical applications. In the present work, we demonstrate that the problem of strong materialabsorption that is inherent to plasmonic systems and metamaterials based on plasmonic components, can beaddressed by utilizing the time dimension. By matching the pulse profile to the actual response of a lossy meta-material, this approach allows to o ff set the e ff ect of the material absorption. The existence of the correspondingsolution relies on the fundamental property of causality, that relates the absorption in the medium to the varia-tions in the frequency-dependent time delay introduced by the material, via the Kramers-Kronig relations. Wedemonstrate that the proposed space-time approach can be applied to a broad range of metamaterial-based andplasmonic systems, from hyperbolic media to metal optics and new plasmonic materials. I. INTRODUCTION
The fields of metamaterials and plasmonics promise a broadrange of exciting applications – from electromagnetic in-visibility and cloaking [1] to negative refraction and super-resolution imaging [2] to subwavelength field localization,confinement and amplification in plasmonic resonators. [3,4] However, the performance of practical plasmonics- andmetamaterials- based applications is severely limited by losses[3] – e.g. it is the material absorption that “confines” the su-perlens to the near-field operation. [5] Despite multiple at-tempts to remove this stumbling block with new materials [6]or incorporating material gain in the composite design [7], the(nearly) “lossless metal” [8] that would allow the evanescentfield control and amplification promised by metamaterial re-search for nearly two decades since the seminal work of J.Pendry, [2] remains an elusive goal.[6]Here we present an alternative approach to address theproblem of optical absorption in metamaterials and plasmon-ics, that builds upon the pioneering work lead by J.-J. Gre ff et[9] that demonstrated a substantial improvement of the super-lens resolution when using time-dependent illumination. Weshow that, by introducing time-variation in the intensity ofthe incident light with the pattern that is defined by the actualresponse of the lossy (meta)material, combined with the time-gating of the transmitted signal, the detrimental e ff ect of theelectromagnetic absorption can be dramatically reduced. Thegeneral existence of such “matching” solution follows fromthe fundamental Kramers-Kronig relations, and the “residual”absorption is only limited by the signal-to-noise ratio in thesystem, and thus ultimately set by the quantum fluctuations.The fundamental impact of material losses in nanopho-tonics goes substantially beyond simply loosing a significantfraction of the signal power. In plasmonic media and metama-terials with plasmonic components, optical absorption leadsto essentially nonuniform attenuation of di ff erent transversewavenumber components. What makes the matters worse fornanophotonics, is that these high-wavenumber fields that areessential for subwavelength light confinement, experience thestronger attenuation. This results in a dramatic deteriorationof the performance of nanophotonic devices even in the con- ditions of the relatively small loss.As an example of this behavior, consider light propagationin a hyperbolic (meta)material, where the real parts of the di-electric permittivity components have opposite signs in twoorthogonal directions. [10–12] In contrast to light in conven-tional dielectrics where the frequency limits the propagationwavelength and the corresponding light focusing and confine-ment, the wavenumbers of propagating fields in hyperbolicmedia are unrestricted by the frequency. [12] As a result, hy-perbolic materials support tightly focused optical beams – seeFig. 1(a). However, as a function of the propagation distancethe relative amplitudes of di ff erent wavenumber componentsin the beam scale (see Methods ) as H k τ (cid:29) ω/ c ∝ exp (cid:40) − Im (cid:34) (cid:114) − (cid:15) τ (cid:15) n (cid:35) · k τ · z (cid:41) , (1)so that the beam looses high- k components at progressivelyfaster rate – leading to the eventual broadening of the beam inthe hyperbolic medium – see Fig. 1.Light in the superlens shows a similar behavior – while theperfect lossless superlens supports all wavenumbers [2], in-troduction of loss e ff ectively suppresses high- k x components[12, 13] – see Fig. 2. With a finite noise that is always presentin the imaging system, relatively weak high-wavenumbercomponents responsible for spatial resolution get lost in thebackground and the resolution is compromised.[5]Note however, that the essential di ff erence in the relativeabsorption of di ff erent wavenumber components, by virtueof causality and its mathematical representation in terms ofKramers-Kronig relations, implies a di ff erence in the corre-sponding phase velocity and the resulting time delays. Asa result, if instead of coming from a continuous-wave (CW)source the incident field forms a pulse train, with the properchoice of pulse profile, the amplitudes of low- and high- k x components can be matched at t > t – see Fig. 3). While de-tection at an earlier time (e.g. t ) will reveal a signal with mis-matched wavenumber components (and resulting real-spacebroadening), time-gating of the output signal near the time t will recover the undistorted high-resolution signal.In order to match the amplitudes of di ff erent k x -componentsat the desired time t , the required pulse profile (and thus itsfrequency spectrum) depends on both the material parameters a r X i v : . [ phy s i c s . op ti c s ] J a n (a) (b)(c) z / λ = 0.001 z / λ = 0.025 z / λ = 0.05 z / λ = 0.1 z / λ = 0.2 - - x / (cid:1) (cid:2) H (cid:3) H FIG. 1. Beam broadening in hyperbolic media. Panel (a): light intensity from a source at the edge of a half-infinite ( z >
0) hyperbolic material(sapphire). The corresponding free-space wavelength is λ = µ m, and the dielectric permittivity components are (cid:15) x = (cid:15) y (cid:39) . + . i , (cid:15) z (cid:39) − . + . i . The subwavelngth source is formed by a metallic mask with a long slit aligned parallel to the y -direction, with the width a =
20 nm. Panel (b): the magnetic field profile at di ff erent distances from the surface of the hyperbolic medium, for the same parameters asthe panel (a). Note the progressive widening of the beams away from the source at x = z =
0. Panel (c): the schematics illustrating theorigin of the degradation of imaging resolution in hyperbolic (meta)materials in the presence of material losses. and the dimensions of the metamaterial / plasmonic system.When the medium satisfies Kramers-Kronig relations, suchsolution always exists and, as we show in the next section, inmost cases is mathematically straightforward. II. RESULTS
In many applications of metamaterials, the observable A out ,in the frequency domain can be related to the original “input” A in via the linear relation A out ( ω ) = c ω − Ω + i γ · A in ( ω ) (2)For example, in the case of subwavelength imaging with ametamaterial superlens, [2] A in represents the original objectpattern, while A out corresponds to the electromagnetic fieldamplitude measured in the image plane – see the Methods sec-tion. In this case, while γ relates to the loss in the metama-terial, the corresponding parameters c and Ω explicitly de-pend on the Fourier wavenumber k τ (see Eqns, (36) - (41) in H in (x)= δ (x) low- k x high- k x high- k x low- k x SL H out (x) Object (Point source) Image
Air Air (a) (b)
FIG. 2. Panel (a): the schematics illustrating the origins of the resolution degradation of the superlens in the presence of material losses.The lossy near-field superlens (SL) creates a blurred image of the object (CW point source) due to uneven attenuation of low- k x and high- k x components. Panel (b): k x -spectrum at the image plane of a lossless superlens ( (cid:15) = −
1, blue dashed line) and a superlens with material loss( (cid:15) ≈ − + . i , orange curve) demonstrating the suppression of high- k x components when illuminated with CW light. The data were obtainedfor a slab (thickness d =
10 nm) of aluminum-doped zinc oxide (AZO) as a superlens creating an image of the p -polarized point light sourcewith the carrier wavelength of 1617 nm corresponding to Re [ (cid:15) ] = − k x is the transverse wavenumber component and k d = π d . Methods ), so that the distinct Fourier components of the orig-inal pattern are disproportionally represented in the observedimage – with the resulting distortions and loss of resolution.However, in the limit γ →
0, at appropriate frequency an ex-act cancellation reduces this “transfer function” to exact unity– corresponding to the behavior of a “perfect lens.” A finiteamount of the e ff ective loss γ in Eqn. (2) will define the actualresolution.In another example, [14] a hyperbolic metamaterial sub-strate is used to generate subwavelength illumination “spots”that can be used for super-resolution imaging within the struc-tured illumination framework, as described in the Methods t tt t z = 0 z = d z E Ex
FIG. 3. The key idea behind the e ff ective loss reduction with pulseshaping. If the illuminating field pulse profile is appropriately cho-sen, with the di ff erence in the time delays accumulated over the prop-agation in the metamaterial, low- k x (red curve) and high- k x (bluecurve) components of the image can be power-matched at some point t in time. section. In this case, A in corresponds to the illumination fieldamplitude at a given wavelength, that is incident on the hyper-bolic metamaterial substrate – see Fig. 1, while A out repre-sents the field at the other surface of the hyperbolic medium.The spatial variation of A out is defined by the coordinate de-pendence of Ω – see Eqn. (25) in Methods . In this exam-ple, the Lorenzian structure of the “transfer function” of Eqn.(2) with the spatial coordinate - dependent Ω corresponds tothe (subwavelength) illumination spot whose central locationdepends on the illumination frequency. The spatial size ofthis localized illumination area, set by the e ff ective materialloss parameter γ (see Eqns. (21),(24) ), ultimately defines theimaging resolution of this structured illumination imaging ap-proach that is based on hyperbolic metamaterials.In the proposed space-time approach, we consider a meta-material system that can be described by the general Eqn. (2),whose “input” A in (such as e.g. the illumination field in animaging system) is modulated with a pulse-shaper, A in ( t ) = A ( t ) · exp ( − i ω t ) , (3)in such a wave that the resulting “input- output” relation in thetime domain A out ( t ) = c e ff ω − Ω + i γ e ff · A in ( t ) (4)has the same mathematical form as (2) but with a new (pos-sibly time-dependent) e ff ective value γ e ff . A substantial re-duction of γ e ff as compared to its original value γ , togetherwith appropriate time-gating of the “output” A out is then anequivalent to using a new class of metamaterials with reducedloss.While somewhat counter-intuitive, this objective can beachieved with a relatively simple pulse shape of the form A N ( t ) = a θ ( t ) t N exp ( − γ t ) , (5)where γ is the material loss parameter as defined in Eqn. (2), N ≥ θ ( t ) is the Heaviside HM (a) (b) (c) FIG. 4. Pulse shaping for structured illumination imaging with hyperbolic media (HM). Panel (a): the schematics of the imaging systemand magnetic field ( B y ) intensity profile of the shaped light source. Panel (b): the magnetic field intensity profiles of the light at the right sidesurface of the slab obtained by illumination with CW light (blue dashed curve) and shaped light (green curve corresponding to the detectiontime t ∗ =
65 ps), normalized to the unity at their respective maxima. The data were obtained for a 3 µ m slab of sapphire, one side of which iscovered with an opaque mask with a small aperture in it that e ff ectively creates a point light source on the surface of the slab. The mask surfaceis illuminated with p -polarized light at the carrier frequency f =
15 THz ( λ = µ m), corresponding to the permittivities (cid:15) e = − . + . i and (cid:15) o = . + . i along the extraordinary and ordinary axes of the sapphire crystal respectively. The pulse profile is shaped accordingto Eq. (5), with the modulation parameter N = f w =
100 GHz,FSR = . function (zero for a negative argument, and unity otherwise).Note that the frequency spectrum of (5) is a simple power law,so that its actual implementation with a practical pulse-shapershould be straightforward.With (5), for A out in the time domain we find A out ( t ) = − i c t · F N [( ω − Ω ) t ] · A in ( t ) , (6)where F N ( x ) = N !( ix ) N + · exp ( ix ) − N (cid:88) k = ( ix ) k k ! (7) = exp ( ix )( ix ) N + (cid:16) N ! − Γ ( N + , ix ) (cid:17) , (8)and Γ ( n , x ) is the incomplete Gamma-function.Using the Pad´e approximation [15] for the function F ( x )(see Methods ), we obtain F N ( x ) (cid:39) N + − ix , (9)which together with (6) yields the desired Eqn. (4), with thee ff ective parameters c e ff = c , (10) γ e ff = N + t . (11)Time-gating the output A out ( t ) at t > / ( N + γ will thenreduce the e ff ective width γ e ff beyond the original value γ thatwas defined by the actual material parameters. Furthermore,as long as the photonic system is adequately described by theclassical Maxwell equations (so that the quantum noise can be neglected), there is no limit on the degree of the reduc-tion of the “e ff ective” loss. As the fundamental level, the pro-posed space-time approach therefore solves the loss problemthat plagued the field of metamaterials for the last decade.However, with the pulse profile (5), time-gating the outputsignal at t (cid:29) /γ implies operating at progressively lowerpowers. As a result, there is a practical limit to the “loss mit-igation” in the proposed space-time approach that is definedby the actual signal-to-noise power ratio SNR in the system.From Eqns. (5) and (11), we obtain γ e ff = γ N + N + log P peak P ( t ) , (12)where P peak is the peak power of the pulse – which defines thelimit to the e ff ective loss reduction in the proposed space-timeapproach γ min = γ log SNR . (13)For super-resolution imaging, this result can also be inter-preted in terms of the e ff ective loss, e.g. the imaginary partof the dielectric permittivity of the (meta)material that wouldallow the same resolution power with CW illumination, as ourapproach: (cid:15) (cid:48)(cid:48) e ff (cid:39) (cid:15) (cid:48)(cid:48) N + N + log P peak P , (14)where (cid:15) (cid:48)(cid:48) corresponds to the actual value of the permittivity.In case of a large signal-to-noise ratio, our approach thereforeo ff ers an intriguing alternative to the search of new and bet-ter metamaterials and plasmonic media – instead of trying to SL (a) (b) (c) FIG. 5. Pulse shaping for imaging for a superlens (SL). Panel (a): the schematics of the imaging system and magnetic field ( H y ) intensityprofile of the shaped light source. Panel (b): the magnetic field intensity of the images obtained by illuminating the target with CW light(blue dashed curve) and with the pulse train “tuned” to the material response (orange and green curves corresponding to the detection times t = .
09 ps and t = .
14 ps respectively), normalized to the unity at their respective maxima. The imaging data were calculated for a 10 nmslab of aluminum-doped zinc oxide (AZO) as a near-field superlens creating an image of two point objects located 20 nm from each other. TheSL is illuminated with p -polarized light source at the carrier frequency f =
185 THz ( λ = (cid:15) ] = −
1. The pulsemodulation parameter N =
3. In the simulation, the superlens is illuminated with a period pulse train, with free spectral range FSR = . f w =
44 THz. reduce the actual material absorption, the same result can beachieved with the temporal modulation of the incident lightthat is “tuned” to the response of the existing media (note thatthe pulse parameter γ in Eqn. (5) is defined by the materialresponse in Eqn. (2) ). III. DISCUSSION
Based on the general linear response formulation of Eqn.(2), our proposed space-time approach can be applied to abroad range of nanophonic systems that involve metamaterialor plasmonic elements. In the first example, we apply it tohyper-structured illumination imaging, [14] using sapphire inits hyperbolic band near 20 µ m – see Fig. 4. In this simu-lation, to adequately represent a practical experimental setup,we furthermore assume that the pulse-shaper is only operatingin a finite bandwidth window, with all the frequency compo-nents of the band-limited signal within its range. With thepulse-shaping of the incident field and the time-gating of thetransmitted light at t ∗ =
65 ps, we achieve the same field local-ization as if the imaginary parts of the dielectric permittivity atboth ordinary and extraordinary axes were e ff ectively reducedby approximately 50% – from Im [ (cid:15) ] ≈ .
29 to Im [ (cid:15) e ff ] ≈ . (cid:15) ] ≈ .
17 to Im [ (cid:15) e ff ] ≈ .
075 in the extraordinary one.As the other example, we consider the application of theproposed space-time approach to super-resolution imagingwith the near-field superlens based on aluminum-doped zincoxide (AZO), operating at the telecommunication wavelengthof 1617 nm – see Fig. 5 . While this superlens is originallyunable to resolve two point objects at 20 nm spacing due tothe material loss of the AZO, using the pulse-shaped illumi- nation with full spectral width f w =
44 THz and free spectralrange FSR = . ff ect transistors [18], Schot-tky diodes [19], or superlattice detectors [20]. For picosecondand sub-picosecond optical pulse measurement that would berequired for implementation of the suggested approach forthe superlens imaging with AZO ( λ = IV. METHODS
In this section, we provide the theoretical background forEqn. (2) that describes electromagnetic field in a range ofnanophotonic systems – from hyperbolic media to plasmonicsystems and negative index metamaterials.
A. Light in hyperbolic media
With the opposite signs of the dielectric permittivitycomponents in two orthogonal directions, hyperbolic me-dia can support TM-polarized electromagnetic fields with thewavenumbers that are only limited by the size of the unit cellof the material. For a uniaxial hyperbolic medium with thepermittivities (cid:15) x = (cid:15) y and (cid:15) z , for the relation between thewavevector k and the frequency ω we find [10] k x + k y (cid:15) z + k z (cid:15) x = ω c , (15)For a thin (subwavelength) illumination slit at the surface ofthe hyperbolic medium, corresponding to the configuration inFig. 1, the TM-polarized magnetic field in the hyperbolic ma-terial, H ( r , t ) = (cid:90) d ω H ω ( x , z ) exp ( − i ω t ) ˆ y (16)where its time-harmonic amplitudes H ω ( x , z ) can be ex-pressed as H ω ( x , z ) = H ω π (cid:90) ∞−∞ dk x exp ( ik x x ) × exp (cid:32) i (cid:113) (cid:15) x ( ω/ c ) − ( (cid:15) x /(cid:15) z ) k x s z (cid:33) (17)where s = sign (cid:34) Im (cid:114) − (cid:15) x (cid:15) z (cid:35) (18)The expression (17) is exact for a natural hyperbolic medium,and is an accurate approximation for a hyperbolic metamate-rial when its unit cell size is on the same order or smaller thanthe width of the illumination slit.Within the absorption distance from a narrow (comparedto the free-space wavelength) illumination field, the integral(17) is dominated by the wavenumbers k x (cid:29) ω/ c . The z -component of the wavevector can then be approximated by k z ≡ (cid:113) (cid:15) x ( ω/ c ) − ( (cid:15) x /(cid:15) z ) k x (cid:39) (cid:112) − (cid:15) x /(cid:15) z | k x | , (19)which corresponds to the linear asymptotic behavior of thedispersion relation (15). The integral (17) can then be calcu-lated analytically, which yields H ω ( x , z ) = H ω π i (cid:88) ± − s (cid:113) − (cid:15) x (cid:15) z z ± x (20) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) x (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) x (cid:1) (cid:1) (a) (b) FIG. 6. The exact values of F N (symbols) and the correspondingPad´e approximation (solid lines), for N = N = F N and F P N , while bluerepresents the corresponding imaginary components. If the frequency bandwidth of the “signal” H ( t ), centeredaround the frequency ω c , is much smaller than the character-istic scale corresponding to a substantial variation of any ofthe components of the permittivity tensor of the hyperbolicmedium, we can approximate s (cid:114) − (cid:15) x (cid:15) z (cid:39) η (cid:48) + η (cid:48) ( ω − ω c ) + i η (cid:48)(cid:48) , (21)which reduces (20) to a special case of our general expression(2), H ω ( x , z ) (cid:39) H ω (cid:88) ± c h ω − Ω ± ( x , z ) + i γ h , (22)where c h = i π z η (cid:48) , (23) γ h = η (cid:48)(cid:48) h η (cid:48) , (24)and Ω ± ( x , z ) = ω c − η (cid:48) η (cid:48) ± x η (cid:48) z . (25) B. Electromagnetic field in the superlens
In its simple realization, [2] the superlens is essentially aparallel slab of a metamaterial with simultaneously negativevalues of the (isotropic) dielectric permittivity (cid:15) and magneticpermeability µ , that “match” the corresponding parameters ofthe surrounding medium (cid:15) , µ : (cid:15) = − (cid:15) , (26) µ = − µ . (27)A perfect resonance of this kind is however unattainable, dueto inevitable finite amount of loss in the metamaterial, leadingto nonzero imaginary parts of (cid:15) and µ .The planar superlens produces a (nearly) perfect imagewhen the separation between the “object”’ and “image” planeis equal to twice the thickness of the superlens d , with thecorresponding transmission coe ffi cient as a function of the in-plane wavenumber k τ , equal to T TM ω ( k τ ) = κ − κ ) d ] (cid:16) + κ(cid:15)κ + (cid:15)κ κ (cid:17) exp (2 κ d ) + (cid:16) − κ(cid:15)κ − (cid:15)κ κ (cid:17) (28) = (cid:15)κ κ(cid:15)κ − κ exp [( κ − κ ) d ] × (cid:88) ± κ − (cid:15)κ ) ± ( κ + (cid:15)κ ) exp ( κ d ) , (29)for TM-polarized fields, and T TE ω ( k τ ) = κ − κ ) d ] (cid:16) + κµκ + µκ κ (cid:17) exp (2 κ d ) + (cid:16) − κµκ − µκ κ (cid:17) (30) = µκ κµκ − κ exp [( κ − κ ) d ] × (cid:88) ± κ − µκ ) ± ( κ + µκ ) exp ( κ d ) , (31)for the TE-polarization, where κ ≡ (cid:113) k τ − ω / c , (32)and κ ≡ (cid:113) k τ − (cid:15)µ ω / c . (33)In the limit (cid:15)/(cid:15) , µ/µ → −
1, we find T ω ( k τ ) → k τ – which implies the formation of a perfect image.Note that the concept of super-resolution relies on the accu-rate representation of high- k Fourier components of the objectpattern, k τ (cid:29) ω/ c . In this limit, for a small loss (cid:15) (cid:48)(cid:48) (cid:28) | (cid:15) | , µ (cid:48)(cid:48) (cid:28) | µ | and the signal bandwidth smaller than the frequencyscale corresponding to a substantial variation of (cid:15) and µ , wefind T TM ω ( k τ ) (cid:39) (cid:88) ± c TM ± ω − Ω TM ± ( k τ ) + i γ TM , (34)and T TE ω ( k τ ) (cid:39) (cid:88) ± c TE ± ω − Ω TE ± ( k τ ) + i γ TE , (35)where c TM ± = ± exp ( − | k τ | d ) d (cid:15) (cid:48) / d ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω = ω c , (36) Ω TM ± ( k τ ) = ω c ∓ − | k τ | d ) d (cid:15) (cid:48) / d ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω = ω c , (37) γ TM = (cid:15) (cid:48)(cid:48) d (cid:15) (cid:48) / d ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω = ω c , (38)and c TE ± = ± exp ( − | k τ | d ) d µ (cid:48) / d ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω = ω c , (39) Ω TE ± ( k τ ) = ω c ∓ − | k τ | d ) d µ (cid:48) / d ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω = ω c , (40) γ TE = µ (cid:48)(cid:48) d µ (cid:48) / d ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω = ω c . (41) As a result, the transmission function of a superlens can alsobe represented by our general expression (2).Note that, for super-resolution imagining in the near field,the desired functionality can be also achieved by the so called“poor man’s superlens,” that is essentially a metallic layerused for TM-polarized light near the plasmon resonance fre-quency that corresponds to Eqn. (26). The correspondingtransmission coe ffi cient, also given by Eqn. (29) but with thevalue of κ taken at µ =
1, in the subwavelength resolution res-olution limit k τ (cid:29) ω/ c then also reduces to Eqn. (34) that canbe treated as a special case of Eqn. (2). C. Pad´e Approximation
For a given function f ( x ) in a specified interval ( a , b ), thePad´e approximant is the rational function whose power seriesexpansions near x = a and x = b agree with the correspondingpower series expansions of f ( x ) to the highest possible order.[15] In our case, we seek to approximate F N ( x ) in the entirerange 0 ≤ x < ∞ , so that with Pad´e approximant of the firstorder, F P N ( x ) = A N B N + C N x , (42)we impose the conditions A N B N = F N (0) , (43) A N C N x = x · lim x →∞ [ x F N ( x )] , (44)which yields A N B N = N , A N C N = i , (45)and therefore F P N ( x ) = N + − ix , (46)which immediately leads to Eqn. (9).In Fig. 6 we compare the exact values of F N for N = N = ff ers an accurate representation of the exact function F ( x ) inthe entire range 0 ≤ x < ∞ . V. ACKNOWLEDGEMENTS
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