Two different versions of an optical theorem for a scattering body embedded inside a lossy background medium are derived in this paper. The corresponding fundamental upper bounds on absorption are then obtained in closed form by elementary optimization techniques. The first version is formulated in terms of polarization currents (or equivalent currents) inside the scatterer and generalizes previous results given for a lossless medium. The corresponding bound is referred to here as a variational bound and is valid for an arbitrary geometry with a given material property. The second version is formulated in terms of the T-matrix parameters of an arbitrary linear scatterer circumscribed by a spherical volume and gives a new fundamental upper bound on the total absorption of an inclusion with an arbitrary material property (including general bianisotropic materials). The two bounds are fundamentally different as they are based on different assumptions regarding the structure and the material property. Numerical examples including homogeneous and layered (core-shell) spheres are given to demonstrate that the two bounds provide complimentary information in a given scattering problem.
This paper reformulates and extends some recent analytical results concerning a new optical theorem and the associated physical bounds on absorption in lossy media. The analysis is valid for any linear scatterer (such as an antenna), consisting of arbitrary materials (bianisotropic, etc.) and arbitrary geometries, as long as the scatterer is circumscribed by a spherical volume embedded in a lossy background medium. The corresponding formulas are here reformulated and extended to encompass magnetic as well as dielectric background media. Explicit derivations, formulas and discussions are also g…
We theoretically study negative refraction of inhomogeneous waves at the interface of lossy isotropic media. We obtain explicit (up to the sign) expressions for the parameters of a wave transmitted through the interface between two lossy media characterized by complex permittivity and permeability. We show that the criterion of negative refraction that requires negative permittivity and permeability can be used only in the case of a homogeneous incident wave at the interface between a lossless and lossy media. In a more general situation, when the incident wave is inhomogeneous, or both media …
We review the properties of nonlinear, multidimensional localized waves whose stationary propagation is sustained by a dynamic equilibrium between self-focusing and nonlinear losses. Their finite-energy versions preserve light bullet behavior well-beyond the characteristic diffraction or dispersion distances, and rebuild after obstacles. There exists a preferential lossy light bullet with maximum intensity and losses, defined solely by the optical properties of the medium, which is the most stable, non-conical localized wave supported by a medium with self-focusing nonlinearity and nonlinear l…
Slow-light enhanced absorption in liquid-infiltrated photonic crystals has recently been proposed as a route to compensate for the reduced optical path in typical lab-on-a-chip systems for bio-chemical sensing applications. A simple perturbative expression has been applied to ideal structures composed of lossless dielectrics. In this work we study the enhancement in structures composed of lossy dielectrics such as a polymer. For this particular sensing application we find that the material loss has an unexpected limited drawback and surprisingly, it may even add to increase the bandwidth for l…
Enhancing the absorption and emission of electromagnetic waves over a broad range of wavelengths is a topic of fundamental and applied interest in photonics and energy research. In the context of light trapping in solar cells, for example, significant interest in the past decade has focused on overcoming limits in the ray-optics regime with nanophotonic structures. However, many such structures, in particular plasmonic structures, or PT-symmetric systems can posses multiple materials with varying values of intrinsic loss. Here, we rigorously determine the effect of parasitic loss on the achiev…
Several dynamical properties such as energy, momentum, angular momentum, and optical helicity have been recently reexamined in dispersive and lossless media. Here, we address a parallel derivation for the optical chirality, extending it so as to include dissipative effects as well. To this end, we first elaborate on the most complete form of the conservation law for the optical chirality, without any restrictions on the nature of the medium. As a result we find a general expression for the optical chirality density both in lossless and lossy dispersive media. Our definition is perfectly consis…
Fundamental upper bounds are given for the plasmonic multipole absorption and scattering of a rotationally invariant dielectric sphere embedded in a lossy surrounding medium. A specialized Mie theory is developed for this purpose and when combined with the corresponding generalized optical theorem, an optimization problem is obtained which is explicitly solved by straightforward analysis. In particular, the absorption cross section is a concave quadratic form in the related Mie (scattering) parameters and the convex scattering cross section can be maximized by using a Lagrange multiplier const…
We analyse the superluminal propagation of narrow-band pulses at resonances in dissipative media. The output waveform is an attenuated, undistorted, time-advanced version of the input which can be interpreted as the result of the interference of two scaled replicas of the input having a positive relative delay. The proposed analysis applies both to the propagation in a passive bulk medium at an electronic resonance and in a dielectric waveguide coupled to a lossy micro-ring resonator. This latter case provides a clear insight of the underlying physical phenomena.
Most transmission and detection channels fail to faithfully support broadband wave packets because of physical limitations, like chromatic dispersion and absorption. We explore the case of lossy detection of ultrashort THz pulses using the widespread electro-optic detection scheme. We demonstrate that one can fully recover the original THz pulse shape, duration and amplitude, using a simple experimental procedure and a reconstruction algorithm which encodes the physical properties of the detection system.
A low loss propagating electromagnetic wave is shown to exist at a gradual interface between two lossy conductive media. The electromagnetic frequency range of this phenomenon may span from UV optics to RF range. In particular, it is demonstrated that such a surface wave may be guided by a seafloor-seawater interface and it may be used in radio communication and imaging underwater. Similar surface waves may also be guided by various tissue boundaries inside a human body. For example, such surface wave solutions may exist at planar interfaces between skull bones and grey matter inside a human h…
Waves entering a spatially uniform lossy medium typically undergo exponential decay, arising from either the energy loss of the Beer-Lambert-Bouguer transmission law or the evanescent penetration during reflection. Recently, exceptional point singularities in non-Hermitian systems have been linked to unconventional wave propagation, such as the predicted extremely spatially broad constant-intensity guided modes. Despite such promises, the possibility of decay-free wave propagation in a purely lossy medium has been neither theoretically suggested nor experimentally realized until now. Here we d…
In this tutorial, we discuss the radiation from a Hertzian dipole into uniform isotropic lossy media of infinite extent. If the medium is lossless, the radiated power propagates to infinity, and the apparent dissipation is measured by the radiation resistance of the dipole. If the medium is lossy, the power exponentially decays, and the physical meaning of radiation resistance needs clarification. Here, we present explicit calculations of the power absorbed in the infinite lossy host space and discuss the limit of zero losses. We show that the input impedance of dipole antennas contains a radi…
We characterize surface-plasmon polaritons at lossy planar interfaces between one dispersive and one nondispersive linear isotropic homogeneous media, i.e., materials or metamaterials. Specifically we solve Maxwell's equations to obtain strict bounds for the permittivity and permeability of these media such that satisfying these bounds implies surface-plasmon polaritons successfully propagate at the interface, and violation of the bounds impedes propagation, i.e., the field delocalizes from the surface into the bulk. Our characterization of surface-plasmon polaritons is valuable for checking v…
The spectral functions are studied in conjunction with the dyadic Green's functions for various media. The dyadic Green's functions are found using the eigenfunction expansion method for homogeneous, inhomogeneous, periodic, lossless, lossy, and anisotropic media, guided by the Bloch- Floquet theorem. For the lossless media cases, the spectral functions can be directly related to the photon local density of states, and hence, to the electromagnetic energy density. For the lossy case, the spectral function can be related to the field correlation function. Because of these properties, one can de…
Several applications, such as optical tweezers and atom guiding, benefit from techniques that allow the engineering of optical fields' spatial profiles, in particular their longitudinal intensity patterns. In cylindrical coordinates, methods such as Frozen Waves allow an advanced control of beams' characteristics, but in Cartesian coordinates there is no analogous technique. Since Cartesian beams may also be useful for applications, we develop here a method to modulate on-demand the longitudinal intensity pattern of any (initially) unidimensional Cartesian beam with concentrated wavevector spe…
In this paper we study the electromagnetic scattering, absorption, and performance bounds for short time modulated pulses that impinge on a time-varying lossy layer that is sandwiched between vacuum and a perfect electric conductor. The electric characteristics of the layer, namely, the conductivity, permittivity, and permeability are assumed to change abruptly or gradually in time. We demonstrate numerically that a time-varying absorbing layer that undergoes temporal switching of its permittivity and conductance can absorb the power of a modulated, ultra-wideband, as well as a quasi-monochrom…
We propose a correction to the conventional Rytov approximation (RA) and investigate its performance for predicting wave scattering under strong scattering conditions. An important motivation for the correction and investigation is to help in the development of better models for inverse scattering. The correction is based upon incorporating the high frequency theory of inhomogeneous wave propagation for lossy media into the RA formulation. We denote the technique as the extended Rytov approximation for lossy media (xRA-LM). xRA-LM significantly improves upon existing non-iterative linear scatt…
This paper discusses and analyzes the quasistatic optimal plasmonic dipole resonance of a small dielectric particle embedded in a lossy surrounding medium. The optimal resonance at any given frequency is defined by the complex valued dielectric constant that maximizes the absorption of the particle under the quasistatic approximation and a passivity constraint. In particular, for an ellipsoid aligned along the exciting field, the optimal material property is given by the complex conjugate of the pole position associated with the polarizability of the particle. In this paper, we employ classica…
On a lossless periodic dielectric structure sandwiched between two homogeneous media, bound states in the continuum (BICs) with real frequencies and real Bloch wavevectors may exist, and they decay exponentially in the surrounding homogeneous media and do not couple with propagating plane waves with the same frequencies and wavevectors. The BICs are of significant current interest, because they give rise to high-$Q$ resonances when the structure or the Bloch wavevector is slightly perturbed. In this paper, the effect of a small material loss on the BICs is analyzed by a perturbation method and…
We give a comprehensive study on regularized approximate electromagnetic cloaking via the transformation optics approach. The following aspects are investigated: (i) near-invisibility cloaking of passive media as well as active/radiating sources; (ii) the existence of cloak- busting inclusions without lossy medium lining; (iii) overcoming the cloaking-busts by employing a lossy layer outside the cloaked region; (iv) the frequency dependence of the cloaking performances. We address these issues and connect the obtained asymptotic results to singular ideal cloaking. Numerical verifications and d…
Magnetic Faraday rotation is widely used in optics and MW. In uniform magneto-optical materials, this effect is very weak. One way to enhance it is to incorporate the magnetic material into a high-Q optical resonator. One problem with magneto-optical resonators is that along with Faraday rotation, the absorption and linear birefringence can also increase dramatically, compromising the device performance. Another problem is strong ellipticity of the output light. We discuss how the above problems can be addressed in the cases of optical microcavities and a slow wave resonators. We show that a s…
Resonant modes in a lossy periodic structure sandwiched between two lossless homogeneous media form bands that depend on the Bloch wavevector continuously and have a complex frequency due to radiation and absorption losses. A complex bound state in the continuum (cBIC) is a special state with a zero radiation loss in such a band. Plane waves incident upon the periodic structure induce local fields that are resonantly enhanced. In this paper, we derive a rigorous formula for field enhancement, and analyze its dependence on the frequency, wavevector and amplitude of the incident wave. For resona…
Coherent perfect absorption (CPA) is an interferometric effect that guarantees full absorption in a lossy layer independently of its intrinsic losses. To date, it has been observed only at a single wavelength or narrow bandwidths, whereupon wavelength-dependent absorption can be ignored. Here we produce CPA over a bandwidth of ~ 60 nm in a 2-um-thick polymer film with a low-doping concentration of an organic laser dye. A planar cavity is designed with a spectral `dip' to accommodate the dye resonant linewidth, and CPA is thus achieved even at its absorption edges. This approach allows realizin…
In this paper we deal with the influence on absorption of the diffusive media characteristics framing the problem in connection with the invariance property (IP) of the mean path length. We show that the IP is an important issue that regulates but not prevent the search of absorption maximization by scattering characteristics. We find that the scattering may increase the absorption or even be detrimental, depending on the geometry of the medium and the conditions of its illumination.
A universal waveguide mass--energy relation for lossy one-dimensional waves in nature
Institute of Estuarine and Coastal Research, School of Ocean Engineering and Technology, Sun Yat-Sen University / Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai)（中山大学海洋工程学院河口海岸研究所 / 南方海洋科学与工程广东省实验室（珠海））；State and Local Joint Engineering Laboratory of Estuarine Hydraulic Technology / Guangdong Provincial Engineering Research Center of Coasts, Islands and Reefs / Guangdong Provincial Key Laboratory of Marine Resources and Coastal Engineering / Guangdong Provincial Key Laboratory of Information Technology for Deep Water Acoustics / Key Laboratory of Comprehensive Observation of Polar Environment (Sun Yat-Sen University), Ministry of Education（教育部河口海岸综合观测重点实验室（中山大学） / 深水声学信息技术广东省重点实验室 / 海洋资源与海岸工程广东省重点实验室 / 岛屿礁滩海岸工程广东省工程研究中心 / 河口水动力技术省部共建国家重点实验室培育基地）；Zhuhai Research Center, Hanjiang National Laboratory（汉江国家实验室珠海研究中心）；School of Marine Sciences, Sun Yat-Sen University（中山大学海洋科学学院）
We extend the ideas of the recently proposed perfect lens [J.B. Pendry, Phys. Rev. Lett. {\bf 85}, 3966 (2000)] to an alternative structure. We show that a slab of a medium with negative refractive index bounded by media of different positive refractive index also amplifies evanescent waves and can act as a near-perfect lens. We examine the role of the surface states in the amplification of the evanescent waves. The image resolution obtained by this asymmetric lens is more robust against the effects of absorption in the lens. In particular, we study the case of a slab of silver, which has nega…
The time reversal symmetry of the wave equation allows wave refocusing back at the source. However, this symmetry does not hold in lossy media. We present a new strategy to compensate wave amplitude losses due to attenuation. The strategy leverages the instantaneous time mirror (ITM) which generates reversed waves by a sudden disruption of the medium properties. We create a heterogeneous ITM whose disruption is unequal throughout the space to create waves of different amplitude. The time-reversed waves can then cope with different attenuation paths as typically seen in heterogeneous and lossy …
We report on the stationary and robust propagation of light beams with rather arbitrary and controllable intensity and dissipation transverse patterns in self-focusing Kerr media with nonlinear absorption. When nonlinear absorption is due to multi-photon ionization at high beam powers in transparent media such as glasses or air, these beams can generate multiple plasma channels with tailored geometries. Their nature and spatial characteristics are discussed in detail, as well as the laws determining their spontaneous formation from coherent superpositions of Bessel beams of different amplitude…
We study the propagation of waves in a set of absorbing subwavelength scatterers positioned on a stealth hyperuniform point pattern. We show that spatial correlations in the disorder substantially enhance absorption compared to a fully disordered structure with the same density of scatterers. The non-resonant nature of the mechanism provides broad angular and spectral robustness. These results demonstrate the possibility to design low-density materials with blackbody-like absorption.
We study phase conjugation in a nonlinear composite slab when the counter propagating pump waves are completely absorbed by means of coherent perfect absorption. Under the undepleted pump approximation the coupling constant and the phase conjugated reflectivity are shown to undergo a substantial increase and multivalued response. The effect can be used for efficient switching of the phase conjugated reflectivity in photonic circuits.
Optical absorption is omnipresent and very often distributed non-uniformly in space. We present a numerical study on the effects of inhomogeneous absorption on transmission eigenchannels of light in highly scattering media. In the weak absorption regime, the spatial profile of a transmission channel remains very similar to that without absorption, and the effect of inhomogeneous absorption can be stronger or weaker than homogeneous absorption depending on the spatial overlap of the localized absorbing region with the field intensity maximum of the channel. In the strong absorption regime, the …
We report a theoretical analysis of total absorption conditions in a structure consisting of a lossy coating layer on top of a specular metal substrate, which is known as a Dallenbach absorber. All possible combinations of complex refractive indices of the coating material and thicknesses of the coating layer providing complete absorption are described for the case of normal incidence of a plane wave. The work of coatings based on epsilon-near-zero (ENZ) materials is analyzed in detail. Simple analytical relations are obtained for the conditions of total absorption. It is shown that the range …
The possibility to excite Surface Plasmon Polaritons (SPPs) at the interface between two media depends on the optical properties of both media and geometrical aspects. Specific conditions allowing the coupling of light with a plasmon-active interface must be satisfied. Plasmonic effects are well described in noble metals where the imaginary part of the dielectric permittivity is often neglected ("perfect medium approximation"). However, some systems exist for which such approximation cannot be applied, hence requiring a refinement of the common SPP theory. In this context, several properties o…
While the absorption of light is ubiquitous in nature and in applications, the question remains how absorption modifies the transmission channels in random media. We present a numerical study on the effects of optical absorption on the maximal transmission and minimal reflection channels in a two-dimensional disordered waveguide. In the weak absorption regime, where the system length is less than the diffusive absorption length, the maximal transmission channel is dominated by diffusive transport and it is equivalent to the minimal reflection channel. Its frequency bandwidth is determined by t…
We show that a weakly absorbing, strongly scattering (white) medium can be made very strongly absorbing at any frequency within its strong-scattering bandwidth by optimizing the input electromagnetic field. For uniform absorption, results from random matrix theory imply that the reflectivity of the medium can be suppressed by a factor ~(l_a/lN^2), where N is the number of incident channels and l,l_a are the elastic and absorption mean free paths respectively. It is thus possible to increase absorption from a few percent to > 99%. For a localized weak absorber buried in a non-absorbing scatteri…
We study the optimal diffusive transmission and absorption of broadband or polychromatic light in a disordered medium. By introducing matrices describing broadband transmission and reflection, we formulate an extremal eigenvalue problem where the optimal input wavefront is given by the corresponding eigenvector. We show analytically that a single wavefront can exhibit strongly enhanced total transmission or total absorption across a bandwidth that is orders of magnitude broader than the spectral correlation width of the medium, due to long-range correlations in coherent diffusion. We find exce…
Surface plasma waves are collective oscillations of electrons that propagate along a metal-dielectric interface. In the last ten years, several groups have reproduced fundamental quantum optics experiments with surface plasmons. Observation of single-plasmon states, waveparticle duality, preservation of entanglement of photons in plasmon-assisted transmission, and more recently, two-plasmon interference have been reported. While losses are detrimental for the observation of squeezed states, they can be seen as a new degree of freedom in the design of plasmonic devices, thus revealing new quant…
A simple analysis is performed for the absorption properties of graphene; sandwiched between two media. For a proper choice of media and graphene doping/gating one can approach 50-100% absorption in the GHz-THz range for the one atom thick material. This absorption is controlled by a characteristic chemical potential which depends only on carrier life-time and the indexes of refraction of the dielectric embedding.
We show that a high-order Bessel beam propagating in a medium with nonlinear absorption is not completely absorbed, but survives in the form of a new propagation invariant vortex beam in which the beam energy and orbital angular momentum are permanently transferred to matter and at the same time refueled by spiral inward currents of energy and angular momentum. Unlike vortex solitons and dissipative vortex solitons, these vortex beams are not supported by specific dispersive nonlinearities (self-focusing or self-defocusing) and do not require gain. Propagation invariance in presence of multipl…
A method for realizing asymmetric (one-way) transmission of discretized light in modulated, linear and purely passive optical lattices is suggested, which exploits the idea of unidirectional coherent perfect absorption. The system consists of a linear photonic lattice of coupled resonators or waveguides, side coupled to a chain of lossy elements, in which light can avoid the occupation of the dissipative sites when propagating in one way, but not in the opposite one. Non-reciprocity requires modulation of the resonator/waveguide parameters, realizing a dissipative optical Aharonov-Bohm diode w…
Polymer compounds from titania-doped polyethylene are fabricated and their linear optical properties characterized by THz-TDS. We show that high concentration of dopants not only enhances the refractive index of the composite material, but also can dramatically raise its absorption coefficient. We demonstrate that the design of Bragg reflectors based on lossy composite polymers depends on finding a compromise between index contrast and corresponding losses. A small absorption value is also shown to be favorable, compared to an ideal lossless reflector, as it enables to smooth the transmission …
An ultrabroad absorber based on double-ring-shaped titanium nitride (TiN) nanoresonators, which can work in high temperatures, is proposed and numerically studied. The absorber with some optimal parameters exhibits an averaged absorption of 94.6% in the range of 200 - 4000 nm (from ultraviolet to mid-infrared) and a band from 200 - 3518 nm having an absorption > 90%. We have demonstrated in detail the physical mechanisms of the ultra-broad absorption, including the dielectric lossy property of TiN material itself in shorter wavelengths and plasmonic resonances caused by the metallic property o…
We elaborate a new solution for the problem of stable propagation of transversely localized vortex beams in homogeneous optical media with self-focusing Kerr nonlinearity. Stationary nonlinear Bessel-vortex states are stabilized against azimuthal breakup and collapse by multiphoton absorption, while the respective power loss is offset by the radial influx of the power from an intrinsic reservoir. A linear stability analysis and direct numerical simulations reveal a region of stability of these vortices. Beams with multiple vorticities have their stability regions too. These beams can then form…
In the presence of arbitrary three-dimensional linear media with material loss and amplification, we present an electromagnetic field quantization scheme for quasinormal modes (QNMs), extending previous work for lossy media [Franke et al., Phys. Rev. Lett. 122, 213901 (2019)]. Applying a symmetrization transformation, we show two fundamentally different ways for constructing a QNM photon Fock space, including (i) where there is a separate operator basis for both gain and loss, and (ii) where the loss and gain degrees of freedom are combined into a single basis. These QNM operator bases are sub…
We study the mode conversion and resonant absorption phenomena occurring in a slab of a stratified anisotropic medium, optical axes of which are tilted with respect to the direction of inhomogeneity, using the invariant imbedding theory of wave propagation. When the tilt angle is zero, mode conversion occurs if the longitudinal component of the permittivity tensor, which is the one in the direction of inhomogeneity in the non-tilted case, varies from positive to negative values within the medium, while the transverse component plays no role. When the tilt angle is nonzero, the wave transmissio…
It has been found that in the media where the dielectric permittivity $ε$ or the magnetic permeability $μ$ is near zero and in transition metamaterials where $ε$ or $μ$ changes from positive to negative values, there occur a strong absorption or amplification of the electromagnetic wave energy in the presence of an infinitesimally small damping or gain and a strong enhancement of the electromagnetic fields. We attribute these phenomena to the mode conversion of transverse electromagnetic waves into longitudinal plasma oscillations and its inverse process. In this paper, we study analogous phen…
In recent works, it has been shown rather generally that the time-reversed process of lasing at threshold realizes a coherent perfect absorber (CPA). In a CPA, a lossy medium in an optical cavity with a specific degree of dissipation, equal in modulus to the gain of the lasing medium, can perfectly absorb coherent optical waves that are the time-reversed counterpart of the lasing field. Here the time-reversed process of lasing is considered in details for a homogeneously-broadened two-level medium in an optical cavity, and the conditions for CPA are derived. It is shown that, owing to the disp…
Study of interaction between high absorption matter and microwave radiated energy is a subject of great importance. Especially, this concerns microwave spectroscopic characterization of biological liquids. Use of effective testing methods to obtain information about physical properties of different liquids on the molecular level is one of the most important problems in biophysics. However, the standard methods based on the microwave resonant techniques are not sufficiently suitable for biological liquids because the resonance peak in a resonator with high-loss liquids is so broad that the mate…
We develop a multiple scattering theory for the absorption of waves in disordered media. Based on a general expression of the average absorbed power, we discuss the possibility to maximize absorption by using structural correlations of disorder as a degree of freedom. In a model system made of absorbing scatterers in a transparent background, we show that a stealth hyperuniform distribution of the scatterers allows the average absorbed power to reach its maximum value. This study provides a theoretical framework for the design of efficient non-resonant absorbers made of dilute disordered mater…
We theoretically and numerically investigate the capability of disordered media to enhance the optical path length in dielectric slabs and augment their light absorption efficiency due to scattering. We first perform a series of Monte Carlo simulations of random walks to determine the path length distribution in weakly to strongly (single to multiple) scattering, non-absorbing dielectric slabs under normally incident light and derive analytical expressions for the path length enhancement in these two limits. Quite interestingly, while multiple scattering is expected to produce long optical pat…
We measured the single-photon detection efficiency of NbN superconducting single photon detectors as a function of the polarization state of the incident light for different wavelengths in the range from 488 nm to 1550 nm. The polarization contrast varies from ~5% at 488 nm to ~30% at 1550 nm, in good agreement with numerical calculations. We use an optical-impedance model to describe the absorption for polarization parallel to the wires of the detector. For lossy NbN films, the absorption can be kept constant by keeping the product of layer thickness and filling factor constant. As a conseque…
The approximate nature of radiative transfer equation (RTE) leads to a bunch of considerations on the effect of "dependent scattering" in random media, especially particulate media composed of discrete scatterers, in the last a few decades, which usually indicates those deviations RTE (combined with ISA) lead to from experimental and exact numerical results due to electromagnetic wave interference. Here we theoretically and numerically demonstrate the effect of dependent scattering on absorption in disordered media consisting of highly scattering scatterers. By making comparison between the in…
