## Abstract

As highlighted by recent articles [Phys. Rev. Lett. 105, 053901 (2010) and Science 331, 889-892 (2011)], the coherent control of narrowband perfect absorption in intrinsic silicon slab has attracted much attention. In this paper, we demonstrate that broadband coherent perfect absorber (CPA) can be achieved by heavily doping an ultrathin silicon film. Two distinct perfect absorption regimes are derived with extremely broad and moderately narrow bandwidth under symmetrical coherent illumination. The large enhancement of bandwidth may open up new avenues for broadband applications. Subsequently, interferometric method is used to control the absorption coherently with extremely large contrast between the maximum and minimum absorptance. Compared with the results in literatures, the thin film CPAs proposed here show much more flexibility in both operation frequency and bandwidth.

©2012 Optical Society of America

## 1. Introduction

Recently, coherent perfect absorption of light is proposed and demonstrated in a planar intrinsic silicon slab when illuminated on both sides by two beams with equal intensities and correct relative phase [1, 2]. Such a device is termed a “coherent perfect absorber” (CPA) and a “time-reversed laser”. The coherent absorption enhancement can also be extended to strong-scattering media [3]. Compared with the perfect absorbers (PAs) based on metamaterials or plasmonic structures [4–7], the new device provides additional tunability of absorption through the interplay of absorption and interference. The coherent control of absorption is potentially useful in transducers, modulators, or optical switches. However, as a time-reversed process of laser, CPA is characterized by narrow bandwidth. It is believed that the bandwidth cannot be increased for broadband applications such as solar photovoltaic or stealth technology.

In this paper, we argue that the bandwidth of a CPA can be rather large if the thickness is very thin and the corresponding material has a specific dispersion resembling that of metal. Heavily doped silicon is used as tunable metal at the terahertz frequency and Drude model is adopted to describe its electromagnetic property. It is shown that the coherent absorption is dependent on both the thickness of the film and the doping concentration of the silicon. Two different regimes of metallic thin film CPA are derived based on the general CPA condition, characterized by extremely broad and moderately narrow bandwidth, respectively. Both the two absorption regimes are realized under symmetrical coherent input, which can be easily obtained using Mach-Zehnder interferometer. Due to the extremely wide bandwidth, such absorption is potentially useful for broadband applications.

## 2. Principle

As illustrated in Fig. 1
, a planar slab with thickness *d* and complex refractive index *n* is illuminated by two coherent beams on both sides at normal incidence. Since the problem is much easier than that in nanostructured surfaces [5], one can solve the Maxwell’s equations directly by using transfer matrix (*T* matrix) or scatter matrix (*S* matrix) [8]. The output beams (*C* and *D*) can be expressed in terms of the input beams (*A* and *B*) through the scatter matrix.

In the case of symmetrical structure, the *S* matrix can be written as:

*r*and

*t*are the reflection and transmission coefficients of a single beam of

*A*or

*B*:

The CPA corresponds to the zeros of the scattering matrix (*C* = *D* = 0). Due to the mirror symmetry of the system we consider, coherent perfect absorption can only be realized for symmetrical inputs (*A* = *B*, *r* + *t* = 0) or antisymmetrical inputs (*A* = -*B*, *r*-*t* = 0). In both conditions, the magnitude of reflection and transmission are equal, implying that the CPAs are exactly beamsplitters when illuminated by a single beam. Using Eqs. (2) and (3), the CPA condition for normal incidence can be obtained:

The $\pm $ sign is corresponding to the symmetrical or antisymmetrical inputs. The transmission coefficient for a single beam illuminating on the slab can be written as:

Obviously, the reflection will introduce a phase shift of $\pi $ for both symmetrical and antisymmetrical CPAs. In contrast, the phase shifts of transmission coefficient for the two types are different, either zero or $\pi $. For two coherent beams satisfying the CPA condition, the transmission of one beam will cancel with the reflection of the other beam, resulting total absorption of all incident energy.

In the previous discussion [1], an infinite number of discrete solutions of Eq. (4) has been be found for *kd*>>1. The bandwidth is defined as the frequency width between the maximum absorption and adjacent minimum absorption and characterized by $\Delta f\approx c/(2nd)$. In this case, the CPA is rather narrowband and referred to as a time-reversed process of laser. Although such narrowband absorbers are potentially useful transducers, modulators, or optical switches, they are not applicable for broadband purpose such as solar photovoltaic.

## 3. Thin film CPA

In fact, the bandwidth of CPA can be very large if *d* is extremely thin (*d*<<λ, |*nkd*|<<1). In this case, the left side of Eq. (4) becomes $\text{1+}inkd$, and the right side can be approximated as $\pm \text{(1-2/n)}$. As |*nkd*| is very small, only plus sign in the right side (symmetric mode) should be chosen and the real and imaginary parts of the refractive index (${n}^{\prime}$and ${n}^{\u2033}$) in Eq. (4) become equal with:

In this case, |*nkd*|<<1 becomes $\sqrt{2kd}$<<1 thus the working regime can be approximated as *kd* <<1 and the required refractive index must be much larger than unit (*n*>>1). Different from the general CPA condition depicted in Eq. (4), the CPA condition for ultrathin film is explicit. Obviously, proper complex refractive index must be chosen to obtain the coherent perfect absorption at a specific frequency. Due to the extremely low quality factor of the thin film, such absorption may be wideband. However, material with specific dispersion characteristics should be used to obtain a broadband CPA since the required complex refractive index is frequency dependent. In the following, we will show that metal is just the natural material for broadband CPAs.

Due to the tunable optical property, heavily doped silicon is used here as artificial metal. The absorption in doped silicon is dominated by free carriers up to far infrared or terahertz frequencies. In this frequency range, the complex dielectric constant can be well described by Drude model [9, 10]:

*N*is the carrier density,

_{c}*e*is the electronic charge, ${\epsilon}_{0}$ is the permittivity of vacuum, and ${m}^{\ast}$is the effective carrier mass taken as 0.26

*m*

_{0}, where

*m*

_{0}is the free electron mass. The electron mobility $\nu $ and $\tau $ are related by $\nu =e\tau /{m}^{\ast}$, and $\nu $ is dependent with the doping material and doping concentration through experiential equation

These parameters can be determined experimentally [11]. The values for Boron doped silicon at room temperatures are ${\nu}_{\text{min}}$ = 44.9cm^{2}/Vs, ${\nu}_{\text{min}}$ = 470.5cm^{2}/Vs, ${N}_{r}$ = 2.23e17 cm^{−3}, $\alpha $ = 0.719. The electron mobility is dependent with temperature as $\nu (T)=v{\left({T}_{0}/T\right)}^{\eta}$, where $\eta $ is about 2.2.

In the regime of very low frequencies ($\omega <<{\tau}^{-1}$), ${\epsilon}_{2}>>{\epsilon}_{1}$ and Eq. (8) becomes ${{n}^{\prime}}^{2}-{{n}^{\u2033}}^{2}+2i{n}^{\prime}{n}^{\u2033}=i{\epsilon}_{2}$. So the real and the imaginary parts of the complex refractive index are of comparable magnitude with

Inserting Eq. (11) into Eq. (7), one can obtain the thickness for CPA at this frequency regime:

where*c*is the speed of light in vacuum. This characteristic length is just the so-called Woltersdorff thickness [12], which quantifies the thickness of a metallic film with maximum absorption for incoherent beam input in the low frequency regime. When

*d*=

*d*, the maximum absorption is 0.5, while the reflection and transmission are 0.25, respectively. For

_{w}*d*<

*d*most of the energy is transmitted; For

_{w}*d*>

*d*most of it is reflected.

_{w}Since the Woltersdorff thickness is independent with frequency, the absorption is very broadband. Generally, this frequency range will cover all the low frequencies up to terahertz. This broadband property is quite different from CPAs based on standard high-Q cavities. Nevertheless, the time reversal process of the broadband CPA can be straightforwardly interpreted using impedance theory [13, 14].

If the working frequency becomes larger and close to the plasmon frequency, the absorption at Woltersdorff thickness will decrease. Fortunately, the absorption can still be near perfect if we changing the thickness of the film. By setting ${\epsilon}_{1}$ in Eq. (9) to be zero, the second characteristic length, which we refer to as Plasmon thickness, can be derived. In this case, the refractive indexes in Eq. (8) become:

The Plasmon thickness can be obtained by combining Eqs. (13) and (7):

As Eq. (13) is correct only for frequency at $\sqrt{{\omega}_{p}^{2}{\epsilon}_{\infty}^{-1}-{\tau}^{-2}}$, the absorption bandwidth is narrow. This kind of absorption is termed Plasmon coherent absorption in the following discussion. When ${\omega}_{p}^{2}{\tau}^{2}={\epsilon}_{\infty}$, the Plasmon thickness is equal to the Woltersdorff thickness and the two absorption couple together. Since the two characteristic lengths have different dependences with the scattering time, the two lengths will depart from each other when $\tau $ change. It should be noted that both the two characteristic lengths are derived under the approximation made in Eq. (7), thus both of them should be in deep-subwavelength scale.

To illustrate the property of the two characteristic lengths, the coherent absorption properties of doped silicon film at symmetrical inputs are calculated using transfer matrix method (TMM) [8] for different thicknesses. The two input beams are assumed to have the same phases in the whole frequency range. The silicon is heavily doped with Boron and the doping concentration is chosen as 4e19cm^{−3}. The corresponding plamon frequency and scattering time are 7.0e14rad/s and 8.1fs. Then the Woltersdorff thickness and Plasmon thickness can be calculated as 151nm and 416nm using Eqs. (12) and (14).

Dependence of the absorption on frequency and the thickness of the film are shown in Fig. 2(a)
. As stated in the literatures [1], there are many frequency-dependent narrow absorption peaks for *kd*>>1. However, in the region where *kd*<<1, the absorption is nearly frequency-independent. In Fig. 2(b), the calculated Woltersdorff thickness and Plasmon thickness are 150nm and 450nm, in good agreement with the theoretical values (151nm and 416nm). The differences arise from the approximation made in Eqs. (12) and (14).

An important property of the Plasmon coherent absorption is the fact that the Plasmon thickness is independent with Plasmon frequency. If the Plasmon frequency is tuned by doping concentration, the Plasmon coherent absorption peak can vary while the thickness of the film keeps as a constant. In Fig. 3 , the coherent absorption curves of a 450nm thick film for different doping concentration are depicted. As we expected, the Plasmon coherent absorption peak can be gradually tuned from zero. In practical application, photoconduction effect may be used to provide the desired tunability of carrier concentration [15]. As highlighted in Fig. 3, the absorption at high frequency region (40THz as an example) can be tuned from near zero to unit.

## 4. Coherent control of absorption

In the metallic CPAs, perfect absorptions are realized at symmetrical inputs, where the two beams have equal phase. In principle, the absorption can be coherently controlled by the phase difference between the two input beams. Once we obtain the required refractive index *n*, the output for each channel with phase modulated input $[1,\mathrm{exp}(i\varphi )]$ can be calculated directly using Eq. (5) and Eq. (6):

*n*is the refractive index satisfying Eq. (4) [1]. By varying the phase difference, the output, or equivalently, the absorption can be controlled coherently.

In section 3, the input beams are assumed to have equal phase at the whole frequency range to achieve perfect absorption. In practical applications, Mach-Zehnder (MZ) geometry is needed to obtain the input phase difference [2]. In such interferometric geometry, broadband beam splitter is needed for broadband CPA. However, the bandwidth of conventional lossless beamsplitter is limited [16]. As we stated in above discussion, a thin metallic film at Woltersdorff thickness can be used as a broadband lossy beamsplitter at normal incidence with a phase shift of $\pi $ between the reflected and transmitted beams. At oblique incidences, the thicknesses for perfect beam splitting are different for TE and TM polarizations. However, the differences can be ignored when the incidence angle is small.

In the MZ geometry, the phase difference between the two arms in air can be stated as$\Delta \varphi =k\Delta l$, where Δ*l* is the path difference between the two arms. In this condition, the output will be frequency dependent with bandwidth of $\Delta f=c/\left(2\Delta l\right)$, which is just the same as that of a thick silicon slab defined in section 2. As shown in Fig. 4
, the coherent absorption of a 150nm thick film can be tuned from unit to near zero at 2.5THz when Δ*l* is increased from 0 to 60μm. As Δ*l* is not very large, the bandwidth is still broadband.

For the CPA at the Plasmon thickness (450nm), Δ*l* is chosen to be near 500μm here to illustrate the narrowband property (Fig. 5
). The absorption curves for symmetrical and antisymmetrical inputs behave as the upper and lower bounds of the coherent absorption. The maximum absorption occurs around 27THz, with a nonzero minimum absorption. As shown in the inset, the absorption curve is periodic with narrow bandwidth of 0.6THz, similar with the intrinsic silicon CPA [2]. The absorption at 27THz can be tuned from near unit to near zero if Δ*l* is tuned from 500μm to 500 ± 5.56μm.

At frequencies larger than 150THz, Drude model cannot be used for doped silicon any more due to the phonon assisted absorption [10]. Nevertheless, various metals can be chosen as the material of CPA at infrared or optical frequencies. For instance, we calculated the coherent absorption of a 17nm thick freestanding tungsten layer in Fig. 6 . The experimental optical constants are used here [17]. As the AC conductivity for tungsten is 1.79e7S/m and the corresponding Woltersdorff thickness is only 0.3nm, the absorption here is coherent Plasmon absorption. However, the absorption bandwidth is much larger than that of Drude metal due to the interband transition [17]. The antisymmetrical absorption is also drawn as a lower bound of the coherent absorption.

Compared with the intrinsic silicon CPA [2], the tungsten CPA here has much larger operation ranges (800nm-1500nm). Moreover, as the coherent absorption of tungsten layer can be very broadband and the spectrum is coincident with that of solar spectrum, one may wonder whether such absorption can be used to harvest solar energy. Unfortunately, the CPA needs Mach-Zehnder geometry with nearly perfect equal arm lengths, which may not applicable for solar energy collection. Nevertheless, this tungsten layer may be used in solar-thermal applications based on multiple-absorption principle [18] since the incoherent absorption is maximized at the given thickness.

In applications, it is useful to have a large contrast between the maximum and minimum absorption. Although the maximum absorption for a general CPA is near unit, the minimum absorption may not be zero. As stated in Eq. (15), the minimum absorption as well as the absorption contrast, defined as 1/(1-*I*_{0}), are dependent on the refractive index. In principle, a larger refractive index will result a larger absorption contrast.

For *kd*>>1, the required refractive index is predominantly real. If the refractive index is large enough, the total absorption can be reduced to near zero. However, the refractive index for normal dielectric material is limited. For instance, the refractive index for intrinsic silicon is 3.6 + 0.0008i at *λ* = 1μm while the minimum absorption is as large as 0.268. Although the intensity contrast can be increased with a nonuniform system as stated in [2], the structure becomes very complex.

However, for the regime where *kd*<<1, the real and imaginary parts of the complex refractive index are much larger. As stated by Eq. (7), the refractive index and contrast are determined by *kd*. As *kd* increases, the minimum absorption will increase and the absorption contrast will decrease. In our simulations, *kd* is 0.0082 for 150nm thick doped silicon at 2.5THz, with a contrast larger than $5\times {10}^{4}$. The value for 17nm tungsten at 300THz is 0.1068, and the corresponding contrast is 250. For 450nm thick doped silicon at 27THz, *kd* is 0.2545 and the contrast becomes only 50.

## 5. Conclusion

In summary, we have broadened the operation range of coherent perfect absorber by using of metallic thin film. It is demonstrated that silicon cannot only be used for narrow band CPA, but also for broadband CPA through heavily doping. Two characteristic lengths are derived, one for ultra broadband absorption and the other for moderately narrowband absorption. As the absorption bandwidth can be dramatically enhanced in such film, it is possible to enhance the performance of solar cells or other broadband absorbers in the future. A disadvantage of the broadband absorber is that the required material loss is very large. This is different from the original CPA concept, where light can be perfectly absorbed even if the intrinsic material absorption is negligible. Finally, Mach-Zehnder geometry is used to demonstrate the coherent modulation of the absorption. It is found that smaller *kd* will result larger contrast between maximum and minimum absorption, which is useful for applications such as transducers, modulators, or optical switches.

## Acknowledgments

This work was supported by 973 Program of China (No. 2011CB301800) and Chinese Nature Science Grant (60825405, 61138002, and 61177013).

## References and links

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**14. **In the impedance theory, the thin film CPA can be approximated as a resistive sheet with $Z=1/({d}_{w}{\sigma}_{0})={Z}_{0}/2$ as the thickness of the slab is much smaller than the skin depth. Here, ${\sigma}_{0}={\omega}_{p}^{2}\tau {\epsilon}_{0}$ is the AC conductivity and ${Z}_{0}=\sqrt{{\mu}_{0}/{\epsilon}_{0}}$ is the impedance of vacuum. Then consider the radiation property of an infinite oscillating current sheet in xy plane. Assuming that the current is $\overrightarrow{J}=K\mathrm{sin}(\omega t)\overrightarrow{x}$, the electric field at z = 0 can be written as: $\overrightarrow{E}=-0.5{\mu}_{0}cK\mathrm{sin}(\omega t)\overrightarrow{x}.$ The effective sheet impedance, defined as *E*/*J*, is -*Z*_{0}/2, which is just in opposite to the thin film CPA condition. Such a radiation can be thought as the time reversed process of the broadband CPA, although the infinite oscillating current sheet is not applicable in practical applications.

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