Topics: Purdue ME 606: Notes on Near-Field Radiation

Purdue ME 606: Notes on Near-Field Radiation

Introduction

In this module, we consider some unique aspects of radiation (and more EM wave propagation more generally) that occur at small length scales. The content here draws from an excellent textbook on nanoscale heat transfer by Zhuomin Zhang, to which the interested reader is referred for more information.

First, we revisit the reflection and transmission of an electromagnetic wave at a planar interface between two media with different optical properties. Considering the figure below, we will first analyze the ‘S’ polarized (or TE=transverse electric) wave. Neglecting the time dependence of the wave, the incident electric field can be written as

$\vec{E}_i = \hat{y} E_ie^{ik_{1z}z+ik_x x}$

where the x-component of the wavevector is the same for the incident, reflected, and transmitted waves. The incident and transmitted polar angles are related by Snell’s law:

$\frac{n_1}{n_2} = \frac{\sin \theta_2}{\sin \theta_1}$

where the n’s are the indices of refraction of the two materials. Also of importance is critical angle above which all incident radiation is reflected (i.e., none is refracted)

$\theta_c = \arcsin \left(\frac{n_2}{n_1} \right)$

Near the interface, the y-component (i.e., out of plane) electric field for an s-polarized wave can be expressed as

$E_y = \left( E_i e^{ik_{1z}z} + E_r e^{-ik_{1z}z} \right)e^{ik_x x}$ for z < 0

$E_y = \left( E_t e^{ik_{2z}z} \right)e^{ik_x x}$ for z > 0

and similar expressions can be derived for the x and z components of the magnetic field $\vec{H}$ .

Near-field Optics

An emerging and important class of nanoscale imaging and patterning tools is based on so-called near-field scanning optical microscopy, or NSOM. This field involves the use of optical elements such as apertures and reflective tips whose characteristic dimensions are smaller than the wavelength of the light with which they interact. In the far field, the resulting manipulated light fields are too diffuse to be useful for most applications. However, in the region very close to the element, the size of the EM field’s extent can be smaller than the wavelength while being concentrated enough to perform useful functions such as patterning photosensitive materials (with feature sizes less than the wavelength of the incident light). This near-field region of concentrated irradition persists only to approximately one wavelength from the element, as shown in the following figure.

Evanescent Waves and Photon Tunneling

We now consider the p-polarized ( TM ) wave of the first figure above. The magnitudes of the wavevectors in the two media can be expressed in terms of material constants as

$k_1^2 = k_x^2 + k_{1z}^2 = \epsilon_1 \mu_1 \omega^2/c_0^2$

$k_2^2 = k_x^2 + k_{2z}^2 = \epsilon_2 \mu_2 \omega^2/c_0^2$

where $ε$ and $μ$ are the relative (to vacuum) permittivity and permeability, and $c 0$ is the speed of light in vacuum.

For p polarization the Fresnel reflection coefficient is

$r_p = \frac{H_r}{H_i} = \frac{k_{1z}/\epsilon_1 - k_{2z} /\epsilon_2}{k_{1z}/\epsilon_1 + k_{2z} /\epsilon_2}$

Also, from above, we know that $k_{2z}^2 = \epsilon_2 \mu_2 \omega^2/c_0^2 - k_x^2$ . Therefore, when $\sqrt{\epsilon_2 \mu_2} < k_xc_0/\omega$ , the angle of refraction is not defined because $k 2 z$ is imaginary. In such cases, we can write

$k 2 z = i η 2$ where $\eta_2 = \sqrt{k_x^2-\epsilon_2 \mu_2 \omega^2/c_0^2}$

When refraction is not allowed, the Fresnel reflection coefficient becomes

$r p = e i δ = e - i 2α$ where $\tan \alpha = \frac{\eta_2/\epsilon_2}{k_{1z}/\epsilon_1}$

Note that the absolute value of the reflection coefficient is unity. The magnetic field in the two media can then be expressed as

$H y = 2 H i e - i α cos(k 1 z z + α)$ for z < 0.

$H_y = 2H_i e^{-i\alpha}\cos (\alpha)e^{-\eta_2z}$ for z > 0.

Clearly, the magnetic field in the incoming medium takes a wave form, with a phase shift of magnitude $α$ . The phase angle of the reflection coefficient is $δ = - 2α$ and is called the Goos-Hanchen phase shift.

Most notably, we see that the field in medium is not zero everywhere, but instead decays to zero exponentially with a characteristic length of $\eta_2^{-1}$ . This decaying field is called an evanescent wave, and if medium 2 is sufficiently thin ( $d~\eta_2^{-1}$ ) (with another medium beneath it) some of the radiation field can cross, or tunnel, through it as depicted in the following illustration.

The figure shows two evanescent waves, each of which decays from the propagating waves in the top and bottom media. In the case of an incident TM (p) wave, the magnetic field in medium 2 is

$H_y = (A e^{ik_{2z}z} + B e^{-ik_{2z}z})e^{ik_x x}$ for 0 < z < d.

After some analysis, the normal component of the Poynting vector in medium 2 can be expressed as

$\langle S_z \rangle = \frac{k_{2z}}{2\omega\epsilon_2\epsilon_0} \left(|A|^2 -|B|^2 \right)$ for $k_{2z}^2 = k_2^2 - k_x^2 > 0$

$\langle S_z \rangle = \frac{-\eta_2}{\omega\epsilon_2\epsilon_0} \Im\left( A B^* \right)$ for $-k_{2z}^2 = k_x^2 - k_2^2 > 0$

The foregoing results can be used to calculate the transmittance through the system because the Poynting vector in medium 3 is the same as that for medium 2, as there is no absorption in medium 2.

The resulting spectral transmittance function takes starkly different forms, depending on whether propagating or evanescent waves exist in medium 2 (i.e., whether the incident angle is less than or greater than the critical angle), as shown in the illustration below. Notably, the propagating wave exhibits oscillating peaks and valleys as the thickness of medium 2 increases. These patterns are formed by the interference of forward and backward propagating waves in medium 2. Conversely, the evanescent wave exhibits a simple exponential decay as the thickness of medium 2 increases. Significantly, photon tunneling is generally negligible for film thicknesses that exceed the incident wavelength.

Near-Field Thermal Radiation

We now consider the problem of thermal radiation between medium1 and medium 3, as depicted in the figure above. Again using the symbol d as the thickness of medium 2, we know that as $d \to\infty$ ,

$q_{13}^{''} = \frac{\sigma T_1^4 - \sigma T_3^4}{1/\epsilon_1 + 1/\epsilon_3 -1}$ as $d \to\infty$

However, when d is comparable to or less than the dominant photon wavelength (as deduced from Wien’s displacement law), we must consider both the propagating and evanescent modes of transmission from medium 1 to 3. In such cases, the effects of polarization and angle of incidence become important, such that, for each polarization, a spectral hemispherical transmission function can be calculated that includes the contributions from evanescent and propagating waves. The resulting analysis is rather involved, and here we present the limiting case of gaps that are much smaller than the dominant photon wavelength (see textbook by Zhang, pp. 389-391). The results suggest that the net heat flux between surfaces 1 and 3 are:

$q^{''}_{13,prop} = \sigma(T_1^4 - T_3^4)$ for propagating waves

$q^{''}_{13,evan} = (n_1^2-1)\sigma(T_1^4 - T_3^4)$ for evanescent waves, with $n 1 = n 3$

$q^{''}_{13,total} = n_1^2 \sigma(T_1^4 - T_3^4)$ for evanescent waves, with $n 1 = n 3$

These results clearly suggest that photon tunneling can enhance radiation exchange between surfaces by a factor up to the square of the refractive index of the emitting material.

A final caveat is, however, in order. A recent paper by Zhang and Basu concludes with the following statement: “At present, a satisfactory second-law interpretation of the near-ﬁeld radiation does not exist.” Thus, the interested reader is both cautioned about the use of near-field radiation predictions and encouraged to pursue new research in this exciting field.

Last modified on 20 Dec, 2008

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Purdue ME 606: Notes on Near-Field Radiation

Table of Contents

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Introduction

Near-field Optics

Evanescent Waves and Photon Tunneling

Near-Field Thermal Radiation