The surface plasma wave (SPW), supported by a conductor–dielectric interface, has an important property: as the wave frequency approaches the surface plasmon resonance, the wavelength and transverse extent of the mode resonantly shrink, confining optical frequency waves to nanodimensions, far below the diffraction limit prescribed for body waves.1,2 This has led to the development of nanoelectronics and nano-devices into a major field12.3.4.–5 of research. Of significant importance is the development of nanolaser or spaser.67.8.9.–10 Noginov et al.11 have recently operated a 531 nm spaser-based nanolaser employing 44 nm diameter nanoparticles with gold core and dye-doped silica shell. Lu et al.12,13 have developed a multilayer plasminic nanolaser using smooth silver film with or nanolayer and gallium nitride nanorod deposited on it. The nanorod comprises a layer of indium gallium nitride that acts as gain medium. The theoretical formalisms of spaser are usually based on quantum density matrix approach.6 However, some employ classical model due to its transparent simplicity. Kumar et al.14 developed a classical analytical formalism of two-layer surface plasmon laser amplifier pumped by a forward biased p–n junction.
In this paper, we present theoretical analysis of optical gain in three surface plasmonic nano-oscillators, (i) a metal film with a nanostrip of gain medium, (ii) a nanohole in a metal film deposited on a gain layer,15 and (iii) a nanoparticle coated with gain medium. These are the configurations experimentally studied in recent years and have promise for building arrays of nanoradiators with desired directivity and power. Here we pursue analytical treatment to reveal physics with some clarity which at times gets masked in numerical simulations.
The oscillators operate near the surface plasmon resonance frequency and the quality factor of the cavity is determined by the free electron collisional damping. In the first oscillator, nanostrip provides for lateral localization of the surface plasmons16 by reducing the phase velocity of the SPW (as compared to the metal–vacuum interface). In the second oscillator, mode is localized around the nanohole. In the third case, space charge oscillations of electron sphere with respect to ion sphere provide a natural oscillator. The field of the oscillator mode stimulates electron–hole recombination in the gain medium that in turn amplifies the mode.
Metal Film—Nanostrip Spaser
Consider a metal film of free electron density , electron effective mass , and lattice dielectric constant . On top of it () lies a nanostrip with , and dielectric constant (cf. Fig. 1). The thickness of the metal film is larger than the skin depth of the SPW, hence it can be treated like a semi-infinite medium (). Grooves are created in the metal at , for longitudinal localization of plasmons.
Had it been an infinite metal–dielectric interface, it would support a SPW with electric field17
For a given , increases with increasing .
When one limits the lateral size of the dielectric layer, the parallel wave number of the SPW on the metal–vacuum interface is
One may take the SPW to be propagating along but with finite . Outside the strip, i.e., on the metal–vacuum interface, the SPW would have the same as inside the strip. However, since , the SPW field outside the strip would be evanescent in . One may take SPW to be localized in and with
For the plasmon nano-oscillator, we may deduce the mode structure as follows. Take , . Assuming , variations of fields as and using and express , , , in terms of . In different regions one may take the , variations of fields to be the same [with , given by Eq. (5)], in compliance with the boundary conditions at and allow to have different values in different media. Thus we write in the region ,
Applying the continuity of , at , we obtain and the dispersion relationFig. 2(a) the real frequency of a surface plasmon eigenmode as a function of normalized width of the strip for typical parameters. The frequency steadily falls off with the width. Figure 2(b) displays the variation of normalized damping rate with normalized frequency. The damping increases rather rapidly as one approaches the surface plasmon resonance frequency.
The energy density of the electromagnetic fields of the plasmon resonator is
Now we allow the dielectric strip to comprise a gain medium (e.g., an optically pumped semiconductor layer) of -width , -width , and -extent , located at . In the conduction and valence bands, the density of states and occupation probability for electrons and holes (of effective masses , ), respectively, are
In a surface plasmon-induced emission process, an electron in energy state in the conduction band goes to energy state in the valence band, recombining with a hole and producing a plasmon of frequency
Equations (15) and (16) give the energy states that participate in the stimulated emission process
The rate of e–h recombination per unit volume per second is proportional to spectral energy density , density of states , the occupation probability of state , the probability of state being vacant,
One may remember that is proportional to the density of states in the valence band . The net SPW energy produced per unit volume per second on integrating over is
Using Eqs. (12) and (21) we obtain
is the growth rate of SPW field energy. The threshold for the SPW growth is given as
This determines the electron and hole density threshold for the growth of surface plasmons.
The Einstein’s coefficient is related to the Einstein’s coefficient for spontaneous emission as , hence
Nanohole-Embedded Metal Film Spaser
Consider a metal film of thickness () and effective relative permittivity given above. It comprises a transverse hole of radius and length with vacuum inside (cf. Fig. 3).
The cylindrical vacuum–metal interface supports an SPW with evanescent fields in and forward backward propagating solutions in . A forward wave SPW field is
The termination of the cylinder at , quantizes and gives standing wave solution to SPW. One may write
The energy density of fields of the plasmon nano-oscillator is
We allow a thin layer of gain medium of thickness placed at . The net SPW energy produced per unit volume per second, , is given by Eq. (20). The rate of energy gain of the SPW oscillator is given by Eq. (21) with
The linear damping rate is of the order of the one given by Eq. (10). Equation (21) gives with growth rate
The threshold value of above which the SPW grows is obtained by putting in Eq. (32). Above the threshold the growth rate is comparable to the earlier case.
Consider a metallic nanoparticle of radius , free electron density , electron effective mass , and lattice permittivity . It is surrounded by a layer of gain medium of thickness and dielectric constant (cf. Fig. 4).
For a moment, let us take and give a displacement to free electrons of the nanoparticles. This creates a space charge field in the overlap region of the free electron sphere and ion sphere. Each electron thus experiences a restoration force , where . The momentum loss per electron per second via collisions is where ν is the collision frequency and is the drift velocity of electrons. Thus the equation of motion for an electron is18
When we allow to have arbitrary value, the Poisson’s equation, governing the space charge field , on writing , gives the solution (for )
Demanding the continuity of , at we obtain
Using the value of given below Eq. (2), we obtain the frequency given by Eq. (35) with replaced by . This is the same result obtained by Bergman and Stockman19 and Noginov et al.20 for the fundamental mode.
The field of the nanoparticle [of dipole moment ] as seen by the gain layer is
Outside the gain layer () the field is given by the above expression with replaced by 1.
The energy stored in the oscillator is
The oscillator field induces e–h recombination producing electromagnetic energy per second as given by Eq. (21) with given by Eq. (35) and
The growth rate of the SPW turns out to be
For , , and , . One percent above the 50% occupation probability, the growth rate .
The surface plasmon eigenmode in the vicinity of SPW resonance, is strongly localized near the metal–dielectric interface with and has resonantly short wavelength. The damping rate of the mode, however, shows no resonant enhancement. A gain medium, within an SPW wavelength from the boundary sustaining SPW, excites the SPW eigenmode. Usually one employs optical pumping to achieve , the condition equivalent of population inversion. For electron and hole occupation probabilities of relevant energy states in the conduction and valence bands exceeding 0.5 each by 1% the growth rate is of the order of .
In the case of metal film loaded with a gain medium nanostrip, frequency can be tuned by varying the permittivity and width of the strip. A drastic reduction in operating frequency can be achieved by shrinking the thickness of the metal film that lowers the SPW resonance frequency.21 However, the present analysis is not valid in that case.
The nanohole oscillator has a gain comparable to the above oscillator at the same level of pumping power flux density. The SPW eigenmode acts as an oscillating dipole and emits far-field radiation. An array of such holes would give a well-collimated beam.
The oscillator comprising a nanoparticle coated with an optically pumped gain medium has operating frequency independent of the radius of the particle. It depends only on its free carrier density, lattice permittivity, and dielectric constant of the surrounding medium. Noginov et al.20 have reported compensation of loss in metal nanoparticles oscillator by gain in interfacing rhodemine 6G dye. With emission cross section of R6G and density of active molecules (corresponding to one molecule per nanoparticles of radius 5 nm) the local gain turns out to be , corresponding to temporal growth which is comparable to our case where e–h recombination leads to growth of plasmons when , i.e., . The quality factor of the oscillators estimated here due to collisional losses appears to be higher than that reported experimentally. Photoabsorption via interband transitions appears to dominate collisional losses. These losses are equivalent to enhanced collision frequency of electrons.
Chuan S. Liu is a leading plasma physicist, credited with groundbreaking contributions to theory of laser plasma interaction and promoting international collaboration in higher education. He joined the University of Maryland in 1975 as professor of physics and served twice as department chair. He also served as vice president for research and dean of the graduate school. From 2003 to 2006, he was president of National Central University, Taiwan. Currently, he is the founding master of the Chao Kuang Piu College.
Vipin K. Tripathi received his master’s in physics from Agra University in 1967 and a PhD from IIT Delhi in 1971. In 1972, he joined the IIT faculty. In 1976, he moved to the University of Maryland and worked in the area of thermonuclear fusion, developing nonlinear theories of RF heating and current drive in tokamak. In 1983, he joined IIT Delhi as professor of physics. He established a leading group in free electron laser and laser plasma interaction.