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Broadband Terahertz Source Based on Photomixing in Laser-Assisted Field Emission with Clusters of Carbon Nanotubes

Terahertz (THz) radiation, which is electromagnetic energy at frequencies in the nominal range of 1011 to 1013 Hz (0.1-10 THz), is being studied for at least 18 different applications (Davies et al., 2002) including urgent needs for medical diagnosis and security applications for the detection of non-metallic concealed weapons, biological and chemical agents, and explosives. However, the present sources of THz radiation present “hurdles” because of their limited tunable bandwidth and output power, and some require fragile, large, and expensive femtosecond lasers or even particle accelerators (Zhang, 2002). Some of the methods that have been used to generate THz radiation are backward wave oscillators with chains of frequency multipliers (Maiwald et al., 2000), the Smith-Purcell Effect (Mross et al., 2003), quantum cascade lasers (Davies et al., 2002), synchrotron radiation from high-energy accelerators (Carr et al., 2002), bulk electro-optic rectification and ultrafast charge transport in semiconductors (Davies et al., 2002), and photomixing in semiconductors (Verghese et al., 1997). For example, photomixing (optical heterodyning) of two lasers at different wavelengths in low-temperature-grown (LTG) GaAs at the feed point of an antenna can generate an output power of only 1 μW at 1 THz, which falls off by 12 dB per octave or 1/F4 at higher frequencies (F), so the power is reduced to 100 pW at 10 THz.


Introduction
Terahertz (THz) radiation, which is electromagnetic energy at frequencies in the nominal range of 10 11 to 10 13 Hz (0.1-10 THz), is being studied for at least 18 different applications (Davies et al., 2002) including urgent needs for medical diagnosis and security applications for the detection of non-metallic concealed weapons, biological and chemical agents, and explosives. However, the present sources of THz radiation present "hurdles" because of their limited tunable bandwidth and output power, and some require fragile, large, and expensive femtosecond lasers or even particle accelerators (Zhang, 2002). Some of the methods that have been used to generate THz radiation are backward wave oscillators with chains of frequency multipliers (Maiwald et al., 2000), the Smith-Purcell Effect (Mross et al., 2003), quantum cascade lasers (Davies et al., 2002), synchrotron radiation from high-energy accelerators (Carr et al., 2002), bulk electro-optic rectification and ultrafast charge transport in semiconductors (Davies et al., 2002), and photomixing in semiconductors (Verghese et al., 1997). For example, photomixing (optical heterodyning) of two lasers at different wavelengths in low-temperature-grown (LTG) GaAs at the feed point of an antenna can generate an output power of only 1 µW at 1 THz, which falls off by 12 dB per octave or 1/F 4 at higher frequencies (F), so the power is reduced to 100 pW at 10 THz.
In field emission an intense electric field of about 5 V/nm is applied to the surface of a metal or other electrical conductor to lower the potential barrier so that electrons are emitted into the surrounding vacuum by quantum tunnelling. The current density for emitters that are made of refractory metals such as tungsten may be as high as 10 9 A/m 2 in steady state, or 10 12 A/m 2 when the applied electric field is pulsed (Gomer, 1993). Some have used field emitters in place of heated filaments as the electron source in microwave triodes (Schwoebel et al., 2005), klystrons (Ryskin et al., 2007), traveling-wave tubes (Makishima et al., 1999;Lin & Lu, 2007), and the closely-related monotron (Yokoo & Ishihara, 1997). Others have used femtosecond lasers to obtain pulses of electrons that are shorter than 70 fs in laser-assisted field emission (Hommelhoff et al., 2006) and analysis shows that there is a fundamental limit at 2 fs (Hagmann, 1998A). However, we have not been able to find any presentation by 23 www.intechopen.com those outside our group that describes the simulation, design, fabrication, or testing of microwave or THz devices that are based on the high-speed that is inherent in field emission.

Photomixing in Laser-Assisted Field Emission
We have pioneered in the study of laser-assisted field emission (LAFE) as a new method to generate microwave and THz radiation with an unusually large tunable bandwidth (Hagmann, 1999A). In effect, a nanoscale ultrafast non-linear optical medium is created in laser-assisted field emission because: (1) the field emitter is much smaller than the wavelength of the incident optical radiation so the potential of the tip oscillates to follow each cycle of this radiation; (2) the current responds to changes in the electric field with a delay of 2 fs (Hagmann, 1998A); and (3) this response is highly nonlinear (Gomer, 1993). LAFE has higher efficiency than non-linear optics materials because the Manley-Rowe relations, which limit the conversion efficiency of three-wave mixing in these materials (Petukhov et al., 1998), do not limit the efficiency in LAFE because of the cascaded processes in which optical radiation releases the stored electrical energy at the output frequencies (Hagmann, 1999B).
We have made rigorous simulations of photomixing in laser-assisted field emission using density functional theory (Hagmann, 1997) with Floquet methods to allow for single-photon and multiphoton processes (Hagmann, 1999B). However, a much simpler procedure will be used here to obtain a closed-form solution that is consistent with the results from the more accurate methods. We begin with the simplified Fowler-Nordheim equation for the current density in field emission (Gomer, 1993).
With optical radiation, where the effects of the photon energy may not be neglected, the major correction to these equations is the effect of a resonance between tunneling electrons and a radiation field (Hagmann, 1995A). This resonance, which is caused by reinforcement of the wave function through virtual photon processes, was later confirmed by others (Mayer & Vigneron, 2000;Mayer et al., 2002Mayer et al., , 2003. Figures 1 and 2 show the gain in the power flux density, which increases E 1 and E 2 in Eqs. (4) and (5), for the resonance with tungsten emitters (Mayer & Vigneron, 2000) and (5, 5) armchair (metallic) CNT emitters (Mayer et al., 2002). Modifying Eqs. (4) and (5) by allowing for the gain G (in dB), as shown in Figs. 1 and 2, the peak value of the mixer current in laser-assisted field emission is given by Our prediction of the resonance (Hagmann, 1995A) and first suggestions for microwave and terahertz devices based on LAFE (Hagmann, 1995B) were made at a time when the general consensus was that the increase in field emission current caused by a laser is a small effect due to heating of the emitter that has a time constant of 10 -1000 µs (Lee & Robbins, 1989). Our measurements of optical rectification in LAFE with a sealed field emitter tube (Leybold AG model 55460) confirm that there is a thermal effect but they also show a non-thermal effect that is in agreement with our analysis and shows the effect of the resonance (Hagmann & Brugat, 1999). We measured a time constant of 300 µs for the decay of the thermal processes, and also found that the cathode-to-anode circuit for the sealed field emitter tube acts as a low-pass filter .
Microwave prototypes have been built and tested using emitters of tungsten and molybdenum (Hagmann, 2004). Two lasers are focused on the emitter to generate current oscillations at their difference frequency by LAFE, as described by Eq. (7). We do not specifically use the emitted electrons as such in our method, but instead we cause the current oscillations to generate a transverse-magnetic (TM) surface wave that propagates on extensions of the emitter (Alonso & Hagmann, 2001). This surface wave is coupled directly to the load or emitted as radiation from antennas that are formed on the emitter (Hagmann, 2004) in order to avoid the low-pass filter in the cathode-to-anode circuit . Our group was the first to suggest the use of surface plasmons to propagate THz radiation on metallic wires (Hagmann, 1998B;Alonso & Hagmann, 1999), which was later implemented by others (Wang & Mittleman, 2004).
It was already shown that photomixing in LAFE can provide an unusually large tunable bandwidth. However, the maximum power that we have obtained with microwave protypes was only 50 pW. This limitation can be understood because the mixer current, which is a few percent of the dc field emission current (30 µA) is coupled to a 50  load or an antenna having a similar value for its radiation resistance. Increasing the size of the emitter is not an effective means to obtain the higher current that is required for greater output power because surface imperfections (Fujieda et al., 2005) cause the dc emitted current to be proportional to A 1/4 , where A is the area of the emitter (Okawa et al., 1988). In the rest of this chapter it will be seen that much greater output power may be obtained without sacrificing the the large tunable bandwidth by using carbon nanotubes (CNT) in LAFE because of a synnergy in which the CNT increase both the dc current and the impedance which is seen by the mixer current.

Carbon Nanotubes as High Impedance Transmission Lines
Equation (7) shows that photomixing in LAFE requires incident electric fields at two different frequencies in order to generate a mixer current at a third frequency which is the difference of the first two frequencies. The output power that is coupled to the TM surface waves that propagate on extensions of the emitter is given by ½ Z 0 I MP 2 , where Z 0 is the (real) characteristic impedance for the transmission line which is a wire extension of the emitter and I MP is the peak value of the mixer current which is given by Eq. (7). Thus, the output power is given by the following equation, and it is proportional to Z 0 which shows the motivation for increasing this impedance in order to obtain greater output power.
The interaction of the mixer current with the impedance Z 0 creates an electric field at the mixer frequency, which has a peak value that is given by E MP = Z 0 I MP . Thus, for consistency, it would be necessary to include this electric field in Eq.
(2), which reduces the current from optical rectification as well as the mixer current. This negative feedback sets an upper limit on the output power which is consistent with the power that is supplied by the two lasers. However, the effect of this negative feedback is generally negligible unless Z 0 exceeds 1 MΩ which does not occur with carbon nanotube transmission lines (Hagmann, 2008).
Others have simulated single-wall (Burke, 2002(Burke, , 2003 and multiwall Sonin, 2001;Bachtold, 2001) carbon nanotubes (SWCNT and MWCNT) on a conducting substrate as transmission lines by using a hybrid approach in which quantum theory is combined with classical distributed element transmission line models (Collin, 1990). This method allows for the effects of the size and shape of the carbon nanotube (CNT) and the insulation separating it from the conducting substrate, as well as the electronic properties of the CNT from quantum theory. These studies show that the characteristic impedance is on the order of 5 k, which may be understood because the distributed kinetic inductance of the CNT is approximately10 4 times the inductance for metals. The high value for the kinetic inductance of MWCNT was verified by measurements, and it has been suggested that CNT on conducting substrates could be used as high-impedance transmission lines in nanoelectronics .
We have extended the previous simulations which were made by others in order to study CNT that are isolated, removed from the substrate, as transmission lines (Hagmann, 2005). The previous work by others was based on the classical distributed element model for a two-conductor transmission line that is formed by the CNT and its image in the substrate, for which the propagation is in a transverse electromagnetic (TEM) wave. However, as a wire is moved away from a conducting surface, there is a transition from TEM propagation (Barlow & Cullen, 1953) to a TM surface wave which is the dominant mode for an isolated wire (Stratton, 1941). Thus, we developed a classical distributed-element model for propagation of the TM surface wave and then added the quantum inductance and the quantum capacitance to this model as it has been done by others (Hagmann, 2005).
Highlights of the results of our simulations of isolated CNT as transmission lines follow: (1) Only the axially-symmetric TM surface wave mode will propagate on an isolated SWCNT. This is consistent with the observation that higher order modes cannot propagate on a metal cylinder having a circumference which is less than one free-space wavelength (Savard, 1967). Single-mode propagation is necessary to prevent dispersion of the propagating electromagnetic fields.
(2) At THz frequencies all of the metallic tubes in an isolated MWCNT can take part in the propagation so that for N shells there are N -1 propagating coaxial modes in addition to the axially-symmetric TM surface wave mode. Thus, in order to prevent dispersion, only SWCNT should be considered for this application.
(3) The axially-symmetric TM surface wave mode in an isolated SWCNT has a characteristic impedance of approximately 44 kΩ. This value is much greater than that for a CNT on a conducting substrate because the capacitance to the substrate has been eliminated. (4) The axially-symmetric TM surface wave mode in an isolated SWCNT has a phase velocity of approximately 2.8 x 10 6 m/s, which is 0.9 percent of the velocity of light in vacuum. The reduced value of the phase velocity is consistent with the increased value of the characteristic impedance. (5) The axially-symmetric TM surface wave mode in an isolated SWCNT has negligible attenuation for propagation over the lengths that will be used in this application. Low ohmic loss is also found with surface waves on a single conducting cylinder, but the energy is loosely bound to the cylinder so it is necessary to keep other objects at a distance to limit radiative loss (Goubau, 1950). Our applications fall under what has been called the "low-loss regime" for propagation on CNT (Burke at al., 2006). on Photomixing in Laser-Assisted Field Emission with Clusters of Carbon Nanotubes 435

Description of the prototypes
In 2004, we began our study of LAFE with single SWCNT and MWCNT, funded by the National Science Foundation (NSF-DMI-0338928). Four custom sealed field emission tubes with CNT were made for us by Xintek (Chapel Hill, NC, U.S.A.), having the same outer structure which is shown in Fig. 3. The copper anode is at the right, and the CNT emitter is mounted on a tungsten wire that is attached to the metal cylinder at the left. Figure 4 shows an SEM image of one of the emitters, taken with a JEOL JEM 6300 before it was placed in one of these tubes.  Most of the measurements of field emission from CNT pertain to the dc characteristics of films (Zhu et al., 1999), arrays (Lim et al., 2001), and composites (Wang et al., 1998), which only provide a statistical evaluation of the properties with large numbers of CNT. Furthermore, the electric field, and thus the total emitted current, is reduced when the CNT are close together (Hii et al., 2006). Careful measurements with individual CNT have shown that the field enhancement factor and work function varies with the structure of the CNT and its surface conditions (Xu et al., 2005). Other measurements with single CNT have shown that field emission is affected by irregularities in the structure of the graphitic sheets , and time-dependent deformations due to reconstruction at the apex increase the electric field at specific sites which are where the field emission occurs (Kuzumaki et al., 2004). Therefore, we chose to begin our studies LAFE with tubes having individual CNT as the emitters.
The two tubes labeled M-1 and M-4 have a single MWCNT as the emitter, and tubes C-3 and C-6 each have a single SWCNT as the emitter. The CNT are in bundles with diameters of 10 to 30 nm that are shaped so that field emission is only from the single CNT at the end of each bundle where the electric field is most intense. A sealed field emitter tube, with an etched tungsten single crystal tip as the emitter (Leybold AG model 55460) was used for comparison in the measurements with the four tubes from Xintek. The characteristics of these five tubes are described in the following sections:

Fowler-Nordheim characterization of the prototypes
First, the dc current-voltage characteristics were measured for each of the five field emission tubes, and then these data were analyzed using the following procedure that is based on the simplified Fowler-Nordheim equation which was shown as Eq. (1).
The measured field emission current I is proportional to the current density J, and the measured voltage V is proportional to the applied electric field E, so Eq. (1) is equivalent to the following expression that may be used with the measured data: Thus, the values of the parameters C and D for each field emission tube may be determined from the slope and intercept of a graph in which the ordinate and abscissa are ln(1/V 2 ) and 1/V, respectively. The values of the parameters A and B in Eq. (1) are calculated from the work functions φ = 4.5 eV for tungsten, and 4.9 eV for graphene in the CNT. Then the following two parameters may be calculated: S = CD 2 /AB 2 ; R = D/B. Here S is called the effective area of the emitter, and R is called the effective radius of the emitter.

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In Table 1, R* is an alternative effective radius of the emitter that is calculated from the effective area of the emitter S, assuming that the shape is a hemisphere. The parameters R and S correspond to the ideal case in which the current density is constant over the area S and zero elsewhere, and the radius of curvature in the area S is equal to R. Leybold states that their emitters of etched single-crystal tungsten have a radius of 100 to 200 nm, which is reasonable agreement with thevalues in the table. The values of R/R * suggest that the field emission from the CNT may have come from an extended length. The unusually large value for tube C-3 suggests that multiple SWCNT may contribute to the current, which is consistent with the SEM images that show the bundle of SWCNT is wider than it is for C-6.

Measurements of mixing at audio frequencies
We have made rigorous quantum simulations of LAFE (Hagmann, 1999B) which show that the radiation from two lasers increases the dc current (optical rectification) and also causes harmonics and mixing terms with frequencies that are given by n 1 f 1 + n 2 f 2 , where f 1 and f 2 are the frequencies of the two lasers and the integers n 1 and n 2 may be positive, zero, or negative. Closed-form expressions have been derived for these terms by using an adiabatic approximation, as was done to obtain Eqs. (1)-(5) which only address optical rectification and mxing at the difference frequency. The terms at frequencies higher than these two are not of immediate interest for photomixing because they would be highly attenuated. However, we made measurements with the five field emission tubes at audio frequencies, where the circuit effects of the sealed tubes do not severely impede the response , to confirm our derivations and demonstrate the action of these devices as mixers.
Two sinusoidal signals, at the frequencies f 1 = 1.67 kHz and f 2 = 1.10 kHz, were superimposed on the high voltage that is fed to a field emission tube, and components of the field emission current at the frequencies f 1 , f 2 , 2f 1 , 2f 2 , f 1 + f 2 , and f 1 -f 2 , as well as the rectified current. The six frequencies correspond to 1.67, 1.10, 3.34, 2.20, 2.77 and 0.57 kHz, respectively. The high voltage path of the measurement circuit consited of the high-voltage power supply, a 100 MΩ ballast resistor, the secondary windings of two transformers, the field emission tube, a dc microammeter, and a 1 MΩ resistor to ground which was a shunt for measuring the current with a digital oscilloscope. The two transformers were used to couple two floating battery-operated Wein bridge oscillators in order to superimpose signals at each of the two frequencies with a potential of 120 V on the dc high voltage. Capacitive shunts across the high-voltage power supply, the ballast resistor, and the dc microammeter were used to simplify the ac equivalent circuit.
With the Leybold tube the currents at the fundamental frequencies f 1 and f 2 were each within 5% of the predicted values, and the rectified current and the currents at each of the other 4 frequencies were each within 10% of the predicted values. However, it was necessary to use fourth order terms ( proportional to the fourth derivative) in the Taylor's series to obtain this accuracy. The rectified current and the currents at the 6 frequencies, were each within a factor of 2 of the predicted values for tubes M-4 and C-6, and within a factor of 3 for tube M-1. However, tube C-3 was too unstable to permit measuring the currents at any of the six frequencies. As noted earlier, it is not possible to clean the CNT, and this causes the parameters in Table 1 to be less reproducable than it is for the Leybold www.intechopen.com tube. We attribute the larger errors in the measurements with the CNT to this effect. Nevertheless, the rectified current and the mixer current were measured with three of the four prototypes using individual CNT, and the values are in reasonable agreement with our analysis.

Measurements of optical rectification
It was not possible to measure photomixing with the five tubes because they do not contain the special structures that are required to output microwave or THz energy in order to make such measurements (Alonso & Hagmann, 2001). However, we did measure the rectified current that is caused by a single laser diode (20 mW, 658 nm), and Eqs. (6) and (7) show that if two lasers with the same power were used the peak value of the mixer current would be two times the rectification current that we measured. Figure 6 is a block diagram showing the experimental configuration that was used to measure optical rectification. The laser diode was amplitude-modulated with a square-wave envelope (TTL) by the pulse generator, and a digitizing oscilloscope was used to measure the time dependence of the response of the current to the laser, as shown in the block diagram. The laser diode was maximally-focused to provide a measured Gaussian profile with a power flux density of approximately 10 7 W/m 2 at the emitter. The four prototype tubes were designed to have a long cylindrical glass window with the emitter on the axis to permit the laser radiation to be normal to the glass for minimum distortion, and to reduce the optical path to limit divergence of the beam. However, tube C-6 could not be used in this test because ripples in the glass envelope prevented proper focusing of the laser radiation. Figure 7 shows the response of the current in tube M-4 as a function of the frequency at which the laser was modulated. The decay in the response is exponential, consistent with the equivalent circuits for the tubes  and the measurement circuit, and it does not show a limitation to the speed of the process of laser-assisted field emission. Table 2 shows values for the time constant τ, and the true value of the rectification current I D , that were determined from the data by least-squares regression. Step in the current, peak-to-peak, A Modulation Frequency, Hz  Table 2. Measurements of optical rectification.
In Table 2, I 0 is the dc field emission current and V 0 is the dc voltage that is applied across each tube. This table shows that the mean increase in the dc current is 6.2 % for the 3 tubes with CNT emitters, as compared with 0.20 % for the Leybold tube. This shows that if two lasers with the same power were used for photomixing, the peak value of the mixer current would be 12 % of the dc current for the tubes with CNT, as compared with 0.40 % for the Leybold tube.
Two papers must be considered because they describe much larger changes in the field emission current being caused by laser radiation. An increase of the field emission current by a factor of 19 was measured when a CW laser (10 mW, 633 nm) was focused on a CNT film (Cheng et al., 2004). However, the time constant for the increase in the current was 6 minutes, and the values are not consistent with our analysis of optical rectification, so it appears that this effect was caused by heat-related processes. An argon ion laser increased the field emission current from a single CNT by a factor of 300, or decreased by a factor of 50, depending on etching with oxygen, but this was also explained as a thermal process . These phenomena that are described in the two papers do not appear to appropriate for generating microwave and terahertz radiation.

Photomixing with Clusters of Carbon Nanotubes
Consider a cluster of N transmission lines that are connected across a single load with an impedance Z L . The Ith transmission line has characteristic impedance Z I , propagation constant β I , length Λ I , and is fed by a constant current source with a peak value of I I . The phase shift that is caused by propagation over the full length of the Ith line is θ I . Figure 8 shows this system for only the two transmission lines for N = 2. Photomixing, caused by the radiation radiation from two lasers that is focused on a field emitter at the free end of the Ith transmission line, generates a mixer current I I which flows through this transmission line. For the special case where each transmission line has the same characteristic impedance Z 0 and the load has a real impedance R, a broadband impedance match to the load occurs when N = Z 0 /R. If the mixer current in each transmission line has a peak value that is equal to I M , then the total current that is delivered to the load is equal to N I M . Thus, the total power that is delivered to the load, P OUT = (N I M ) 2 R/2 = I M 2 Z 0 2 /2R.
For example, let Z 0 = 44 k from Section 3 of this chapter, with R = 50 Ω, so that the total number of CNT in the cluster is given by N = 880. Assume that the dc field emission current from each CNT is I 0 = 10 μA. Then the the total dc field emission current is 8.8 mA and from section 4.4 the mixer current in each CNT is I M = 0.12 I 0 , or 1.2 µA peak. The total mixer current in the load would be 1.06 mA peak, for an output power of 28 μW. This is 14 dB greater than that from photomixing in LTG GaAs (Verghese et al., 1997), but the most significant advantage of photomixing with clusters of CNT is the possibility of obtaining a much greater tunable bandwidth.
More generally, when there is not a perfect impedance match, reflected waves propagate on each transmission line and the voltage and current on the Ith transmission line are given by But this voltage is related to the total current through the load by V L = Z L I L. , which leads to a matrix equation that may be solved to determine the A and B coefficients in Eqs. (11)- (14). For the special case where the characteristic impedance and phase delay of each transmission line are equal to Z 0 and θ, respectively, and the mixer current is the same in each line, the power that is delivered to the load is γ multiplied by what it would be for a single field emitter, where the multiplying factor γ is given by  Figure 6 shows the multiplying factor γ as a function of the number and the phase delay of the transmission lines, where Z 0 = 1000 Z L . The horizontal line for N = 1000 corresponds to the broadband match for N = Z 0 /R which has already been described. The following behavior is also seen from Eq. (15) and illustrated in Fig. 6: (1) For small values of θ, γ = N 2 , regardless of the values of Z 0 and Z L , because the N currents feed directly to the load. This is, it is best to have short transmission lines.
(2) For θ = 90 0 , corresponding to quarter-wavelength transmission lines, γ = (Z 0 /Z L ) 2 , regardless of the value of N, because the load projects an impedance Z 0 2 /NZ L to each field emitter.
(3) When Z 0 > Z L , if N > Z 0 /Z L the gain decreases from N 2 to (Z 0 /Z L ) 2 as θ varies from 0 o to 90 o .
(4) When Z 0 > Z L ,, if N < Z 0 /Z L the gain increases from N 2 to (Z 0 /Z L ) 2 as θ varies from 0 o to 90 o .

Consideration of some of the design requirements:
(1) The lengths of the SWCNT must be less than one-quarter of the wavelength at the mixer frequency, 7.5 µm at 10 THz, so the mixer currents are generated in phase at the free ends of the SWCNT. The spread in the lengths must not exceed 100 nm at 10 THz so the mixer currents add in phase after propagating to the load on the SWCNT.
(2) The common junction for the SWCNT may be the rounded end of a wire having a radius as small as 30 nm, depending on the size of the SWCNT (Verema et al., 2000).
(3) The contact resistance between the SWCNT and the wire (Tersoff, 1999) must be mitigated by adding gold (Dockendorf et al., 2007) or nickel (Ribaya et al., 2008), or by other means (Kim et al., 2008) at the common junction.
(4) The wire from the common junction may be used as a transmission line or a traveling wave antenna (Alonso & Hagmann, 2001) as we have previously done in microwave prototypes (Hagmann, 2004). The tunable bandwidth is limited only by the transmission line or the antenna so that the ratio of the upper frequency limit to the lower frequency limit could be 10:1 or even somewhat greater. This will be treated in more detail in the following section.
(5) The SWCNT will fan out radially from their common junction during field emission because of electrostatic forces (Kim et al., 2006). (6) Discrepencies in measurements of luminescence of CNT (Bonard et al., 1998;Rinzler et al., 1995) suggest that the mechanism for electron emission may shift from field emission to thermal processes at high currents so it may be necessary to limit the current to avoid the slower thermal effects.

Carbon nanotube transmission lines
The wire from the common junction could be used as a CNT transmission line to propagate energy to the load. In this case the characteristic impedance of the transmission line must be equal to the load impedance or be tapered to provide a broadband impedance match to the load. In section 3 of this chapter we noted that others have shown that CNT on a conducting substrate are TEM transmission lines with a characteristic impedance of approximately 5 kΩ, and isolated SWCNT have a characteristic impedance of approximately 44 kΩ. However, a much lower impedance, such as 50 Ω, is required because the output power varies inversely with the impedance.

Other types of transmission lines
The wire from the common junction could also be used as another type of transmission line to propagate energy to the load. Again, the characteristic impedance of the transmission line must be equal to the load impedance or be tapered to provide a broadband impedance match to the load. Our group has studied the use of single metallic wires to propagate THz radiation as TM surface waves with enhanced confinement because of surface plasmons (Hagmann, 1998B;Alonso & Hagmann, 1999), and used metallic wires for this purpose in a microwave prototype (Hagmann, 2004). However, the characteristic impedance of this type of transmission line is approximately equal to the impedance of free-space (≈ 377 Ω). A much lower impedance is requried in order to provide greater output power. One possibility is the parallel-plate plasmonic transmission line which can provide an impedance of 50 Ω for TEM mode propagation at THz frequencies with a cut-off frequency of zero Hz (Ghamsari et al., 2008). The common junction of the CNT could be attached to one plate of this transmission line with the other plate grounded.

Carbon nanotube antennas
The wire from the common junction could be used as a CNT antenna to generate THz radiation. In this case the radiation resistance and reactance of the antenna would constitute the load impedance. The electrical conductivity of CNT is several times larger than copper, but the diameter is small so the resistive lossses are high (Burke et al., 2006). Thus, it may be practical to use short CNT in dipoles, but structures that must be much larger than a wavelength, such as traveling wave antennas and other types of antennas having a large bandwidth, would have too much loss.

Other types of antennas
The wire from the common junction could be used as another type of antenna to generate THz radiation. Again, the radiation resistance and reactance of the antenna would constitute the load impedance. A variety of different types of antennas have been used at THz frequencies including dipole and bow tie (Yano et al., 2005), spiral (Verghese et al, 1997), and log-periodic structures (Mendis et al., 2005). We have also studied the zigzag antenna for broadband applications of LAFE at THz frequencies (Alonso et al., 2001).
Parallel-plate transmission lines have been used with lenses to obtain highly focused THz radiation comparable to that expected for a 3-D optical element in free-space (Dai et al., 2004). This method shows promise for coupling energy from the common junction because it would be possible to obtain an impedance of 50 Ω for TEM mode propagation at THz frequencies with a cut-off frequency of zero Hz (Ghamsari et al., 2008). As was already noted in section 6.2, the common junction of the CNT could be attached to one plate of this transmission line with the other plate grounded.

Conclusions
Photomixing in laser-assisted field emission shows considerable promise as a means to generate microwave or THz radiation with an extremely large tunable bandwidth. However, our simulations and measurements with microwave prototypes show that the output power is quite limited because of the small value for the dc field emission current. This chapter presents further simulations, and measurements with prototypes in which the field emission is from individual CNT, which suggest that it may be able to obtain an output power of 10 μW over a tunable bandwidth ratio of at least 10:1. We are continuing to develop such advanced sources with the objective of providing new devices that are needed for the many different applications of THz radiation (Davies et al., 2002). We also hope that this effort will continue to lead to a more fundamental understanding of the process of quantum tunneling (Hagmann, 1992;Hagmann et al, 1993;Hagmann, 1995A).