The optical transmission through a subwavelength hole in a metal film is usually very small, and the beam radiated from its exit aperture is very broad. However, the transmission may be increased by orders of magnitude, and the output beam sharply narrowed, when the tiny hole is surrounded by a properly designed periodic structure, which may take the form of an array of grooves or indentations on the metal surface. We have shown that these dramatic effects are due to the excitation of a leaky surface plasmon mode by the periodic structure on the metal film. Following this understanding, we introduce a simple but effective leaky wave antenna model, which we use to further explain and to quantify these dramatic effects. Particular attention is given to optimizing the structure to achieve maximum radiation at broadside, which offers a significant challenge in view of the open stopband in precisely the broadside direction.
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 A very interesting and initially puzzling effect at optical frequencies relates to the transmission of power through a single subwavelength hole in a metal plate. When this very tiny hole is made in a smooth metal plate the transmission through the plate is extremely small, as expected from small-aperture theory (Bethe hole theory) [Bethe, 1944; Collin, 1991]. It was found experimentally, however, that when a periodic array of indentations or grooves is imposed on the initially smooth surface surrounding the entrance to the hole, the transmission of power through the hole is increased dramatically, by up to several orders of magnitude [Grupp et al., 1999; Thio et al., 2000, 2001, 2002].
 The radiation from the exit side of the hole takes the form of a broad beam that is spread over a wide range of angles. When a similar periodic structure is placed on the exit face of the hole, rather than the entrance face, it was found experimentally [Lezec et al., 2002] that the radiation from the tiny hole can be formed into a narrow beam with its maximum at any desired angle, even broadside. When such a periodic structure is placed on both the entrance and exit faces, a combination of both effects occurs, so that the output power from the subwavelength hole is greatly increased and the radiation occurs in the form of a sharply directive beam.
 These initially surprising measured results were at first difficult to understand, but phenomenological theories were proposed in physics journals which go a long way in explaining the major features of this behavior [Lezec et al., 2002; Martin-Moreno et al., 2003]. However, these theories do not correctly explain the actual mechanism that produces the effects. We have correctly identified the actual physical mechanism, and have developed a theory based on leaky waves [Oliner and Jackson, 2003] that not only explains all the effects fully but also readily tells us how to design the structure in order to optimize the power transmitted through the hole.
 At optical frequencies, the dielectric constant of a metal is negative. For metals that have low loss at optical frequencies, such as silver, the permittivity can be approximated as a negative real number. As a result, a smooth metal-air interface can support an electromagnetic surface wave, which is called a surface plasmon mode. This surface wave is largely confined to the metal-air interface, decaying exponentially in both directions away from the interface. All of the early papers on these effects recognized that these surface plasmon modes play a key role in the explanation of these effects. If a periodic array of grooves (or indentations) is placed on such a surface, and if the period is large enough relative to the wavelength, the surface wave becomes a leaky wave. A basic paper in 1959 [Oliner and Hessel, 1959] was the first to demonstrate that a surface wave would become a leaky wave under such conditions, and that this leaky wave then radiates power away at an angle determined by the period, the wavelength, and the dispersion properties of the leaky wave. Our theory [Oliner and Jackson, 2003] first expands the periodic structure leaky wave field into space harmonics and focuses on the n = −1 space harmonic, which is the only radiating space harmonic. It then employs the values of β and α of the leaky wave to explain the radiation performance in a clear fashion, but it also shows quantitatively how those values can be used to optimize the design of the periodic structure. The development of Oliner and Jackson  also explains the theory in terms of a band structure (or Brillouin) diagram, which then demonstrates even more clearly that the physical mechanism is due to a traveling wave that radiates all along the length of the periodic structure. In the present paper, we are interested primarily in the radiation from the tiny hole when the periodic structure is placed on the exit face. However, the leaky wave theory applies both to when the periodic structure is placed on the exit face (which produces a sharply directive radiated beam), and to when the periodic structure is located on the entrance face (which then captures the incoming radiation and funnels it to the hole, thereby greatly enhancing the power transmitted through it). These two cases are directly related, as can be shown by reciprocity. The discussion in section 2 addresses these aspects of the periodic structure locations.
 Two novel aspects are stressed in this paper.
 1. The first aspect is the creation of a simple and effective leaky wave model. The theory referred to above showed us that the basic behavior of the periodic structure in these optical phenomena is similar to that in a periodic microwave leaky wave antenna. Typically, such an antenna is constructed by placing a periodic structure on the surface of a dielectric slab. We can therefore create a leaky wave antenna model if we employ the proper corresponding parameters, such as choosing the dielectric constant of the dielectric slab to have a negative value appropriate to the optical wavelength involved. The details of the model are presented in section 3, and we show later in this paper how we have used this model to obtain accurate numerical values for the performance properties of a corresponding idealized canonical optical structure.
 2. The second aspect is how to overcome the open stopband at broadside. The first step in this contribution is to point out that a special problem arises when the radiated beam needs to be pointed in the broadside direction. Although measured results [Lezec et al., 2002] were obtained for radiation at various angles, the broadside direction is clearly the preferred one. The problem for radiation at broadside is the presence of an open stopband in the vicinity of the broadside direction, so that, for an ideal periodic structure of infinite length, the radiation from the leaky wave goes exactly to zero when the beam is chosen to scan exactly at broadside [Hessel, 1969; Oliner, 1993]. This “open stopband” is a well-known phenomenon. Despite this fact, none of the papers already published on these subwavelength hole phenomena has indicated any recognition of this problem. We show here that by choosing a leaky wave phase constant that corresponds to an appropriate scan angle slightly away from the broadside direction, a maximum power density in the broadside direction can be achieved. This procedure is discussed in section 5, and numerical results in this connection are presented in section 6.
 The radiation patterns are calculated separately for the total field radiated from the small hole with the surrounding periodic structure present, and from the leaky wave on the periodic structure alone, to demonstrate that the narrow-beam radiation from the small hole is indeed due to a leaky wave. The methods used for these pattern calculations are described in section 4.
2. Entrance and Exit Enhancement Effects
 The structures with which we are concerned are based on a subwavelength-size hole in a layer of metal at optical wavelengths. The metal is chosen to be silver because its loss is relatively small at these wavelengths. The dramatic effects we are discussing occur when a suitable periodic structure, in the form of an array of grooves or indentations, is placed on either the entrance face or the exit face of the hole. These two cases are illustrated in Figures 1a and 1b.
 Leaky wave theory can be used to explain both the entrance and exit enhancement effects, and these are discussed separately. First, it is discussed below how the periodic structure on the exit face causes the radiation from the exit aperture to assume the form of a narrow beam, with a greatly increased power density in the broadside direction. The total power radiated by the exit aperture is about the same as that without the periodic structure, however.
 Next, it is discussed how the periodic structure on the entrance face causes the field in the entrance aperture to be greatly increased. This is due to a leaky plasmon mode on the entrance face that collects the fields scattered by each unit cell of the periodic structure on the entrance face and coherently adds them together at the entrance aperture. This increased field in the entrance aperture in turn enhances, by the same factor, the field that propagates through the hole to the exit face, and hence the amplitude of the radiated field in the exit region. The periodic structure on the entrance face does not change the shape of the radiated beam in the exit region, however. Finally, some comments are presented as to what to expect when the same periodic structure is placed on both the entrance and exit faces.
2.1. Periodic Structure on the Exit Face
 When the exit face of the subwavelength hole is flat and smooth, the power exiting from the hole radiates in a broad beam, over a wide range of angles. When a suitably designed periodic structure is placed on the exit face, surrounding the hole, the wide beam is transformed into a narrow beam, sharply aimed into some selected direction, usually broadside. As indicated in section 1, the field at the output end of the hole excites a surface plasmon mode on the metal exit face, and the periodic structure on that face changes the bound (nonradiating) surface plasmon mode into a leaky plasmon mode, which propagates along the periodic structure away from the tiny hole, radiating as it propagates. The plasmon mode propagates radially outward from the hole in all directions, but mainly in the E plane directions (i.e., along ± x, assuming the electric field in the slot is polarized in the x direction), since it is a TMz wave. The leaky plasmon mode possesses a complex wave number, β − jα, where the attenuation constant α represents the power radiated per unit length as the leaky plasmon mode travels along the periodic surface. (The complex wave number β − jα is actually a function of the propagation angle ϕ due to the periodic structure, but, as mentioned, the E plane direction is the most important, and henceforth it is assumed that the wave number corresponds to propagation in this direction.) The angle of the radiation can be selected by choosing appropriately the ratio of the period to the wavelength. More details are presented in section 5.
 When the beam is designed to radiate at broadside, the narrow beam with a high directivity causes the power density to be very high in the broadside direction. To quantify the increased power density transmitted in the broadside direction, a factor R may be introduced, defined as the ratio of the power density radiated at broadside by the hole with the periodic structure present on the exit face, to the power density radiated at broadside by the same hole without the periodic structure on the exit face. Assuming that the power radiated at broadside is mainly due to the leaky plasmon mode, the R factor can be approximated as
= total power radiated by the hole on the exit face (which is approximately the same with or without the periodic structure on the exit face);
directivity at broadside of the radiation pattern from the hole on the exit face without the periodic structure on the exit face;
power launched in the plasmon mode, which gets radiated by the leaky plasmon mode;
directivity at broadside of the radiation pattern from the hole on the exit face, when the periodic structure is present on the exit face. This pattern is mainly due to the leaky plasmon mode.
The ratio Pp/Pa is fairly small since only a fraction of the total power from the exit aperture is launched into the plasmon mode, with a typical value being 0.5. (This number depends on the radiation pattern level of the magnetic dipole source and the field configuration of the surface plasmon mode.) The rest of the total power gets radiated directly into space in the exit region in the form of a wide beam. However, the ratio DLW/Da may be very large because of the highly directive leaky wave beam that is radiated in the exit region. Hence the factor R can be very large for a highly directive leaky wave beam. Quantitative values for these constituents of the radiated power are discussed in section 6.
2.2. Periodic Structure on the Entrance Face
 On the entrance face the incoming plane wave excites a leaky plasmon mode that coherently collects power from the periodic array and directs it toward the entrance aperture. This results in a strongly increased field at the entrance aperture. To quantify this effect, reciprocity may be used [Harrington, 2001]. In the reciprocity analysis, an electric “source” dipole is placed in the aperture of the hole. A “testing” electric dipole is placed in the far field in the broadside direction (the direction of interest). Reciprocity implies that the far field radiated by the source dipole in the aperture is the same as the electric field inside the aperture when illuminated by the incident field coming from the testing dipole, which is essentially a plane wave (since the testing dipole is in the far field). Therefore the factor by which the periodic structure enhances the radiated far field from the exit aperture in the broadside direction is the same as the factor by which the field in the entrance aperture is increased, assuming the same periodic structure exists on the entrance and exit faces. The radiated far-field amplitude is increased by a factor of R1/2 (since the factor R was defined in terms of power). Therefore the field in the entrance aperture is also increased by the same factor of R1/2 when the periodic structure is placed on the entrance face.
2.3. Periodic Structure on Both Faces
 When the same periodic structure is placed on both the entrance and exit faces, both enhancement effects occur simultaneously. The field in the entrance aperture is increased by a factor R1/2, which means that the field in the exit aperture is also increased by the factor R1/2. Furthermore, the far-zone electric field radiated in the broadside direction by a fixed-amplitude electric field on the exit aperture is increased by R1/2 because of the increased directivity of the exit beam. Thus, together, the two effects imply that the total far-zone electric field radiated by the exit aperture in the broadside direction is increased by a factor R. Hence the total increase in the level of the radiated power density from the exit aperture in the broadside direction is a factor of R2. This explains why the enhanced transmission effect is very dramatic when the periodic structure is placed on both sides of the metal film.
3. Leaky Wave Antenna Model
 As mentioned in section 1, the physical mechanism that produces both the enhanced transmission through the tiny hole and the sharply directive radiated beam from the hole is basically the same as that which governs the radiation from microwave leaky wave antennas. In all these cases, that mechanism is the leaky wave that is produced by the presence of the periodic structure. A leaky wave antenna model can therefore be created if the proper corresponding parameters are chosen. A suitable model was adopted and then used in this paper to obtain quantitative information on the radiation properties of a typical subwavelength hole surrounded by a periodic structure at optical wavelengths. The geometry of the model employed in this paper is presented in Figure 2, with a three-dimensional view and a side view shown in Figures 2a and 2b, respectively.
 The metal film consisting of silver corresponds to a dielectric layer of thickness (or height) h, with a dielectric constant ɛr that depends on the optical wavelength employed. We chose a wavelength of 400 nm, so that ɛr ≅ −4.5 [Johnson and Christy, 1972]. Although the actual periodic structure consists of an array of periodic grooves or indentations on the surfaces of the metal film, a simpler model is chosen for ease of analysis. First of all, the periodic structure is placed on only one surface (the exit face) because we are concerned here with the radiation into a sharply directive beam in the broadside direction. The second simplification in the model is that the array of discontinuities consists of a planar two-dimensional array of perfectly conducting metal patches on one surface of the metal film, as shown in Figure 2. This situation is fictitious since perfectly conducting patches do not exist at optical frequencies. However, the structure captures the relevant physics of the problem, since the directive beam effect is due to the fact that there is a periodic structure on the exit face of the metal film, and it is of secondary importance what the periodic structure is, at least for explaining the fundamentals of the behavior. As seen in Figure 2a, the patches have length L and width W, and the period in the x direction is a.
 The next step is to close off the subwavelength hole and the exit aperture, and to replace the exit aperture by a magnetic current, as shown in Figure 2b. (This replacement is exact for a perfectly conducting film, and approximate for a good conductor.) The magnetic current may be approximated as an infinitesimal magnetic dipole, since the aperture is small compared to a wavelength. Assuming that the electric field in the aperture is polarized in the x direction, the magnetic dipole will be in the y direction. The magnetic dipole excites a surface plasmon mode that propagates outward radially from the dipole. This dipole source will launch a TMz surface plasmon mode that has a cos ϕ angular dependence, so that most of the power of the mode is launched in the E plane direction (along the x axis). When a properly designed periodic structure is placed on the exit face, the surface plasmon mode is converted to a leaky plasmon mode, which radiates from the n = −1 space harmonic (Floquet mode) [Marcuvitz, 1956; Oliner and Hessel, 1959; Tamir and Oliner, 1963a, 1963b; Hessel, 1969; Tamir, 1969; Zucker, 1969; Oliner, 1993]. The real part β−1LW of the wave number of the −1 space harmonic for the plasmon mode propagating in the x direction is β−1LW = β0LW − 2π/a, where β0LW is the fundamental phase constant of the leaky plasmon mode propagating on the periodic structure in the x direction, and a is the period in the x direction. By properly choosing the period so that β−1LW ≈ 0, the leaky plasmon mode radiates a narrow beam in the broadside direction provided α is small. (A further discussion of how the beam is optimized near broadside is given in sections 5 and 6). The creation of the narrow beam at broadside in the exit region is thus very similar to the manner in which two-dimensional leaky wave antennas radiate at microwave frequencies [Zhao et al., 2001, 2002].
4. Radiation Pattern Calculations
 A numerically accurate moment method procedure is used to calculate the radiation pattern of the magnetic dipole surrounded by the periodic structure, as shown in Figure 2. Once the pattern is obtained, the R factor may be calculated. The magnetic dipole is assumed to be in the y direction, corresponding to an electric field in the aperture that is polarized in the x direction. The radiation pattern of the leaky plasmon mode by itself may also be calculated as discussed below, in order to compare with the exact pattern. This demonstrates that the narrow-beam pattern of the magnetic dipole surrounded by the periodic structure is indeed due to the leaky plasmon mode.
4.1. Exact Pattern Calculation
 The far-field pattern of the magnetic dipole is found by reciprocity, which involves calculation of the magnetic field Hy at the dipole location, due to an incident plane wave. A periodic moment method solution is used to calculate this field [Zhao, 2003]. In this solution procedure the currents on the patch elements are found under the plane wave excitation. The electric field integral equation (EFIE) is reduced using a Galerkin method to a matrix equation of the form [Z][I] = [V]. The patch currents are then obtained from the matrix solution, and from this the field Hy at the aperture location is then determined, which is equivalent (by reciprocity) to a calculation of the far-field pattern.
4.2. Leaky Wave Pattern Calculation
 In order to calculate the radiation pattern due to the leaky plasmon mode alone, the complex propagation wave number of the mode is first found from a characteristic equation, which comes from the same moment method type of analysis described above. The characteristic equation is obtained by setting the determinant of the matrix [Z] to be zero, and this characteristic equation yields the complex propagation wave number for the leaky plasmon mode propagating in the x direction, kxLW = β0LW − jαLW.
 The leaky wave pattern is obtained by summing the pattern contributions from each patch element, assuming that the currents on the patch elements are determined solely by the leaky plasmon mode. Because of the high directivity, the element pattern is ignored, and only the array factor is calculated. Let In denote the current on patch n, on the center row of the periodic structure, that is, the row that is along the x axis. (All other rows are neglected for simplicity, in calculating the E plane pattern. This approximation is justifed by the fact that the leaky plasmon mode is excited primarily in the E plane direction, along the x axis.) By symmetry, I−n = In. The array pattern factor is
On the basis of the leaky plasmon mode with a complex wave number in the x direction, the currents are (to within a constant of proportionality) In = e, n > 0 and In = e, n < 0. Therefore the array factor is
Defining ζ1 = k0 sin (θ) − kxLW, ζ2 = k0 sin (θ) + kxLW, and z1 = ej(aζ1), z2 = e−j(aζ2), the array factor then becomes
Summing the geometric series gives
Hence the array factor for the leaky wave pattern becomes
Using the above expression, an approximate pattern based solely on the leaky plasmon mode can be obtained. The degree to which this pattern agrees with the exact pattern is a measure of how important the leaky plasmon mode is. This agreement will be demonstrated in section 6.
5. Design Considerations for Radiation at Broadside
 According to periodic structure theory, the leaky plasmon mode will radiate via the n = −1 space harmonic. The beam angle θ−1 from broadside is given to a very good approximation by
provided α ≪ β−1LW. Hence the period a (or the wavelength) can be adjusted to achieve the desired beam angle. With the model discussed above, we have chosen to keep the wavelength the same, and to vary the period.
 If the periodic structure is absent, the surface plasmon mode on the smooth surface is a bound surface wave for which there is a simple closed-form solution for the propagation wave number βp [Raether, 1988]. It is
where ɛr is the relative permittivity of the silver. (This solution assumes an infinitely thick film, but it is an excellent approximation if the film thickness is many penetration depths, as it is in the measurements.) Although the silver film is lossy at optical frequencies, the loss is small, so the relative permittivity may be approximated as a negative real number (which is ∼4.5 in our example).
 When the periodic structure is present, it produces a loading on the surface plasmon mode in two ways. The first way is that it causes the surface plasmon mode to become leaky, so that the wave number of the n = −1 space harmonic becomes complex, as β−1LW − jα, where the attenuation constant α is a measure of the radiation per unit length along the periodic structure. The angle θ−1 of the radiation from broadside is then approximately given by
The second way is to alter the value of the phase constant of the fundamental (n = 0) space harmonic β0LW, where
and a is the period of the periodic structure. Depending on the amount of the loading that the periodic structure exerts on the surface plasmon mode, the values of β0LW and βp may be close to each other or rather different. In general, they are fairly close to each other, but sufficiently different so that β0LW should be calculated separately if accurate results are required.
 It is well known that in the immediate neighborhood of broadside, an open stopband is encountered [Hessel, 1969; Oliner, 1993]. An open stopband is encountered when β0LWa = 2π, and, from (10), β−1LW = 0. From (9) we would then expect that θ−1, the angle of radiation, would occur exactly at broadside. However, because of the open stopband, the value of α goes exactly to zero there. Thus, in the ideal case, no power is radiated at exactly broadside. At this “stopband null” point, the leaky plasmon mode becomes a standing wave instead of a traveling wave, so that no power can be radiated [Hessel, 1969; Oliner, 1993].
 If the wavelength is kept constant, but the period is modified, one finds that just below the stopband null point the value of α increases sharply and goes through a maximum before dropping again. (If the period is kept constant but the wavelength is varied, the sharp rise in α occurs just after the null point, rather than before it.) The large value of α does not represent radiated power, however, but a reactive value caused by power being returned to the source (the small hole). Figure 3 in section 6 illustrates this behavior quantitatively, in context of how one must proceed in order to obtain a beam with maximum radiation at broadside.
 When the periodicity is such that β−1LW ≠ 0, there are always two beams that are radiated, each originating from one half of the structure, due to those parts of the leaky plasmon mode traveling in the ± x directions from the aperture. It will be demonstrated in section 6 that at the optimum point of maximum broadside radiation the two beams merge together so that the overall pattern appears as a single beam with a maximum at broadside, and is on the verge of splitting.
 In all of the results a silver substrate is used at a wavelength of 400 nm (the frequency is 7.5 × 1014 Hz). At this wavelength, the relative permittivity of the silver is approximately ɛr = −4.5. The loss is neglected in the radiation pattern calculations. The substrate thickness h is chosen as 10 times the penetration depth of silver, which is 3 × 10−8 m at this wavelength. Hence a 300 nm film thickness is used. The parameters of the periodic patch structure are L = 140 nm, W = 50 nm, b = 90 nm. The period a in the x direction is variable, and is specified in each result reported here. This is the parameter that is varied in order to achieve an optimum power density radiated at broadside. Two design procedures are described below, which yield two different values of the period a.
 This design procedure is the correct one to use, and is very accurate. The optimized value of period a in this procedure is based on a numerical search to find the value of the period that yields the maximum radiated power density at broadside. The resulting value of the period in our case is a = 377 nm.
6.1.2. Design Procedure 2: Approximate Design Using the Surface Plasmon Mode Wave Number
 This procedure yields an approximate result, and was described in section 5. It employs the phase constant βp for the surface plasmon mode on a smooth silver film, which is calculated easily using (8). Then βp, which is valid in the absence of the periodic structure, is substituted in (10) for β0LW, which is valid in the presence of the periodic structure, but is much harder to calculate. Next, we ignore the stopband effect and assume that the radiation is occurring at broadside, so that β−1LW in (10) becomes zero, and the value of period a follows directly. Thus this procedure allows us to easily calculate an approximate value for a, but the result will be in error. For angles away from broadside, the error produced by this procedure may be tolerable, but, as we see later, the error is not acceptable when we are near to broadside. The resulting value of the period is a = 353 nm in our example.
6.2. Wave Number Considerations
 A plot of α/k0 versus β0LWa for a range of values in the neighborhood of broadside is shown in Figure 3. The qualitative behavior of this plot has been described in section 5, where it is pointed out that an open stopband occurs in the neighborhood of broadside. This open stopband is characterized by a stopband null for α that appears at β0LWa = 2π, and by a rapid rise and then a fall in the values of α just before this null. As explained in section 5, the large values of α over most of this range represents a reactive response and not an increase in radiation.
 We may note that three points are identified on this curve in Figure 3, two of which correspond to the two design procedures described just above. The third point corresponds to the open stopband null itself, at β0LWa = 2π.
 It is difficult to know from these wave number values alone what sort of radiation behavior should correspond to the various portions of this curve, except of course from the null point itself, where there is no radiation from the leaky plasmon mode. We have therefore calculated the radiation patterns directly for each of these cases.
6.3. Radiation Patterns
 The normalized radiated power patterns for each of these cases are calculated in the manner described in section 4. The patterns are normalized by the power radiated from the same unit strength magnetic dipole in free space, and then expressed in dB. All of the patterns are in the E plane (ϕ = 0), which is the plane that is most influenced by the leaky plasmon mode. The patterns are discussed individually below.
6.3.1. Numerically Optimized Design
 We have explained earlier that the radiated beam becomes very narrow when a suitable periodic structure is placed on the exit face. For the sharply directive beam, part of the power emerging from the subwavelength hole on the exit face (or the magnetic dipole in the leaky wave model) radiates directly in a broad pattern, and part goes into a surface plasmon mode that is guided by the periodic surface around the tiny hole (or magnetic dipole). The periodic structure modifies the phase constant of the surface plasmon mode, and, because of the relation between the period and the wavelength, the surface plasmon mode becomes leaky and radiates power around the broadside direction. Employing the accurate analysis mentioned above, we have calculated the radiation pattern obtained from the small hole for several different cases.
 A typical radiation pattern for radiation at broadside is presented in Figure 4, corresponding to the numerically optimized case for which a = 377 nm, and the other dimensions are shown in Figure 2. Actually, Figure 4 shows polar plots of three radiation patterns: (1) the pattern for the direct source radiation from the magnetic dipole in Figure 2, which is the pattern for the magnetic dipole on the silver film without the periodic structure; (2) the pattern due to the periodic structure, which is obtained by subtracting the direct source field from the total radiation field; and (3) the total radiation pattern.
Figure 4 shows clearly that without the periodic structure the pattern is broad and not peaked at broadside. The field from the periodic structure shows a typical leaky wave pattern, with a highly directive narrow beam at broadside. This field agrees well with the total field, demonstrating that the periodic structure is responsible for the enhanced directivity at broadside.
 From the numerical values in Figure 4, we find that the power density at the peak of the radiation pattern due to the leaky plasmon mode is 24 dB (or about 250 times) greater than the corresponding value for the direct radiation from the hole (magnetic dipole) at broadside. We therefore see that the effect of the periodic array on the exit face is to increase the power density at the peak of the radiation pattern by about 250 times. Correspondingly, by reciprocity, if the periodic array would be placed on the entrance face it would enhance the power transmitted through the small hole by about 250 times more than would be the case if the periodic array were absent. When the periodic arrays are placed on both the entrance and exit faces, the power density at the peak of the output radiation pattern is increased by about (250)2, or about 6 × 104. These numbers have been optimized for this idealized case; in practice, the result would be less dramatic but very impressive nevertheless.
Figure 5 shows, on an expanded scale, a rectangular plot of the fields due to the periodic structure and to the field calculated from the leaky wave array factor (equation (6)) for this same case. Both plots are normalized to 0 dB at the maximum. It is seen that the two patterns agree very well, verifying the assumption that the sharp beam radiated by the periodic structure is mainly due to a leaky plasmon mode.
6.3.2. Approximate Design Using the Wave Number of the Surface Plasmon Mode
Figure 6a shows the E plane patterns for the design that employs the wave number of the bound surface plasmon mode on a smooth silver film, where a = 353 nm. The patterns are shown on an expanded rectangular scale, as in Figure 5. The direct source pattern remains the same as the one in Figure 5 (since this field is not affected by the periodic structure), while the total pattern near broadside exhibits a split beam with two peaks. Clearly this pattern shows that this design, which does not take the loading from the periodic structure into account, and also ignores the stopband effect, produces an undesirable beam. Although this simple procedure can produce a decent approximation under certain conditions, it is seen to be unsuitable for angles near to broadside because of the rapid wave number changes in that region, as seen in Figure 3. In spite of this, Figure 6b shows that there is good agreement between the pattern due to the periodic structure and that due to the leaky wave array factor pattern, demonstrating that the leaky plasmon mode is responsible for the pattern, even when the design is not optimum.
6.3.3. Discussion of Optimization
Figure 7 shows the E plane radiation patterns for three different periods, for values of the period close to the numerically optimized value of 377 nm. It is seen that the numerically optimized case corresponds to the value of period a that causes the beam at broadside to be on the verge of splitting into two beams. For a period a = 376 nm (Figure 7a), which is slightly smaller than the optimum value, the beam remains at broadside, but the power density at broadside is a little less than optimum. For a period a = 378 nm (Figure 7c), that is slightly larger than optimum, the beam has begun to split, so that there is a dip in the power density level at broadside. This effect can be understood by realizing that the radiation pattern from the leaky plasmon mode always consists of two beams, one coming from each part of the periodic structure on either side of the magnetic dipole along the x axis. For a small period, such as a = 353 nm corresponding to the plasmon wave number design, these two beams will point in different directions, causing the overall pattern to have a dual-peaked nature, as seen in Figure 6. As the period increases, each beam moves toward broadside. At the stopband null condition, a = 360 nm, the two beams would both point exactly at broadside, but there is no radiation from either one. As the period is further increased, the beams continue to move and begin to scan away from broadside. At the optimum period a = 377 nm (Figure 7b), the two beams are pointing in different directions, on opposite sides of broadside, but the directions are close enough together relative to the beam widths so that the overall pattern appears as a single merged beam, with a maximum radiated power density at broadside. As the period continues to increase beyond the optimum point, the two beams move sufficiently apart so that the overall pattern once again exhibits a split beam, and this results in a drop in the radiated power density at broadside.
6.3.4. Finite-Size Structures
 The simple array factor expression obtained previously in equation (6) can be modified to predict the pattern of a finite-size structure, for which there are N patch elements on either side of the aperture in the E plane dimension. In this case, equation (6) is simply modified by using an upper limit of N instead of infinity. With a finite-size periodic structure, sidelobes will arise in the radiation pattern, and the power density at broadside will decrease (the results are omitted here).
 The power transmitted through a very tiny hole in a metal plate at optical wavelengths is known to be extremely small. Also, the beam radiated from the exit face of that hole is very broad, with the radiation spreading out over a wide range of angles. When a suitable periodic array of grooves or indentations is placed around the tiny hole on its entrance face, it has been demonstrated by measurements that the power transmitted through the hole can be increased dramatically, by as much as two orders of magnitude. When a similar array is placed on the exit face, measurements have shown that the broad beam can be transformed into a sharply directive narrow beam, in which the power density at the center of the beam is increased by as much as two orders of magnitude.
 We have provided a complete theoretical explanation for these dramatic effects, where the key feature is that the periodic structure modifies the surface plasmon mode excited by the hole and transforms it into a leaky plasmon mode. The leaky plasmon mode is then viewed as a sum of space harmonics, of which only the n = −1 space harmonic is radiating. A more complete discussion is presented in sections 1 and 2 in this paper. Section 2 also contains a discussion based on reciprocity, which shows that the factor by which the field in the entrance aperture is enhanced is exactly the same as the factor by which the radiated field at broadside from the exit aperture is enhanced, assuming that the same periodic structure is placed on both faces.
 The leaky wave mechanism that underlies the explanation of the enhancement effects associated with the subwavelength hole at optical wavelengths is basically the same as the one that governs the radiation from microwave leaky wave antennas. A leaky wave antenna model can therefore be created if the proper corresponding parameters are chosen. The details of this model are shown in Figure 2, and are discussed in section 3. The model provides a somewhat simplified structure that allows some ease of analysis, but nevertheless captures all the relevant physics. It is used in section 6 to obtain much quantitative insight into the performance properties of the subwavelength hole structure.
 This paper addresses two major challenges. The first, which we have discussed above, is the creation of a suitable simplified leaky wave model that permits us to obtain reliable and accurate quantitative data and also additional physical insight. For example, a major contribution to both physical understanding and quantitative information relates to the radiation patterns in Figure 4. The curves in Figure 4 show that there are two sources of radiation from the exit face. One is the direct radiation from the small hole, which produces a broad beam, and is what would be obtained if the periodic structure were absent, and the other is the radiation from the periodic structure, which causes the surface plasmon mode to become leaky and provides the sharp beam. These results also demonstrate that the ratio of the contributions at broadside from these two sources is about 24 dB (or about 250 in power). The enhancement of the peak power density in the radiation pattern due to the presence of the periodic structure is therefore about 250 times in this example. Other interesting quantitative results are presented in section 6.
 The other major challenge, which is discussed in detail in sections 5 and 6, is the fact that there exists an open stopband in the vicinity of broadside, from which we desire a peak in the radiation pattern. The challenge is how to design the structure so as to maximize the radiation at broadside since we cannot do it directly. We need to choose the period of the periodic structure so that the maximum power density is radiated at broadside in the leaky wave antenna model. The optimum period corresponds to designing the structure so that the n = −1 space harmonic of the radiating leaky plasmon mode radiates at an angle close to, but not exactly equal to, broadside. When the period is chosen so that the n = −1 space harmonic radiates exactly at broadside, the structure is operating at a stopband null, where the radiated power from the leaky plasmon mode drops to zero. As discussed in section 6, and illustrated in Figure 7, an optimum power density at broadside occurs when the phase constant of the n = −1 space harmonic is chosen slightly away from broadside, corresponding to the overall pattern being on the verge of splitting into two separate beams.
 Finally, leaky wave theory can be used to obtain the radiation pattern when the periodic structure is of finite size, to study the effects of truncation. This is useful for predicting the size of the periodic structure that is necessary to avoid serious truncation effects and a consequent decrease in the transmission enhancement.
 This work was partially supported by the state of Texas Advanced Technology Program. The authors would also like to thank Tineke Thio for important discussions on the topic of plasmons and enhanced transmission.