Plasmonic and magnetoplasmonic interferometry for sensing

The use of plasmonic and magnetoplasmonic interferometers as refractometric-based sensors is studied theoretically, and their performance compared to that of the most commonly used plasmonic sensing technique, the surface plasmon resonance (SPR) sensor. The analysis is based on the evolution with the refractive index of the physical parameters involved for each kind of sensor, as well as the behaviour of the actual measured quantity. Two kinds of sensing configurations, two-dimensional surface and three-dimensional bulk, are considered, and the spectral dependence of the three systems is also taken into account. We show that, although the plasmonic interferometer and the SPR system are based on the same physical parameter, namely the surface plasmon polariton (SPP) wavevector, the first offers better properties as its final sensitivity depends on the interferometer size and can thus be increased by its enlargement. For the magnetoplasmonic interferometer, on the other hand, a new physical parameter participates—the modulation of the SPP wavevector induced by an external applied magnetic field. This new parameter has a higher sensitivity to the refractive index than the SPP wavevector, so monitoring it can lead to sensors with increased properties.


Introduction
Surface plasmon resonances (SPRs) are commonly used for sensing due to their high sensitivity to changes occurring at the interface in which they take place. Although in previous years there have been several advancements in the use of metallic nanoparticles supporting localized surface plasmons for sensing applications [1][2][3], traditionally the term SPR sensors denotes those based on thin films with propagating surface plasmon polaritons (SPPs). In fact, SPR sensors are one of the most popular sensing methods, commercialized by several companies, and they are applied in many areas [4,5], label-free biosensing being one of the most attractive [6]. Nowadays, the main goals in the development of SPR sensors lie within the improvement of the sensitivity and the limit of detection [7], as well as miniaturization [8]. Within this context, some variations of the standard SPR technique have been developed, introducing, for example, magnetic field [9] or photonic waveguides [10].
Interferometry is known as a very sensitive and reliable technique for different applications, and in particular for sensing and biosensing [11,12]. It constitutes an important route for developing compact integrated sensors, such as Mach-Zehnder configurations in silicon [13] or polymer [14]. Interferometry based sensors have already been compared to SPR techniques, and the former have demonstrated higher sensitivity in most situations [11]. However, SPR sensors are still highly competitive, taking into account the ease of their use and the extended knowledge of immobilization protocols in gold. Recently, both interferometry and plasmonics have been put together in the so-called plasmonic interferometers [15][16][17]. In fact, plasmonic interferometry for sensing has also been demonstrated, both theoretically and experimentally, in different configurations [18][19][20][21][22][23][24]. Although these works show that the plasmonic interferometers offer a good performance as sensors, a direct comparison with the traditional SPR configuration has not been carried out yet. On the other hand, modulation techniques are usually applied to increase the signalto-noise ratio in small signals and can be employed to increase the limit of detection and the sensitivity for different sensing systems. In particular, for SPR systems, mechanical [25], phase [26] or magnetic field modulations [9,27] have been implemented, demonstrating an increase in sensitivity compared to non-modulated configurations. Plasmonic interferometers also allow the introduction of modulated configurations, such as all-optical [16] or electrooptical [28]. Nevertheless, these modulations are originated at the dielectric material, and are therefore not suitable for sensing applications since the dielectric constitutes the sensing material or analyte. Magnetically modulated plasmonic interferometers have also been demonstrated [29], and in this case the active material is a ferromagnetic metal, which avoids the aforementioned problem. Moreover, the magnetic modulation of the SPP presents quite a large dependence on the refractive index of the dielectric material [30], which indicates that magnetoplasmonic (MP) interferometers are a promising tool for sensor development.
Following the path inspired by these results, in this paper we propose the use of MP interferometers as sensing devices, and we theoretically analyse their sensitivity compared to plasmonic (non-magnetic) interferometers. Moreover, we first carry out a detailed comparison of the latter with the traditional SPR technique.

General description of sensing layers and methodology
As mentioned in the introduction, three sensing techniques will be compared: SPR, plasmonic interferometry and MP interferometry. All the analysis has been carried out on a numerical basis using a transfer matrix formalism that includes magneto-optical effects [31,32].
In figure 1 we show schematically the three compared sensing techniques. SPR (figure 1(a)) is based on exciting a surface plasmon in a thin metallic layer (usually ∼50 nm Au layer) by means of attenuated total reflection (ATR) configuration and measuring the corresponding minimum in the reflectivity R. Both interferometries (figures 1(b) and (c)), on the other hand, require a thick metallic layer (usually ∼200 nm thick) and a defect such as a groove to launch the SPP. They are based on the existence of interferences between the SPP and the light directly transmitted through a slit cutting the metal film. In a plasmonic interferometer the metallic film is a noble metal (such as gold) and we measure directly the intensity of these interferences I o , while for MP interferometry a thin ferromagnetic film (such as Co) is inserted to allow magnetic modulation, and we measure the magnetically modulated interferences intensity I mp .

Metal thickness optimization
As can be seen in figure 1 and has been briefly explained above, the three sensors use different metallic layers. To perform a fair comparison, we have theoretically optimized the thickness of the metallic layers for each configuration and wavelength using the same noble metal, Au, in the three cases. The optimized parameters appear in table 1. For conventional SPR, we have chosen SF10 glass (n SF10 = 1.73) as substrate and we have calculated the Au layer thickness that provides the optimum SPP excitation, which is that providing the reflectivity closer to zero under ATR configuration at the SPP angle excitation. The obtained Au thickness values range from 32 to 50 nm. In the case of plasmonic interferometry, we need an optically opaque gold layer, so an appropriate thickness is 200 nm Au layer [33] independently of the wavelength. Finally for the MP interferometer, we look for the maximum magnetically induced modulation, which is controlled by the thickness and position of the Co layer. The optimum position of the Co layer is as close to the surface as possible, but this will cause oxidation so we have fixed it at 10 nm, which is a reasonable value to prevent oxidation in sensing aqueous environment. The complete metallic multilayer for the MP interferometers is then 180 nm Au/Y nm Co/10 nm Au, where the thickness of the cobalt layer varies from 5 to 10 nm.

Sensing layer
Sensing can be performed in different environments depending on the analyte. One of the most commonly used ones, mainly in biosensing, is an aqueous solution. Although each analyte and sensing protocol requires a specific geometry of the sensing layer, in our theoretical study we have chosen two generic situations that reproduce two of the most common configurations: twodimensional ('surface') immobilization of biorecognition elements [7] and absorption of the analyte in a three-dimensional matrix ('bulk') [7,34]. Figure 2(a) represents the theoretically simulated 'bulk' configuration that consists of a 80 nm layer of hydrated carboxymethyl dextran, an hydrogel used for biosensing with gold in SPR and whose refractive index in aqueous solution is 1.36 [5,35]. A sensing experiment will be then simulated as a uniform 80 nm medium whose refractive index is varied from its original refractive index n 0 = 1.36 up to  On the other hand, the 'surface' configuration, shown in figure 2(b), consists of a 2 nm selfassembled monolayer of thiol (a standard ligand with n = 1.5) with a uniform 4 nm overlayer whose refractive index will vary from 1.33 (water) up to 1.333, mimicking the adhesion of molecules such as streptavidin to the thiol layer.

Methodology
Finally, the three analysed sensing techniques use different sensor outputs, which makes their comparison subtle. We will focus on the sensitivity of the sensor, i.e. the dependence of the measured output, O, with the refractive index of the dielectric medium, n. In a plasmonic system, the sensitivity can be described as [6,36] In this expression, together with the sensor output and n, it also appears that the physical parameter is being modified by the change in the refractive index, which in a plasmonic sensor usually corresponds to the SPP wavevector, k sp . In fact, (1) shows that the sensitivity S can be decomposed in two terms: the variation of the actual output measured in the experiment with the physical parameter; and the variation of this physical parameter with the refractive index. The first term is dependent on the measurement method, whereas the other term depends on the used materials and geometry and the properties of the associated SPP. In order to differentiate the effect on the sensitivity ascribed to the measuring technique from those related to the properties of the physical parameter, in our comparison we will analyse separately the second term and the complete dependence of O regarding n for the three sensors. Moreover, as SPP properties depend on the wavelength, we will perform this comparison for different wavelengths.

Dependence of k sp on n
We will start our analysis comparing the SPR performance with that of the plasmonic interferometer. Both sensors are based on SPPs, which are evanescent waves that propagate along a metal-dielectric interface and are defined by its wavevector, k sp . For an interface of two semi-infinite materials, k sp is expressed as [37] where k 0 is the wavevector of light in vacuum, n diel the refractive index of the dielectric material and ε m the dielectric constant of the metal. When the refractive index of the dielectric material (our sensing layer) changes from n 0 to n, the physical parameter involved in both techniques is therefore the SPP wavevector, whose modification can be expressed as where n = n − n 0 . Equation (2) can only be applied to an interface composed of two semi-infinite materials, but given the evanescent nature of the SPP it is a good approximation for the thick metallic layer of the plasmonic interferometer. However, for the thin layer composing the SPR sensor, the SPP is affected by the presence of the substrate and its wavevector is slightly different (no analytical expression for k sp can be obtained in this case, it has to be calculated numerically). As the wavevectors are different for the thin and thick metal film cases, the sensitivity of k sp for the two sensing systems may also differ. Moreover, we have analysed both the 'surface' and 'bulk' configurations in order to investigate possible differences in the SPP spreading across the sensing layer between the two systems, which would affect their sensitivity differently for the two situations. Besides, being the dielectric constant of metals dispersive, k sp and consequently its sensitivity also depend on wavelength.
The analysis of the sensitivity of the SPP wavevector ( ∂k sp ∂n = k sp n for small n) for both techniques as a function of the wavelength, for both 'bulk' and 'surface' configurations, is shown in figure 3. It can be seen that the two techniques present a quite close sensitivity in both 'surface' and 'bulk' configurations. The 'bulk' configuration is more sensitive than the 'surface' one, as expected [7] since the spreading of the SPP is much larger than the sensed region for the 'surface' system. However, the important result here is that the sensitivity is the same for the two analysed techniques in each sensing system, meaning that both metallic layers are equally appropriate for sensing bulk changes or surface ones. Regarding the wavelength dependence, both techniques show the same behaviour: the SPP wavevector change with n increases for lower wavelengths, as the SPP wavevector does [7,37].
In summary, the physical parameter involved in both SPR and plasmonic interferometry, k sp , shows the same sensitivity in both cases. However the final sensitivity of a sensing technique also depends on how the variation of k sp is translated into a variation of the measured parameter O, as stated by (1). In the following we will thus analyse the sensitivity of O.

Dependence of O with n: SPR
The SPR technique, as explained briefly in section 2, measures the reflectivity R when exciting the SPP in ATR configuration. To do this, a semi-circular prism is usually set in contact with the metal, as shown in figure 1(a). For the surface plasmon to be excited, the angle of incidence of the light, θ, must fulfil k sp = k 0 n p sin θ , with n p being the refractive index of the prism (n p = 1.73 in our simulations). Therefore, when measuring the reflectivity as a function of θ, a minimum appears when the SPP is excited.
In a sensing experiment, the position of this minimum changes when n varies, due to the associated modification of the plasmon wavevector. In figure 4(a) we present a series of reflectivity curves, calculated in an exact way by our transfer matrix formalism, for different refractive indexes of the sensing layer at a given wavelength λ 0 = 633 nm. The change in the reflectivity minimum with n can be clearly seen.
A typical procedure in a SPR experiment is to monitor the reflectivity under varying conditions that modify n (e.g. with and without analyte) for a fixed set of wavelength and angle We would like to note here that measuring the variation of the reflectivity (which is the method explained here) is not the only procedure for SPR. It can also be done otherwise, by measuring the shift in the angle of minimum reflectivity, or by analysing the reflectivity as a function of the wavelength for a given angle, and measuring the wavelength shift of the minimum of the reflectivity. However, there are studies that show that the final sensitivity of the SPR technique is independent of the measurement procedure [36], being in all cases affected by the experimental noise in the same manner. Thus the only differences when comparing the different procedures are due to the precision of the apparatus. As we cannot take this into account in a theoretical paper, we are going to limit our analysis to the above described reflectivity variation procedure for SPR, without loss of generality. To obtain the maximum sensitivity with this procedure, it is necessary to carefully choose the angle of measurement, θ m : the maximum R is obtained for the angle that maximizes the slope of the R versus θ curve (see figure 4(a)).
To obtain insight on the evolution of the reflectivity with n, we can approximate the resonance dip by a Lorentzian curve [36,38] (this approximation holds quite accurately when |Re(ε m )| n 2 diel and the metal has low losses, i.e. Im(ε m ) |Re(ε m )|). The reflectivity around the SPP dip can be written as where γ i and γ r are the SPP absorption and radiation damping coefficients, respectively [38], and are responsible of the SPP dip width. The angle of measurement can be obtained after equation (5) by looking for the angle of maximum slope Thus, R (≡ (∂ R/∂n) n for small n) at θ m is given by In the case when γ i = γ r , it can be seen from (5) that the reflectivity value at the minimum is zero, which is also the condition for optimum SPP excitation [38] that we have employed to define the optimum metal thickness for SPR sensors (see section 2.1). Then, R can be rewritten as These two last equations, although they are not exact, indicate that the sensitivity of the SPR, R/ n, is related to both the sensitivity of k sp and a term related to the width of the reflectivity dip.

Dependence of O with n: plasmonic interferometry
There are several geometries to implement a plasmonic interferometer, such us using parallel slits [18,19], parallel slit and groove [15-17, 21, 23, 24], tilted groove and slit [20,22,29,30,33]. . . etc. Nevertheless all the configurations are intrinsically equivalent. In our simulations, the employed geometry is based on a tilted groove-slit pair (see figure 1(b)), and is similar to that described in [29,30,33]. When radiative light illuminates the interferometer, the plasmon is launched at the groove and it travels towards the slit, where it decouples again into radiation. This radiation, together with part of the incident light, traverse the slit and produce interferences at the back side of the interferometer. The transmitted intensity at a given position in a plasmonic interferometer can then be described as [29,30] I o = I r + I sp + 2 I sp I r cos(k sp d + ϕ 0 ), where I r and I sp are constants that account for the intensity of the two interfering waves, d is the distance travelled by the plasmon (i.e. the slit-groove distance at that point) and ϕ 0 is an arbitrary phase that depends on the light to plasmon conversion and vice versa. . The contrast of this interferogram is maximized when I r and I sp have the same value, which can be achieved by tuning the position of the incident light regarding the slit-groove pair and the efficiency of the SPP excitation at the groove [39,40]. When the refractive index, and therefore the SPP wavevector, change, the interferogram is shifted.
Similarly to the SPR system, both changes in the intensity at a fixed position and shifts in the position of the minima-maxima can be employed as the sensor output O without affecting the intrinsic sensitivity of the system. In this paper, we have selected the variation with the refractive index of the transmitted intensity I o at a fixed slit position as the quantity monitored for sensing Again, in order to attain the highest sensitivity, the optimum position for measuring the changes of I o is that of the maximum slope of the I o versus d curve. If we neglect the attenuation of the SPP while propagating (i.e. if we consider the imaginary part of the plasmon wavevector to be very small), the variation of I o due to the change of the refractive index in the sensing layer, normalized to the amplitude of the measured interferogram, can be written to first order approximation as As can be seen in equation (11), the sensitivity of the plasmonic interferometer, S = I norm o / n, is proportional to k sp × d, i.e. to the sensitivity of k sp times the distance travelled by the SPP. As this distance can be chosen when fabricating the device, in this sensor we have an external method to increase its sensitivity (see figure 4(b)). The distance travelled by the plasmon can be expressed in terms of the SPP propagation length, L sp , which corresponds to the distance travelled by the plasmon so that its intensity becomes attenuated in a 1/e factor [37]. For a plasmonic interferometer, where the Au layer is very thick, the SPP damping is due only to metal absorption, so that L sp = 1/(2γ i ) [38]. By expressing d in units of L sp , d = f × L sp , (11) becomes Plasmonic interferometry proves to be more sensitive than SPR for long enough d, because its sensitivity increases linearly with the distance travelled by the plasmon (equations (11) and (12)). The plasmonic interferometer sensitivity surpasses that of the SPR sensor for d 1.5L sp in all the analysed spectral range relevant for sensing (λ 0 600 nm). In fact, this result can also be obtained by comparing equations (8) and (12). Moreover, this slit-groove distance is small enough to keep a good contrast in the interferogram. We estimate that the maximum reasonable value of d is that which provides a minimum interferogram contrast of 0.3 in order to clearly resolve the oscillations. Assuming I r = I sp , this results in d 2L sp . This implies that, by choosing the appropriate slit-groove distance, the performance of plasmonic interferometers for sensing can beat that of the conventional SPR.

Comparison of the sensitivity for SPR and plasmonic interferometry
Moreover, figure 5 also shows that the sensitivity increases with the wavelength for both kinds of sensors. This behaviour, opposite to the results presented in figure 3, reflects the fact that the final sensitivity of a sensor depends not only on the sensitivity of the physical parameter but also on other factors associated with the measured quantity. So, in the case of the SPR sensor, the width of the reflectivity resonance dip also influences S (see equations (7) and (8)). Since this width decreases strongly with the wavelength, the sensitivity of a SPR system increases at higher wavelengths even though the sensitivity of the physical parameter k sp decreases (figure 3). Regarding plasmonic interferometry, the SPP propagation length L sp augments rapidly with the wavelength, and being S proportional to the distance travelled by the plasmon (equations (11) and (12)), this counteracts the tendency of k sp .

Evolution of k m with n
The MP interferometer, as briefly mentioned in section 2, is based on magnetically modulating the plasmon wavevector [29,30]. For that, a thin ferromagnetic metallic layer is introduced within the Au thick film, as shown in figure 1(c). This ensures the presence of a sizable magnetic effect and that the plasmon propagating distance is not dramatically reduced due to the higher absorption of Co. When a magnetic field is applied to the MP interferometer in the direction parallel to the surface and perpendicular to the SPP propagation direction, the SPP wavevector is modified as follows [29]: where k sp (n, M) and k sp (n, 0) denote the SPP wavevectors in the presence or absence of sample magnetization, respectively, and k m (n, M) represents the magnetic field induced modification of the SPP wavevector. Although the absolute value of k m is four orders of magnitude smaller than that of k sp [41], it can be easily measured when using a plasmonic interferometric configuration and inverting synchronously an external magnetic field strong enough to magnetically saturate the Co layer [29,30,41]. When the refractive index changes, both k sp and k m are modified and therefore the two physical parameters should be taken into account when analysing the performance of the MP interferometer as a sensor. The general expression for S in this case becomes Analogously to k sp , the variation of k m when the refractive index of the sensing layer is modified is defined as Figure 6 shows the evolution of the relative changes of k sp and k m with the wavelength for the plasmonic and the MP interferometers (note that here we need to compare the relative changes of both quantities, and not their absolute values, as these ones differ in several orders of magnitude as mentioned above). The comparison of both quantities offers an interesting result: k m /k m is one order of magnitude bigger than k sp /k sp . This is due to the fact that the magnetic field induced modulation of the SPP wavevector has a strong dependence on the fourth power of the refractive index of the dielectric [30,41]. As a consequence, it is to be expected that a sensing technique relying on the measurement of the variations of k m exceeds in sensitivity that based on variations of k sp .
From figure 6 it can also be seen that, for both 'surface' and 'bulk' sensing configurations, the relative sensitivity of the magnetic parameter surpass by one order of magnitude that of the purely plasmonic one. Regarding the spectral behaviour, except for the smallest wavelength value shown, k m /k m decreases with the wavelength but in a slower manner than k sp , and indeed for the 'bulk' configuration it can be considered as nearly constant.

Dependence of O with n: magnetoplasmonic interferometry
From the previous analysis of k m , the MP interferometer seems a promising candidate for developing a sensor. However, care has to be taken in selecting the appropriate measured quantity. As mentioned above, a magnetic field applied to the MP interferometer modifies the SPP wavevector, which induces a shift in the transmitted intensity interferogram (see (9)). If the magnetic field is reversed, both the SPP wavevector modification and the shift take place in the opposite direction. Therefore, when applying an alternating magnetic field, the transmitted intensity I o is modulated synchronously with it. This modulation of the intensity, named magnetoplasmonic intensity, I mp , can be expressed in first approximation as if we neglect again the imaginary part of k sp , and considering k m d 1 [29,30,41]. We would like to note that, for the optimization of the MP interferometer described in section 2.1, the quantity we have maximized is I mp : as this quantity is proportional to k m × d, we have indeed maximized the product k m × L sp (as L sp limits the maximum value of d that can be used).
One option for using the MP interferometer as a sensor would be the monitorization of I mp at a fixed slit point when the refractive index changes. The variation of this intensity, normalized to its initial amplitude (≡ I MAX mp (n 0 ) − I min mp (n 0 ) = 8 I sp √ I r k m (n 0 ) d), can be expressed in first order as where tan γ = k sp dk m k m . Equation (17) shows that in fact the MP interferometer is more sensitive than an equivalent plasmonic one. Nevertheless, we should analyse this statement very cautiously. As in the case of the plasmonic interferometer for I o , the sensitivity of I mp depends on the product k sp × d, and therefore the sensitivity can be boosted by increasing the slit-groove distance. This distance is however limited by the SPP propagation distance and, as discussed above, values of d longer than 2L sp will result in too reduced a contrast for comfortable measurements. If we impose this limit of d 2L sp , the sensitivity of the plasmonic interferometer surpasses that of the MP one as the presence of Co in the metallic trilayer, highly absorbent, strongly reduces the value of L sp . This is shown in figure 7, where the sensitivity of the plasmonic interferometer, relying on the measurement of I norm o , is compared with that of the MP interferometer based on the monitorization of I norm mp . The sensitivity of a plasmonic interferometer with d = L sp is higher than that of the MP one with d = L sp or even d = 2L sp due to the smaller value of L sp in this last case (see inset in the figure with the evolution of L sp with the wavelength for the two kind of interferometers). Furthermore, even for a MP interferometer with d equal to the value of L sp corresponding to the pure Au layer composing the plasmonic one, the sensitivity of I norm mp is comparable or smaller than that of I norm o (compare light blue dotted line and black solid line in figure 7). This is due to the fact that the introduction of the Co layer in the MP interferometer also reduces the value of k sp , therefore losing the increase provided by the extra term k m /k m in (17).
The alternative of implementing the MP interferometer by monitoring the shifts in the I mp interferogram is not useful for taking advantage of the higher sensitivity of k m with respect to k sp either, as the shift is only related to k sp (see (16)). A last option that could offer the advantage of higher sensitivity for the MP interferometer is based on the direct monitorization of k m . For that, both the transmitted intensity without a magnetic field applied and the modulated intensity  have to be collected for a full oscillation, and by normalizing I mp to the amplitude of the full oscillation of I o and dividing by the slit-groove distance, the value of k m is obtained [30,41]. This full process should be repeated when the refractive index of the sensing layer changes to obtain the new value of k m . This option, although feasible, requires more signal processing and time acquisition, so it could impose some time restrictions in the sensing experiments. However, the higher sensitivity of k m to the refractive index variations could compensate for this drawback.

Conclusions
To summarize, the sensitivity of three plasmonic-based sensing devices have been compared: standard SPR technique, plasmonic interferometry and MP interferometry. Every system has been considered at its best material implementation, and both the sensitivity of the physical parameter on which the method is based and the actual sensor output have been studied. Table 2 provides a summary of all the relevant parameters for each sensor. Both SPR and plasmonic interferometry rely on the modification of the SPP wavevector under refractive index variations. Given that the physical system supporting each of them is very similar, i.e. Au films, there are not significative differences regarding the sensitivity of the SPP wavevector in both cases, neither in a 'bulk' sensing configuration or in a 'surface' one. When analysing the measured output, however, it has been shown that the plasmonic interferometer can surpass the SPR in sensitivity. This is due to the fact that the sensitivity of the plasmonic interferometer is proportional to the slit-groove distance, so by increasing this distance the sensitivity can be enhanced. Even though the slit-groove distance is enlarged to improve the plasmonic interferometer sensitivity, it is still kept on the order of a few tens of microns, which allows its integration in a chip for the development of miniaturized sensors. Regarding the MP interferometer, two physical parameters are involved in a sensing experiment: the modification of the SPP wavevector and the variation of the magnetic field driven SPP wavevector modulation. This last quantity has a stronger dependence with the refractive index than k sp , so a sensor based on it would provide a higher performance. However, in order to take advantage of this, the monitored output quantity in a MP interferometer has to be directly the processed value of k m .
Regarding the wavelength behaviour, the sensitivity of both the SPR and the plasmonic interferometer increases with wavelength even though the variations of k sp with the refractive index behave in the opposite manner. This is because of the reduced losses of the SPP at higher wavelengths, which generate narrower reflections dips in SPR sensors and allow longer values of d in the plasmonic interferometers. For the MP interferometer, on the other hand, the use of smaller wavelengths would be advantageous as k m , which is the most adequate output here, decreases with the wavelength.