1. Introduction

Nanophotonic structures are inherently local or point-like, often part of a complex structure that cannot be divided into smaller pieces, as is the case for photonic crystals [1, 2], and do not necessarily have an output in the far-field. In fact, the localization of the optical electric field to the nanoscale implies that at least some of the field components do not propagate into the far-field [3]. In many applications of nanophotonic structures, it is necessary to have control over the light-matter interaction and the light propagation on an ultrafast timescale, for example in optical communication and all optical processing, where the drive to achieve higher bit-rates necessitates the use of short light pulses, and therefore integrated optical components such as ultrafast switches [2, 4, 5]. Another example is the investigation of ultrafast light-matter interaction on the nanoscale [6], where nanophotonic structures could be used to bring the ultrafast optical characterization techniques to the nanoscale [7]. It is therefore essential to have an instrument that can characterize the light propagation and localization in nanophotonic structures on an ultrafast timescale and do so in the near-field rather than in the propagating far-field.

Near-field scanning optical microscopy (NSOM) has been used to characterize the near-field of photonic structures [8–16] and also time-resolved information has been obtained [8, 10, 14, 15]. An NSOM not only provides a sub-wavelength spatial resolution, but, more importantly, it measures the evanescent field on the surface of the structure, and thereby it allows characterization of field distributions in parts of the structure where there is no emission into the far-field.

In order to completely characterize a photonic structure, however, one needs to measure not just the optical field intensity, but the but the full time or frequency dependent amplitude and phase of the light field, as well as the polarization. The frequency dependent amplitude and phase can be measured using spectral interferometry [17], i.e. frequency domain interferometry, which is based on measuring the spectrum of the sum of the signal pulse and a reference pulse. As in any interferometric method, one can selectively map a specific polarization component of the field by choosing the polarization of the reference field. Knowing the spectral amplitude and phase of the electric field means that we obtain information about the time-domain electric field in the structure, since these are related by a Fourier transform. This then gives information on the interaction between the light and the sample on an ultrafast (femtosecond) timescale. Nanophotonic structures have recently been characterized locally and on an ultrafast timescale, with methods where the spectral phase and amplitude were retrieved, but with signal detection in the far-field [7,18–20]. Of the time-resolved NSOM methods, only those based on measuring a field cross correlation between the signal and a reference by scanning the delay [10, 13, 14] have so far retrieved the spectral phase and amplitude, that is the full spectral near-field. That method is however slow, and therefore also sensitive to problems with drift, and relatively difficult to implement [17], at least in the visible and NIR. The alternative method, developed by Balistreri et al.[8], which is based on scanning the probe across a larger area of the sample, does not retrieve the full time-domain field. This since the measurement is not equivalent to a cross correlation in the presence of sample dispersion [21]. For that method the group velocity can be obtained in most cases, provided that the sample is long enough, but the dispersion can only be retrieved from modelling in some simple cases [10, 21], and also requires a number of assumptions.

In this paper we discuss the combination of an NSOM with crossed beam spectral interferometry to enable measurements of the spectral amplitude and phase in the near-field, for both guided and localized fields, which overcomes the disadvantages of both those methods. The key advantage of combining NSOM with spectral interferometry is that it allows the characterization of the full field at individual probe positions. As a result of this, the method only requires measurements at two points on the sample (the point of interest and a reference point to correct for the relative phase of the two interferometer branches) to determine the dispersion properties. This is in contrast to the time-resolved interferometric NSOM methods that require scanning across a large (several tens of wavelengths long) area of the sample [8, 16]. This not only dramatically reduces the measurement time and problems with drift, but is particularly useful where it is not possible to scan a large area as is the case when the structure is complex and aperiodic, or when the information at a single point is of interest. A prime example of this is single quantum emitters in cavities and other nanophotonic structures, where it is essential to characterize the near-field of the photonic structure precisely where the emitter is located.

Measuring the spectral phase and amplitude furthermore reveals when there are multiple modes excited in the structure via the spectral fringes resulting from interference between multiple pulses. As in Refs. [15, 22, 23], in order to actually retrieve separate values for the refractive indices of the individual modes, it is necessary that the refractive indices are different enough in relation to the length of the structure that is measured.

Here we use a combination of crossed beam spectral interferometry and NSOM to characterize guided modes in a ridge waveguide structure. We retrieve the spectral amplitude and phase from a single camera frame with an integration time of 70 ms for a signal power of 50–100 pW. We use the method to obtain the group velocity as well as the wavelength dependent refractive index, i.e. the dispersion, locally in the structure.

2. Experimental

A schematic of the experimental setup is shown in Fig. 1. A linearly polarized broadband laser pulse (FWHM 25 nm) from a Ti:Sapphire laser was coupled into the waveguide structure. A home-built NSOM with a fiber probe was used to measure the local evanescent field on top of the sample. We used a tip-scanning NSOM with uncoated etched fiber probes. The NSOM used shear force feedback to control the probe-sample distance, with the probe attached to a quartz crystal tuning fork. We used a closed loop piezo scanner to allow accurate positioning of the probe on the waveguide. The signal was then detected either with heterodyne detection in a Mach-Zehnder interferometer or with crossed beam spectral interferometry [24].

 

Fig. 1 a) Illustration of the principle of the crossed beam spectral interferometry. b)The experimental setup, combining crossed beam spectral interferometry and a Mach-Zehnder interferometer. The output from the NSOM probe can be switched between the SEA TADPOLE spectrometer and the Mach-Zehnder interferometer. FC=fiber coupler, ref-STP=reference for SEA TADPOLE measurement, refMZ=reference for Mach-Zehnder interferometer, AOM=acousto-optical modulator, P=polarization controller, LIA=lock-in amplifier.

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We used a version of crossed beam spectral interferometry, SEA TADPOLE, (Spatially Encoded Arrangement for Temporal Analysis by Dispersing a Pair of Light E-fields) [25, 26] for detecting the light. As illustrated in Fig. 1(a) and described in more detail below, this gives a spectrally resolved image of the phase that is encoded in the crossed beams. SEA TAD-POLE is a fiber based method, which makes it suitable for combination with a fiber based NSOM. Since crossed beam spectral interferometry has the phase information encoded in spatial fringes and zero delay between reference and signal it avoids the high requirements on the spectral resolution found in co-linear spectral interferometry [26], and its fiber based implementation STARFISH [27]. Spectral interferometry is, furthermore, a linear method [17], where the weak signal can be amplified by the stronger reference beam and therefore it lends itself well to combining with NSOM measurements where the signals are inherently weak.

The signal from the NSOM probe was combined with a reference pulse from the same laser. The reference pulse was passed via a delay stage to make the signal and reference pulse overlap in time with near zero relative delay in a spectrometer adapted for SEA TADPOLE, see Fig. 1(a). The two laser pulses were made to cross at an angle in the vertical direction by mounting the end of the NSOM fiber probe and a fiber carrying the reference pulse parallel to each other in the focal plane of a lens. The two collimated crossing beams were then dispersed by a grating and focused using a cylindrical lens, resulting in a spectrally resolved fringe pattern which was detected on a camera.

The fringe pattern is described by

The reference spectrum can easily be measured, and the reference spectral phase remains the same for all points on the sample. Therefore, by making measurements at two points on the sample and subtracting the phases of the interference terms the reference spectral phase cancels. The difference in spectral phase Δφ_sample(ω) thus obtained describes the effect of the piece of sample in between the two measurement points on the electric field of the light, which is also the argument of the complex transfer function H(ω) [17, 28].

If it is necessary to absolutely determine the field at a single point, rather than to measure it relative to a reference point, for example for shaping the nanoscale field [29] or in coherent control experiments, we can do this by completely characterizing the reference field using a self-referenced method such as frequency resolved optical gating (FROG), thus obtaining the reference spectral phase as well as the spectrum. The detected signal field is the field after the NSOM probe, so we have to characterize the NSOM probe as well, which can in principle be done. The dispersion of the NSOM probe is mainly from the length of fiber, but the tapered end can also have additional dispersion, at least for coated probes [30]. We emphasize that the full optical field is retrieved from a single camera frame, so that the speed of the method is in principal limited by the maximum frame transfer rate of the camera, making it suitable for shaping the nanoscale field using an iterative algorithm.

The camera used in the experiments was an uncooled 8-bit CMOS camera. Despite this we could retrieve the data, that is the complete spectral field in one point, with a less than 100ms integration time for about 50–100 pW signal output from the NSOM probe. This is in stark contrast to the tens of minutes that were required to scan a large area along the waveguide as required for detection with the Mach-Zehnder interferometer. A camera with lower noise or more bits would allow detection with even shorter integration times. With the camera we used here, the reference intensity could not be increased further to increase the interferometric amplification due to the limited dynamic range of the camera.

To map the spatial extent of the optical field in the waveguide, and for comparison with the SEA TADPOLE measurements, we detected the phase and amplitude of the NSOM signal by heterodyne detection in a Mach-Zehnder interferometer, similar to Refs. [8, 21], see Fig. 1. The signal and reference beam were frequency shifted, by 80.0 and 80.04 MHz respectively, by acousto-optic modulators (AOMs). The NSOM signal was mixed with the reference in a 50:50 single mode fiber coupler, and the interference signal was detected on a photodetector. The reference beam was passed through a variable delay stage to overlap the NSOM signal and the reference pulse in time at the detector. The phase and amplitude of the signal from the detector was detected at the difference frequency between the two AOMs (40kHz) using a dual output lock-in amplifier. The phase of the signal can be used to obtain the refractive index by measuring the fringe spacing in the phase along the propagation direction, and thus the effective center wavelength of the propagating light [8].

For both interferometers, the two arms of the interferometer were approximately dispersion matched by inserting a length of fiber in the reference branch equal in length to the NSOM probe. This maximizes the signal in the Mach-Zehnder interferometer [21] and for the SEA TADPOLE detection it ensures that the maximum spectral phase difference is smaller than the upper limit for measurement, which is determined the spectral resolution of the spectrometer [26].

The sample was a SiON (refractive index n = 1.486) ridge waveguide on a Si substrate with an 8 μm thick SiO₂ thermal oxide layer. The waveguide had a width of 2.2 μm and a height of 850 nm. The waveguide supported only a single TE mode over the entire band width of the laser. The polarization of the laser was chosen parallel to the substrate, exciting only the TE mode. We modelled the effective refractive index of the waveguide for a number of wavelengths across the pulse spectrum using a mode solver based on the wave-matching-method [31]. The structure proved somewhat difficult to simulate reliably, as the mode has an effective refractive index close to that of the substrate. This is mostly a problem for calculating the dispersion of the structure since this is small.

3. Results and discussion

The spatial extent of the interference signal, as measured by the Mach-Zehnder interferometer, is show in Fig. 2(a), and the simultaneously acquired topography is shown in Fig. 2(b). The light is clearly confined to the waveguide. The width of the interferogram in the propagation direction is given by the bandwidth of the signal and the reference pulse and their relative dispersion, and is not a measurement of the actual pulse length [21]. The large height of the ridge waveguide made it necessary to use a low scan speed in the NSOM due to the fragile fiber probes, and therefore we measured with a fairly large spacing between the scan lines. This is clearly an issue for the measurements with heterodyne detection in the Mach-Zehnder interferometer, but not really for the SEA TADPOLE measurements since those do not require scanning, highlighting the usefulness of a method that does not require scanning a large area to obtain information. The spurious signal outside the waveguide in the bottom of the image is due to the fact that when moving onto the waveguide the tip is in close proximity to the edge of the waveguide, and since the fiber probe was uncoated, it picks up light propagating inside the waveguide also slightly above the end of the tip.

 

Fig. 2 a) The optical amplitude of the pulse in the waveguide, measured by heterodyne detection in the Mach-Zehnder interferometer. The white line is the amplitude across the center of the pulse. b) Topography of the waveguide. The colour scale is graded in μm. c) The interferogram from a SEA TADPOLE measurement in a single point on top of the waveguide. The interferogram is obtained in a single camera frame with an integration time of 70 ms. d) The retrieved phase (thick black line) from the interferogram in c), equivalent to the difference in spectral phase Δφ(ω) between the reference and signal field and the spectrum (thin red line) S(ω) of the signal field.

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Figure 2(c) shows the SEA TADPOLE interferogram from a single point on the waveguide. This demonstrates that we have an instrument that can measure the spectral amplitude and phase for a light pulse propagating in a photonic structure in a single point. The full optical field is obtained from a single camera image, in contrast to the measurements with the Mach-Zehnder interferometer that requires tens of minutes per frame. Here the integration time was 70ms, and the signal power from the NSOM probe was about 50–100pW. Figure 2(d) shows the phase retrieved from a Fourier transform along the pixel columns of the image (thick black line). This is the the difference in spectral phase Δφ(ω) between the reference and signal field. The spectral phase difference shows a slight tilt due to the fact that the reference and signal pulse arrived with a small time separation. We have corrected for the small linear phase term resulting from the frequency dependence of the fringe spacing, see Eq. 1 and [26]. Figure 2(d) also shows the spectrum of the signal calculated from the Fourier transform (thin red line).

To illustrate how this can be used to measure the group velocity, we made SEA TADPOLE measurements at an arbitrarily chosen reference point on the wave guide, and then at probe positions 2.5, 7.5, 12.5, 17.5, and 22.5 μm further along the waveguide. The difference in spectral phase between the reference point and the individual probe positions is plotted in Fig. 3(a). It is clear that the spectral phase has a first order term (the tilt of the phase), which increases with distance in a linear fashion. This illustrates that the pulse position shifts in time as the pulse propagates along the waveguide. There is almost no curvature in the spectral phase indicating that the amount of dispersion over these short distances is small, in agreement with modelling. The oscillations in the spectral phase are spectral fringes, from interference between the pulse propagating in the waveguide and an additional much weaker pulse. This is in fact how multiple pulses due to excitation of multiple modes can be detected with this method. Since the waveguide only supports one TE mode, however, we attribute the weaker pulse to scattering from the sample. Since the NSOM probe was uncoated, it is quite sensitive to scattered light, and the problem could probably be reduced by using metal-coated NSOM probes. The presence of the additional pulse was also seen as spectral fringes in the optical amplitude (data not shown).

 

Fig. 3 (a) The change in spectral phase between a reference point and probe positions 2.5, 7.5, 12.5, 17.5, and 22.5 μm further along on the waveguide. (b) The intensity of the electric field of the pulse in the time domain for the five probe positions and the reference point. The fifth pulse is drawn in a contrasting colour (red) to make the pulse structure easier to see. The variations in signal level and thus height of the pulses are due to unevenness of the sample surface. (c) The position of the center of the pulse versus probe position (crosses). The line is a linear fit to the data. The group index calculated from the slope of the fitted line is n_g = 1.51.

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The group delay τ_g(ω) can be calculated from the spectral phase as

However, since there are oscillations in the spectral phase, rather than fitting a polynomial to the data to extract the group velocity, we recovered information on changes in the pulse position in the time domain by Fourier transforming the spectral field constructed from the spectral amplitude and the phase differences [18]. The time-domain data is shown in Fig. 3(b). The position of the center of the pulse versus the probe position is plotted in Fig. 3(c). The data was in good agreement with a linear dependence of the pulse position on the probe position (solid line in Fig. 3(c)). The slope of the line gives the group velocity and the group index n_g of the waveguide. The retrieved group index is n_g = 1.51, which is somewhat higher than suggested by the model (1.465), but still in reasonable agreement considering the difficulties in modelling the dispersion accurately. The pulse does not reshape substantially, illustrating again the small dispersion on these length scales. Furthermore, since the pulse has not reshaped, the shift in the center position of the pulse is a good measure for the group velocity. Since the time-domain data in Fig. 3(c) was constructed from the spectral phase difference between two measurement points, rather than the actual spectral phase, the apparent width of the pulse does not correspond to the actual pulse width, which is probably much larger.

The change in spectral phase with probe position is related to the effective refractive index, and thus the phase velocity, at a specific frequency as

 

Fig. 4 (a) The spectral phase at 9 probe positions at 244 nm intervals. (b) The calculated refractive index as a function of wavelength. Each point is obtained from a linear fit to the 9 phases measured for each wavelength.

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In order to measure the effective refractive index, we must move the probe in increments smaller than the wavelength. This since the phase retrieved from the interferogram is the same for a spectral phase φ and φ + 2π. These measurements must be made faster than any drift in the interferometer, since the drift will shift the spatial fringes on the camera, and thus change the retrieved phase. This is less of an issue for measuring the higher order phase terms, such as for retrieving the group velocity. Fast drift within the 70 ms integration time of an individual camera frame will smear out the fringes, but slower drift is not a problem since drift between two frames does not matter at all, since this only changes the position of the fringes, not their tilt or curvature.

4. Conclusion

In conclusion, we have demonstrated that by combining NSOM with crossed beam spectral interferometry we enable a local measurement of the spectral phase and amplitude of light propagating in photonic structures. This corresponds to a characterization of the light propagation on an ultrafast timescale. We demonstrate the method by characterizing a broadband laser pulse propagating in a SiON ridge waveguide. We measure the refractive index as a function of wavelength, i.e. the dispersion, and the group index of the waveguide. The data is in good agreement with modelling. The method only requires measurements in the single point of interest and a reference point on the sample to measure the dispersion properties of the sample, and the spectral phase and amplitude is obtained in a single camera frame. This is in contrast to previous time-resolved interferometric NSOM methods that require scanning across a large area of the sample [8] or scanning the delay line [10]. This not only reduces the measurement time substantially, but is particularly useful where it is not possible to scan a large area as is the case for complex or non-periodic structures or characterization of highly localized fields. With this method, measurements of the higher order phase terms, related to group velocity and dispersion, are robust against drift in the interferometer, in contrast to scanning methods.

The fact that the spectral phase and amplitude is obtained in a single camera frame, in combination with that the method measures in a single point suggests that this method would be useful in experiments where the aim is to tailor the field on the nanoscale, or even in a single point, and on a femtosecond timescale [29].