To determine the slow light properties of the crystal, the amplitude and phase responses are needed.
,out prob
E G E (6.1) where Eprob(ω) is the frequency domain amplitude of the probe field, in our case a Gaussian pulse with 1 μs full-width half maximum (FWHM)
2 0 0, exp / prob I t I t ,
1 2 ln 2 s , (6.2)
2 0 0 0 0, exp / / 2 , prob E t E t E I (6.3)
1
' ' ', 2 i t prob prob E E t e dt
(6.4) and G(ω) is the complex frequency domain amplitude response function
,
i
G T e (6.5) If the intensity transmission spectrum is known, the phase of the response function can be calculated from the Hilbert transformation (so-called Hilbert phase):
1 ln
' '. ' T d
(6.6) The last formula is equivalent to Kramers-Krönig relation as discussed in156 and is quite straightforward to implement numerically to estimate the slow light effect for an arbitrary spectral hole shape.Fig. 6.2 shows the amplitude response functions for several different spectral hole filters used in our experiment along with the spectrum of the Gaussian input pulse used to illuminate the tissue sample. Here red, green and blue correspond to the transmission spectra T
for spectral hole burned with 1 MHz, 500 kHz and 250 kHz scanning respectively, and the black curve is the Gaussian pulse spectrum. As can be seen only the 1 MHz spectral hole passes the entire spectrum of the Gaussian pulse. However the group delay time (or light speed), which is related to the slope of the Hilbert phase ∆θ(ω), calculated with equation (6.6) (circles in Fig. 6.2) is greatest for the narrowest spectral hole. This time-bandwidth tradeoff is familiar in slow light and is optimized by using an optically thick material.Time domain intensity profile at the output of the Pr:YSO crystal was calculated by:
2
2 , i t out in I t E t
E G e d (6.7) Fig. 6.3 shows this calculated time response for the spectral holes in Fig. 6.2, using the same color coding (red for 1 MHz, green for 500 kHz, and blue for 250 kHz). Again the black curve corresponds to the Gaussian input pulse.Figure 6.2 | Spectral hole shape and corresponding phase change. Black dashed line,
amplitude spectrum of input Gaussian pulse; red, green and blue lines correspond to transmission spectrum of spectral hole burning for 1 MHz, 500 kHz and 250 kHz respectively; red, green and blue circles correspond to phase change over frequency with 1 MHz, 500 kHz and 250 kHz burning.
2. Compare Slow Light Theory to Experiment
To verify that these calculated slow light delays are correct for diffuse light we performed an experiment, as illustrated in Fig. 6.1a. Here, diffuse light was generated by back-illuminating a sample (opposite side of the PSHB crystal) with laser light and the scattered light emerging from the front of the sample was directed through a Pr:YSO crystal into a photomultiplier (PMT) detector. Prior to the light slowing experiments, spectral holes were engraved into the Pr:YSO crystal using pulse sequences similar to those by different group102,157. In particular these spectral holes were burned using a very slow scanning speed (10 kHz/μs) to avoid coherent radiation158. The resulting spectral hole transmission profiles are the actual traces shown in Fig. 6.2. The sample used to generate
Figure 6.3 | Slow light effect with PSHB. a, Experimental data for 1 μs FWHM
Gaussian pulse propagation through spectral holes burned with 1 MHz, 500 kHz and 250 kHz chirping. b, Numerical simulation for Gaussian input pulse delay with different hole burning width.
diffused light is a tissue phantom of dimensions (x×y×z = 25×70×70 mm3) where x is along the light propagation direction. It is made from 10% porcine skin gelatin and 1% Intralipid (reduced scattering coefficient μs‘ = 10 cm-1). The diffuse light emerging from this phantom was then directed into the Pr:YSO SHB crystal with dimensions (x×y×z = 12×10×10 mm3), and the resulting slowed light was detected with the PMT.
The slowing of diffused light is shown in Fig. 6.3a. As seen, there is good agreement between these experimental delays and the simulation results in Fig. 6.3b. After passing through the crystal, the Gaussian pulse spread increased from 1 μs FWHM to ~2 μs FWHM even for the longest 17 μs delay, meaning this slowing technique can be used to make UOT images with minimal distortion. Note that the ‗time-delay bandwidth product‘ is large in Fig. 6.3 (up to 7), as expected based on the ~33 dB optical depth of the crystal159.
3. UOT Signal Bandwidth Estimate
In UOT experiment, the optical illumination should be matched to the ultrasound pulse duration to minimize laser exposure. To this end Fig. 6.4 shows the bandwidth calculation for a 2-cycle ultrasound pulse at an ultrasound frequency of 2.3MHz. The ultrasound pulse temporal shape is shown in Fig. 6.4a. The sound pressure amplitude spectrum produced by this pulse is shown in Fig. 6.4b. As can be seen the 2-cycle pulse gives a sound pressure bandwidth of 1.6 MHz FWHM. The spectrum of the ultrasound modulated light is close to that of the ultrasound pressure except the peak is slightly shifted (<10%) toward lower frequencies, but the linewidth is nearly unchanged 160.
To ensure a linear response between UOT signal and its corresponding slow down part, the illumination pulse was chosen to be much longer than the 2-cycle ultrasound pulse because of limited time-delay bandwidth product. Of course in actual applications, these could be more closely matched to reduce laser exposure. In particular we chose a 10 μs FWHM super-Gaussian pulse (of order 3). This illumination pulse has a ~80 kHz bandwidth and is therefore negligible.
Fig. 6.5 shows UOT signals for 2-cycle ultrasound pulses filtered through different spectral hole widths including the same as used above (red, green , blue) = (1 MHz, 500 kHz, 250 kHz) and an additional spectral hole width of 100 kHz (pink curve). As before, the black curve shows the super-Gaussian input pulse. A new curve, the light blue curve shows the UOT signal when no ultrasound is applied. This is the first key result of this chapter. From this blue curve it is seen that there is considerable light leakage during the on-time of the illumination pulse. However at later times, when the
delayed UOT signals appear (for example the pink one corresponding to 100 kHz hole burning), there is negligible background noise. Thus slow light has successfully eliminated the loss of filter discrimination due to polarization leakage and any other light leakage.
Also from Fig. 6.5 it is seen that the narrower spectral hole gives more delay for the UOT signal but at the expense of lower transmission. This is in agreement with other slow light experiments. Here we note that even though the light and ultrasound pulses were synchronized, the ultrasound had to propagate a distance of ~7 cm from the transducer to the focus. Assuming a sound speed in water is 1.5 mm/μs, the ultrasound pulse reaches its focus with a delay of 47 μs. This time delay was built into the trigger pulse for the laser illumination so that the ultrasound arrived at the center of the laser illumination pulse.