1. Introduction

The control of velocity of light is a foundation of both fundamental physics and optical applications. Superluminal or negative group velocity, that is fast light, has been attracting interest since this effect is seemingly against causalities [1–3]. Slow light has shown its potentialities in many optical applications, including optical buffer in optical data manipulation, switching and nonlinear optics, etc [4,5]. One method to control the group velocity of light is to use material resonances. Fast light appears in sharp absorption lines of atomic vapor, or in anomalous dispersions induced by gain-assisted resonance lines [1,2]. Slow light also appears in atomic systems. Specifically, in an electromagnetically induced transparency (EIT) system, quantum interference induced by a strong coherent light creates ultra fine spectral structures in refractive index, and the resultant linear dispersion has been used in the drastic reduction of the light velocity [6]. This technique has been developed for coherent optical information storage or the freezing of light [7,8]. An alternative and most promising approach for dispersion engineering may be material engineering to fabricate microstructures. Velocity of light can be slowed down drastically near the band edge in photonic crystals and coupled resonator optical waveguides [9,10]. Fast and slow light has also been observed in microspheres and microrings coupled with waveguides. Furthermore, with a system involving coupled resonator-induced transparency consisting of two microspheres, the classical analogy of extremely slow light in EIT has been demonstrated [11]. A new theoretical mechanism for stopping light as a classical analogy to EIT has been proposed using a waveguide side coupled to optical resonators [12]

A challenging extension of fast and slow light techniques may be all optical advancement and delay of two-dimensional optical images. All-optical delay of images has been realized using a relatively flat dispersion between two absorption lines in Cs atoms [13]. Furthermore, storage and retrieval of a transverse image has also been demonstrated in an atomic vapor using a technique based on EIT [14,15]. Experiments so far reported on the fast and slow light in photonic structures may be restricted to single transverse mode lasers. Here, we report optical image propagation experiments in a structured dispersion, developing our previous experiments on the fast and slow light propagation in a microsphere system [16]. We employ a Fabry–Perot resonator in reflection geometry as a parallel processing system to propagate two dimensional optical images. Two dimensional images encoded in pulses of 32ns were stored in the image resonator, and either advanced, −6.0ns, or delayed, 10.9ns, using the relevant dispersions relation. Compared with material dispersions in atomic resonances, structural dispersions have significant advantages. The photonic structures are free from

Doppler motion, which can degrade the image quality [14,15]. In contrast to the atomic system, photonic structures have more flexibility in terms of design space, and hence the available frequencies are not restricted to the intrinsic and very limited discrete atomic transitions. For example, in a color image, we need at least three colors (RGB). In the EIT system, optical information are stored in the ground level sublevel coherence, while in the image resonators, images are stored as optical waves. It could preserve not only the phase information but also all quantum characteristics of images, such as squeezing and entanglement.

2. Modeling

Figure 1 shows a schematic illustration of the experimental setup. For the image resonator, we employ a Fabry–Perot cavity consisting of two mirrors with reflectivity $R_i=r_i{r}_i*=r_i\text{'}{r}_i\text{'}*$, where $i=1,2$for the first and second mirrors, respectively, in a reflection geometry. Note that this configuration is topologically homotopy to our previous experimental setup of microsphere fiber-taper system for the observation of fast and slow light [16]. The electric field of the incoming and outgoing two-dimensional image is represented as $A_0(x,y,\nu )$and $A(x,y,\nu )$, respectively, where ν is frequency, and the x and y axes describe the image plane. The reflected image from the resonator is then

 

Fig. 1 Schematic illustration of image propagation through an image resonator in reflection geometry. IR is the image resonator, EO is an electro-optic modulator, M is a mirror, HM is a half mirror, and IICCD is image-intensifier CCD-array camera

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Figure 2(a) and (c) show calculated reflection and transmission intensities using Eq. (2), and 2(b) and (d) are the relevant phase-shifts as a function of the detuning frequency, respectively. In the transmission geometry (Fig. 2 (d)), the derivative of the phase-shift is always positive as the frequency increases; hence, normal dispersion and the relevant slow light are expected. In the reflection geometry, we obtain both anomalous and normal dispersions depending on $R_1$and $R_2$. In the under-coupling condition, i.e., $R_1$>$R_2$, a negative gradient appears in the dispersion curve within the resonance dip. Such a slope results in anomalous dispersion, and fast light is expected. On the other hand, under the over-coupling condition, i.e., $R_1$<$R_2$, the phase increases monotonically as the frequency of the laser light is increased. In the transmission geometry, every time the frequency increases across the resonance, the transmitted phase-shift jumps by π radians, whereas in the reflection geometry, under the over-coupling condition, this phase jump is $2\pi$ radians. The group delay time for a wave packet to pass the cavity is readily calculated as$\tau =\partial \theta (\omega )/\partial \omega$.

 

Fig. 2 Calculated, (a) reflected and (c) transmitted intensities as a function of the detuning frequency. (b) The reflected and (d) transmitted phase-shift as a function of the detuning frequency. The parameters used in the calculations were $R_1$ = 0.8, L = 0.25m, and $R_2$ = 0.65, 0.79, and 0.90, for the solid (red), dashed (green), and dotted (blue) lines, respectively. The detuning frequency is normalized such that the free spectral range is equal to π.

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Next, we analyze the propagation of the image through the resonator on the basis of Eq. (1) and take the diffraction effects into account. Figure 3 shows examples of numerical calculations of the images of double rectangular slits propagated through the resonator. Figure 3(a) and (b) are the transmitted images at the on-resonance ($\varphi (\nu )=0$) and off-resonance ($\varphi (\nu )=\pi /2$) conditions, and Fig. 3(c) and (d) are the reflected images, in the under-coupling condition, at the on-resonance and off-resonance frequencies, respectively. It can be seen that, overall, the images are clearly propagated through the image resonator. In Fig. 3(a), the edge region of the transmitted image is slightly rounded, which is attributed to diffraction. Under the off-resonance condition, shown in Fig. 3(b), striped structures appear in the slit patterns, also attributed to diffraction. The reflected image in Fig. 3(c) shows that the edge region of the slits is increased while the intensity of the central regions is decreased. Figure 3(e) shows the reflected image under the critical-coupling condition, i.e. $R_1=R_2$ [16]. The reflected intensity tends to zero at the resonance frequency, as the direct and multiple reflected components cancel. The edge regions of the image are, however, subject to the diffraction effect, and the phase changes rapidly as a function of position. The destructive interference is not perfect, which results in the edge filtering effect. A similar effect appears in the transmission geometry at the off-resonance frequency when the mask size matches with the diffraction effect.

 

Fig. 3 Numerical simulations of images propagated through the image resonator. (a) Transmitted image under the on-resonance condition and (b) transmitted image under the off-resonance condition. (c) Reflected image under the on-resonance condition and (d) reflected image under the off-resonance condition. In (a)–(d), $R_1$ = 0.9, $R_2$ = 0.7, and L = 0.2m. (e) Reflected image under the critical-coupling (on-resonance) condition; $R_1$ = 0.7, $R_2$ = 0.7 and L = 0.2m. (f) The double rectangular slit pattern, with dimensions of 0.0275 × 0.04 m. The white lines in each figure show the horizontal cross-section.

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3. Experiments and results

Now we move on to the combined experimental demonstrations of image propagation and the temporal advancement and delay. We used a 532-nm second harmonic of a continuous wave Nd³⁺ Yttrium Aluminum Garnet (YAG) laser with a linewidth of 1kHz as a light source. The frequency of the laser light was thermally tuned by the cavity length control, and the mode-hop-free tuning range was over 10GHz. Almost Gaussian transform-limited temporal pulses were generated using an electro-optic (EO) modulator. The duration of the pulses was $t_p$=32ns, the EO modulator was operated at a repetition rate of 100kHz, and the incident power was 10 mW. The laser beam was expanded using two lenses (not shown in Fig. 1) to a diameter of 3 cm, and an amplitude mask in the shape of “2” was inserted into the collimated beam. This pattern was selected because it has both straight and curved lines. The transmitted and reflected images were detected using a cooled charge-coupled device (CCD) array camera with an image intensifier to achieve single-photon-counting level detection. For measurement of the temporal profiles of the transmitted and reflected images, we used a photomultiplier tube and a digital storage oscilloscope. In some of the measurements, a streak camera with a time resolution of 10ps was also used to confirm the time responses.

The images were propagated through the resonator, and are shown in Fig. 4 ; Fig. 4(a), (b), (g), and (h) are transmitted images, whereas Fig. 4(c) and (d) are reflected images under the over-coupling condition (slow image on the resonance), and Fig. 4(e) and (f) are reflected images in the under-coupling condition (fast image on the resonance). We see that all the images are propagated through the resonator clearly. The transmitted image is bright under the on-resonance condition, whereas the reflected image is bright under the off-resonance condition. Some characteristics seen in the simulated data (shown in Fig. 3), such as the fine striped strictures due to diffraction, are also seen in the experimental data. In Fig. 4(g) and (h), the image size was reduced ( ×0.5) so that the diffraction effect is enhanced, and we observe that the edge region of the image is enhanced in the transmitted image under the off-resonance condition. This effect may be used as an edge filter. In Fig. 4(i), the propagated image was fed into an image-intensifying CCD array camera, and observed in a photon-counting mode. The laser beam was attenuated significantly using filters so that the average photon number inside the cavity is of the order of less than one. Images consist of many pixels. We may detect very weak images using sophisticated photon counting technologies, and the detected signal can be electrically delayed. However, in this process, much information is irreversibly lost at the time of photon counting. On the other hand, the experimental result shown in Fig. 4(i), indicates that even very weak optical image of the order of a single photon energy level inside the cavity can be advanced or delayed, preserving information on the image as the superposition of all possible pixels or transverse modes.

 

Fig. 4 Experimental observation of images propagated through the image resonator. (a), (b), (g), and (h) are the transmitted images, and (c), (d), (e), and (f) are reflected images. The top row and (h) are observed under the on-resonance condition;, the second row and (g) are under the off-resonance condition. In (a), (b), (c), and (d), $R_1$=0.85, $R_2$=0.95 (over coupling), L=0.3m. In (e) and (f) $R_1$=0.7, $R_2$=0.6 (under coupling), L=0.3m. In (g) and (h), the mask was replaced with a reduced sized mask. In (i), the image propagated through the image resonator was detected in photon-counting mode.

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Figure 5(a) shows temporal profiles of the reflected images. The origin of the time axis, i.e., τ = 0, is taken at the pulse peak observed at the off-resonance frequency. Note that the profiles are normalized to see the advancement or delay. The pulse peak of the image arrives at ${\tau }_{Run}=$−6.0ns in the under-coupling condition, whereas the peak arrives at ${\tau }_{Rov}=$+10.9ns under the over-coupling condition. This experimental observation, therefore, indicates that we can observe both advanced and delayed images, i.e. fast and slow images in a single image resonator system by controlling the coupling strength. Figure 5(b) shows the temporal profile of the transmitted image, where the pulse peak is delayed by ${\tau }_T=$+8.1ns. The observed delay times show good agreement with calculations from Eq. (2); the simulated times were ${\tau }_{Run}=$−5.2ns, ${\tau }_{Rov}=$+11.5ns and ${\tau }_T=$+5.6ns. The temporal profiles remained almost identical when the mask pattern was removed.

 

Fig. 5 (a) Experimental observation of the temporal profiles of the reflected images. The solid black line is far from resonance, the dotted red and dashed blue lines are the under-coupling ($R_1$ =0.85, $R_2$ =0.75, L =0.3m) and over-coupling ($R_1$ =0.85, $R_2$ =0.95, L =0.3m) conditions, respectively. (b) Temporal profiles of the transmitted pulses. The solid line is the off-resonance condition (superluminal), and the dashed green line is the on-resonance condition. ($R_1$=0.85, $R_2$=0.95, L=0.3m) The inset shows the position where the temporal profile was observed.

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As an alternative approach for the analysis of the diffraction, we may employ a Fourier transform analysis of the image. We denote the Fourier transform of incoming image to be $A_0(k_x,k_y,\nu )$ and that of the outgoing image be $A(k_x,k_y,\nu )$. The propagated image can be represented as $A(k_x,k_y,\nu )=T(k_x,k_y,\nu )A_0(k_x,k_y,\nu )$, where $T(k_x,k_y,\nu )$ is the amplitude transfer function of an image cavity, similar to that of imaging lens system [17]. We can also define the bandwidth of $T(k_x,k_y,\nu )$ and calculate the temporal and spatial bandwidths for the cavity used to generate the images shown in Fig. 4(a) and (b) as $\Delta \nu =$15.2 MHz and $\Delta k=$4.0×10³ m⁻¹, respectively. Typical bandwidths of the injected image are $\delta \nu =$0.441$/t_p$ = 13.8MHz, where $t_p=$32ns, and $\delta k=$3.1×10³ m⁻¹. Therefore, the main temporal and spatial components of the image lie within the bandwidth of the image resonator used. This means that the temporal and spatial profiles do not suffer significant broadening or deformations [18], in good accordance with the experimental results shown in Fig. 4(a)–(d) and Fig. 5(a) and (b).

4. Conclusion

In summary, we have both modeled and experimentally demonstrated image resonator storage of two-dimensional images, and the image can be either advanced or retarded using the relevant dispersion relation. Note that the delay -bandwidth product, which is the delay time measured in units of pulse duration, may be restricted less than one, as long as we use the single stage static resonator. The single resonator is a basic unit as is a single atom in gaseous system, hence we may develop linear array of coupled image resonator structures to achieve a large delay-bandwidth product. Alternative approach to achieve the break-through of the fundamental trade off between the bandwidth and the optical delay may be dynamical tuning of the photo structures [19]. The diffraction effect, which could be one of the fundamental limits for image resonators, may also be overcome by introducing waveguide geometries. One possible structure is a bundle of optical fibers imaging plates. Alternative structures include fiber Bragg gratings or three-dimensional photonic crystals. The image resonators, as well as other photonic structures to store images, are useful in image processing, including image buffer, image filter, nonlinear imaging and image sensors.