Open Access
Add to BibSonomy
Abstract
Fast and slow light media exploit a steep frequency dependence in their index of refraction in order to advance or delay a modulated signal. Here we observe a qualitatively similar advance and delay from an optical phase-sensitive amplifier (PSA). Unlike in the case of slow and fast light, this effect is due to a redistribution of power between imbalanced signal sidebands, and the advance or delay is dependent on the optical phase of the input. The PSA adds energy and also changes the frequency spectrum of the input. We show that the advances and delays observed in a PSA implemented using four-wave mixing in a warm rubidium vapor are consistent with the expected behavior of an ideal PSA.
1. Introduction
The phenomena of fast and slow light has attracted considerable attention over the last few decades [1–3]. A modulated signal entering such a medium can be decomposed into a carrier and sidebands. In addition to any overall phase shift of all of the components, the sidebands can be phase shifted relative to the carrier resulting in an apparent signal advance or delay. This fast and slow light phenomenon raises interesting questions about the transfer of information, particularly in the quantum limited regime [4]. Here we present a system where a similar signal advance and delay can be observed, and which has the additional feature of redistributing energy between the sidebands. We examine a modulated light wave entering an optical phase-sensitive amplifier (PSA) and find that, depending on the sideband amplitudes and the optical phase of the carrier, the modulation can be advanced or delayed in a way reminiscent of that observed in conventional fast and slow light media. Unlike a conventional fast or slow light medium, the advance or delay in this case depends on both the optical phase and the exact sideband structure of the input signal.
Phase sensitive amplifiers are of interest because of their ability to noiselessly amplify or deamplify a signal [5,6]. The first demonstration of a continuous wave PSA was by Ou et al. [7] and since then PSAs have been shown to have potential utility in quantum communication [8], quantum information processing [9], and quantum imaging [10]. Due to their ability to beat the 3 dB noise figure limit [11] and to regenerate signals [12], developing a robust fiber-based PSA has become an area of active research in classical optical communication [13,14].
2. Model
The PSA under consideration has two strong pump beams and a weak probe beam as inputs, as shown in Fig. 1(a). The beams have optical phases of $\theta _{\textrm {pump1}}$, $\theta _{\textrm {pump2}}$, and $\theta _{\textrm {pr}}$ which contribute to the total PSA phase defined as
which determines the gain of the amplifier.
Fig. 1. (a) The phase-sensitive amplifier requires three inputs: two pump beams at optical frequencies $\omega _{\textrm {pump}1}$ and $\omega _{\textrm {pump}2}$, and a probe beam at an optical frequency $\omega _{\textrm {pr}}$. These three beams have phases $\theta _{\textrm {pump1}}$, $\theta _{\textrm {pump2}}$, and $\theta _{\textrm {pr}}$, respectively. Here we study the effect the PSA has on the probe beam. (b) The probe is amplitude modulated at a frequency $\Omega$ so that an intensity modulation can be observed on an oscilloscope. It is then split into a signal that is sent through the PSA and a reference that is not. After the PSA, the AC portions of the modulated intensity of the two beams are compared and the gain and delay of the signal beam are determined.
Here we study the output of the PSA when the weak probe has a slow amplitude modulation at a frequency $\Omega$ (Fig. 1(b)). The modulation creates sidebands at a frequency $\Omega$ above and below the carrier frequency. We examine below the effect of imbalance between those sidebands on the output of the PSA.
The electric field amplitudes of the input carrier, positive sideband, and negative sideband will be
The equations that describe the gain of an ideal phase-sensitive amplifier can be characterized by the gain matrix [15],
The carrier then is amplified according to
If the sideband amplitudes are assumed to be equal ($a_+=a_-=a$) and small, so higher order terms in $a$ can be ignored, the total input and output intensities are
If the input signal has one sideband only, so that $E_{\textrm {-,in}}=0$ in Eq. (2), it can be seen from Eq. (5) that the PSA will generate the other sideband so that both sidebands will be present in the output field. Furthermore, the output intensity of the PSA becomes
3. Experiment
The PSA was created via a four-wave mixing (FWM) process in a warm rubidium vapor cell in a similar way to that described in Ref. [10]. At low gains, this configuration has been shown to operate near the fundamental noise limits set by quantum mechanics. Figure 2(a) shows the atomic energy level diagram and the double-lambda scheme used in the FWM process. The probe ($\omega _{\mathrm {pr}}$) is either amplified via a photon from each pump ($\omega _{\mathrm {pump_1}},\omega _{\mathrm {pump_2}}$) being converted to two probe photons or deamplified via two probe photons being converted into two pump photons. The maximum gain of the PSA ($G_{\mathrm {PSA}}$) is determined by many parameters including the intensities in the pump beams, the one-photon detuning ($\Delta$), the two-photon detuning ($\delta$), the temperature of the cell, and the phase matching angle between the probe and pump beams in the cell. The data presented here was collected with the PSA tuned to a gain of $G_{\mathrm {PSA}}=2$.
Fig. 2. (a) The level diagram and relevant detunings of the $^{85}$Rb D1 line used in the phase-sensitive amplifier. The detunings used in this experiment are $\Delta = 1500$ MHz and $\delta = 4$ MHz. (b) The experimental setup used to create the PSA. AOM = acousto-optic modulator, PMF = polarization maintaining fiber, ND = neutral density filter, TA = tapered amplifier, BS = Beamsplitter (nonpolarizing), PBS = Polarizing Beam Splitter, PD = photodiode.
The experimental setup is shown in Fig. 2(b). All beams in the experiment are derived from a Ti:Sapphire laser that is tuned to the probe frequency. Each pump frequency is created by double-passing a high-frequency acousto-optic modulator (AOM) to frequency-shift a beam, one up 3040 MHz, one down 3040 MHz, in order to set the two-photon detuning relative to the ground state splitting of $^{85}$Rb. These two beams are then each amplified via a semiconductor tapered amplifier (TA), sent through single mode, polarization maintaining optical fibers (PMF$_2$, PMF$_3$) in order to spatially filter them, and crossed at an angle of 0.4$^\circ$ through the Rb cell to act as pump beams.
The signal beam is amplitude modulated at $\Omega =3$ MHz by diffracting out some light via an 80 MHz AOM. The undiffracted beam is sent through a single-mode polarization maintaining fiber (PMF$_1$). A variable neutral-density (ND) filter is placed right after PMF$_1$ in order to control the optical power going to the rest of the experiment. The sidebands on the probe are observed on an RF spectrum analyzer by picking off a small portion of the probe and beating it against another beam that has been frequency-shifted by a second 80 MHz AOM. Thus the carrier and sidebands are observed centered around a frequency of $\sim$80 MHz determined by the exact RF driving frequency of this second AOM. Due to a non-uniformity in the frequency response of the photodiode and amplifier used to detect the beat signal, measurements of relative sideband heights need to be normalized.
Right before the Rb cell the probe beam is split into two beams, one to be used as a reference which falls directly on a photodiode, the other is sent through the Rb cell as the signal beam. The AOM that is modulating the probe beam is aligned to minimize phase modulation using the same technique described in Ref. [16].
The signal and pump beams are combined on a polarizing beam splitter (PBS) and the signal beam is aligned to bisect the pump beams at the center of the cell in order to satisfy the phase-matching condition. After the Rb cell the pump beams are filtered out using another PBS and the probe beam is collected on an identical detector as the reference beam.
The probe beam is amplitude modulated at 3 MHz with a 20% modulation depth. The probe has an optical power of 80 µW and a $1/e^2$ diameter of 250 µm at the center of the Rb cell. Each pump beam has an optical power of 100 mW and a $1/e^2$ diameter of 550 µm at the center of the Rb cell.
The 80 MHz AOM that is modulating the probe beam at 3 MHz has frequencies of 77 MHz, 80 MHz, and 83 MHz all present in the driving RF signal. Since different diffraction angles result from different driving frequencies, a spatial dependence in sideband amplitude would be expected in the first order beam. If the beam has a curved wavefront inside the AOM, each portion of the beam would be optimally diffracted at a different driving frequency. Therefore a spatial dependence to the sidebands would be imparted even on the zeroth order beam. This spatial dependence can be exploited to imbalance the sidebands of the signal beam by slightly misaligning the the input to PMF$_1$. As the horizontal control of the fiber alignment is adjusted, the relative heights of the sidebands as measured on the spectrum analyzer can be changed. The variable ND filter after PMF$_1$ is adjusted so that the optical power going to the experiment is the same for each fiber alignment.
The oscilloscope records the modulated signal from both the signal and reference beams. Sinusoids are fitted to each and the gain of the PSA is determined by comparing the amplitudes of the signal and reference. The advance or delay of the signal relative to the reference beam is determined by comparing the phases of the two fits.
4. Results
Data is collected by letting the phases of all three beams drift and sampling one millisecond long time traces. This is a short enough time so that the phases do not noticeably drift during a trace. Gain is plotted versus delay, and over time the phases of the probe and pumps drift randomly over all possible PSA phases and a plot like those shown in Figs. 3(a)–(d) builds up. Notice that for a given sideband imbalance, the intensity modulation envelope may be either advanced or delayed, depending on the optical phase of the carrier relative to the pumps. In general there are two choices of optical phase that give the same gain but correspond in one case to modulation advance and in the other to modulation delay. The gain versus delay plots and corresponding spectra for four different sideband imbalances, adjusted by changing the alignment of PMF$_1$, are shown in Fig. 3. The maximum gain of the PSA does not change, only the delay and advance of the signal.
Fig. 3. (a-d) Gain and delay data taken for four different alignments of PMF$_1$. The first alignment is when the beam is coupled into PMF$_1$ optimally; there is minimal delay and minimal sideband imbalance. The next three alignments are such that one sideband is coupled-in more than the other, therefore producing an apparent advance or delay in the signal. (e-h) The unnormalized spectrum of the light after PMF$_1$ is shown underneath its corresponding gain versus delay plot. For each subsequent fiber alignment, the sideband imbalance increases and the PSA produces a larger apparent advance and delay. The gray dashed lines are at the peak of each sideband to illustrate the imbalance. The carrier peak has a height of -34 dBm for each case. Since the advance and delay in plot (a) was minimized, the slight sideband imbalance in (e) is assumed to be due to a nonuniform frequency response of the detection system. For subsequent comparison of experimental results to the model in section 2, the normalization factor derived by assuming these sidebands to be equal is applied to all sideband measurements. The normalized sideband imbalance, $s=a_+/a_-$, from the model described in Sec. 2, is indicated in each panel.
For each gain versus delay plot shown in Fig. 3, the model described in Sec. 2 is used to predict the gain versus delay plot expected from a PSA with a maximum gain of $G=2$ and the corresponding measured sideband imbalance. The data and corresponding model prediction for each trial is shown in Fig. 4. We find that the observed phase dependent advances and delays are consistent with the operation of an ideal PSA, as assumed in the model in Sec. 2. If the PSA itself were to have a differential gain or loss for the two sidebands an advance and delay would be expected even in the case of balanced input sidebands.
Fig. 4. Gain and delay data (points) from the four cases in Fig. 3 are shown with their corresponding prediction (solid lines). The prediction for each different data set was generated by using the normalized sideband imbalance, $s=a_+/a_-$, in the model described in Sec. 2.
While the data presented above examines the phase advance and delay for the cw modulation of a beam, we have observed similar gain versus delay plots for a probe beam modulated with a 500 ns full-width half-maximum gaussian pulse having a continuum of sidebands within the gain bandwidth of the PSA. We used pulses that explicitly had their frequency bandwidth within the gain bandwidth of the PSA for this because we anticipate that, while broader-bandwidth pulses could be used, the frequency components outside of this bandwidth would not experience either the same gain, or a similar phase-dependent advance or delay. Thus, we would expect that a pulse would experience severe distortions under these conditions.
5. Conclusions
We have presented observations of a modulated signal beam passing through a phase-sensitive optical amplifier that display an advance or delay of the modulation envelope that resembles that produced by passage through a fast or slow light medium. Instead of only imparting phase shifts on the sidebands, however, the apparent advance and delay from a phase-sensitive amplifier is a result of the amplification and redistribution of power between unbalanced sidebands and depends on the optical phase of the carrier. This changes the spectrum of the amplifier output relative to that of the input signal. We have shown that data taken using an optical phase-sensitive amplifier implemented using four-wave mixing in warm rubidium vapor matches well to the theoretical predictions of a simple model assuming an ideal phase-sensitive amplifier. We would like to emphasize that the present observations are of small advances and delays on relatively slow modulations or broad pulses, but the effect could be more noticeable in high-gain or broadband amplifier systems, such as fiber amplifiers [12,13] and could potentially distort signals. Since our implementation of a phase-sensitive amplifier has been shown to operate near the noise limits set by quantum mechanics [10], an interesting future avenue of work would involve studying how information propagates through this phase sensitive amplifier, in a similar way to that presented in Ref. [4] for conventional fast and slow light media.
Funding
Air Force Office of Scientific Research (FA9550-16-1-0423).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. R. W. Boyd, “Slow and fast light: Fundamentals and applications,” J. Mod. Opt. 56(18-19), 1908–1915 (2009). [CrossRef]
2. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in ultracold atomic gases,” Nature 397(6720), 594–598 (1999). [CrossRef]
3. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301(5630), 200–202 (2003). [CrossRef]
4. J. B. Clark, R. T. Glasser, Q. Glorieux, U. Vogl, T. Li, K. M. Jones, and P. D. Lett, “Quantum mutual information of an entangled state propagating through a fast-light medium,” Nat. Photonics 8(7), 515–519 (2014). [CrossRef]
5. J. A. Levenson, P. Grangier, I. Abram, and T. Rivera, “Reduction of quantum noise in optical parametric amplification,” J. Opt. Soc. Am. B 10(11), 2233 (1993). [CrossRef]
6. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26(8), 1817–1839 (1982). [CrossRef]
7. Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Quantum Noise Reduction in Optical Amplification,” Phys. Rev. Lett. 70(21), 3239–3242 (1993). [CrossRef]
8. J. A. Levenson, I. Abram, T. Rivera, P. Fayolle, J. C. Garreau, and P. Grangier, “Quantum Optical Cloning Amplifier,” Phys. Rev. Lett. 70(3), 267–270 (1993). [CrossRef]
9. S. Fossier, E. Diamanti, T. Debuisschert, R. Tualle-Brouri, and P. Grangier, “Improvement of continuous-variable quantum key distribution systems by using optical preamplifiers,” J. Phys. B: At., Mol. Opt. Phys. 42(11), 114014 (2009). [CrossRef]
10. N. V. Corzo, A. M. Marino, K. M. Jones, and P. D. Lett, “Noiseless Optical Amplifier Operating on Hundreds of Spatial Modes,” Phys. Rev. Lett. 109(4), 043602 (2012). [CrossRef]
11. W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett. 35(22), 1954–1955 (1999). [CrossRef]
12. K. Croussore and G. Li, “Phase Regeneration of NRZ-DPSK Signals Based on Symmetric-Pump Phase-Sensitive Amplification,” IEEE Photonics Technol. Lett. 19(11), 864–866 (2007). [CrossRef]
13. Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011). [CrossRef]
14. T. Umeki, T. Kazama, O. Tadanaga, K. Enbutsu, M. Asobe, Y. Miyamoto, and H. Takenouchi, “PDM Signal Amplification Using PPLN-Based Polarization-Independent Phase-Sensitive Amplifier,” J. Lightwave Technol. 33(7), 1326–1332 (2015).
15. G. Grynberg, A. Aspect, and C. Fabre, Introduction to quantum optics: from the semi-classical approach to quantized light (Cambridge University, 2010).
16. T. Li, B. E. Anderson, T. Horrom, K. M. Jones, and P. D. Lett, “The effect of input phase modulation to a phase-sensitive optical amplifier,” Opt. Express 24(17), 19871 (2016). [CrossRef]
