Abstract

Configurable narrow bandwidth filters are indispensable components in optical communication networks. Here, we present an easily-integrated compact tunable filtering based on polarization-coupling process in a thin periodically poled lithium niobate (PPLN) in a reflective geometry via the transverse electro-optic (EO) effect. The structure, composed of an in-line polarizer and a thinned PPLN chip, forms a phase-shift Solc-type filter with similar mechanism to defected Bragg gratings. The filtering effect can be dynamically switched on and off by a transverse electric filed. Analogy of electromagnetically induced transparency (EIT) transmission spectrum and electrically controllable group delay is experimentally observed. The mechanism features tunable center wavelength in a wide range with respect to temperature and tunable optical delay to the applied voltage, which may offer another way for optical tunable filters or delay lines.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Sharp optical dispersion at resonances in atomic or photonic systems has great applications, but the associated absorption inevitably deteriorates the transmission. Electromagnetically induced transparency (EIT) effect provides a profound mechanism for realizing steep dispersion within a rendered transparent window in the otherwise opaque resonance, opening great opportunities in all-optical buffering, processing, optical storage, and enhanced nonlinear optics [1–4]. To overcome many of the limitations of EIT on extreme experimental conditions, bandwidth, and decoherence, its classical counterpart or equivalent phenomenon in optics, i.e., EIT-like effect, has been proposed in a variety of systems and based on different mechanisms. Passive structures, such as phase-shifted Bragg gratings [5], photonic crystals (PhCs) [6], coupled-resonator optical waveguides (CROWs) [7], coupled whispering-gallery-mode (WGM) microresonators [8, 9], exhibit EIT-like transmission spectra either by introducing a defect mode in the bandgap or by interfering two closely located optical resonances. However, their operational wavelength and delay tunability requires precise control of resonances, mostly via temperature or mechanical control. Novel mechanisms in active systems, such as optomechanically induced transparency (OMIT) [10,11] and Brillouin scattering induced transparency (BSIT) [12], also exhibit extremely narrow EIT-like spectra. Because optomechanical/acoustical modes have a much smaller frequency than light. The EIT-like spectrum features a very narrow window, where enhanced dispersion associated with it is exploited for slow or even storage of light [13]. However, they only operate at a very narrow frequency range (MHz or less) and lack of wideband wavelength tunability. Still, these devices suffers from complexity in controlling. Practical issues, like system complexity, limited bandwidth, operational wavelength tunability, and time delay controllability, still hinder them from direct applications in optics. Thus, there is an imperative to address all of these issues.

Electro-optic (EO) effect in periodically poled ferroelectric crystals has been widely used for polarization coupling using quasi-phase matching (QPM) schemes. For example, periodically poled lithium niobate (PPLN) under transverse electric field mimics that of a folded Solc filter, and has been utilized as narrow-band tunable filters [14], ultra-precision polarization generator [15], and optical delay lines [16]. Moreover, PPLN with a defect has also been proposed to realize optical diodes [17]. Similar to the configuration in phase-shifted Bragg gratings, a defect embedded PPLN (placed in between two orthogonal polarizers) has also been reported to manifest an EIT-like transmission spectrum, as reported in [18] and [19]. However, with a low number of domains, the observed dual-wavelength filter had a relatively wide bandwidth. Modified dispersion and group delay was theoretically predicted but too small to be experimentally observed. Besides, the use of bulky crystal also requires high driving voltage. Whereas, for EO applications, there is a high demand to achieve low voltage drive, either by using thinned crystals or waveguide structures [20].

Here, Solc-type filtering based on polarization coupling in a thinned PPLN crystal is experimentally investigated. With only 0.3 mm in the transverse dimension, the half-wave voltage is reduced to below 100 V. Analogy of EIT transmission spectrum in the PPLN based Solc-type filter in a reflective geometry is observed. The mechanism features tunable central wavelength in a wide range versus temperature and controllable optical delay versus the applied external electrical field. An advantage is that the transmission (and dispersion) can be controlled by the applied electric field. The device is dynamically addressed to obtain different tunable group velocity with near unity of transmission. The proposed scheme represents a significant step towards reconfigurable optical delay lines for optical communications, processing, and computing technology.

2. Theoretical model

The polarization evolution of light in PPLN under transverse EO effect can be modeled using Jones matrix. We define the coordinate system as: the light propagation direction is along the X axis of the PPLN, a transverse electric filed is applied along the Y axis, and Z axis is the optical axis (c axis). The transfer matrix of the i-th domain is then expressed as

Mi=R(θ)WiR(θ)=[cosθsinθsinθcosθ][eiδϕ/200eiδϕ/2][cosθsinθsinθcosθ].

Here, θ is the rotation angle of the optical axis about the X axis in the presence of a transverse electric field (Y direction), and θγ51E/(1/ne21/no2), where E is the transverse electric field intensity, γ51 is the EO coefficient, and ne, no are the extraordinary and ordinary refractive indices, respectively. For positive and negative domains in PPLN, there is a sign flip in θ due to the modulation of the EO coefficient by domain reversal. δϕ = 2π(neno)d/λ is the phase retardation in one domain, where d is the domain length and λ is the wavelength in vacuum. For wavelengths that experience λ phase shift in one domain, the required d is the same with that under the QPM condition. The transmittance matrix of the device is given by T=Πi=1NMi, where N is the number of domains that the optical wave propagates through. Then, the final transmittance can easily be obtained.

3. Experiment and discussion

The experimental setup is depicted in Fig. 1(a). Light from a tunable laser (telecom C band) is modulated by a lithium niobate (LN) amplitude modulator (10 GHz), before passing through a 10/90 coupler. A polarization controller (PC) is used to adjust its polarization. A circulator is used to divert the reflected from the PPLN to another port (the output port). The device is composed of a thin PPLN, a temperature controller, and an in-line polarization beam splitter (PBS), as marked by the dashed rectangle. A segment of polarization maintaining (PM) fiber is used in between the two components (Blue line). A photograph of the device is given in Fig. 1(b). An amplified spontaneous emission (ASE) source is launched into the device through the coupler when measuring the transmission spectrum. The other arm from the 10/90 coupler is used as reference during the transmission and group delay measurement.

 

Fig. 1 Phase-shifted Solc-type filter based on PPLN in a reflective geometry. (a) Schematic of the experimental setup. (b) Photograph of the device components, compared with a coin. (c) Beam profile after passing the PPLN in the far field.

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The dimension of the PPLN sample is 25(x) × 0 3(y) × 10(z) mm3, which is thin in the transverse direction (Y). Electrodes are coated on the Y-surfaces of the PPLN and connected to a voltage supplier (Range: [0, 100] V). The temperature of the sample is controlled at an accuracy of 0.1°C. The poling period is 20.6 μm, corresponding to a nominal QPM wavelength of 1550 nm at 20 degree (Sellmeier equation). The QPM period is determined by Λ = λ/(none), where λ is the wavelength in vacuum, no and ne are the ordinary and extraordinary refractive indices, respectively. The domain number is approximately 2500 in a single pass and doubled in a reflective geometry. Surfaces of the collimator and the PPLN are anti-reflection coated to reduce insertion loss.

We characterized the transmission spectrum, temperature tuning, EO tuning, and group delay properties of the device in both single-pass and reflective geometries. During the experiment, highly linearly polarized input is guaranteed in order to achieve a high extinction ratio in filtering. In the polarization coupling, both vertically and horizontally polarized input shows the same characteristics. The transmission characteristics of PPLN in the single-pass geometry is investigated by replacing the reflective mirror (Mirror) with a parallel polarizer [The PBS cubic is not shown in Fig. 1(a)], and monitoring the transmitted spectrum or power. The spatial output is shown in Fig. 1(c), whose Gaussian beam profile is in good quality. The insertion loss, compared with the power after the circulator, is only 0.2 dB. In this circumstance, the configuration is a PPLN based Solc-type filter when a transverse electric field is applied. Figure 2(a) shows the sinc function shaped transmission of this Solc-type filter. The full-width at half-maximum (FWHM) of the spectrum is 1.1 nm, measured at a resolution of 0.03 nm. The maximum extinction ratio of the Solc filter is measured to be 32 dB. Since the PPLN is thin in the transverse dimension, a reduced half-wave voltage of 93 V is obtained, as shown in Fig. 2(b). This is in accordance with the theoretical prediction value of 96.4 V and is much smaller than previous demonstration in thick samples [14–19]. Besides, the center wavelength can be tuned by the temperature in a wide range of over 30 nm covering the telecom C band, as shown in Fig. 2(c). The excellent linear dependence of operational wavelength on temperature is favorable in practical applications.

 

Fig. 2 (a) The PPLN based Solc-type transmission characteristics in single-pass geometry. The applied voltage is 90 V. (b) The transmission at the central wavelength varied with the applied voltage. (c) The central wavelength of Solc filtering varied with temperature.

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In the reflective geometry in Fig. 1(a), light passing through the PPLN is reflected back by a concave mirror (Mirror), passes the PPLN again in the backward direction, and is then coupled back to the in-line PBS. The input and output light of the device thus has the same polarization. A near-π phase shift is introduced during the reflection at the last domain. It is equivalent to a twice-long PPLN sample with a defect of twice domain units long embedded in the middle [18]. The phase shift induced by the defect during the polarization coupling opens a narrow window in the opaque transmission at the central wavelength, similar to the effect in a phase-shifted Bragg grating. Due to large dispersion in this passband, it exhibits large group delay for slow light.

The advantage of the reflective scheme is that the domain number is doubled, the resonance bandwidth is thus narrower and the drive voltage can be reduced. The device insertion loss in this geometry is mainly attributed to the back-coupling efficiency to the collimator in the return path, which leads to an overall loss of 3 dB. Figure 3 shows the measured EIT-like spectra of the device versus the applied voltages at 47.7°C. The FWHM of the transparent widow in the EIT-like spectrum is 0.5 nm. Dispersion control is of importance in many optical systems and devices. The tunable transmission spectrum means that its dispersion can also be tuned via the transverse EO effect, i.e., tunable group delay can be obtained by applying different voltages. The effect can be turned on and off just by switching the applied voltage. This is an advantage in applications.

 

Fig. 3 The transmission of the PPLN in the reflective geometry, showing tunable EIT-like spectra under different applied voltages.

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In measuring the group velocity delay, the amplitude of the pump light is modulated at 10 GHz, see Fig. 1(a). The output (reflected light) and reference light is simultaneously monitored by two fast photodiodes (PDs), with a 3-dB bandwidth of 10 GHz. A high-speed oscilloscope (Agilent 91604a, analog bandwidth 16 GHz) is used to observe the relative time delay between the two signal. As comparison, we conducted the optical delay experiment both in a Bragg grating and our PPLN device.

In the reflective Bragg grating scheme, the PPLN device is replace by a 1-cm long Bragg grating to investigate tunable group delay on the edge of its photonic gap. The center wavelength of the Bragg grating is 1549.6 nm with a FWHM of 0.6 nm. The temporal signal traces measured at the photonic gap edge are shown in Fig. 4(a). Different temporal delay is achieved by red-shifting the input wavelength away from the photonic gap center. The optical delay on the edge of photonic gap suffers from decreased reflection for large optical delay. A delay of 15.6 ps was observed with only a normalized transmission of ∼ 10% at 1549.95 nm. The dilemma is the same in phase-shifted Bragg gratings, where transmission can be compromised for large group delays.

 

Fig. 4 (a) Experimentally measured temporal delay in a reflective Bragg grating at different wavelengths at the photonic gap edge of 1549.85 nm. (b) Experimentally measured temporal delay in PPLN in reflective geometry controlled by external voltage.

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In our device in the reflective geometry, the group delay is measured at different applied voltages at the input wavelength of 1544.5 nm. The spectrum of the modulated waves falls in the rendered transparency window in the EIT-like spectrum of the device. Figure 4(b) shows the observed temporal traces of signal at different group delays with respect to the applied voltage. A 6.0-ps delay was observed at the applied voltage of 63 V (Blue curve). The transmission spectrum corresponds to the red curve in Fig. 3. And the group delay is 3.0 ps when the applied voltage is 30 V. This is consistent with the theoretical prediction [18]. The relationship between the tunable group delay time and the applied voltages was not recorded due to the resolution limit of the electronic equipment, but the tunability of the group delay is apparent. One critically important aspect is that the transmission is kept nearly constant with different delays as the applied voltage increases. This can be referred from Fig. 3 that the transmission at the center wavelength is kept near unity under different applied voltages. This would be an advantage over other schemes for implementing tunable slow light.

It is worth mentioning that the observed characteristics have been investigated using both polarization, showing the same features. Besides, the central wavelength of EIT-like spectrum with respect to temperature is the same with that in the Solc-type filter configuration, as shown in Fig. 2(c). Although EO effect has near instantaneous response time (ps scale), the electrical response of an EO device is actually capped by that of its electronic circuits. The upper limit of the electrical response frequency of our device is measured to be approximately 2.0 MHz when driven by square wave signals, because of large capacity of the electrodes. The capacity constant is measured to be 50 pF, which caps the upper limit of the RC circuit.

Although, the demonstrated temporal delays in the phase-shifted PPLN in the reflective geometry is still not significant in our experiment, the proposed mechanism is obviously working. We believe that the proposed concept may offer a new way for implementing electrically tunable optical tunable filters or delay lines. In principle, with lithium niobate ridge waveguide structures, the drive voltage can be further reduced to several voltages and the response time can also be improved using microelectrodes.

4. Conclusion

In summary, we have presented a phase-shifted Solc-type filter based on thin PPLN in a reflective geometry via transverse EO effect. Tunable EIT-like transmission spectrum is electrically controlled, while the central transparent window keeps highly constant transmission. The mechanism features tunable center wavelength in a wide range with respect to temperature and controllable optical delay to the applied external electrical field, which may provide a new way for tunable optical filters or delay lines.

Funding

National Natural Science Foundation of China (NSFC) (11604206, 11734011); National Key R&D Program of China (2017YFA0303700); Shanghai Education Development Foundation (SHEDF) (16CG08); The Foundation for Development of Science and Technology of Shanghai (17JC1400400).

References and links

1. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]  

2. R. L. W. Novikova and Y. Xiao, “Electromagnetically induced transparency-based slow and stored light in warm atoms,” Laser Photonics Rev. 6(3), 333–353 (2012). [CrossRef]  

3. G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013). [CrossRef]   [PubMed]  

4. S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107 (1990). [CrossRef]   [PubMed]  

5. S. Longhi, D. Janner, G. Galzerano, G. Valle, D. Gatti, and P. Laporta, “Optical buffering in phase-shifted fibre grating,” Electron. Lett. 41, 1075–1077 (2005). [CrossRef]  

6. X. D. Yang, M. B. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102, 173902 (2009). [CrossRef]   [PubMed]  

7. C. W. Neff, L. M. Andersson, and M. Qiu, “Coupled resonator optical waveguide structures with highly dispersive media,” Opt. Express 15(16), 10362–10369 (2007). [CrossRef]   [PubMed]  

8. C. Zheng, X. S. Jiang, S. Y. Hua, L. Chang, G. Y. Li, H. B. Fan, and M. Xiao, “Controllable optical analog to electromagnetically induced transparency in coupled high-Q microtoroid cavities,” Opt. Express 20, 18319–18325 (2012). [CrossRef]   [PubMed]  

9. Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006). [CrossRef]   [PubMed]  

10. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010). [CrossRef]   [PubMed]  

11. C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012). [CrossRef]   [PubMed]  

12. J. H. Kim, M. C. Kuzyk, K. W. Han, H. L. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Physics 11, 275–280 (2015). [CrossRef]  

13. C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015). [CrossRef]   [PubMed]  

14. X. F. Chen, J. H. Shi, Y. P. Chen, Y. M. Zhu, Y. X. Xia, and Y. L. Chen, “Electro-optic Solc-type wavelength filter in periodically poled lithium niobate,” Opt. Lett. 28, 2115–2117 (2003). [CrossRef]   [PubMed]  

15. K. Liu, J. H. Shi, and X. F. Chen, “Linear polarization-state generator with high precision in periodically poled lithium niobate,” Appl. Phys. Lett. 94, 101106 (2009). [CrossRef]  

16. K. Liu, W. J. Lu, Y. P. Chen, and X. F. Chen, “Active control of group velocity by use of folded dielectric axes structures,” Appl. Phys. Lett. 97, 071104 (2010). [CrossRef]  

17. Q. Wang, F. Xu, Z. Y. Yu, X. S. Qian, X. K. Hu, Y. Q. Lu, and H. T. Wang, “A bidirectional tunable optical diode based on periodically poled LiNbO3,” Opt. Express 18, 7340–7346 (2010). [CrossRef]   [PubMed]  

18. X. S. Song, F. Xu, and Y. Q. Lu, “Electromagnetically induced transparency-like transmission in periodically poled lithium niobate with a defect,” Opt. Lett. 36, 4434–4436 (2011). [CrossRef]   [PubMed]  

19. G. H. Shao, J. Song, Y. P. Ruan, G. X. Cui, and Y. Q. Lu, “Tunable dual-wavelength filter and its group delay dispersion in domain-engineered lithium niobate,” AIP Adv. 6, 125034 (2016). [CrossRef]  

20. I. Mhaouech, V. Coda, G. Montemezzani, M. Chauvet, and L. Guilbert, “Low drive voltage electro-optic Bragg deflector using a periodically poled lithium niobate planar waveguide,” Opt. Lett. 41, 4174–4177 (2016). [CrossRef]   [PubMed]  

References

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  1. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
    [Crossref]
  2. R. L. W. Novikova and Y. Xiao, “Electromagnetically induced transparency-based slow and stored light in warm atoms,” Laser Photonics Rev. 6(3), 333–353 (2012).
    [Crossref]
  3. G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013).
    [Crossref] [PubMed]
  4. S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107 (1990).
    [Crossref] [PubMed]
  5. S. Longhi, D. Janner, G. Galzerano, G. Valle, D. Gatti, and P. Laporta, “Optical buffering in phase-shifted fibre grating,” Electron. Lett. 41, 1075–1077 (2005).
    [Crossref]
  6. X. D. Yang, M. B. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102, 173902 (2009).
    [Crossref] [PubMed]
  7. C. W. Neff, L. M. Andersson, and M. Qiu, “Coupled resonator optical waveguide structures with highly dispersive media,” Opt. Express 15(16), 10362–10369 (2007).
    [Crossref] [PubMed]
  8. C. Zheng, X. S. Jiang, S. Y. Hua, L. Chang, G. Y. Li, H. B. Fan, and M. Xiao, “Controllable optical analog to electromagnetically induced transparency in coupled high-Q microtoroid cavities,” Opt. Express 20, 18319–18325 (2012).
    [Crossref] [PubMed]
  9. Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
    [Crossref] [PubMed]
  10. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
    [Crossref] [PubMed]
  11. C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
    [Crossref] [PubMed]
  12. J. H. Kim, M. C. Kuzyk, K. W. Han, H. L. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Physics 11, 275–280 (2015).
    [Crossref]
  13. C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
    [Crossref] [PubMed]
  14. X. F. Chen, J. H. Shi, Y. P. Chen, Y. M. Zhu, Y. X. Xia, and Y. L. Chen, “Electro-optic Solc-type wavelength filter in periodically poled lithium niobate,” Opt. Lett. 28, 2115–2117 (2003).
    [Crossref] [PubMed]
  15. K. Liu, J. H. Shi, and X. F. Chen, “Linear polarization-state generator with high precision in periodically poled lithium niobate,” Appl. Phys. Lett. 94, 101106 (2009).
    [Crossref]
  16. K. Liu, W. J. Lu, Y. P. Chen, and X. F. Chen, “Active control of group velocity by use of folded dielectric axes structures,” Appl. Phys. Lett. 97, 071104 (2010).
    [Crossref]
  17. Q. Wang, F. Xu, Z. Y. Yu, X. S. Qian, X. K. Hu, Y. Q. Lu, and H. T. Wang, “A bidirectional tunable optical diode based on periodically poled LiNbO3,” Opt. Express 18, 7340–7346 (2010).
    [Crossref] [PubMed]
  18. X. S. Song, F. Xu, and Y. Q. Lu, “Electromagnetically induced transparency-like transmission in periodically poled lithium niobate with a defect,” Opt. Lett. 36, 4434–4436 (2011).
    [Crossref] [PubMed]
  19. G. H. Shao, J. Song, Y. P. Ruan, G. X. Cui, and Y. Q. Lu, “Tunable dual-wavelength filter and its group delay dispersion in domain-engineered lithium niobate,” AIP Adv. 6, 125034 (2016).
    [Crossref]
  20. I. Mhaouech, V. Coda, G. Montemezzani, M. Chauvet, and L. Guilbert, “Low drive voltage electro-optic Bragg deflector using a periodically poled lithium niobate planar waveguide,” Opt. Lett. 41, 4174–4177 (2016).
    [Crossref] [PubMed]

2016 (2)

G. H. Shao, J. Song, Y. P. Ruan, G. X. Cui, and Y. Q. Lu, “Tunable dual-wavelength filter and its group delay dispersion in domain-engineered lithium niobate,” AIP Adv. 6, 125034 (2016).
[Crossref]

I. Mhaouech, V. Coda, G. Montemezzani, M. Chauvet, and L. Guilbert, “Low drive voltage electro-optic Bragg deflector using a periodically poled lithium niobate planar waveguide,” Opt. Lett. 41, 4174–4177 (2016).
[Crossref] [PubMed]

2015 (2)

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

J. H. Kim, M. C. Kuzyk, K. W. Han, H. L. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Physics 11, 275–280 (2015).
[Crossref]

2013 (1)

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013).
[Crossref] [PubMed]

2012 (3)

R. L. W. Novikova and Y. Xiao, “Electromagnetically induced transparency-based slow and stored light in warm atoms,” Laser Photonics Rev. 6(3), 333–353 (2012).
[Crossref]

C. Zheng, X. S. Jiang, S. Y. Hua, L. Chang, G. Y. Li, H. B. Fan, and M. Xiao, “Controllable optical analog to electromagnetically induced transparency in coupled high-Q microtoroid cavities,” Opt. Express 20, 18319–18325 (2012).
[Crossref] [PubMed]

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (3)

K. Liu, W. J. Lu, Y. P. Chen, and X. F. Chen, “Active control of group velocity by use of folded dielectric axes structures,” Appl. Phys. Lett. 97, 071104 (2010).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Q. Wang, F. Xu, Z. Y. Yu, X. S. Qian, X. K. Hu, Y. Q. Lu, and H. T. Wang, “A bidirectional tunable optical diode based on periodically poled LiNbO3,” Opt. Express 18, 7340–7346 (2010).
[Crossref] [PubMed]

2009 (2)

X. D. Yang, M. B. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102, 173902 (2009).
[Crossref] [PubMed]

K. Liu, J. H. Shi, and X. F. Chen, “Linear polarization-state generator with high precision in periodically poled lithium niobate,” Appl. Phys. Lett. 94, 101106 (2009).
[Crossref]

2007 (1)

2006 (1)

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

2005 (2)

S. Longhi, D. Janner, G. Galzerano, G. Valle, D. Gatti, and P. Laporta, “Optical buffering in phase-shifted fibre grating,” Electron. Lett. 41, 1075–1077 (2005).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

2003 (1)

1990 (1)

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107 (1990).
[Crossref] [PubMed]

Andersson, L. M.

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Bahl, G.

J. H. Kim, M. C. Kuzyk, K. W. Han, H. L. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Physics 11, 275–280 (2015).
[Crossref]

Chang, L.

Chauvet, M.

Chen, X. F.

K. Liu, W. J. Lu, Y. P. Chen, and X. F. Chen, “Active control of group velocity by use of folded dielectric axes structures,” Appl. Phys. Lett. 97, 071104 (2010).
[Crossref]

K. Liu, J. H. Shi, and X. F. Chen, “Linear polarization-state generator with high precision in periodically poled lithium niobate,” Appl. Phys. Lett. 94, 101106 (2009).
[Crossref]

X. F. Chen, J. H. Shi, Y. P. Chen, Y. M. Zhu, Y. X. Xia, and Y. L. Chen, “Electro-optic Solc-type wavelength filter in periodically poled lithium niobate,” Opt. Lett. 28, 2115–2117 (2003).
[Crossref] [PubMed]

Chen, Y. L.

Chen, Y. P.

K. Liu, W. J. Lu, Y. P. Chen, and X. F. Chen, “Active control of group velocity by use of folded dielectric axes structures,” Appl. Phys. Lett. 97, 071104 (2010).
[Crossref]

X. F. Chen, J. H. Shi, Y. P. Chen, Y. M. Zhu, Y. X. Xia, and Y. L. Chen, “Electro-optic Solc-type wavelength filter in periodically poled lithium niobate,” Opt. Lett. 28, 2115–2117 (2003).
[Crossref] [PubMed]

Coda, V.

Cui, G. X.

G. H. Shao, J. Song, Y. P. Ruan, G. X. Cui, and Y. Q. Lu, “Tunable dual-wavelength filter and its group delay dispersion in domain-engineered lithium niobate,” AIP Adv. 6, 125034 (2016).
[Crossref]

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Dong, C. H.

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[Crossref] [PubMed]

Fan, H. B.

Fan, S. H.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

Field, J. E.

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107 (1990).
[Crossref] [PubMed]

Fiore, V.

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[Crossref] [PubMed]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

Fu, W.

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Galzerano, G.

S. Longhi, D. Janner, G. Galzerano, G. Valle, D. Gatti, and P. Laporta, “Optical buffering in phase-shifted fibre grating,” Electron. Lett. 41, 1075–1077 (2005).
[Crossref]

Gatti, D.

S. Longhi, D. Janner, G. Galzerano, G. Valle, D. Gatti, and P. Laporta, “Optical buffering in phase-shifted fibre grating,” Electron. Lett. 41, 1075–1077 (2005).
[Crossref]

Gavartin, E.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Guilbert, L.

Guo, G. C.

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Halfmann, T.

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013).
[Crossref] [PubMed]

Han, K. W.

J. H. Kim, M. C. Kuzyk, K. W. Han, H. L. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Physics 11, 275–280 (2015).
[Crossref]

Harris, S. E.

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107 (1990).
[Crossref] [PubMed]

Heinze, G.

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013).
[Crossref] [PubMed]

Hu, X. K.

Hua, S. Y.

Hubrich, C.

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013).
[Crossref] [PubMed]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107 (1990).
[Crossref] [PubMed]

Janner, D.

S. Longhi, D. Janner, G. Galzerano, G. Valle, D. Gatti, and P. Laporta, “Optical buffering in phase-shifted fibre grating,” Electron. Lett. 41, 1075–1077 (2005).
[Crossref]

Jiang, X. S.

Kim, J. H.

J. H. Kim, M. C. Kuzyk, K. W. Han, H. L. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Physics 11, 275–280 (2015).
[Crossref]

Kippenberg, T. J.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Kuzyk, M. C.

J. H. Kim, M. C. Kuzyk, K. W. Han, H. L. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Physics 11, 275–280 (2015).
[Crossref]

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[Crossref] [PubMed]

Kwong, D. L.

X. D. Yang, M. B. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102, 173902 (2009).
[Crossref] [PubMed]

Laporta, P.

S. Longhi, D. Janner, G. Galzerano, G. Valle, D. Gatti, and P. Laporta, “Optical buffering in phase-shifted fibre grating,” Electron. Lett. 41, 1075–1077 (2005).
[Crossref]

Li, G. Y.

Lipson, M.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

Liu, K.

K. Liu, W. J. Lu, Y. P. Chen, and X. F. Chen, “Active control of group velocity by use of folded dielectric axes structures,” Appl. Phys. Lett. 97, 071104 (2010).
[Crossref]

K. Liu, J. H. Shi, and X. F. Chen, “Linear polarization-state generator with high precision in periodically poled lithium niobate,” Appl. Phys. Lett. 94, 101106 (2009).
[Crossref]

Longhi, S.

S. Longhi, D. Janner, G. Galzerano, G. Valle, D. Gatti, and P. Laporta, “Optical buffering in phase-shifted fibre grating,” Electron. Lett. 41, 1075–1077 (2005).
[Crossref]

Lu, W. J.

K. Liu, W. J. Lu, Y. P. Chen, and X. F. Chen, “Active control of group velocity by use of folded dielectric axes structures,” Appl. Phys. Lett. 97, 071104 (2010).
[Crossref]

Lu, Y. Q.

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

Mhaouech, I.

Montemezzani, G.

Neff, C. W.

Novikova, R. L. W.

R. L. W. Novikova and Y. Xiao, “Electromagnetically induced transparency-based slow and stored light in warm atoms,” Laser Photonics Rev. 6(3), 333–353 (2012).
[Crossref]

Povinelli, M. L.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

Qian, X. S.

Qiu, M.

Rivière, R.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Ruan, Y. P.

G. H. Shao, J. Song, Y. P. Ruan, G. X. Cui, and Y. Q. Lu, “Tunable dual-wavelength filter and its group delay dispersion in domain-engineered lithium niobate,” AIP Adv. 6, 125034 (2016).
[Crossref]

Sandhu, S.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

Schliesser, A.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Shakya, J.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

Shao, G. H.

G. H. Shao, J. Song, Y. P. Ruan, G. X. Cui, and Y. Q. Lu, “Tunable dual-wavelength filter and its group delay dispersion in domain-engineered lithium niobate,” AIP Adv. 6, 125034 (2016).
[Crossref]

Shen, Z.

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Shi, J. H.

K. Liu, J. H. Shi, and X. F. Chen, “Linear polarization-state generator with high precision in periodically poled lithium niobate,” Appl. Phys. Lett. 94, 101106 (2009).
[Crossref]

X. F. Chen, J. H. Shi, Y. P. Chen, Y. M. Zhu, Y. X. Xia, and Y. L. Chen, “Electro-optic Solc-type wavelength filter in periodically poled lithium niobate,” Opt. Lett. 28, 2115–2117 (2003).
[Crossref] [PubMed]

Song, J.

G. H. Shao, J. Song, Y. P. Ruan, G. X. Cui, and Y. Q. Lu, “Tunable dual-wavelength filter and its group delay dispersion in domain-engineered lithium niobate,” AIP Adv. 6, 125034 (2016).
[Crossref]

Song, X. S.

Valle, G.

S. Longhi, D. Janner, G. Galzerano, G. Valle, D. Gatti, and P. Laporta, “Optical buffering in phase-shifted fibre grating,” Electron. Lett. 41, 1075–1077 (2005).
[Crossref]

Wang, H. L.

J. H. Kim, M. C. Kuzyk, K. W. Han, H. L. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Physics 11, 275–280 (2015).
[Crossref]

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[Crossref] [PubMed]

Wang, H. T.

Wang, Q.

Weis, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Wong, C. W.

X. D. Yang, M. B. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102, 173902 (2009).
[Crossref] [PubMed]

Xia, Y. X.

Xiao, M.

Xiao, Y.

R. L. W. Novikova and Y. Xiao, “Electromagnetically induced transparency-based slow and stored light in warm atoms,” Laser Photonics Rev. 6(3), 333–353 (2012).
[Crossref]

Xu, F.

Xu, Q. F.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

Yang, X. D.

X. D. Yang, M. B. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102, 173902 (2009).
[Crossref] [PubMed]

Yu, M. B.

X. D. Yang, M. B. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102, 173902 (2009).
[Crossref] [PubMed]

Yu, Z. Y.

Zhang, Y. L.

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Zheng, C.

Zhu, Y. M.

Zou, C. L.

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

AIP Adv. (1)

G. H. Shao, J. Song, Y. P. Ruan, G. X. Cui, and Y. Q. Lu, “Tunable dual-wavelength filter and its group delay dispersion in domain-engineered lithium niobate,” AIP Adv. 6, 125034 (2016).
[Crossref]

Appl. Phys. Lett. (2)

K. Liu, J. H. Shi, and X. F. Chen, “Linear polarization-state generator with high precision in periodically poled lithium niobate,” Appl. Phys. Lett. 94, 101106 (2009).
[Crossref]

K. Liu, W. J. Lu, Y. P. Chen, and X. F. Chen, “Active control of group velocity by use of folded dielectric axes structures,” Appl. Phys. Lett. 97, 071104 (2010).
[Crossref]

Electron. Lett. (1)

S. Longhi, D. Janner, G. Galzerano, G. Valle, D. Gatti, and P. Laporta, “Optical buffering in phase-shifted fibre grating,” Electron. Lett. 41, 1075–1077 (2005).
[Crossref]

Laser Photonics Rev. (1)

R. L. W. Novikova and Y. Xiao, “Electromagnetically induced transparency-based slow and stored light in warm atoms,” Laser Photonics Rev. 6(3), 333–353 (2012).
[Crossref]

Nat. Commun. (1)

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Nat. Physics (1)

J. H. Kim, M. C. Kuzyk, K. W. Han, H. L. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Physics 11, 275–280 (2015).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. Lett. (4)

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013).
[Crossref] [PubMed]

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107 (1990).
[Crossref] [PubMed]

X. D. Yang, M. B. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102, 173902 (2009).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

Science (2)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 Phase-shifted Solc-type filter based on PPLN in a reflective geometry. (a) Schematic of the experimental setup. (b) Photograph of the device components, compared with a coin. (c) Beam profile after passing the PPLN in the far field.
Fig. 2
Fig. 2 (a) The PPLN based Solc-type transmission characteristics in single-pass geometry. The applied voltage is 90 V. (b) The transmission at the central wavelength varied with the applied voltage. (c) The central wavelength of Solc filtering varied with temperature.
Fig. 3
Fig. 3 The transmission of the PPLN in the reflective geometry, showing tunable EIT-like spectra under different applied voltages.
Fig. 4
Fig. 4 (a) Experimentally measured temporal delay in a reflective Bragg grating at different wavelengths at the photonic gap edge of 1549.85 nm. (b) Experimentally measured temporal delay in PPLN in reflective geometry controlled by external voltage.

Equations (1)

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M i = R ( θ ) W i R ( θ ) = [ cos θ sin θ sin θ cos θ ] [ e i δ ϕ / 2 0 0 e i δ ϕ / 2 ] [ cos θ sin θ sin θ cos θ ] .

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