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Abstract
Optical interferometer has played an important role in optics. Up to now, many kinds of interferometers have been realized and found their applications. In this letter, we experimentally construct an interferometer by using parametric amplifier as a wave splitter and beam splitter as a wave combiner. We make measurements of interference fringes and explore the relationships between the interference visibility of the interferometer and various system parameters, such as the gain of the parametric amplifier, the one-photon detuning and the temperature of the Rb-85 vapor cell.
© 2017 Optical Society of America
1. Introduction
As the basis of precision measurement, optical interferometers [1] have made significant contributions in the development of physics. One of the most important optical interferometers is the Michelson interferometer [2] which was designed by A.A. Michelson and E. W. Morley in the late 19th century. It has provided evidence for Einstein s special relativity. Even in the present, the Michelson interferometer is an indispensable tool for the Laser Interferometer Gravitational-Wave Observatory (LIGO) [3–5] project. A transient gravitational-wave signal from a 410 megaparsec (1.3 billion light years) distant merger of two black holes has been successfully detected by the detectors of the LIGO in 2015 [6].
Generally, an optical interferometer is mainly composed of two parts. One part is the wave splitter which divides an input beam into two arms, the other part is the wave combiner which recombines them. Traditional optical interferometer such as the Mach-Zehnder interferometer [7, 8] is combined by two linear beam splitters. Recently, by using two parametric amplifiers (PAs) which are based on nondegenerate four-wave mixing (NDFWM) process with a double-Λ configuration [9–22] in two hot Rb-85 atomic vapor cells, our group successfully constructed the SU(1,1) interferometer [23–27]. The SU(1,1) interferometer also has been studied by several groups [28, 29]. In this letter, by replacing the wave combiner (the second PA) of the SU(1,1) with linear beam splitter, we experimentally implement an interferometer with a PA (frequency degenerate four-wave mixing (DFWM)) as the wave splitter and a beam splitter as the wave combiner. Compared to the PA, the beam splitter is more concise in configuration. Thus, this interferometer is more concise compared to the nonlinear interferometer [23, 24]. Due to the involving of both nonlinear PA and linear beam splitter, we could call it a hybrid interferometer. Then, we study the dependence of its interference visibility on different system parameters, such as the gain of the PA, the one-photon detuning and the temperature of the Rb-85 vapor cell.
2. Degenerate four-wave mixing
The PA exploited in our experiment is based on DFWM in Rb-85 vapor cell. In order to obtain the seed beam, one only needs to split a small portion of the pump beam using a linear beam splitter. On the contrary, the seed beam of the NDFWM is obtained by double passing a small portion of pump beam through a GHz level acoustic-optic modulator. This is determined by the double-Λ energy-level configuration of their NDFWM process. Therefore, the DFWM can be realized more easily than NDFWM due to its frequency degeneracy. As shown in Fig. 1(a), a weak signal beam with the same frequency as the pump is crossed with the strong pump beam in the center of the Rb-85 vapor cell at a small angle. During this process, the seed (signal) field is amplified and a phase conjugate (idler) field is generated in the meantime. On account of the phase-matching condition, the pump-idler angle is identical with the pump-signal angle on the other side of the pump. Due to the energy conservation, the idler beam has the same frequency as both the signal beam and pump beam. This is different from the NDFWM as shown in Fig. 1(b), where the idler beam has a 6.08 GHz frequency shift from the signal beam.
Fig. 1 Comparison between the scheme of DFWM and the scheme of NDFWM. (a) The scheme of DFWM where all the three output beams have the same frequency. (b) The scheme of NDFWM where all the three output beams have different frequencies. Is,out, Ii,out, Ip,out, the intensities of the output beams; ωs, ωi, ωp, the frequency of the output beams.
In order to prove that the above mentioned DFWM can be used as a PA for constructing the hybrid interferometer, we study the relationships between the power of two output ports and different system parameters, such as the seed power, the pump power, the one-photon detuning and the temperature of the Rb-85 vapor cell. Firstly, we set the pump power at 900 mW measured by optical power and energy meter (PM100D), maintain the temperature of Rb-85 vapor cell at 120 °C by using a digital temperature controller (E5EC-RR2ASM-820), set the one-photon detuning at 0.9 GHz and change the seed power from 18 μW to 105 μW. As shown in Fig. 2(a), we find that the output power of the signal and idler beams approximately have linear dependences on the seed power. And the difference between the power of signal and idler beams is approximately equal to the input seed power. Secondly, while keeping the seed power at 50 μW, the temperature of Rb-85 vapor cell at 120 °C, the one-photon detuning at 0.9 GHz and varying the pump power from 300 mW to 800 mW, we record the power of the signal and idler beams at the output. As shown in Fig. 2(b), the singal and idler power of two output ports increase with the increasing of the pump power. The power difference between the signal and idler beams almost keeps constant and is approximately equal to the input seed power. Thirdly, we fix the seed power at 5 μW, the pump power at 1 W, the temperature of Rb-85 vapor cell at 120 °C and scan the one-photon detuning from 0.2 GHz to 1.8 GHz. The result is shown in Fig. 2(c). When the one-photon detuning is 0.7 GHz, the system gain reaches its maximum of about 32. The system gain is determined by calculating the ratio between the output signal power and the input seed power. Finally, we vary the temperature of the Rb-85 vapor cell from 100 °C to 120 °C and set the seed power at 5 μW, the pump power at 1 W, the one-photon detuning at 0.9 GHz to study the influence of the temperature. As shown in Fig. 2(d), it clearly shows that the power of both the two output ports increase as the temperature increases. Based on all the experimental results as shown in Fig. 2, we can see that DFWM can work as a PA and therefore it can be used to construct the hybrid interferometer involving both the nonlinear PA and linear beam splitter.
Fig. 2 The relationships between the power of two output ports of DFWM and various system parameters. (a) The relationship between the power of two output ports (signal beam, blue dot and idler beam, green diamond) and the seed power. (b) The relationship between the power of two output ports (signal beam, blue dot and idler beam, green diamond) and the pump power. (c) The power of two output ports (signal beam, blue dot and idler beam, green diamond) as a function of the one-photon detuning. (d) The power of two output ports (signal beam, blue dot and idler beam, green diamond) as a function of the temperature of the Rb-85 vapor cell. Both (c) and (d) figures include the traces for the gain of the system (red square).
3. Experimental setup
Our experimental layout for realizing the hybrid interferometer is shown in Fig. 3(a). The PA is based on the DFWM process in a Rb-85 vapor cell discussed above. We use a Ti: sapphire laser as our laser system. The frequency of the Ti: sapphire laser is blue detuned 0.9 GHz from the Rb-85 D1 line (5S1/2, F = 2 → 5P1/2). A polarization beam splitter (PBS) is used to divide the beam into two. The strong one is injected into Rb-85 vapor cell as the pump beam. The weak one is also injected into the Rb-85 vapor cell as the seed beam. The Rb-85 vapor cell is 12 mm long and the temperature of the Rb-85 vapor cell is stabilized at 120 °C. The waist of pump beam is about 650 μm, and the waist of seed beam is about 320 μm. Combined by a Glan-Laser polarizer (GL), they are crossed in the center of the Rb-85 vapor cell at the angle of about 7 mard. The residual pump beam is eliminated by a Glan-Thompson polarizer (GT) with an extinction ratio of 105:1 after the Rb-85 vapor cell. During this process, the signal (seed) beam is amplified and an idler beam is generated on the other side of the pump. These two beams have the same frequency as the pump beam. With this DFWM based PA, we have realized the wave splitting process for the hybrid interferometer. Then, by using a piezo-electric transducer (PZT), we introduce a phase shift ϕ on the generated idler field which can change the phase difference between the amplified signal field and generated idler field, and thus change the interference fringe. It is worth pointing out that the displacement induced by the scanning of the PZT is at the level of 10 μm. It is much less than the size of the beam. Therefore, the scanning of the PZT wouldn’t change the alignment of our hybrid interferometer. In order to realize the wave combining process, we combine the output signal and idler beams by a linear beam splitter (BS). This is due to the frequency degeneracy of the signal and idler fields from the DFWM. Then, the two output ports of the BS are detected by two photodetectors (D1, D2).
Fig. 3 Experimental setup. (a) HWP, half wave plate; PBS, polarization beam splitter; GL, Glan-Laser polarizer; GT, Glan-Thompson polarizer; PZT, piezo-electric transducer; BS, beam splitter; D1, D2, photodetectors; OS, oscilloscope. (b) Energy level diagram of Rb-85 D1 line for DFWM. Δ, one-photon detuning.
4. Results and discussions
Figure. 4 shows the typical interference fringes of the two output ports of the hybrid interferometer detected by D1, D2 respectively. The interference visibilities of the D1 and D2 output ports are almost identical. The interference fringes of the two output ports are out of phase with the scanning of the PZT. Under the above experimental conditions, the interference visibility of D1 output port is about 96.0%, and the interference visibility of D2 output port is about 95.4%.
Fig. 4 Typical interference fringes detected by D1, D2 photodetectors respectively. (a) D1 output. (b) D2 output.
Since interference visibility is a direct measure of how perfectly the two interfering waves can cancel due to destructive interference and gives the degree of coherence between the waves [30]. Interference visibility is also important for precision measurement where the imperfect interference visibility can quadratically reduce the detection efficiency [31] and thus reduce the signal to noise ratio of the measurement. Therefore, it is worth to study the dependence of interference visibilities on the different system parameters, such as the gain of the PA, the one-photon detuning and the temperature of the Rb-85 vapor cell.
Firstly, in order to explore the relationship between the interference visibility and the gain of the PA, we vary the pump power from 300 mW to 1100 mW, which ranges the gain from 1.5 to 26. In this case, we keep the frequency of the pump beam unchanged at blue detuned about 0.9 GHz from the D1 line Rb-85 (5S1/2, F = 2 → 5P1/2), which is called one-photon detuning (Δ). The seed beam has the same frequency as the pump, and its power is fixed at about 5 μW. The temperature of Rb-85 vapor cell is kept at 120 °C. As shown in Fig. 5(a), the blue dot line is the relationship between the interference visibility and the gain for D1 output port and the green diamond line is for the D2 output port. Both of the interference visibilities of the D1 and D2 output ports increase as the gain increases. The two visibilities are almost identical and can be above 90% for the gain larger than about 12.
Fig. 5 The dependence of the interference visibilities on various system parameters. (a) The interference visibilities of the two output ports as a function of the gain of the PA. (b) The relationship between the interference visibilities of the two output ports and the one-photon detuning. (c) The relationship between the interference visibilities of the two output ports and the temperature of the Rb-85 vapor cell. Both (b) and (c) figures include the traces for the gain of the PA.
Secondly, we investigate how the interference visibility depends on the one-photon detuning of the PA. In order to study this effect, we set the seed power at 5 μW, the pump power at 1 W, the temperature of Rb-85 vapor cell at 120 °C and scan the one-photon detuning from 0.2 GHz to 1.8 GHz by changing the frequency of the pump beam. As shown in Fig. 5(b), the blue dot line is the relationship between the interference visibility and the one-photon detuning for D1 output port and the green diamond line is for the D2 output port. The red square line represents the gain of the PA. We find that the interference visibilities of D1 and D2 output ports maintain above 90% for the range of the one-photon detuning from 0.7 GHz to 1.2 GHz. And when the one-photon detuning is 0.9 GHz, the interference visibility reaches its maximum of about 95%.
Thirdly, we fix the seed power at 5 μW, the pump power at 1 W, the one-photon detuning at 0.9 GHz and vary the temperature of the Rb-85 vapor cell from 100 °C to 120 °C. The result is shown in Fig. 5(c). The interference visibilities of D1 and D2 output ports increase with the increasing of the temperature. Within the range of the temperature from 115 °C to 120 °C, the interference visibilities for both output ports are all about 90%.
The frequency degeneracy of our hybrid interferometer ensure that the signal and idler fields experience same levels of self-focusing and optical losses. Therefore, the visibility of our hybrid interferometer will not be affected by these two experimental imperfections. However, the visibility of nonlinear interferometer [23, 24] can be affected by these two experimental imperfections due to that the signal and idler fields at different frequencies experience different levels of self-focusing and optical losses. Therefore, our hybrid interferometer is more robust to experimental imperfections.
5. Conclusion
In conclusion, by using the nonlinear PA which is based on the DFWM as a wave splitter and the linear BS as a wave combiner, we have constructed a hybrid interferometer, and explored the dependences of interference visibilities on different system parameters. We find that the gain of PA can influence the interference visibilities, and the interference visibilities increase with the increasing of the gain. We also study the relationships between interference visibilities and other system parameters, such as the one-photon detuning and the temperature of the Rb-85 vapor cell. And we found that the interference visibilities can be above 90% in broad ranges of system parameters. Our hybrid interferometer is more concise and more easily to be realized in terms of both the wave splitter and wave combiner, compared with the nonlinear interferometer [23, 24]. Therefore, our hybrid interferometer is more suitable for finding application in precision measurement when the amplification, robustness and conciseness are required.
Funding
The National Natural Science Foundation of China (NSFC) (91436211, 11374104, 10974057); Natural Science Foundation of Shanghai (17ZR1442900); Program for New Century Excellent Talents in University (NCET) (NCET-10-0383); Shu Guang project (11SG26); Shanghai Pujiang Program (09PJ1404400); National Basic Research Program of China (2016YFA0302103); SRFDP (20130076110011); Program of Introducing Talents of Discipline to Universities (B12024); Program of State Key Laboratory of Advanced 207 Optical Communication Systems and Networks (2016GZKF0JT003).
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