High time–bandwidth product and high repetition rate period signal generation based on spectral hole burning crystal Xiurong Ma,1,2 Song Wang,1,2,* Yuqing Liang,1,2 and Yunlong Shan1,2 1
Engineering Research Center on Communication Devices (Ministry of Education), School of Computer and Communication Engineering, Tianjin University of Technology, Tianjin 300384, China
2
Tianjin Key Laboratory of Film Electronic and Communication Device, School of Electronic Information Engineering, Tianjin University of Technology, Tianjin 300384, China *Corresponding author: [email protected] Received 19 January 2015; revised 25 February 2015; accepted 25 February 2015; posted 26 February 2015 (Doc. ID 232586); published 27 March 2015
This paper proposes an approach for the generation of high time–bandwidth product (TBP) and high repetition rate pulse compression period signal. The complex spectral grating is created through a reference pulse and multiple programming pulses with different start frequencies. As the multiple probe chirped pulses with different start frequencies interact with the complex spectral gratings, a high TBP and repetition rate period signal is thus generated. This technique has the potential to generate a time–bandwidth product of 105 when the repetition rate reaches up to tens of GHz. At the end of this paper, two simulation results of pulse compression period signal with 4 × 105 TBP and 20 GHz repetition rate are presented. © 2015 Optical Society of America OCIS codes: (320.5520) Pulse compression; (320.5540) Pulse shaping. http://dx.doi.org/10.1364/AO.54.002891
1. Introduction
Pulse compression signals with large time– bandwidth products (TBPs) have been extensively employed in modern radar systems to improve the range resolution. Based on traditional methods, a pulse compression signal is usually generated in the electrical domain by virtue of digital electronics, but some limitations still exist, such as small time– bandwidth products [1]. In order to solve the problems concerning electronic methods, optical technologies have been investigated recently to generate large time–bandwidth product signals. Through the employment of fiberoptic devices, a large time–bandwidth product microwave arbitrary waveform can also be generated
1559-128X/15/102891-06$15.00/0 © 2015 Optical Society of America
based on optical pulse shaping. For example, with an optical phase modulator which is incorporated in one arm of the Mach–Zehnder interferometer (MZI), a phase-coded pulse can be generated by beating two dispersed optical pulses in an unbalanced MZI [2]. Otherwise, a large time–bandwidth product signal should be created based on optical pulse shaping through the use of a single spatially discrete chirped fiber Bragg grating [3]. In the above-mentioned methods, the TBP of the signal is less than 100. Spectral hole burning (SHB) material has been widely used in signal processing [4] and spectrum analysis [5]. It is possible to use SHB-material-based photonic arbitrary waveform generation (S2-PAWG) to generate a signal with a high time–bandwidth product [6]. This approach is implemented based on the application of two temporally overlapping linear frequency chirps (TOLFCs) and multiple probe pulses which interact with the SHB material [7–9]. 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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For the purpose of obtaining a high TBP signal, a larger initial frequency shift is needed by probe pulse, which is not conducive to increasing the TBP of the signal. The signal is also generated by using multiple TOLFCs and a probe pulse which interacts with the SHB material [10]. Compared with [7–9], the same TBP can reach a smaller initial frequency shift. However, the repetition rate of the signal decreases. In this paper, as the SHB material, thulium-doped yttrium aluminum garnet (Tm:YAG) is used to produce a pulse compression period signal that has the potential to achieve a quite large time–bandwidth product of 105 and a high repetition rate of tens of GHz. This method is implemented by virtue of the application of a reference pulse, multiple programming pulses, and probe pulses which interact with the SHB material. Compared with [10], this approach can obtain a higher repetition frequency signal. At the end of this paper, two simulation results of pulse compression period signals with 4 × 105 TBP and 20 GHz repetition rate are presented. 2. Principle A.
Pulse Compression
Figure 1 demonstrates a timing diagram for the generation of a compressed pulse. Here, two TOLFCs are spatially overlapped within a SHB material [7]. The chirp rates of the two TOLFCs are represented by α1 (reference pulse) and α2 (programming pulse), respectively, and the start frequencies of the two TOLFCs are f s1 and f s2 , respectively. The duration of the two TOLFCs is represented by τc. The spectral overlap of the two TOLFCs defines the bandwidth of the spectral grating, B, which can be described as B τc · α2 − f s1 − f s2 ;
(1)
where B ≤ Binhom , and Binhom is the bandwidth of the S2 inhomogeneous broadening absorption spectrum.
For Tm:YAG, the inhomogeneously broadened linewidth is 40 GHz. Therefore, the temporal profiles of the two TOLFCs can be listed as follows: 1 2 E1 t A1 exp i2π f s1 t α1 t ; 2
(2)
1 E2 t A2 exp i2π f s2 t α2 t2 ; 2
(3)
where A1 and A2 express the amplitudes of the two TOLFCs, respectively. As a result, the spectrum of the spectral grating can be summarized as Gf E1 f E2 f 2πf s2 − f 2 2πf s1 − f 2 0 0 i A1 A2 exp −i 2α2 2α1 π i signα2 − signα1 ; 4
(4)
where the spectrum amplitudespof two the TOLFCs p are expressed as A01 A1 ∕ α1 , A02 A2 ∕ α2 , respectively. In general, the delay between two chirps τD and an instantaneous frequency f can be shown as follows:
1 1 f − f s2 − : s1 τD f f − f s1 α1 α2 α1
(5)
After the spectral grating is produced, a probe laser scans the whole bandwidth by virtue of the chirp α2 rate of α3 αα21−α (α3 ≫ α2 ), and the FWHM temporal 1 duration τcp of this compressed pulse is ≈1∕B, so the probe pulse can be expressed as 1 2 ; E3 t A3 exp i2π f s3 t α3 t 2
(6)
where A3 represents the amplitude of the probe chirp, and f s3 represents the start frequency of the probe pulse. As a result, the temporal profile of a compressed pulse Ecp t can be described as Z Ecp t
f c B2 f c −B2
E1 f E2 f E3 f expi2πf tdf
f f f K · exp i2πf c t − − s3 − s2 s1 α3 α2 α1 f f f ; · sin c Bπ t − − s3 − s2 s1 α3 α2 α1 Fig. 1. Timing diagram for the generation of a compressed pulse. 2892
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where
(7)
f2 f2 f2 K A · B · exp −i s3 − i s2 i s1 2α3 2α2 2α1 π · exp i signα3 signα2 − signα1 ; 4 p A03 A3 ∕ α3 , A A01 A02 A03 is determined by the amplitudes of all those three pulses as well as the chirp rates, and f c f s1 B∕2 is the central frequency of the spectral grating. B.
Signal Generation Using Multiple Probe Pulses
When the probe pulse is the frequency shift resulting from Δf s3 , Eq. (7) can also be written as f s3 f s2 f s1 0 Ecp t K 2 · exp i2πf c t − − − α3 α2 α1 f s3 Δf s f s2 f s1 − · sin c Bπ t − − α2 α1 α3 Δf (8) Ecp t − s3 ; jα3 j where f2 f2 f Δf s3 − i s2 i s1 K 2 A · B · exp −i s3 2α3 2α2 2α1 π · exp i signα3 signα2 − signα1 : 4 For Δf s3 ≪ f s3, one can think K 2 ≈ K. In practice, Eq. (8) is identical with Eq. (7) except for a Δτ3 Δf s3 ∕jα3 j delay. As shown in Fig. 2, if an additional chirped probe pulse that is the frequency shift resulting from Δf s3 is sent relative to the original chirped probe pulse, an additional compressed pulse that is delayed by Δτ3 relative to the original compressed pulse will thus be created [7]. So, multiple compressed pulses can be achieved by emitting multiple probe pulses with frequency shift Δf s3 ; moreover, there is also an interval Δτ3 among all
Fig. 2. Timing diagram for the compressed interval.
compressed pulses. The amplitude of the compressed pulse is controllable as well, which is in proportion with the amplitude of the probe pulse. If one wishes to create a signal constituted by N such compressed pulses, N different start frequencies and amplitude probe pulses should be adopted. In essence, the signal is created with a set of compressed pulses with controllable intervals and amplitudes. As is shown in Fig. 3, the signal is generated by N probe pulses with the start frequency interval Δf s3 . The time width of the signal is T 3 N · Δτ3 N · Δf s3 ∕jα3 j, and the repetition rate of the signal is f r3 1∕Δτ3 jα3 j∕Δf s3. C.
Signal Generation Using Multiple Programming Pulses
If the programming pulse is the frequency shift by Δf s2 , Eq. (7) can also be expressed as f f f E00cp t K 3 · exp i2πf c t − − s3 − s2 s1 α3 α2 α1 f s2 Δf s f s3 f s1 − · sin c Bπ t − − α3 α1 α2 Δf (9) Ecp t − s2 : jα2 j In the same way as Section 2.B, one obtains K 3 ≈ K. It is observed that Eq. (9) is also same as Eq. (7) except for a delay Δτ2 Δf s2 ∕jα2 j, which demonstrates that a complex spectral grating is produced by multiple programming pulses with various amplitudes and start frequencies. When this complex grating is probed by a probe pulse, the signal will thus be generated. According to Fig. 4, the complex spectral grating is produced by N programming pulses with the start frequency interval Δf s2 . The signal can be generated if this complex spectral grating is probed by a probe pulse [8]. Hence, the time width of the signal is T 2 M · Δτ2 M · Δf s2 ∕jα2 j, and the repetition rate of the signal is f r2 1∕Δτ2 jα2 j∕Δf s2 . Provided that M N and Δf s2 Δf s3 , the time width ΔT 2 will be α3 ∕α2 times as much as the time width ΔT 3 . The increase in the time width of the
Fig. 3. Time diagram for signal generation using multiple probe pulses. 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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Fig. 4. Time diagram for signal generation using multiple programming pulses.
signal can also be viewed as an increase in the TBP of the signal. While the time width increases by α3 ∕α2 times, the TBP of the signal will increase α3 ∕α2 times as well. However, when the repetition rate of the signal f r2 decreases by α3 ∕α2 times, the TBP will increase by α3 ∕α2 times. The repetition rate f r3 is α3 ∕α2 times as much as the repetition rate f r2 . D. Period Signal Generation Using Multiple Programming Pulses and Probe Pulses
Provided that the programming pulse is the frequency shift by Δf s2, and the probe pulse is the frequency shift by Δf s3, Eq. (7) can be written as E000 cp t
f Δf s3 K 23 · exp i2πf c t − − s3 α3 f Δf s2 f s1 − s2 α1 α2 f s2 Δf s2 f s3 Δf s3 f s1 − · sin c Bπ t − − α1 α2 α3 Δf s2 Δf s3 − : (10) Ecp t − jα2 j jα3 j
Fig. 5. Time diagram for signal generation using multiple programming pulses and multiple probe pulses.
P pulses M The interval i1 Ecpi1 t will be generated. PM among compressed pulses i1 Ecpi1 t is Δτ2 P Δf s2 ∕jα2 j. M compressed pulses M i1 Ecpi2 t are generated by the probe pulse E32 t, which is the frequency shift by Δf s3 relative to the probe pulse E P31Mt. The interval among all compressed pulses i1 Ecpi2 t is Δτ 2 Δf s2 ∕jα2 j and the interval between compressed pulses Ecpi1 t and compressed pulses Ecpi2 t is Δτ3 Δf s3 ∕jα3 j. If this complex spectral grating is probed by N probe pulses E3k t (k 1; 2…; N), with the start interval PN P frequency Δf s , M × N compressed pulses M E i1 k1 cpik t will be generated. Compressed pulse Ecpik t is generated through the programming pulse E2i t and probe pulse E3k t, which can be expressed as Z Ecpik t
f c B2 f c −B2
E1 f E2i f E3k f expi2πf tdf : (13)
In the same way as Section 2.B, one obtains K 23 ≈ K. Based on Eq. (10), the delay of Eq. (10) relative to Eq. (7) can be expressed as Δτ23 Δτ2 Δτ3 Δf s2 ∕jα2 j Δf s3 ∕jα3 j:
(11)
As shown in Fig. 5, in order to create a complex spectral grating, a reference pulse E1 t and M programming pulses E2i t (i 1; 2; …M), with the start frequency interval Δf s2 , are temporally and spatially overlapped within the SHB. Hence, the complex spectral grating can be written as GM f
M X i1
E1 f · E2i f :
(12)
As shown in Fig. 6, while the complex grating is probed by the probe pulse E31 t, M compressed 2894
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Fig. 6. Time diagram for the compressed pulses that were generated by multiple programming pulses and multiple probe pulses.
P PN In fact, the compressed pulse M i1 k1 Ecpik t is a period signal with period of ΔT Δf s ∕jα2 j. In one period, the interval among compressed pulses PN E k1 cpik t is Δτ Δf s ∕jα3 j. The time width of the signal is T 23 M · Δf s2 ∕jα2 j, and the repetition rate of the signal is f p23 1∕Δτ3. Compared with the method which employs multiple probe pulses, the TBP is α3 ∕α2 times larger, and B · T 23 B · T 3 · α3 ∕α2 , meanwhile, the repetition rate remains unchanged, f p23 f p3 , M N and Δf s2 Δf s3 . Compared with the method which employs multiple probe pulses, the TBP remains unchanged, B · T 23 B · T 2 , the repetition rate is α3 ∕α2 times larger, and f p23 f p2 · α3 ∕α2 . Based on the above analysis, using multiple programming pulses and multiple probe pulses can generate the signal with higher TBP and repetition rate it can be concluded that signal with higher TBP and repetition rate can be generated by virtue of multiple programming pulses and multiple probe pulses. The upper limit of the TBP is determined by the product of Binhom and the coherence time of the SHB crystal. As for Tm:YAG, the maximum of the TBP can reach up to 105 . The upper limit of the repetition rate is limited by the frequency shift of the modulator. The maximum of the repetition rate can amount to tens of GHz.
that the frequency errors of the chirp pulse must be kept less than the inverse of the temporal duration of the chirp pulse. 4. Simulation Results
As shown in Fig. 8, ten periods of a 13-bit Barker code are generated with the setting described in Fig. 5. Figure 8 shows one period of a 13-bit Barker code (1,1,1,1,1,0,0,1,1,0,1,0,1). The chirps used by reference pulse and programming pulses to produce the spectral grating are different: α1 4.04 × 1013 Hz∕s and α2 4 × 1013 Hz∕s. The start frequency of the reference pulse is set as f s1 80 MHz for the reference pulse and f s2i 63 MHz i − 1 · 4 MHz for programming pulses; meanwhile, the duration of the reference pulse and programming pulses is set as τc 1 ms. According to the relationship among the three chirps, since the 13 probe pulses can be α3 α1 α2 ∕α2 − α1 1015 Hz∕s, the duration of probe chirp τp is set as 50 μs, the start frequency of the probe pulses is set as f s3k 40.063 GHz k − 1 · 50 kHz, the amplitude of probe pulses is (1,1,1,1,1,0,0,1,1,0,1,0,1), the bandwidth of this Barker code is 40 GHz, the period of this Barker code is 1 μs and the time width of this Barker code is T 23 10 μs. In one period, the time width of these
3. Period Signal Generation Device
Figure 7 shows the period signal generation device. The red solid arrows indicate the optical path and the blue dotted arrows indicate the electronic path. The laser beam with chirp rate α2 is split in two arms. The reference pulse E1 t is created by AOM1, which offers a chirp rate difference α1 − α2 . N programming pulses E2i t are produced by AMO2, which created the frequency-shifted Δf s2 copies of the original programming pulse E2 t. E1 t and E2i t are simultaneously generated from the same source. When E1 t and E2i t are focused into the cryogenically cooled SHB material (Tm:YAG), they created the complex grating in the SHB material absorption profile. Then M probe pulses E3k t are produced by AMO1, which created the frequency-shifted Δf s3 copies of the original programming pulse E3 t. The time ordering of E1 t and E3 t is controlled by AOM1 and AWG. When E3k t pass through the SHB material, the period signal is generated. This signal is detected by a high-speed photodetector. It should be noted
Fig. 7. Experimental setup. Definitions of the acronyms are as follows: beam splitter, B.S.; arbitrary waveform generator, AWG; acousto-optic modulator, AOM; photodetector, PD.
Fig. 8. Ten periods of a 13-bit Barker code (1,1,1,1,1,0,0,1, 1,0,1,0,1). 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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triangle wave signal is 4 × 105 and the repetition rate of this triangle wave signal is 20 GHz. 5. Conclusion
In this paper, a photonic echo process for the generation of a high TBP and repetition rate period signal is explored. By using a reference pulse and multiple programming pulses with different start frequencies, the paper presents the procedure to produce the complex spectral grating in an inhomogeneous-broadened absorber. When multiple probe-chirped pulses with different start frequencies are input to complex spectral gratings, a high TBP and repetition rate period signal will thus be generated. This method can produce a pulse compression signal with a time–bandwidth product reaching up to 105 . At the end of this paper, two simulations are performed, which demonstrate a 40 GHz pulse compression period signal with a 4 × 105 time–bandwidth product and 20 GHz repetition rate. References
Fig. 9. Ten periods of a triangle wave signal that were created with nine compressed pulses per period.
compressed pulses is 25 ps, which are spaced by 50 ps. The TBP of this Barker code signal is 4 × 105 and the repetition rate of this triangle wave signal is 20 GHz. Figure 9 shows ten periods of a triangle wave signal that were created by nine compressed pulses per period. The parameters of the reference pulse and 10 programming pulses are the same as those of the previous simulation. The 9 probe pulses can be described as α3 α1 α2 ∕α2 − α1 1015 Hz∕s, so the duration of probe chirp τp is set as 50 μs, and the start frequency of the 9 probe pulses is set as f s3k 40.063 GHz k − 1 · 50 kHz. The amplitude of the 9 probe pulses is increased gradually. The bandwidth of this triangle wave signal is 40 GHz. The period of this triangle wave signal is 1 μs and the time width of it is T 23 10 μs. In one period, the time width of these compressed pulses is 25 ps and they are spaced by 50 ps. The TBP of this
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1. H. D. Griffiths and W. J. Bradford, “Digital generation of high time–bandwidth product linear FM waveforms for radar altimeters,” in Radar and Signal Processing, IEE Proceedings F (IET Digital Library, 1992), Vol. 139, pp. 160–169. 2. H. Chi and J. Yao, “An approach to photonic generation of high-frequency phase-coded RF pulses,” IEEE Photon. Technol. Lett. 19, 768–770 (2007). 3. C. Wang and J. Yao, “Large time–bandwidth product microwave arbitrary waveform generation using a spatially discrete chirped fiber Bragg grating,” J. Lightwave Technol. 28, 1652–1660 (2010). 4. L. Youzhi, A. Hoskins, F. Schlottau, K. H. Wagner, C. Embry, and W. R. Babbitt, “Ultrawideband coherent noise lidar rangeDoppler imaging and signal processing by use of spatial-spectral holography in inhomogeneously broadened absorbers,” Appl. Opt. 45, 6409–6420 (2006). 5. M. Colice, F. Schlottau, and K. H. Wanger, “Broadband radiofrequency spectrum analysis in spectral-hole-burning media,” Appl. Opt. 45, 6393–6408 (2006). 6. Z. W. Barber, M. Tian, R. R. Reibel, and W. R. Babbitt, “Optical pulse shaping using optical coherent transients,” Opt. Express 10, 1145–1150 (2002). 7. R. R. Reibel, T. Chang, M. Tian, and W. R. Babbitt, “Optical linear sideband chirp compression for microwave arbitrary waveform generation,” in IEEE International Topical Meeting on Microwave Photonics (IEEE, 2004). 8. T. Chang, M. Tian, Z. W. Barber, and W. R. Babbitt, “Numerical modeling of optical coherent transient processes with complex configurations—II. Angled beams with arbitrary phase modulations,” J. Lumin. 107, 138–145 (2004). 9. C. J. Renner, R. R. Reibel, M. Tian, T. Chang, and W. R. Babbitt, “Broadband photonic arbitrary waveform generation based on spatial-spectral holographic materials,” J. Opt. Soc. Am. B 24, 2979–2987 (2007). 10. V. Damon, V. Crozatier, T. Chaneliere, J.-L. Le Gouet, and I. Lorgere, “Broadband photonic arbitrary waveform generation using a frequency agile laser at 1.5 μm,” J. Opt. Soc. Am. B 27, 524–530 (2010).
Engineering Research Center on Communication Devices (Ministry of Education), School of Computer and Communication Engineering, Tianjin University of Technology, Tianjin 300384, China
2
Tianjin Key Laboratory of Film Electronic and Communication Device, School of Electronic Information Engineering, Tianjin University of Technology, Tianjin 300384, China *Corresponding author: [email protected] Received 19 January 2015; revised 25 February 2015; accepted 25 February 2015; posted 26 February 2015 (Doc. ID 232586); published 27 March 2015
This paper proposes an approach for the generation of high time–bandwidth product (TBP) and high repetition rate pulse compression period signal. The complex spectral grating is created through a reference pulse and multiple programming pulses with different start frequencies. As the multiple probe chirped pulses with different start frequencies interact with the complex spectral gratings, a high TBP and repetition rate period signal is thus generated. This technique has the potential to generate a time–bandwidth product of 105 when the repetition rate reaches up to tens of GHz. At the end of this paper, two simulation results of pulse compression period signal with 4 × 105 TBP and 20 GHz repetition rate are presented. © 2015 Optical Society of America OCIS codes: (320.5520) Pulse compression; (320.5540) Pulse shaping. http://dx.doi.org/10.1364/AO.54.002891
1. Introduction
Pulse compression signals with large time– bandwidth products (TBPs) have been extensively employed in modern radar systems to improve the range resolution. Based on traditional methods, a pulse compression signal is usually generated in the electrical domain by virtue of digital electronics, but some limitations still exist, such as small time– bandwidth products [1]. In order to solve the problems concerning electronic methods, optical technologies have been investigated recently to generate large time–bandwidth product signals. Through the employment of fiberoptic devices, a large time–bandwidth product microwave arbitrary waveform can also be generated
1559-128X/15/102891-06$15.00/0 © 2015 Optical Society of America
based on optical pulse shaping. For example, with an optical phase modulator which is incorporated in one arm of the Mach–Zehnder interferometer (MZI), a phase-coded pulse can be generated by beating two dispersed optical pulses in an unbalanced MZI [2]. Otherwise, a large time–bandwidth product signal should be created based on optical pulse shaping through the use of a single spatially discrete chirped fiber Bragg grating [3]. In the above-mentioned methods, the TBP of the signal is less than 100. Spectral hole burning (SHB) material has been widely used in signal processing [4] and spectrum analysis [5]. It is possible to use SHB-material-based photonic arbitrary waveform generation (S2-PAWG) to generate a signal with a high time–bandwidth product [6]. This approach is implemented based on the application of two temporally overlapping linear frequency chirps (TOLFCs) and multiple probe pulses which interact with the SHB material [7–9]. 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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For the purpose of obtaining a high TBP signal, a larger initial frequency shift is needed by probe pulse, which is not conducive to increasing the TBP of the signal. The signal is also generated by using multiple TOLFCs and a probe pulse which interacts with the SHB material [10]. Compared with [7–9], the same TBP can reach a smaller initial frequency shift. However, the repetition rate of the signal decreases. In this paper, as the SHB material, thulium-doped yttrium aluminum garnet (Tm:YAG) is used to produce a pulse compression period signal that has the potential to achieve a quite large time–bandwidth product of 105 and a high repetition rate of tens of GHz. This method is implemented by virtue of the application of a reference pulse, multiple programming pulses, and probe pulses which interact with the SHB material. Compared with [10], this approach can obtain a higher repetition frequency signal. At the end of this paper, two simulation results of pulse compression period signals with 4 × 105 TBP and 20 GHz repetition rate are presented. 2. Principle A.
Pulse Compression
Figure 1 demonstrates a timing diagram for the generation of a compressed pulse. Here, two TOLFCs are spatially overlapped within a SHB material [7]. The chirp rates of the two TOLFCs are represented by α1 (reference pulse) and α2 (programming pulse), respectively, and the start frequencies of the two TOLFCs are f s1 and f s2 , respectively. The duration of the two TOLFCs is represented by τc. The spectral overlap of the two TOLFCs defines the bandwidth of the spectral grating, B, which can be described as B τc · α2 − f s1 − f s2 ;
(1)
where B ≤ Binhom , and Binhom is the bandwidth of the S2 inhomogeneous broadening absorption spectrum.
For Tm:YAG, the inhomogeneously broadened linewidth is 40 GHz. Therefore, the temporal profiles of the two TOLFCs can be listed as follows: 1 2 E1 t A1 exp i2π f s1 t α1 t ; 2
(2)
1 E2 t A2 exp i2π f s2 t α2 t2 ; 2
(3)
where A1 and A2 express the amplitudes of the two TOLFCs, respectively. As a result, the spectrum of the spectral grating can be summarized as Gf E1 f E2 f 2πf s2 − f 2 2πf s1 − f 2 0 0 i A1 A2 exp −i 2α2 2α1 π i signα2 − signα1 ; 4
(4)
where the spectrum amplitudespof two the TOLFCs p are expressed as A01 A1 ∕ α1 , A02 A2 ∕ α2 , respectively. In general, the delay between two chirps τD and an instantaneous frequency f can be shown as follows:
1 1 f − f s2 − : s1 τD f f − f s1 α1 α2 α1
(5)
After the spectral grating is produced, a probe laser scans the whole bandwidth by virtue of the chirp α2 rate of α3 αα21−α (α3 ≫ α2 ), and the FWHM temporal 1 duration τcp of this compressed pulse is ≈1∕B, so the probe pulse can be expressed as 1 2 ; E3 t A3 exp i2π f s3 t α3 t 2
(6)
where A3 represents the amplitude of the probe chirp, and f s3 represents the start frequency of the probe pulse. As a result, the temporal profile of a compressed pulse Ecp t can be described as Z Ecp t
f c B2 f c −B2
E1 f E2 f E3 f expi2πf tdf
f f f K · exp i2πf c t − − s3 − s2 s1 α3 α2 α1 f f f ; · sin c Bπ t − − s3 − s2 s1 α3 α2 α1 Fig. 1. Timing diagram for the generation of a compressed pulse. 2892
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where
(7)
f2 f2 f2 K A · B · exp −i s3 − i s2 i s1 2α3 2α2 2α1 π · exp i signα3 signα2 − signα1 ; 4 p A03 A3 ∕ α3 , A A01 A02 A03 is determined by the amplitudes of all those three pulses as well as the chirp rates, and f c f s1 B∕2 is the central frequency of the spectral grating. B.
Signal Generation Using Multiple Probe Pulses
When the probe pulse is the frequency shift resulting from Δf s3 , Eq. (7) can also be written as f s3 f s2 f s1 0 Ecp t K 2 · exp i2πf c t − − − α3 α2 α1 f s3 Δf s f s2 f s1 − · sin c Bπ t − − α2 α1 α3 Δf (8) Ecp t − s3 ; jα3 j where f2 f2 f Δf s3 − i s2 i s1 K 2 A · B · exp −i s3 2α3 2α2 2α1 π · exp i signα3 signα2 − signα1 : 4 For Δf s3 ≪ f s3, one can think K 2 ≈ K. In practice, Eq. (8) is identical with Eq. (7) except for a Δτ3 Δf s3 ∕jα3 j delay. As shown in Fig. 2, if an additional chirped probe pulse that is the frequency shift resulting from Δf s3 is sent relative to the original chirped probe pulse, an additional compressed pulse that is delayed by Δτ3 relative to the original compressed pulse will thus be created [7]. So, multiple compressed pulses can be achieved by emitting multiple probe pulses with frequency shift Δf s3 ; moreover, there is also an interval Δτ3 among all
Fig. 2. Timing diagram for the compressed interval.
compressed pulses. The amplitude of the compressed pulse is controllable as well, which is in proportion with the amplitude of the probe pulse. If one wishes to create a signal constituted by N such compressed pulses, N different start frequencies and amplitude probe pulses should be adopted. In essence, the signal is created with a set of compressed pulses with controllable intervals and amplitudes. As is shown in Fig. 3, the signal is generated by N probe pulses with the start frequency interval Δf s3 . The time width of the signal is T 3 N · Δτ3 N · Δf s3 ∕jα3 j, and the repetition rate of the signal is f r3 1∕Δτ3 jα3 j∕Δf s3. C.
Signal Generation Using Multiple Programming Pulses
If the programming pulse is the frequency shift by Δf s2 , Eq. (7) can also be expressed as f f f E00cp t K 3 · exp i2πf c t − − s3 − s2 s1 α3 α2 α1 f s2 Δf s f s3 f s1 − · sin c Bπ t − − α3 α1 α2 Δf (9) Ecp t − s2 : jα2 j In the same way as Section 2.B, one obtains K 3 ≈ K. It is observed that Eq. (9) is also same as Eq. (7) except for a delay Δτ2 Δf s2 ∕jα2 j, which demonstrates that a complex spectral grating is produced by multiple programming pulses with various amplitudes and start frequencies. When this complex grating is probed by a probe pulse, the signal will thus be generated. According to Fig. 4, the complex spectral grating is produced by N programming pulses with the start frequency interval Δf s2 . The signal can be generated if this complex spectral grating is probed by a probe pulse [8]. Hence, the time width of the signal is T 2 M · Δτ2 M · Δf s2 ∕jα2 j, and the repetition rate of the signal is f r2 1∕Δτ2 jα2 j∕Δf s2 . Provided that M N and Δf s2 Δf s3 , the time width ΔT 2 will be α3 ∕α2 times as much as the time width ΔT 3 . The increase in the time width of the
Fig. 3. Time diagram for signal generation using multiple probe pulses. 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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Fig. 4. Time diagram for signal generation using multiple programming pulses.
signal can also be viewed as an increase in the TBP of the signal. While the time width increases by α3 ∕α2 times, the TBP of the signal will increase α3 ∕α2 times as well. However, when the repetition rate of the signal f r2 decreases by α3 ∕α2 times, the TBP will increase by α3 ∕α2 times. The repetition rate f r3 is α3 ∕α2 times as much as the repetition rate f r2 . D. Period Signal Generation Using Multiple Programming Pulses and Probe Pulses
Provided that the programming pulse is the frequency shift by Δf s2, and the probe pulse is the frequency shift by Δf s3, Eq. (7) can be written as E000 cp t
f Δf s3 K 23 · exp i2πf c t − − s3 α3 f Δf s2 f s1 − s2 α1 α2 f s2 Δf s2 f s3 Δf s3 f s1 − · sin c Bπ t − − α1 α2 α3 Δf s2 Δf s3 − : (10) Ecp t − jα2 j jα3 j
Fig. 5. Time diagram for signal generation using multiple programming pulses and multiple probe pulses.
P pulses M The interval i1 Ecpi1 t will be generated. PM among compressed pulses i1 Ecpi1 t is Δτ2 P Δf s2 ∕jα2 j. M compressed pulses M i1 Ecpi2 t are generated by the probe pulse E32 t, which is the frequency shift by Δf s3 relative to the probe pulse E P31Mt. The interval among all compressed pulses i1 Ecpi2 t is Δτ 2 Δf s2 ∕jα2 j and the interval between compressed pulses Ecpi1 t and compressed pulses Ecpi2 t is Δτ3 Δf s3 ∕jα3 j. If this complex spectral grating is probed by N probe pulses E3k t (k 1; 2…; N), with the start interval PN P frequency Δf s , M × N compressed pulses M E i1 k1 cpik t will be generated. Compressed pulse Ecpik t is generated through the programming pulse E2i t and probe pulse E3k t, which can be expressed as Z Ecpik t
f c B2 f c −B2
E1 f E2i f E3k f expi2πf tdf : (13)
In the same way as Section 2.B, one obtains K 23 ≈ K. Based on Eq. (10), the delay of Eq. (10) relative to Eq. (7) can be expressed as Δτ23 Δτ2 Δτ3 Δf s2 ∕jα2 j Δf s3 ∕jα3 j:
(11)
As shown in Fig. 5, in order to create a complex spectral grating, a reference pulse E1 t and M programming pulses E2i t (i 1; 2; …M), with the start frequency interval Δf s2 , are temporally and spatially overlapped within the SHB. Hence, the complex spectral grating can be written as GM f
M X i1
E1 f · E2i f :
(12)
As shown in Fig. 6, while the complex grating is probed by the probe pulse E31 t, M compressed 2894
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Fig. 6. Time diagram for the compressed pulses that were generated by multiple programming pulses and multiple probe pulses.
P PN In fact, the compressed pulse M i1 k1 Ecpik t is a period signal with period of ΔT Δf s ∕jα2 j. In one period, the interval among compressed pulses PN E k1 cpik t is Δτ Δf s ∕jα3 j. The time width of the signal is T 23 M · Δf s2 ∕jα2 j, and the repetition rate of the signal is f p23 1∕Δτ3. Compared with the method which employs multiple probe pulses, the TBP is α3 ∕α2 times larger, and B · T 23 B · T 3 · α3 ∕α2 , meanwhile, the repetition rate remains unchanged, f p23 f p3 , M N and Δf s2 Δf s3 . Compared with the method which employs multiple probe pulses, the TBP remains unchanged, B · T 23 B · T 2 , the repetition rate is α3 ∕α2 times larger, and f p23 f p2 · α3 ∕α2 . Based on the above analysis, using multiple programming pulses and multiple probe pulses can generate the signal with higher TBP and repetition rate it can be concluded that signal with higher TBP and repetition rate can be generated by virtue of multiple programming pulses and multiple probe pulses. The upper limit of the TBP is determined by the product of Binhom and the coherence time of the SHB crystal. As for Tm:YAG, the maximum of the TBP can reach up to 105 . The upper limit of the repetition rate is limited by the frequency shift of the modulator. The maximum of the repetition rate can amount to tens of GHz.
that the frequency errors of the chirp pulse must be kept less than the inverse of the temporal duration of the chirp pulse. 4. Simulation Results
As shown in Fig. 8, ten periods of a 13-bit Barker code are generated with the setting described in Fig. 5. Figure 8 shows one period of a 13-bit Barker code (1,1,1,1,1,0,0,1,1,0,1,0,1). The chirps used by reference pulse and programming pulses to produce the spectral grating are different: α1 4.04 × 1013 Hz∕s and α2 4 × 1013 Hz∕s. The start frequency of the reference pulse is set as f s1 80 MHz for the reference pulse and f s2i 63 MHz i − 1 · 4 MHz for programming pulses; meanwhile, the duration of the reference pulse and programming pulses is set as τc 1 ms. According to the relationship among the three chirps, since the 13 probe pulses can be α3 α1 α2 ∕α2 − α1 1015 Hz∕s, the duration of probe chirp τp is set as 50 μs, the start frequency of the probe pulses is set as f s3k 40.063 GHz k − 1 · 50 kHz, the amplitude of probe pulses is (1,1,1,1,1,0,0,1,1,0,1,0,1), the bandwidth of this Barker code is 40 GHz, the period of this Barker code is 1 μs and the time width of this Barker code is T 23 10 μs. In one period, the time width of these
3. Period Signal Generation Device
Figure 7 shows the period signal generation device. The red solid arrows indicate the optical path and the blue dotted arrows indicate the electronic path. The laser beam with chirp rate α2 is split in two arms. The reference pulse E1 t is created by AOM1, which offers a chirp rate difference α1 − α2 . N programming pulses E2i t are produced by AMO2, which created the frequency-shifted Δf s2 copies of the original programming pulse E2 t. E1 t and E2i t are simultaneously generated from the same source. When E1 t and E2i t are focused into the cryogenically cooled SHB material (Tm:YAG), they created the complex grating in the SHB material absorption profile. Then M probe pulses E3k t are produced by AMO1, which created the frequency-shifted Δf s3 copies of the original programming pulse E3 t. The time ordering of E1 t and E3 t is controlled by AOM1 and AWG. When E3k t pass through the SHB material, the period signal is generated. This signal is detected by a high-speed photodetector. It should be noted
Fig. 7. Experimental setup. Definitions of the acronyms are as follows: beam splitter, B.S.; arbitrary waveform generator, AWG; acousto-optic modulator, AOM; photodetector, PD.
Fig. 8. Ten periods of a 13-bit Barker code (1,1,1,1,1,0,0,1, 1,0,1,0,1). 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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triangle wave signal is 4 × 105 and the repetition rate of this triangle wave signal is 20 GHz. 5. Conclusion
In this paper, a photonic echo process for the generation of a high TBP and repetition rate period signal is explored. By using a reference pulse and multiple programming pulses with different start frequencies, the paper presents the procedure to produce the complex spectral grating in an inhomogeneous-broadened absorber. When multiple probe-chirped pulses with different start frequencies are input to complex spectral gratings, a high TBP and repetition rate period signal will thus be generated. This method can produce a pulse compression signal with a time–bandwidth product reaching up to 105 . At the end of this paper, two simulations are performed, which demonstrate a 40 GHz pulse compression period signal with a 4 × 105 time–bandwidth product and 20 GHz repetition rate. References
Fig. 9. Ten periods of a triangle wave signal that were created with nine compressed pulses per period.
compressed pulses is 25 ps, which are spaced by 50 ps. The TBP of this Barker code signal is 4 × 105 and the repetition rate of this triangle wave signal is 20 GHz. Figure 9 shows ten periods of a triangle wave signal that were created by nine compressed pulses per period. The parameters of the reference pulse and 10 programming pulses are the same as those of the previous simulation. The 9 probe pulses can be described as α3 α1 α2 ∕α2 − α1 1015 Hz∕s, so the duration of probe chirp τp is set as 50 μs, and the start frequency of the 9 probe pulses is set as f s3k 40.063 GHz k − 1 · 50 kHz. The amplitude of the 9 probe pulses is increased gradually. The bandwidth of this triangle wave signal is 40 GHz. The period of this triangle wave signal is 1 μs and the time width of it is T 23 10 μs. In one period, the time width of these compressed pulses is 25 ps and they are spaced by 50 ps. The TBP of this
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