- Email: [email protected]
Spectral hole burning for pulse repetition frequency analysis
ARTICLE IN PRESS
Journal of Luminescence 127 (2007) 129–134 www.elsevier.com/locate/jlumin
Spectral hole burning for pulse repetition frequency analysis Max Colice, Jingyi Xiong, Kelvin Wagner Department of Electrical & Computer Engineering, Optoelectronic Computing Systems Center, University of Colorado, Boulder, CO 80309-0425, USA Available online 3 March 2007
Abstract We demonstrate an optical processor based on spectral hole burning (SHB) that maps the carrier frequency into the time domain and the pulse repetition frequency (PRF) into the spatial domain by illuminating an SHB crystal with a signal beam that is scanned by a tilting mirror across a slice of the crystal. This time-to-space mapping makes it possible to measure signal envelopes with a resolution of 1=T 1 ¼ 100 Hz. A signal with a pulsed envelope engraves a vertical absorption grating with a spatial periodicity given by the product of the PRF and the scan velocity. Reading the grating, which the crystal stores for up to T 1 , with a collimated beam yields orders diffracted at angles proportional to the PRF, which are Fourier-transformed to produce spots displaced from the DC position by distances proportional to the PRF. Increasing the PRF increases the grating periodicity, causing the diffracted spots to move away from the DC position. r 2007 Elsevier B.V. All rights reserved. PACS: 42.79.Hp; 42.40.My Keywords: Spatial–spectral holography; Spectral hole burning; Microwave photonics
1. Introduction Recent work on spectrum analysis using spectral hole burning (SHB) exploits the large inhomogeneous bandwidth of SHB crystals to capture broadband signals with unity probability of intercept [1–4]. At present, spectrum analysis over bandwidths of more than 10 GHz using SHB are single-shot measurements with resolutions limited by the linewidth of the lasers used to read and write the holes. Improving the resolution below 25 kHz will require ultrastable lasers, crystals cooled to below 4 K, and new techniques to stabilize chirped lasers [5,6]. Obtaining subkilohertz resolution of spectral features usually requires looking at coherent effects, such as photon echoes, instead of spectral holes. Here, we demonstrate a processor that detects pulsed signals with a resolution of 100 Hz, which is well below the homogeneous linewidth. We do this with a single-shot technique that transforms slow time into space,
Corresponding author.
E-mail address: [email protected] (M. Colice). 0022-2313/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2007.02.052
a trick that does not require extraordinary laser stabilization or crystal-cooling measures. Transforming time into space for optical processing is an old idea that plays on the inherent spatial parallelism of optics and the Fourier-transform properties of coherent light. The traveling acoustic waves in acousto-optic deflectors (AODs), for example, turn RF modulation into spatial index modulations. Illuminating the resulting index grating with a collimated laser beam produces beams diffracted at angles proportional to the RF frequency. Fourier-transforming the diffracted beams produces spots whose positions are, in turn, proportional to the diffraction angle. The spots in the Fourier plane of the AOD are a scaled version of the RF modulation power spectrum mapped into the spatial domain [7]. Some signals, such as those emitted by pulse-Doppler radars, are difficult to analyze because they usually consist of high-frequency carriers pulsed at kilohertz rates. For example, adequately sampling a carrier at 1 GHz pulsed at 10 kHz requires at least a million channels. Pulse repetition frequency (PRF) analyzers based on acousto-optic tripleproduct processors project the carrier frequency into one
ARTICLE IN PRESS 130
M. Colice et al. / Journal of Luminescence 127 (2007) 129–134
spatial dimension and the PRF into the other such that the total number of channels is the product of the space– bandwidth products of the two acousto-optic modulators (AOMs) used in the processor [8,9]. Acoustic wave generation and propagation limit these systems to signal bandwidths of less than 1 GHz and about 106 spectral channels, assuming a space–bandwidth product of 103 for each of the pair of acousto-optic devices in the processor. SHB crystals can process much higher bandwidth signals using the spectral degree of freedom that they possess in addition to the spatial degrees of freedom used by conventional optical processors. As we demonstrated previously, SHB crystals are capable of storing more than 20,000 channels in the spectral domain over bandwidths of 20 GHz or more [1]. SHB crystals also accommodate spatially varying signals for use in spectrum analysis [10] and time-integrating correlators [11] with space–bandwidth products of over 103 . In this experiment, an SHB crystal records the PRF and carrier frequency of a modulated laser beam as a spatial–spectral absorption grating. Spatially scanning the beam across a slice of the crystal engraves a spatial absorption grating into the crystal at the spectral bin corresponding to the carrier frequency. Interrogating the grating’s spectral location with a chirped laser beam yields the carrier frequency, and measuring the grating’s spatial periodicity yields the PRF. Scanning a spot with a diameter of 50 mm and a confocal length equal to or greater than the crystal thickness across a crystal 10 mm in width gives a space–bandwidth product of about 500. Combining this with the 200,000 channels in the spectral domain gives us 108 total channels for processing pulsed signals with carrier frequencies up to 20 GHz.
2. Theory Illuminating a single spot in an SHB crystal with a modulated signal beam engraves spectral holes into the absorption profile of the material at frequencies corresponding to the signal spectrum [1–4]. These holes persist in the crystal for the excited state lifetime and can be mapped by probing the absorption profile with a lowpower chirp. As the chirp sweeps across a hole, the transmitted intensity increases in proportion to the hole depth, which is itself proportional to the strength of the corresponding component of the signal power spectrum. Mapping the instantaneous chirp frequency to the temporal coordinate of the transmitted power provides a measurement of the signal power spectrum, jSðktÞj2 , blurred by a Lorentzian whose width is 1=T 2 , the crystal’s homogeneous linewidth. The inhomogeneous linewidth, 1=T 2 , determines the maximum signal bandwidth, and the homogeneous linewidth determines the resolution limit. For Tm3þ :YAG crystals with 1=T 2 20 GHz and 1=T 2 100 kHz, this translates to a spectrum analyzer time–bandwidth product of more than 105 .
Mapping space into time with a scanning mirror causes the SHB crystal to record the pulse rate of high-frequency carriers as spatial–spectral gratings. As in previous spectrum analysis experiments [1–4], a modulated signal beam illuminates the crystal, but the mirror scans the signal beam to record the signal’s time/frequency history in a slice of the crystal instead of integrating the signal spectrum in a single spot. The crystal records the carrier frequency in the spectral domain, as before, while the PRF is mapped into the vertical dimension of the slice, as shown in Fig. 1. The spot’s movement does induce a symmetric Doppler spread in the carrier frequency equal to the spot size divided by the apparent spot velocity, or several kilohertz, which is much smaller than the spectral sensitivity of the crystal. A signal that blinks on and off at the PRF engraves a vertical absorption grating with a spatial periodicity given by the product of the PRF and the scan velocity. The optical system and the PRF band of interest determine the scan time, which should not exceed the excited state lifetime of T 1 ¼ 10 ms. Illuminating the grating, which the crystal stores for up to T 1 , with a collimated beam yields orders diffracted at angles proportional to the PRF. Fourier-transforming the diffracted beams with a lens produces spots displaced from the DC position by distances proportional to the PRF. A 1-dimensional (1D) detector in the Fourier plane of the grating records the positions of these spots, yielding a measure of the PRF. Increasing the PRF increases the grating’s spatial frequency, shifting the diffracted spots away from the DC position. Because SHB crystals perform spectral discrimination, they can record a set of spectrally multiplexed gratings in a single slice. The different carriers each write their own absorption grating, all of which lie in the same slice of the crystal, as shown in Fig. 2. Interrogating the gratings with a collimated, chirped beam produces a measurement of the PRFs in the band of interest, as shown in Fig. 3. As the chirp sweeps in frequency, it reads the spatial absorption gratings one spectral bin at a time. A time-domain detector records the PRF measurement taken by the 1D detector
z
SHB Crystal
F 1
|S(ω0)|2
y θ
s(t)
V
s(t - y/V)
Fig. 1. Recording process for spatial–spectral holograms. A scanning _ mirror steers a modulated signal beam with an angular velocity y, producing a spot focused in the SHB crystal that travels at a velocity _ 1 . A signal that pulses on and off (as indicated by the dashes in the V ¼ yF crystal) writes a set of spectral holes (right) at the modulation carrier frequency. The period of this spatial–spectral absorption grating is determined by the signal PRF.
ARTICLE IN PRESS M. Colice et al. / Journal of Luminescence 127 (2007) 129–134
Scanning Mirror Stabilized Laser τPRF
EOM
spatial–spectral absorption grating that we write as
SHB Crystal
Lens θ
Lens
aðy; o; tÞ ¼ fa0 þ Z½gðyÞ aðt y=V ÞLðo o0 Þ
V=θF1
exp½ðt y=V Þ=T 1 g rectðy=hÞ,
F1
Signal
Λ=VτPRF Time
Fig. 2. Spatial–spectral grating write process for multiple pulsed signals. A laser beam propagates through an electro-optic modulator (EOM) driven by a set of carrier frequencies pulsed at different rates. Scanning and focusing the modulated beam in the crystal record a set of spectrally multiplexed gratings with periods given by the product of the scan velocity and the different pulse periods.
Lens Ramp
Lens
1-D Detector Array
Chirped Laser
Spot Size d = 100 μm Scan Velocity V = 0.3 m/s Focal Length f = 500 mm Angle φ = 0.002 rad Position Δy = 1-3 mm
Δy
φ
Fig. 3. Spatial–spectral grating interrogation process with various experimental parameters. A collimated beam illuminates the grating, producing diffracted orders that are focused to spots in the Fourier plane of the lens F 2 . As the beam chirps in frequency (top to bottom), it interrogates the gratings in different spectral bins. The detector records the spot position for each chirp frequency to give a measure of the PRF at different carrier frequencies.
array at the instantaneous chirp frequency. Tracking the spot positions and the instantaneous chirp frequency over time produces a record of the PRFs at different carrier frequencies. To get an analytic expression for the output at the detector plane, we consider a laser beam modulated with a signal sðtÞ ¼ aðtÞ expðio0 tÞ that illuminates an SHB crystal, where o0 is the carrier frequency and aðtÞ is the much slower envelope modulation. The crystal records a spectral hole centered at the carrier frequency and whose shape is given by 2
2
2
Lðo o0 Þ ¼ ð1=T 2 Þ =½ð1=T 2 Þ þ ðo o0 Þ .
(1)
_ The scanning mirror rotates with an angular velocity y, tilting the reflected beam at the same angular velocity, as shown in Fig. 1. Focusing the scanning beam in the crystal with a lens of focal length F 1 produces a spot with a Gaussian envelope gðyÞ that moves through a slice of _ 1 . The resulting signal, the crystal with a velocity V ¼ yF which has an envelope gðyÞ aðy VtÞ, produces a
ð2Þ
where indicates a convolution, a0 is the background absorption, Z is the writing efficiency, and h is the scan height. The exponential decay accounts for the excited state lifetime, T 1 . A chirped, collimated laser beam illuminates the grating at an angle in x to produce a set of diffracted beams. Fourier-transforming the diffracted beams with a lens produces diffracted spots whose displacements are a function of the PRF. Using a lens of focal length F 2 produces the Fourier-plane output ZZ E 0 exp½iðpkt2 þ os tÞ oð f y ; tÞ ¼ aðy; o; tÞ expði2pf y yÞ do dy,
SHB Crystal
131
ð3Þ
where E 0 is field amplitude of the chirped beam, k is the chirp rate in Hz/s, os is the chirp start frequency, and f y is the spatial coordinate in the Fourier plane. The double integral in Eq. (3) is a convolution in o and a Fourier transform in y. The convolution integral in Eq. (3) shows that the chirped beam only senses the grating when the chirp experiences the changes in absorption created by the signal beam. At the moment when the instantaneous chirp frequency sweeps through the bottom of the spectral hole (i.e., when pkt þ os ¼ o0 ), the chirped beam diffracts off the grating. The diffracted field in the Fourier plane is oð f y Þ / f1 GðvÞVAðVvÞ ½VT 1 =ð1 i2pVT 1 vÞg sincðhvÞ,
ð4Þ
where Gð f y Þ is the Fourier transform of gðyÞ, AðoÞ is the spectrum of aðtÞ, v ¼ f y =ðlF 2 Þ, and l is the laser wavelength. A sinusoidal envelope, for example, produces sidebands whose distance from the DC peak is given by the envelope modulation frequency—the PRF. Increasing the PRF moves the sidebands away from the DC order up to the limit imposed by Gð f y Þ, and decreasing the PRF moves them back. Similarly, as the instantaneous chirp frequency moves away from the signal frequency, the diffraction grows weaker and quickly disappears. 3. Experiment We perform PRF analysis in a 1.0% Tm3þ : YAG crystal that is 10 10 1 mm and antireflection coated on both faces, as shown in Fig. 4. Illuminating Tm3þ :YAG with laser light at 793.3 nm excites the 3 H6 to 3 H4 transition in Tm3þ , which has an inhomogeneous linewidth of about 20 GHz and a homogeneous linewidth of about 100 kHz. The excited state, which has a lifetime of hundreds of microseconds, decays preferentially to the 3 F4 level, which has a lifetime of about 10 ms [12]. We exploit this relatively long bottleneck state lifetime to write and read the
ARTICLE IN PRESS M. Colice et al. / Journal of Luminescence 127 (2007) 129–134
Fig. 4. Experimental setup for PRF analysis. A pulsed signal drives the write AOM, producing a pulsed beam that is steered by the galvo-scanning mirror to produce a beam that blinks on and off as it moves in the crystal plane (inset, left). Illuminating the resulting spatial–spectral grating with a brief pulse from the write AOM produces a diffracted spot (inset, right) whose position varies as a function of the grating period.
spatial–spectral absorption gratings used to find the carrier and PRF. An external-cavity diode laser stabilized to a spectral feature in a different spot in the crystal [13] generates the scanned and chirped beams that act as signal and read beams, respectively. A portion of the laser beam is siphoned off for laser stabilization purposes and the remainder is amplified with a tapered amplifier. The amplifier output, about 20 mW, propagates through a polarization-maintaining fiber to the experiment before being split into s^-polarized read and write beams, as shown in Fig. 4. The two beams illuminate a pair of AOMs, which are driven with two outputs from an arbitrary waveform generator (AWG). The 1 order beams from the AOMs write and read the grating. A scanning mirror steers the signal beam over a little less than 1 at a scan rate of 100 Hz (i.e., one scan period per excited state lifetime). Once the mirror reaches the bottom of its scan range, it resets to the top of its range and begins its next downward scan. A 200 mm focal-length lens Fourier-transforms the scanning beam into a moving spot focused in the volume of the crystal. The spot, which has a diameter of 100 mm, moves at about 0.3 m/s over a distance of about 3 mm for a space–bandwidth product of 30. The signal beam illuminates the grating at an angle in x after being focused in x and collimated in y using a cylindrical lens with a focal length of 150 mm in combination with the 200 mm focal-length spherical lens mentioned above. The beam is about 100 mm wide and 3 mm tall in the crystal. Because the write and read beams are tilted with respect to each other, the processor can record and interrogate gratings simultaneously. A 500 mm focal-length lens Fourier-transforms the beams diffracted by the grating. A razor blade blocks the DC and 1 orders in the crystal Fourier plane, which is imaged onto a 2D CCD using a 2 telescope not shown in Fig. 4.
In the simplest case, a pulsed carrier modulates the write AOM. A single pulse at the same carrier frequency drives the second AOM, producing a brief pulse of light that illuminates the grating. The diffracted beam is Fouriertransformed before being detected by the CCD. Driving the write AOM with more complicated waveforms produces spectrally multiplexed gratings. Chirping or stepping the frequency of the read pulse allows us to interrogate gratings in different spectral bins at different times. The grating write and read processes are synchronized to the mirror scan period, although the recording and interrogation processes can occur asynchronously. Collecting data at the output takes more careful timing. Interrogating the CCD too infrequently produces CCD traces containing the PRFs at multiple carrier frequencies. Conversely, interrogating the CCD too frequently produces outputs with poor signal-to-noise ratios (SNRs). Optimal processing occurs when we interrogate the CCD once for each carrier frequency. The noise characteristics of the CCD and the grating diffraction efficiency determine the minimum CCD integration time while the excited state lifetime fixes the maximum CCD integration time. The ratio of minimum to maximum integration times sets the number of carrier frequencies that our system can process. This number is usually much lower than the fundamental limit given by the number of spectral bins in the SHB crystal. 4. Results and discussion Fig. 5 shows the position of the focused diffracted beam as a function of the PRF. Our results show that we can distinguish 30 different PRFs with a resolution of 100 Hz and even see the second-order diffraction just below the dotted line. We could extend the PRF bandwidth upward by increasing the scan velocity or focusing the signal beam to a tighter spot, provided that the confocal length remains equal to or greater than the crystal thickness. For example, 3.0 2.5 Displacement (mm)
132
Variance Estimation Region
2.0 1.5 1.0 0.5 0.0 0.5
1.0
1.5 2.0 2.5 Pulse Repetition Frequency (kHz)
3.0
3.5
Fig. 5. Diffracted spot positions for single-carrier PRF cuing. The secondorder diffraction follows a path just below the dashed line.
ARTICLE IN PRESS M. Colice et al. / Journal of Luminescence 127 (2007) 129–134
1.0 Both Gratings 0.8 Amplitude (a.u)
focusing to a spot diameter of 50 mm in a crystal less than 5 mm thick and increasing the scan range to 10 mm would increase the space–bandwidth product to 200 and increase the PRF bandwidth to 20 kHz with a resolution of 100 Hz. Raster scanning a 50 mm spot in a 2D pattern across a 10 mm 10 mm crystal in 10 ms would increase the PRF bandwidth to 4 MHz while maintaining 100 Hz resolution in a folded spectrum [14]. We cannot, however, extend the PRF analysis bandwidth to lower frequencies for two reasons. The first is a fundamental limit: the excited state lifetime is 10 ms, so a PRF at 100 Hz will produce a grating with a single period before a 1=e decay. Unfortunately, we face a more immediate limit. The DC order produces enormous sidelobes in the Fourier plane, inhibiting our ability to detect weak diffractions. Interrogating a spatial grating with a continuous wave (cw) beam causes scatter from the fluctuating sidelobes to accumulate in the CCD, obscuring the diffracted spot. Using a brief read pulse reduces accumulation, and better apodization should also suppress these sidelobes, making it possible to interrogate the grating with chirped cw beams. For some reason, these sidelobes fluctuate wildly and unexpectedly at temperatures below 5 K, but the fluctuations stop when the crystal temperature rises to 6 K. Phonon broadening at temperatures above 8 K erases the grating. To produce Fig. 5, we record the spot on the 2D CCD and integrate the resulting frame along x. To compute the SNR, we excise tracks along the composite image corresponding to the first- and second-order peak locations and compute a weighted average background trace, which we subtract from the data. The fluctuating sidelobes make this subtraction imperfect, so we compute the SNR by using the variance in each slice of the wedge marked by the dashed line. For a write beam power of 1 mW and a read beam power beam of 1–5 mW, a read pulse between 250 ms and 1 ms long gives a median SNR of about 21 dB. The CCD limits the dynamic range to about 30 dB, which is usually enough for PRF measurements—the strength of the envelope is not as important as knowing that the carrier is being modulated. Fig. 6 shows the diffracted spots from a pair of spectrally multiplexed gratings. The first grating is at an RF carrier frequency of 100 MHz and a PRF of 1 kHz. The second grating is at an RF carrier frequency of 110 MHz and a PRF of 1.67 kHz. Fig. 6 shows the diffracted spots from the first grating, the second grating, and both sequentially within a single CCD integration period. By timing the CCD line readouts to coincide with the read pulses, we can read the gratings sequentially and capture the result on an oscilloscope. The SNR is about 13 dB, simply because the coupled-mode response of the write AOM causes the diffraction efficiency to fall as the number of modes squared, hence the roughly 6 dB drop in SNR. Programming the write AOM waveform more intelligently can recover some of this drop, and increasing the optical power would also improve the SNR.
133
110 MHz Grating
0.6
0.4
0.2
100 MHz Grating
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Displacement (mm) Fig. 6. Diffracted peaks for spectrally multiplexed gratings with different spatial periods. Interrogating the first grating at 100 MHz produces one peak, interrogating the second grating at 110 MHz produces a shifted peak, and interrogating both sequentially in a single CCD integration period produces both peaks.
5. Conclusion By mapping slow time into space, we can resolve signal envelopes with a resolution of 100 Hz without any extraordinary laser stabilization or cooling the crystal to temperatures below 4 K. To our knowledge, the only other way to achieve spectral resolution on this scale in SHB media is by operating in the photon echo regime. Increasing the system’s space–bandwidth product by focusing to a smaller spot would increase both the maximum detectable PRF and the number of PRF channels while maintaining 100 Hz resolution for 10 ms raster scan times. Eliminating or suppressing scatter from the main beam will improve the dynamic range, which is essential to increasing the number of spectrally multiplexed channels. Our demonstration shows that spectral multiplexing is possible, but the SNR must be increased to make it practical. Acknowledgements We thank S. Pappert of the DARPA AOSP program for funding, and K. Merkel and Z. Cole of S2Corp for generously loaning us a laser stabilization circuit. M. Colice also thanks the NSF IGERT OSEP program for financial support. References [1] M. Colice, F. Schlottau, K. Wagner, Appl. Opt. 45 (2006). [2] R.K. Mohan, et al., in: IEEE International Topical Meeting on Microwave Photonics, Ogunquit, ME, USA, 2004. [3] F. Schlottau, M. Colice, K.H. Wagner, W.R. Babbitt, Opt. Lett. 30 (2005) 3003. [4] G. Gorju, V. Crozatier, I. Lorgere´, J.-L. Le Goue¨t, F. Bretenaker, IEEE Photonics Technol. Lett. 17 (2005) 2385.
ARTICLE IN PRESS 134
M. Colice et al. / Journal of Luminescence 127 (2007) 129–134
[5] G. Gorju, V. Crozatier, V. Lavielle, I. Lorgere´, J.-L. Le Goue¨t, F. Bretenaker, Eur. Phys. J. 30 (2005) 175. [6] V. Crozatier, G. Gorju, F. Bretenaker, J.-L. Le Goue´t, I. Lorgere´, C. Gagnol, E. Ducloux, J. Lumin. (2006). [7] T. Turpin, Proc. IEEE 69 (1981) 80. [8] D.C. Hartup, W.T. Rhodes, in: D.R. Pape (Ed.), Proceedings of SPIE on Advances in Optical Information Processing IV, vol. 1296, 1990. [9] D.C. Hartup, W.T. Rhodes, H.F. Engler, A.K. Garrison, in: D.R. Pape (Ed.), Proceedings of SPIE on Advances in Optical Information Processing V, vol. 1704, AP, NJ, 1992.
[10] V. Lavielle, F. De Seze, I. Lorgere´, J.-L. Le Goue¨t, J. Lumin. 107 (2004) 75. [11] F. Schlottau, K.H. Wagner, J. Lumin. 107 (2004) 90. [12] G. Armagan, A.M. Buoncristiani, B. Di Bartolo, Opt. Mater. 1 (1992) 11. [13] N.M. Strickland, P.B. Sellin, Y. Sun, J.L. Carlsten, R.L. Cone, Phys. Rev. B 62 (2000) 1473. [14] C.E. Thomas, Appl. Opt. 5 (1966) 1782.
Journal of Luminescence 127 (2007) 129–134 www.elsevier.com/locate/jlumin
Spectral hole burning for pulse repetition frequency analysis Max Colice, Jingyi Xiong, Kelvin Wagner Department of Electrical & Computer Engineering, Optoelectronic Computing Systems Center, University of Colorado, Boulder, CO 80309-0425, USA Available online 3 March 2007
Abstract We demonstrate an optical processor based on spectral hole burning (SHB) that maps the carrier frequency into the time domain and the pulse repetition frequency (PRF) into the spatial domain by illuminating an SHB crystal with a signal beam that is scanned by a tilting mirror across a slice of the crystal. This time-to-space mapping makes it possible to measure signal envelopes with a resolution of 1=T 1 ¼ 100 Hz. A signal with a pulsed envelope engraves a vertical absorption grating with a spatial periodicity given by the product of the PRF and the scan velocity. Reading the grating, which the crystal stores for up to T 1 , with a collimated beam yields orders diffracted at angles proportional to the PRF, which are Fourier-transformed to produce spots displaced from the DC position by distances proportional to the PRF. Increasing the PRF increases the grating periodicity, causing the diffracted spots to move away from the DC position. r 2007 Elsevier B.V. All rights reserved. PACS: 42.79.Hp; 42.40.My Keywords: Spatial–spectral holography; Spectral hole burning; Microwave photonics
1. Introduction Recent work on spectrum analysis using spectral hole burning (SHB) exploits the large inhomogeneous bandwidth of SHB crystals to capture broadband signals with unity probability of intercept [1–4]. At present, spectrum analysis over bandwidths of more than 10 GHz using SHB are single-shot measurements with resolutions limited by the linewidth of the lasers used to read and write the holes. Improving the resolution below 25 kHz will require ultrastable lasers, crystals cooled to below 4 K, and new techniques to stabilize chirped lasers [5,6]. Obtaining subkilohertz resolution of spectral features usually requires looking at coherent effects, such as photon echoes, instead of spectral holes. Here, we demonstrate a processor that detects pulsed signals with a resolution of 100 Hz, which is well below the homogeneous linewidth. We do this with a single-shot technique that transforms slow time into space,
Corresponding author.
E-mail address: [email protected] (M. Colice). 0022-2313/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2007.02.052
a trick that does not require extraordinary laser stabilization or crystal-cooling measures. Transforming time into space for optical processing is an old idea that plays on the inherent spatial parallelism of optics and the Fourier-transform properties of coherent light. The traveling acoustic waves in acousto-optic deflectors (AODs), for example, turn RF modulation into spatial index modulations. Illuminating the resulting index grating with a collimated laser beam produces beams diffracted at angles proportional to the RF frequency. Fourier-transforming the diffracted beams produces spots whose positions are, in turn, proportional to the diffraction angle. The spots in the Fourier plane of the AOD are a scaled version of the RF modulation power spectrum mapped into the spatial domain [7]. Some signals, such as those emitted by pulse-Doppler radars, are difficult to analyze because they usually consist of high-frequency carriers pulsed at kilohertz rates. For example, adequately sampling a carrier at 1 GHz pulsed at 10 kHz requires at least a million channels. Pulse repetition frequency (PRF) analyzers based on acousto-optic tripleproduct processors project the carrier frequency into one
ARTICLE IN PRESS 130
M. Colice et al. / Journal of Luminescence 127 (2007) 129–134
spatial dimension and the PRF into the other such that the total number of channels is the product of the space– bandwidth products of the two acousto-optic modulators (AOMs) used in the processor [8,9]. Acoustic wave generation and propagation limit these systems to signal bandwidths of less than 1 GHz and about 106 spectral channels, assuming a space–bandwidth product of 103 for each of the pair of acousto-optic devices in the processor. SHB crystals can process much higher bandwidth signals using the spectral degree of freedom that they possess in addition to the spatial degrees of freedom used by conventional optical processors. As we demonstrated previously, SHB crystals are capable of storing more than 20,000 channels in the spectral domain over bandwidths of 20 GHz or more [1]. SHB crystals also accommodate spatially varying signals for use in spectrum analysis [10] and time-integrating correlators [11] with space–bandwidth products of over 103 . In this experiment, an SHB crystal records the PRF and carrier frequency of a modulated laser beam as a spatial–spectral absorption grating. Spatially scanning the beam across a slice of the crystal engraves a spatial absorption grating into the crystal at the spectral bin corresponding to the carrier frequency. Interrogating the grating’s spectral location with a chirped laser beam yields the carrier frequency, and measuring the grating’s spatial periodicity yields the PRF. Scanning a spot with a diameter of 50 mm and a confocal length equal to or greater than the crystal thickness across a crystal 10 mm in width gives a space–bandwidth product of about 500. Combining this with the 200,000 channels in the spectral domain gives us 108 total channels for processing pulsed signals with carrier frequencies up to 20 GHz.
2. Theory Illuminating a single spot in an SHB crystal with a modulated signal beam engraves spectral holes into the absorption profile of the material at frequencies corresponding to the signal spectrum [1–4]. These holes persist in the crystal for the excited state lifetime and can be mapped by probing the absorption profile with a lowpower chirp. As the chirp sweeps across a hole, the transmitted intensity increases in proportion to the hole depth, which is itself proportional to the strength of the corresponding component of the signal power spectrum. Mapping the instantaneous chirp frequency to the temporal coordinate of the transmitted power provides a measurement of the signal power spectrum, jSðktÞj2 , blurred by a Lorentzian whose width is 1=T 2 , the crystal’s homogeneous linewidth. The inhomogeneous linewidth, 1=T 2 , determines the maximum signal bandwidth, and the homogeneous linewidth determines the resolution limit. For Tm3þ :YAG crystals with 1=T 2 20 GHz and 1=T 2 100 kHz, this translates to a spectrum analyzer time–bandwidth product of more than 105 .
Mapping space into time with a scanning mirror causes the SHB crystal to record the pulse rate of high-frequency carriers as spatial–spectral gratings. As in previous spectrum analysis experiments [1–4], a modulated signal beam illuminates the crystal, but the mirror scans the signal beam to record the signal’s time/frequency history in a slice of the crystal instead of integrating the signal spectrum in a single spot. The crystal records the carrier frequency in the spectral domain, as before, while the PRF is mapped into the vertical dimension of the slice, as shown in Fig. 1. The spot’s movement does induce a symmetric Doppler spread in the carrier frequency equal to the spot size divided by the apparent spot velocity, or several kilohertz, which is much smaller than the spectral sensitivity of the crystal. A signal that blinks on and off at the PRF engraves a vertical absorption grating with a spatial periodicity given by the product of the PRF and the scan velocity. The optical system and the PRF band of interest determine the scan time, which should not exceed the excited state lifetime of T 1 ¼ 10 ms. Illuminating the grating, which the crystal stores for up to T 1 , with a collimated beam yields orders diffracted at angles proportional to the PRF. Fourier-transforming the diffracted beams with a lens produces spots displaced from the DC position by distances proportional to the PRF. A 1-dimensional (1D) detector in the Fourier plane of the grating records the positions of these spots, yielding a measure of the PRF. Increasing the PRF increases the grating’s spatial frequency, shifting the diffracted spots away from the DC position. Because SHB crystals perform spectral discrimination, they can record a set of spectrally multiplexed gratings in a single slice. The different carriers each write their own absorption grating, all of which lie in the same slice of the crystal, as shown in Fig. 2. Interrogating the gratings with a collimated, chirped beam produces a measurement of the PRFs in the band of interest, as shown in Fig. 3. As the chirp sweeps in frequency, it reads the spatial absorption gratings one spectral bin at a time. A time-domain detector records the PRF measurement taken by the 1D detector
z
SHB Crystal
F 1
|S(ω0)|2
y θ
s(t)
V
s(t - y/V)
Fig. 1. Recording process for spatial–spectral holograms. A scanning _ mirror steers a modulated signal beam with an angular velocity y, producing a spot focused in the SHB crystal that travels at a velocity _ 1 . A signal that pulses on and off (as indicated by the dashes in the V ¼ yF crystal) writes a set of spectral holes (right) at the modulation carrier frequency. The period of this spatial–spectral absorption grating is determined by the signal PRF.
ARTICLE IN PRESS M. Colice et al. / Journal of Luminescence 127 (2007) 129–134
Scanning Mirror Stabilized Laser τPRF
EOM
spatial–spectral absorption grating that we write as
SHB Crystal
Lens θ
Lens
aðy; o; tÞ ¼ fa0 þ Z½gðyÞ aðt y=V ÞLðo o0 Þ
V=θF1
exp½ðt y=V Þ=T 1 g rectðy=hÞ,
F1
Signal
Λ=VτPRF Time
Fig. 2. Spatial–spectral grating write process for multiple pulsed signals. A laser beam propagates through an electro-optic modulator (EOM) driven by a set of carrier frequencies pulsed at different rates. Scanning and focusing the modulated beam in the crystal record a set of spectrally multiplexed gratings with periods given by the product of the scan velocity and the different pulse periods.
Lens Ramp
Lens
1-D Detector Array
Chirped Laser
Spot Size d = 100 μm Scan Velocity V = 0.3 m/s Focal Length f = 500 mm Angle φ = 0.002 rad Position Δy = 1-3 mm
Δy
φ
Fig. 3. Spatial–spectral grating interrogation process with various experimental parameters. A collimated beam illuminates the grating, producing diffracted orders that are focused to spots in the Fourier plane of the lens F 2 . As the beam chirps in frequency (top to bottom), it interrogates the gratings in different spectral bins. The detector records the spot position for each chirp frequency to give a measure of the PRF at different carrier frequencies.
array at the instantaneous chirp frequency. Tracking the spot positions and the instantaneous chirp frequency over time produces a record of the PRFs at different carrier frequencies. To get an analytic expression for the output at the detector plane, we consider a laser beam modulated with a signal sðtÞ ¼ aðtÞ expðio0 tÞ that illuminates an SHB crystal, where o0 is the carrier frequency and aðtÞ is the much slower envelope modulation. The crystal records a spectral hole centered at the carrier frequency and whose shape is given by 2
2
2
Lðo o0 Þ ¼ ð1=T 2 Þ =½ð1=T 2 Þ þ ðo o0 Þ .
(1)
_ The scanning mirror rotates with an angular velocity y, tilting the reflected beam at the same angular velocity, as shown in Fig. 1. Focusing the scanning beam in the crystal with a lens of focal length F 1 produces a spot with a Gaussian envelope gðyÞ that moves through a slice of _ 1 . The resulting signal, the crystal with a velocity V ¼ yF which has an envelope gðyÞ aðy VtÞ, produces a
ð2Þ
where indicates a convolution, a0 is the background absorption, Z is the writing efficiency, and h is the scan height. The exponential decay accounts for the excited state lifetime, T 1 . A chirped, collimated laser beam illuminates the grating at an angle in x to produce a set of diffracted beams. Fourier-transforming the diffracted beams with a lens produces diffracted spots whose displacements are a function of the PRF. Using a lens of focal length F 2 produces the Fourier-plane output ZZ E 0 exp½iðpkt2 þ os tÞ oð f y ; tÞ ¼ aðy; o; tÞ expði2pf y yÞ do dy,
SHB Crystal
131
ð3Þ
where E 0 is field amplitude of the chirped beam, k is the chirp rate in Hz/s, os is the chirp start frequency, and f y is the spatial coordinate in the Fourier plane. The double integral in Eq. (3) is a convolution in o and a Fourier transform in y. The convolution integral in Eq. (3) shows that the chirped beam only senses the grating when the chirp experiences the changes in absorption created by the signal beam. At the moment when the instantaneous chirp frequency sweeps through the bottom of the spectral hole (i.e., when pkt þ os ¼ o0 ), the chirped beam diffracts off the grating. The diffracted field in the Fourier plane is oð f y Þ / f1 GðvÞVAðVvÞ ½VT 1 =ð1 i2pVT 1 vÞg sincðhvÞ,
ð4Þ
where Gð f y Þ is the Fourier transform of gðyÞ, AðoÞ is the spectrum of aðtÞ, v ¼ f y =ðlF 2 Þ, and l is the laser wavelength. A sinusoidal envelope, for example, produces sidebands whose distance from the DC peak is given by the envelope modulation frequency—the PRF. Increasing the PRF moves the sidebands away from the DC order up to the limit imposed by Gð f y Þ, and decreasing the PRF moves them back. Similarly, as the instantaneous chirp frequency moves away from the signal frequency, the diffraction grows weaker and quickly disappears. 3. Experiment We perform PRF analysis in a 1.0% Tm3þ : YAG crystal that is 10 10 1 mm and antireflection coated on both faces, as shown in Fig. 4. Illuminating Tm3þ :YAG with laser light at 793.3 nm excites the 3 H6 to 3 H4 transition in Tm3þ , which has an inhomogeneous linewidth of about 20 GHz and a homogeneous linewidth of about 100 kHz. The excited state, which has a lifetime of hundreds of microseconds, decays preferentially to the 3 F4 level, which has a lifetime of about 10 ms [12]. We exploit this relatively long bottleneck state lifetime to write and read the
ARTICLE IN PRESS M. Colice et al. / Journal of Luminescence 127 (2007) 129–134
Fig. 4. Experimental setup for PRF analysis. A pulsed signal drives the write AOM, producing a pulsed beam that is steered by the galvo-scanning mirror to produce a beam that blinks on and off as it moves in the crystal plane (inset, left). Illuminating the resulting spatial–spectral grating with a brief pulse from the write AOM produces a diffracted spot (inset, right) whose position varies as a function of the grating period.
spatial–spectral absorption gratings used to find the carrier and PRF. An external-cavity diode laser stabilized to a spectral feature in a different spot in the crystal [13] generates the scanned and chirped beams that act as signal and read beams, respectively. A portion of the laser beam is siphoned off for laser stabilization purposes and the remainder is amplified with a tapered amplifier. The amplifier output, about 20 mW, propagates through a polarization-maintaining fiber to the experiment before being split into s^-polarized read and write beams, as shown in Fig. 4. The two beams illuminate a pair of AOMs, which are driven with two outputs from an arbitrary waveform generator (AWG). The 1 order beams from the AOMs write and read the grating. A scanning mirror steers the signal beam over a little less than 1 at a scan rate of 100 Hz (i.e., one scan period per excited state lifetime). Once the mirror reaches the bottom of its scan range, it resets to the top of its range and begins its next downward scan. A 200 mm focal-length lens Fourier-transforms the scanning beam into a moving spot focused in the volume of the crystal. The spot, which has a diameter of 100 mm, moves at about 0.3 m/s over a distance of about 3 mm for a space–bandwidth product of 30. The signal beam illuminates the grating at an angle in x after being focused in x and collimated in y using a cylindrical lens with a focal length of 150 mm in combination with the 200 mm focal-length spherical lens mentioned above. The beam is about 100 mm wide and 3 mm tall in the crystal. Because the write and read beams are tilted with respect to each other, the processor can record and interrogate gratings simultaneously. A 500 mm focal-length lens Fourier-transforms the beams diffracted by the grating. A razor blade blocks the DC and 1 orders in the crystal Fourier plane, which is imaged onto a 2D CCD using a 2 telescope not shown in Fig. 4.
In the simplest case, a pulsed carrier modulates the write AOM. A single pulse at the same carrier frequency drives the second AOM, producing a brief pulse of light that illuminates the grating. The diffracted beam is Fouriertransformed before being detected by the CCD. Driving the write AOM with more complicated waveforms produces spectrally multiplexed gratings. Chirping or stepping the frequency of the read pulse allows us to interrogate gratings in different spectral bins at different times. The grating write and read processes are synchronized to the mirror scan period, although the recording and interrogation processes can occur asynchronously. Collecting data at the output takes more careful timing. Interrogating the CCD too infrequently produces CCD traces containing the PRFs at multiple carrier frequencies. Conversely, interrogating the CCD too frequently produces outputs with poor signal-to-noise ratios (SNRs). Optimal processing occurs when we interrogate the CCD once for each carrier frequency. The noise characteristics of the CCD and the grating diffraction efficiency determine the minimum CCD integration time while the excited state lifetime fixes the maximum CCD integration time. The ratio of minimum to maximum integration times sets the number of carrier frequencies that our system can process. This number is usually much lower than the fundamental limit given by the number of spectral bins in the SHB crystal. 4. Results and discussion Fig. 5 shows the position of the focused diffracted beam as a function of the PRF. Our results show that we can distinguish 30 different PRFs with a resolution of 100 Hz and even see the second-order diffraction just below the dotted line. We could extend the PRF bandwidth upward by increasing the scan velocity or focusing the signal beam to a tighter spot, provided that the confocal length remains equal to or greater than the crystal thickness. For example, 3.0 2.5 Displacement (mm)
132
Variance Estimation Region
2.0 1.5 1.0 0.5 0.0 0.5
1.0
1.5 2.0 2.5 Pulse Repetition Frequency (kHz)
3.0
3.5
Fig. 5. Diffracted spot positions for single-carrier PRF cuing. The secondorder diffraction follows a path just below the dashed line.
ARTICLE IN PRESS M. Colice et al. / Journal of Luminescence 127 (2007) 129–134
1.0 Both Gratings 0.8 Amplitude (a.u)
focusing to a spot diameter of 50 mm in a crystal less than 5 mm thick and increasing the scan range to 10 mm would increase the space–bandwidth product to 200 and increase the PRF bandwidth to 20 kHz with a resolution of 100 Hz. Raster scanning a 50 mm spot in a 2D pattern across a 10 mm 10 mm crystal in 10 ms would increase the PRF bandwidth to 4 MHz while maintaining 100 Hz resolution in a folded spectrum [14]. We cannot, however, extend the PRF analysis bandwidth to lower frequencies for two reasons. The first is a fundamental limit: the excited state lifetime is 10 ms, so a PRF at 100 Hz will produce a grating with a single period before a 1=e decay. Unfortunately, we face a more immediate limit. The DC order produces enormous sidelobes in the Fourier plane, inhibiting our ability to detect weak diffractions. Interrogating a spatial grating with a continuous wave (cw) beam causes scatter from the fluctuating sidelobes to accumulate in the CCD, obscuring the diffracted spot. Using a brief read pulse reduces accumulation, and better apodization should also suppress these sidelobes, making it possible to interrogate the grating with chirped cw beams. For some reason, these sidelobes fluctuate wildly and unexpectedly at temperatures below 5 K, but the fluctuations stop when the crystal temperature rises to 6 K. Phonon broadening at temperatures above 8 K erases the grating. To produce Fig. 5, we record the spot on the 2D CCD and integrate the resulting frame along x. To compute the SNR, we excise tracks along the composite image corresponding to the first- and second-order peak locations and compute a weighted average background trace, which we subtract from the data. The fluctuating sidelobes make this subtraction imperfect, so we compute the SNR by using the variance in each slice of the wedge marked by the dashed line. For a write beam power of 1 mW and a read beam power beam of 1–5 mW, a read pulse between 250 ms and 1 ms long gives a median SNR of about 21 dB. The CCD limits the dynamic range to about 30 dB, which is usually enough for PRF measurements—the strength of the envelope is not as important as knowing that the carrier is being modulated. Fig. 6 shows the diffracted spots from a pair of spectrally multiplexed gratings. The first grating is at an RF carrier frequency of 100 MHz and a PRF of 1 kHz. The second grating is at an RF carrier frequency of 110 MHz and a PRF of 1.67 kHz. Fig. 6 shows the diffracted spots from the first grating, the second grating, and both sequentially within a single CCD integration period. By timing the CCD line readouts to coincide with the read pulses, we can read the gratings sequentially and capture the result on an oscilloscope. The SNR is about 13 dB, simply because the coupled-mode response of the write AOM causes the diffraction efficiency to fall as the number of modes squared, hence the roughly 6 dB drop in SNR. Programming the write AOM waveform more intelligently can recover some of this drop, and increasing the optical power would also improve the SNR.
133
110 MHz Grating
0.6
0.4
0.2
100 MHz Grating
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Displacement (mm) Fig. 6. Diffracted peaks for spectrally multiplexed gratings with different spatial periods. Interrogating the first grating at 100 MHz produces one peak, interrogating the second grating at 110 MHz produces a shifted peak, and interrogating both sequentially in a single CCD integration period produces both peaks.
5. Conclusion By mapping slow time into space, we can resolve signal envelopes with a resolution of 100 Hz without any extraordinary laser stabilization or cooling the crystal to temperatures below 4 K. To our knowledge, the only other way to achieve spectral resolution on this scale in SHB media is by operating in the photon echo regime. Increasing the system’s space–bandwidth product by focusing to a smaller spot would increase both the maximum detectable PRF and the number of PRF channels while maintaining 100 Hz resolution for 10 ms raster scan times. Eliminating or suppressing scatter from the main beam will improve the dynamic range, which is essential to increasing the number of spectrally multiplexed channels. Our demonstration shows that spectral multiplexing is possible, but the SNR must be increased to make it practical. Acknowledgements We thank S. Pappert of the DARPA AOSP program for funding, and K. Merkel and Z. Cole of S2Corp for generously loaning us a laser stabilization circuit. M. Colice also thanks the NSF IGERT OSEP program for financial support. References [1] M. Colice, F. Schlottau, K. Wagner, Appl. Opt. 45 (2006). [2] R.K. Mohan, et al., in: IEEE International Topical Meeting on Microwave Photonics, Ogunquit, ME, USA, 2004. [3] F. Schlottau, M. Colice, K.H. Wagner, W.R. Babbitt, Opt. Lett. 30 (2005) 3003. [4] G. Gorju, V. Crozatier, I. Lorgere´, J.-L. Le Goue¨t, F. Bretenaker, IEEE Photonics Technol. Lett. 17 (2005) 2385.
ARTICLE IN PRESS 134
M. Colice et al. / Journal of Luminescence 127 (2007) 129–134
[5] G. Gorju, V. Crozatier, V. Lavielle, I. Lorgere´, J.-L. Le Goue¨t, F. Bretenaker, Eur. Phys. J. 30 (2005) 175. [6] V. Crozatier, G. Gorju, F. Bretenaker, J.-L. Le Goue´t, I. Lorgere´, C. Gagnol, E. Ducloux, J. Lumin. (2006). [7] T. Turpin, Proc. IEEE 69 (1981) 80. [8] D.C. Hartup, W.T. Rhodes, in: D.R. Pape (Ed.), Proceedings of SPIE on Advances in Optical Information Processing IV, vol. 1296, 1990. [9] D.C. Hartup, W.T. Rhodes, H.F. Engler, A.K. Garrison, in: D.R. Pape (Ed.), Proceedings of SPIE on Advances in Optical Information Processing V, vol. 1704, AP, NJ, 1992.
[10] V. Lavielle, F. De Seze, I. Lorgere´, J.-L. Le Goue¨t, J. Lumin. 107 (2004) 75. [11] F. Schlottau, K.H. Wagner, J. Lumin. 107 (2004) 90. [12] G. Armagan, A.M. Buoncristiani, B. Di Bartolo, Opt. Mater. 1 (1992) 11. [13] N.M. Strickland, P.B. Sellin, Y. Sun, J.L. Carlsten, R.L. Cone, Phys. Rev. B 62 (2000) 1473. [14] C.E. Thomas, Appl. Opt. 5 (1966) 1782.