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Femtosecond laser range finders based on traditional cross correlation method and frequency resolved dispersion compensation method
Optics Communications 316 (2014) 179–189
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Femtosecond laser range finders based on traditional cross correlation method and frequency resolved dispersion compensation method Fu Yang, Jing Zhang, Yage Zhan n Department of Physics, Donghua University, No. 2999, North Renmin Road, Shanghai 201620, China
art ic l e i nf o
a b s t r a c t
Article history: Received 22 August 2013 Received in revised form 21 November 2013 Accepted 22 November 2013 Available online 16 December 2013
In order to realize high range accuracy, two laser range finder systems using broadband femtosecond lasers as optical sources are discussed and compared in detail. One system uses a traditional cross correlation method, and the other system uses a frequency resolved dispersion compensation method. Better than 0.65 μm distance accuracy and ambiguity range of pulse-to-pulse separation can be realized in both systems. In the system using the traditional cross correlation method, submicron step resolution of the scanning system should be maintained to yield an authentic cross correlation graph. While in the system using the frequency resolved dispersion compensation method, no scanning system is required as the signal is spectrally resolved into several channels and the process is finished in software. In the system using the frequency resolved dispersion compensation method, the influence of channel number is much more significant than that of channel spacing and individual channel bandwidth. And the channel spacing should be a multiple integral of the laser repetition rate to facilitate the measurement; otherwise the optical path difference must be controlled to less than a pulse-to-pulse separation. Both systems are well suited to the range lidar requiring submicron distance accuracy, and can be applied in the future space missions. & 2013 Elsevier B.V. All rights reserved.
Keywords: Laser range finder Lidar Remote sensing and sensors Ultrafast lasers Dispersion compensation
1. Introduction Laser range finder plays an important role in the field of laser remote sensing. Based on a satellite or airborne platform, threedimensional map of the detected area can be achieved, which provides a powerful tool for the planetary exploration, position, navigation, and disaster relief. As a passion for precision, researchers worldwide are seeking a way to realize higher range resolution. The development of mode locking and frequency stabilization technique in recent years has commercialized the femtosecond laser frequency combs [1–3]. It is natural to extend the light source to range measurement since the broad spectrum supports high range resolution. The lidar system with femtosecond laser frequency combs usually adopts a configuration similar to that of Michelson's interferometer. In terms of the detection method, Cui et al. [4] and Matsumoto et al. [5] adopt the crosscorrelation method; a fine scanning system must be equipped in the system, and the target distance is obtained through the maximum value of the cross correlation. This is the most traditional ranging method using the femtosecond laser, and an ambiguity range of pulse-to-pulse distance can be realized. Xia and Zhang [6,7] use a dispersive interferometer method; target
n
Corresponding author. E-mail address: [email protected] (Y. Zhan).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.11.043
distance can be calculated from the data recorded in the oscilloscope (OSC) or optical spectrum analyzer (OSA). The measurement principle is convenient to apply; however, the dynamic range is restricted by the bandwidth of the OSC, which is 8.7 mm in the experiment by using 7 GHz bandwidth OSC [7]. Swann and Newbury [8] employ the frequency resolved dispersion compensation method, which is an extremely attractive method; the coherent signal is spectrally resolved into N channels, phase compensation is implemented to each channel and then summed up to realize range detection. However, details of the principle are missed in the paper, and the reader is confused. Coddington et al. [9] apply a pair of stabilized femtosecond laser frequency combs of slightly different frequencies so the ambiguity range can be extended greatly. The ambiguity range is 30 km in the paper which is the longest value reported in the literature until now. As two phase locked femtosecond lasers are required, the system is both expensive and complicated. Lepetit et al. [10], Rovati et al. [11], Doloca et al. [12], and Pesatori et al. [13] measure the distance by the phase delay of the returning optical pulse train with respect to the outgoing one. Lepetit et al. [10] and Rovati et al. [11] use optical coherent detection methods and spectrally resolve the coherent signal to the line CCDs. Phase information induced by a signal range is obtained. The difference is that Rovati et al. [11] replace the femtosecond laser by a low cost multi-longitudinal mode laser. Doloca et al. [12] and Pesatori et al. [13] use independent photodetectors to measure the signal pulse and reference pulse;
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phase difference is judged in an electric domain. The difference between them is that Doloca et al. [12] evaluate the phase shift of two distinct comb frequencies. A coarse measurement is performed by a comb line of 100 MHz, and a fine measurement is performed by a comb line of 11.4 GHz; the two results are combined to obtain range information. Pesatori et al. [13] expand the time interval corresponding to the phase delay by downconverting beat notes from high harmonics of the pulse repetition rate (up to 10 GHz) to several kilohertz in the electric domain; this way both high range resolution (based on 10 GHz) and convenient phase measurement (based on the time period of several kilohertz) can be realized. As the target range is calculated by the phase difference, the target distance will be affected by any phase disturbance, such as changes in temperature, pressure, humidity and so on. It means that the phase detection method is exact strictly with the operation environment. In this paper, for the first time to our knowledge, the detailed theory of the frequency resolved dispersion compensation method used in the femtosecond ranging system is illustrated. And it is also compared with the traditional cross correlation method to show its advantages. This paper is arranged as follows. Section 1 presented the introduction part. In Section 2 the system adopting the traditional cross correlation method is described. The principle, ambiguity range and distance accuracy, influence of the optical path difference and the spectral width, and the requirement of the scanning system's step resolution are included. In Section 3 the system using the frequency resolved dispersion compensation method is described. The principle, ambiguity range and distance accuracy, the requirement of optical delay line's adjustment range, and the influence of the channel number, channel spacing, and channel bandwidth are given. Section 4 presents the conclusion.
2. Femtosecond laser range finder based on the traditional cross correlation method 2.1. Principle A schematic of the femtosecond laser range finder based on the traditional cross correlation method is shown in Fig. 1. The optical source adopts phase-stabilized mode-locked femtosecond laser. This source emits a broad spectrum of light that comprises a set of narrow individual optical lines in frequency space separated by the laser repetition rate. A beam splitter divides the light into two
parts. One part serves as a local oscillator (LO), and the other part serves as the main oscillator (MO). The LO is in the non-varying arm, and light is reflected by a retro-reflector. The optical path of the MO can be varied with a scanning system. The two beams combine with each other, and are then detected by a photodetector. The electric field of LO and MO is expressed, respectively, as ELO and ESIG in the following equations: N
ELO ðtÞ ¼ ∑ a1;n ðωn Þ exp ðiωn tÞ
ð1Þ
n¼1 N
ESIG ðtÞ ¼ ∑ a2;n ðωn Þ exp ðiωn tÞ
ð2Þ
n¼1
where a1;n ðωn Þ and a2;n ðωn Þ are the signal spectra of LO and MO respectively; ωn ¼ ω0 þnωrep , ωrep ¼ 2πf rep is the angular repetition rate, and ω0 is the offset frequency due to the difference of group velocity and phase velocity. After traversing different paths x1 and x2 in air, the two signals can be expressed as ELO ðt; x1 Þ and ESIG ðt; x2 Þ in the following equations: N
ELO ðt; x1 Þ ¼ ∑ a1;n ðωn ÞRLO exp ðiωn t kω x1 þ ϕ1;n Þ
ð3Þ
n¼1 N
ESIG ðt; x2 Þ ¼ ∑ a2;n ðωn ÞRSIG exp ðiωn t kω x2 þ ϕ2;n Þ
ð4Þ
n¼1
where kω ¼ nω ω=c is the wave vector; nω is the refractive index in air deduced by the Ciddor equation [14], which is closely related to the system environment, such as temperature, atmospheric pressure, humidity and so on; RLO and RSIG represent the transmission losses in the LO and MO paths; ϕ1;n and ϕ2;n are the phase offsets in the two arms. As the femtosecond pulses are too fast to respond, the output of the detector shown in Eq. (5) is proportional to the average value of the two beams cross correlation signal: I ω ¼ ℜ ½ELO ðt; x1 Þ þESIG ðt; x2 Þ½ELO ðt; x1 Þ þ ESIG ðt; x2 Þn ð5Þ where ℜ is the proportion factor. Assuming x ¼ x2 x1 , ϕ ¼ ϕ2;n ϕ1;n , and substituting Eqs. (3) and (4) into Eq. (5), we can get the following equation: N h i I ω ðxÞ ¼ ℜ ∑ j a1;n ðωn ÞRLO j 2 þ j a2;n ðωn ÞRSIG j 2 n¼1 N
þ 2 ∑ j a1;n ðωn ÞRLO :a2;n ðωn ÞRSIG j cos ðkω x þ ϕÞ n¼1
ð6Þ
As shown in Eq. (6), I ω varies with the optical path difference x. Eq. (6) consists of three items. The first two items are DC signals.
Fig. 1. Schematic of the femtosecond laser range finder based on the traditional cross correlation method.
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The influence of optical path difference x is reflected in the last item. For convenience, set ϕ as a constant. Assuming that the optical spectrum is Gauss shaped, the normalized spectrum and the dispersion situation of the optical source are shown in Fig. 2, using the system and environment parameters shown in Table 1. As shown in Fig. 2, the refractive index difference in the spectral range is 0.1 ppm. The optical path difference can be controlled accurately with a scanning system, like a piezo-electric system. The AC output of the detector is shown in Fig. 3 when the optical path difference is in the range of ð 100 μm; 100 μmÞ. It is observed that the closer to the zero optical path difference, the stronger the detector's output. The target distance is obtained through seeking the scanning range corresponding to the maximum value of the detector's output. 2.2. Ambiguity range and distance accuracy As the LO and MO come from the same light source, we can regard a1;n ðωn Þ ¼ a2;n ðωn Þ, when the system environment stays the same. The target distance is contained in the third term of Eq. (6). Expanding the third term in Eq. (6), the following equation can be obtained: h n ω x n ω x ω1 1 ω2 2 þ ϕ þ cos þϕ I ef f ¼ Q cos c c n ω x n ω x i ωn n ωN N þ ϕ þ⋯ þ cos þϕ ð7Þ þ ⋯ cos c c where Q ¼ 2ja1;n ðωn ÞRLO jja2;n ðωn ÞRSIG j. between two adjacent items is phadif f ¼
nωn ωn x nωðn 1Þ ωn 1 x c c
The
phase
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in refractive index, the dispersion should also be considered due to the wide spectrum. Setting the step resolution of the scanning system at 200 nm, 50 nm, and 25 nm, the difference between mlrep and the maximum value of the cross correlation after algorithm extraction [15] is shown in Fig. 4, when the system parameters are the same as in Table 1. As shown in Fig. 4, the maximum distance deviations from mlrep are 2.5 μm, 0.65 μm and 0.65 μm corresponding to the step resolutions of 200 nm, 50 nm, and 25 nm. When the step resolution is sparse, higher distance accuracy is obtained through Table 1 System and environment parameters. Parameter
Description
Value
T P h λc λBW f rep RLO RSIG
Environment temperature Atmosphere pressure Air humidity Center wavelength Spectral full width at half maximum Pulse repetition rate
20 1C 1 atm 50% 1550 nm 40 nm 1 GHz
Optical path attenuation Signal path attenuation
0.5 0.5
difference ð8Þ
As the refractive index differs a little in the spectrum, set nωn ¼ nωðn 1Þ nc ng , where nc is the refractive index of the center wavelength, and ng is the group refractive index. The pulse-to-pulse separation is lrep ¼ c=ng f rep . When the optical path difference x is an integral multiple of lrep ,phadif f ðng x=cÞðωn ωn 1 Þ ¼ 2πm, where m is an integral number, all frequency combs in the spectrum are coherent to enhance, and a local maximum value in the cross correlation is produced. So the ambiguity range based on the traditional cross correlation method is the pulse-topulse separation. Since the scanning system has a finite step size, it is impossible to position the optical path difference x at an exact value of an integral multiple of lrep . Besides, although there is a little difference
Fig. 3. Normalized cross correlation when the optical path difference is in the range of ð 100 μm; 100 μmÞ with a step resolution of 50 nm.
Fig. 2. Normalized optical spectrum (left), and dispersion of the light source (right).
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Fig. 4. Distance deviation versus variable optical path difference under the step resolutions of 200 nm, 50 nm and 25 nm.
smaller step resolution. However, the distance accuracy remains the same when the step resolution is small enough. It can be explained by the fact that there does exist an offset between the real maximum value of the cross correlation and mlrep owing to the dispersion problem in the wide spectrum, and this offset value is contained in the distance accuracy. So the distance accuracy of the femtosecond laser range finders based on the traditional cross correlation method is within 0.65 μm. 2.3. Influence of optical path difference Adopting the parameters in Table 1, the phase difference within the frequency combs is shown in Fig. 5, and the corresponding normalized cross correlation is shown in Fig. 6 when the optical path difference is 0, 1500lrep , 3000lrep , and 4500lrep . As illustrated in Fig. 5, when the optical path difference is zero, the phase of every comb is in strict conformance. With the increment of optical path difference, the phase difference of each frequency comb gradually becomes evident, so there is a reduction in the maximum value of the cross correlation as shown in Fig. 6.
Fig. 5. Phase difference between the frequency combs under the optical path difference of 0, 1500lrep , 3000lrep , and 4500lrep .
2.4. Influence of spectrum width In order to manifest the influence of the spectrum width, set the FWHM (full width at half maximum) of the spectrum at 40 nm and 4 nm respectively; other parameters are the same as in Table 1.
Normalized cross correlation graphs are shown in Fig. 7 when the optical path difference is in the range of ð 700 μm; 700 μmÞ, which is carried out by the scanning system.
F. Yang et al. / Optics Communications 316 (2014) 179–189
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Fig. 6. Cross correlation under the optical path difference of 0, 1500lrep , 3000lrep , and 4500lrep .
Fig. 7. Normalized cross correlation when the FWHM of the light spectrum is 40 nm and 4 nm.
It is observed that the cross correlation width of the 4 nm FWHM is 10 times that of the 40 nm FWHM. As illustrated in chapter 3 of reference [16], the permitted maximum optical path difference for interference is λc 2 =λBW , which is in inverse proportion to the spectral width. Similar to this, a narrower spectrum shows broader cross correlation width.
2.5. Requirement of the scanning system's step resolution The target distance, which is represented by the optical path difference, is contained in the third term of Eq. (6). For an authentic cross correlation graph, the step resolution of the scanning system Δx should be controlled, namely the phase
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Fig. 8. Schematic of femtosecond laser range finder based on the frequency resolved dispersion compensation method.
variance kω Δx in the third term of Eq. (6) should be kept low enough to ensure sufficient sample numbers in one cycle. Taking the optical center frequency as an example, the step resolution should be less than 155 nm to make sure that at least 10 points are sampled in one cycle. In order to extract a maximum value in the cross correlation, the scanning range needs to contain at least one pulse-to-pulse distance. The pulse-to-pulse distance lrep is nearly 300 mm using the 1 GHz repetition rate in Table 1. It means that the optical path difference should be varied 106 times with submicron step resolution to complete one time target distance measurement.
3. Femtosecond laser range finder based on the frequency resolved dispersion compensation method 3.1. Principle A schematic of the femtosecond laser range finder based on the frequency resolved dispersion compensation method is shown in Fig. 8. The laser is of phase-stabilized mode-locked femtosecond type. A coupler splits the light into two parts, namely LO and MO paths. The LO path has a variable delay line, whose delay time can be controlled accurately. Light in the MO path enters into a circulator, and is then transmitted into the target through a telescope. The return light from the target is collected by the same telescope, and mixed with the LO in a coupler. The monostatic setup simplifies the optical adjustment greatly. Then the mixed coherent signal is spectrally resolved into N channels by using a spectral filter, such as arrayed waveguide grating, dense wavelength division multiplexing and so on. The N channels signals are detected by photon detectors and sampled separately, phase compensation is implemented independently to each channel in software, and then summed together to produce a range image. The signal can also be expressed as Eq. (7), where x is the physical optical path difference between two arms. Assuming that the spacing between channels is Δf , and p frequency combs are contained in each channel, the spectrum width contained in each channel after filtering is FBW ¼ pf rep . If channel spacing is Δf ¼ 1000 GHz, spectrum width after filtering is FBW ¼ 50 GHz, and channel number is N ¼ 7, the normalized spectrum after spectrum filtering is shown in Fig. 9.
Fig. 9. Normalized spectrum when the channel spacing is 1000 GHz, individual channel bandwidth is 50 GHz, and channel number is 7.
Fig. 10. Final signal after dispersion compensation when the sum of physical optical path difference and compensated optical path is in the range of ð 10 μm; 610 μmÞ.
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After spectrum filtering, the signal of nth and n þ1th channel can be expressed as follows: I n ¼ Q cos ðkn1 x þ ϕÞ þ cos ðkn2 x þ ϕÞ þ ⋯ þ cos ðknp x þ ϕÞ ; 1 rn r ðN 1Þ I n þ 1 ¼ Q cos ðkðn þ 1Þ1 x þ ϕÞ þ cos ðkðn þ 1Þ2 x þ ϕÞ þ ⋯ ð10Þ þ cos ðkðn þ 1Þp x þ ϕÞ where 2πΔf ¼ ðωn þ 1 ωn Þ. Ifnωn ¼ nωðn 1Þ nc ng , the corresponding phase difference between the adjacent channels can be illustrated as follows: phaadj ¼ kðn þ 1Þi x kni x ¼
2πng Δf 2πx x ½n f nni f ni c c ðn þ 1Þi ðn þ 1Þi
ð11Þ
185
In essence, dispersion compensation is phase compensation. It is implemented by delaying a variable optical path xi through software. The phase compensated to the nth channel is nð2πng Δf =cÞxi . The signal of nth and n þ1th channels after dispersion compensation can be expressed as follows: I n;com ¼ Q cos ðkn1 x þ ϕ þnψÞ þ cos ðkn2 x þ ϕ þ nψÞ þ ⋯ þ cos ðknp x þ ϕ þ nψÞ (
I n þ 1;com ¼ Q
cos ½kðn þ 1Þ1 x þ ϕ þ ðn þ 1Þψ þ cos ½kðn þ 1Þ2 x þ ϕ þ ðn þ 1Þψ þ ⋯ þ cos ½kðn þ 1Þp x þ ϕ þ ðn þ 1Þψ
)
ð12Þ where ψ ¼ ð2πng Δf =cÞxi , and 1 r n r ðN 1Þ. The final signal is obtained after summing up the N channels compensated signals: N
I sig ¼ ∑ I n;com ; n¼1
1rnrN
ð13Þ
The corresponding phase difference between the adjacent channels after dispersion compensation can be illustrated as follows: phaadj;com ¼ kðn þ 1Þi x kni x þ ψ Fig. 11. Range resolutions in the frequency resolved dispersion compensation system.
2πng Δf ðx þ xi Þ c
ð14Þ
lΔf ¼ c=ng Δf is the range resolution corresponding to the filter spacing. When x þ xi ¼ mlΔf , the corresponding phase difference
Fig. 12. Distance deviation versus variable optical path difference for the step resolution of 200 nm, 50 nm and 25 nm when the filter spacing is 1000 GHz, channel number is 7, and individual filter bandwidth is 100 GHz.
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between the adjacent channels phaadj;com 2πm, the channels are coherent to each other, and a nearly maximum value in I sig is obtained. Taking the specifications in Table 1 and the parameters according to Fig. 9, the final signal after dispersion compensation is shown in Fig. 10 when the sum of physical optical path difference
and compensated optical path x þ xi is in the range of ð 10 μm; 610 μmÞ. As the channel spacing isΔf ¼ 1000 GHz, the range resolution corresponding to the channel spacing is about 300 μm. So there is a peak value in the position near 0 μm, 300 μm and 600 μm, which is consistent with the above principle.
Fig. 13. Final signal under the pulse repetition rate of 20 GHz (left) and 30 GHz (right) when the channel spacing is 1000 GHz, channel number is 7, and individual filter bandwidth is 200 GHz.
Fig. 14. Final signals for the channel number of 3, 7, 11 and 15 when the filter spacing is 500 GHz, and individual filter bandwidth is 200 GHz.
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3.2. Range resolutions, ambiguity range and range accuracy in the femtosecond laser range finder based on the frequency resolved dispersion compensation method Range resolution is proportional to the inverse bandwidth. The range resolution corresponding to the channel spacing Δf islΔf p ð1=Δf Þ, range resolution corresponding to the individual filter bandwidth FBW is lFBW p ð1=FBWÞ, and range resolution corresponding to the full optical spectrum BW is lBW p ð1=BWÞ. The pulse to pulse separation is proportional to the inverse value of laser repetition rate f rep , namely lrep p ð1=f rep Þ. The relation among them is shown in Fig. 11. As illustrated, according to the sizes of above bandwidths, lBW olΔf o lFBW olrep . By adjusting the compensated optical path xi in software, the response (narrow peaks in Fig. 11) is scanned over the range under the dashed envelope. A sharp response peak is repeated at every range increment lΔf ¼ c=ng Δf . However, the signal intensity corresponding to these peaks is modulated by the dashed envelope shown in Fig. 11. The original ambiguity range adopting this method is lΔf . It can be increased by improving the scanning range of compensated optical path xi to experience a complete pulse-to-pulse separation lrep . The target distance is obtained through seeking xi corresponding to the maximum value of the final signal. In this way, the range ambiguity can be increased to pulse-to-pulse separation lrep . In order to compare with the range accuracy adopting the traditional cross correlation method, the step resolution of xi is
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also set at 200 nm, 50 nm, and 25 nm. The difference between mlrep and the maximum value of the final signal after algorithm extraction [15] is shown in Fig. 12, when the system parameters are the same as in Table 1, the channel spacing is Δf ¼ 1000 GHz, channel number is N ¼ 7, and individual filter bandwidth is FBW ¼ 100 GHz. As shown in Fig. 12, the maximum distance deviations from mlrep are 1.1 μm, 0.65 μm and 0.65 μm corresponding to the step resolutions of 200 nm, 50 nm, and 25 nm. When the step resolution is sparse, higher distance accuracy is obtained through smaller step resolution. However, the distance accuracy remains the same when the step resolution is small enough. It can be explained by the fact that there does exist an offset between the real maximum value of the cross correlation and mlrep owing to the dispersion problem in the wide spectrum, and this offset value is already contained in the distance accuracy. So the distance accuracy adopting the frequency resolved dispersion compensation method is within 0.65 μm. In other words, the ambiguity range and distance accuracy of the femtosecond laser range finders based on the frequency resolved dispersion compensation method are the same as those of the system adopting traditional cross correlation. The optical path difference in the system adopting the traditional cross correlation method is varied by hardware. An electric engine and fine translation stage as well as control and feedback system should all be incorporated with the system. However, in the system adopting the frequency resolved dispersion compensation
Fig. 15. Final signals for the channel spacing of 500 GHz, 1000 GHz, 1500 GHz, and 2000 GHz when the channel number is 5 and the individual filter bandwidth is 100 GHz.
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method, the optical path difference is varied by software; no electric engine and fine translation stage as well as control and feedback system are needed, which simplifies the system structure and improves the system reliability. 3.3. Requirement of the adjustment range of the variable delay line As discussed in Section 2.2, when the optical path difference x ¼ mlrep , where m is an integer, all frequency combs in the spectrum are coherent to enhance, and a nearly maximum value in the cross correlation is produced. However, in the frequency resolved dispersion compensation system, if x ¼ mlrep , only the frequency combs in the individual filtered channel are coherent to enhance. Coherence between channels is decided by ð2πng Δf = cÞ ðx þ xi Þ, as shown in Eq. (14). If Δf ¼ mf rep , ð2πng Δf =cÞðx þ xi Þ is an integral multiple of 2π when x þ xi ¼ mlrep . It means that peaks are repeated at every range increment of lrep . Otherwise, only one peak can be obtained when x þ xi ¼ 0. To demonstrate intuitively, except the laser repetition rate, the system parameters are the same as in Table 1. The normalized final signals of f rep ¼ 20 GHzðΔf ¼ mf rep Þ and f rep ¼ 30 GHz ðΔf a mf rep Þ are given in Fig. 13 when filter channel number is N ¼ 7, individual filter bandwidth is FBW ¼ 200 GHz, channel spacing is Δf ¼ 1000 GHz, and x þ xi is in the range of ð 7:5 μm; 16:5 mmÞ, namely two times integral multiple of lrep is obtained. In the left figure, when x þ xi ¼ mlrep (circle marked), the maximum values appear. It means that when the filter spacing
Δf is an integral multiple of laser repetition rate f rep , measurement can be accomplished by delaying the optical path in software in a scope of pulse-to-pulse separation lrep ; the system has no requirement on the length of variable delay line. Thus, the optical variable delay line can be removed. In the right figure, there is only one maximum when x þxi ¼ 0 (circle marked); at other mlrep ðm a0Þ (circle marked), no maximum appears. It means that when the filter spacing Δf is not an integral multiple of laser repetition rate f rep , the length of the variable delay line should be adjusted to make sure that the physical optical path difference meets the demand jxj r lrep . As a result, the relation of the channel spacing and laser repetition rate should be controlled to facilitate the measurement. 3.4. Influence of channel number, channel spacing, and individual filter bandwidth 3.4.1. Influence of channel number Using the parameters in Table 1, and filter spacing Δf ¼ 500 GHz, individual filter bandwidth FBW ¼ 200 GHz, x þxi is in the range of ð 20 μm; 1220 μmÞ, namely three times integral multiple of lrep is obtained ðm ¼ 0; 1; 2Þ; the normalized final signals are shown in Fig. 14 when channel numbers are 3, 7, 11, and 15. As illustrated, the larger the channel number, the sharper the final normalized signal, namely the better the final signal. It can be explained by the fact that when the filter spacing and individual filter bandwidth are the same, more energies are combined with
Fig. 16. Final signals for the individual filter bandwidth of 100 GHz, 200 GHz, 400 GHz, and 800 GHz when the channel number is 7 and channel spacing is 1000 GHz.
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increased channel number, like the main peak of multiple-slits diffraction being much brighter than that of double slit interference. However, in the real system, the number of channels cannot be increased unrestrictedly. The complexity of the system should also be considered. 3.4.2. Influence of channel spacing Using parameters in Table 1, channel number N ¼ 5, and individual filter bandwidth FBW ¼ 100 GHz, the normalized final signals are shown in Fig. 15 when the channel spacing Δf is 500 GHz, 1000 GHz, 1500 GHz, and 2000 GHz. As shown in Fig. 15, the narrower the channel spacing, the sharper the final normalized signal, namely the better the final signal. It can be explained by the fact that the spectrum strength decreases with departure from the spectrum center; the narrower the channel spacing, the stronger the combined energy to form the final signal, so the better the final signal. However, compared with the effect of channel number, the influence of channel spacing is much weaker. 3.4.3. Influence of filter bandwidth Using parameters in Table 1, channel number is N ¼ 7, and filter spacing Δf ¼ 1000 GHz, the normalized final signals are shown in Fig. 16 when individual filter bandwidth FBW is 100 GHz, 200 GHz, 400 GHz, and 800 GHz. As shown, the modulation effect at integral multiples of lΔf , which is the range resolution corresponding to the filter spacing Δf , is much more obvious with the increase of individual filter bandwidth FBW because the larger the individual filter bandwidth FBW, the smaller the range resolution lFBW corresponding to FBW, and the intensity of mlΔf is modulated by the range window of lFBW as shown in Fig. 11. Except the modulation effect, no apparent difference can be illustrated in Fig. 16. 4. Conclusions Two laser range finder systems using broadband femtosecond lasers as optical sources are discussed and compared in detail. One system uses the traditional cross correlation method, and the other system uses the frequency resolved dispersion compensation method. Better than 0.65 μm range accuracy and ambiguity range of pulse-to-pulse distance can be realized in both systems. In the system using the traditional cross correlation method, the main peak value of the cross correlation is reduced with the increase of the optical path difference; the FWHM of the cross correlation's width is decreased with the increase of spectral
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width; submicron step resolution of the scanning system is required to yield an authentic cross correlation graph. In the system adopting frequency resolved dispersion compensation method, no scanning system is needed; all process is finished in software; the channel spacing should be a multiple integral of the laser repetition rate to facilitate the measurement; otherwise the optical path difference must be controlled to less than a pulse-to-pulse separation; the influence of channel number is much more significant than the channel spacing and individual filter bandwidth; of course, the channel number ought to be considered in combination with the system complexity. Using the wide spectrum of the femtosecond light source, both systems can be applied in the field of range detection with submicron accuracy, while the system stability can be guaranteed because of the adoption of mature communication band devices.
Acknowledgments The authors acknowledge the support of National Natural Science Foundation of China (61108067 and 11005017), the central project in Colleges and Universities, and the constructive comments from the reviewers and editors.
References [1] D.J. Jones, S.A. Diddams, J.K. Ranka, A. Stentz, R.S. Windeler, J.L. Hall, S.T. Cundiff, Science 288 (2000) 635. [2] A. Bartels, C.W. Oates, L. Hollberg, S.A. Diddams, Opt. Lett. 29 (2004) 1081. [3] W.C. Swann, J.J. McFerran, I. Coddington, N.R. Newbury, I. Hartl, M.E. Fermann, P.S. Westbrook, J.W. Nicholson, K.S. Feder, C. Langrock, M.M. Fejer, Opt. Lett. 31 (2006) 3046. [4] M. Cui, R.N. Schouten, N. Bhattacharya, S.A. van den Berg, J. Eur. Opt. Soc. (2008)08003. [5] H. Matsumoto, X. Wang, K. Takamasu, T. Aoto, Appl. Phys. Express 5 (2012) 046601. [6] Haiyun Xia, Chunxi Zhang, Opt. Express 18 (2010) 4118. [7] H. Xia, C. Zhang, Opt. Lett. 34 (2009) 2108. [8] W.C. Swann, N.R. Newbury, Opt. Lett. 31 (2006) 826. [9] I. Coddington, W.C. Swann, L. Nenadovic, N.R. Newbury, Nat. Photon. 3 (2009) 351. [10] L. Lepetit, G. Ch´eriaux, M. Joffre, J. Opt. Soc. Am. B 12 (1995) 2467. [11] L. Rovati, U. Minoni, M. Bonardi, F. Docchio, J. Opt. 29 (1998) 121. [12] N.R. Doloca, K. Meiners-Hagen, M. Wedde, F. Pollinger, A. Abu-Zeid, Meas. Sci. Technol. 21 (2010) 115302. [13] A. Pesatori, M. Norgia, C. Svelto, M. Zucco, M. Stupka, A. De Marchi, IEEE Trans. Instrum. Meas. 61 (2012) 1536. [14] K.P Birch, M.J Downs, Metrologia 30 (1993) 155. [15] V.K. Jain, W.L. Collins, D.C. Davis, IEEE Trans. Instrum. Meas. 28 (1979) 113. [16] Jingzhen Li, Handbook of OpticsShanxi Science and Technology PressXi'an (2010) 129.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Femtosecond laser range finders based on traditional cross correlation method and frequency resolved dispersion compensation method Fu Yang, Jing Zhang, Yage Zhan n Department of Physics, Donghua University, No. 2999, North Renmin Road, Shanghai 201620, China
art ic l e i nf o
a b s t r a c t
Article history: Received 22 August 2013 Received in revised form 21 November 2013 Accepted 22 November 2013 Available online 16 December 2013
In order to realize high range accuracy, two laser range finder systems using broadband femtosecond lasers as optical sources are discussed and compared in detail. One system uses a traditional cross correlation method, and the other system uses a frequency resolved dispersion compensation method. Better than 0.65 μm distance accuracy and ambiguity range of pulse-to-pulse separation can be realized in both systems. In the system using the traditional cross correlation method, submicron step resolution of the scanning system should be maintained to yield an authentic cross correlation graph. While in the system using the frequency resolved dispersion compensation method, no scanning system is required as the signal is spectrally resolved into several channels and the process is finished in software. In the system using the frequency resolved dispersion compensation method, the influence of channel number is much more significant than that of channel spacing and individual channel bandwidth. And the channel spacing should be a multiple integral of the laser repetition rate to facilitate the measurement; otherwise the optical path difference must be controlled to less than a pulse-to-pulse separation. Both systems are well suited to the range lidar requiring submicron distance accuracy, and can be applied in the future space missions. & 2013 Elsevier B.V. All rights reserved.
Keywords: Laser range finder Lidar Remote sensing and sensors Ultrafast lasers Dispersion compensation
1. Introduction Laser range finder plays an important role in the field of laser remote sensing. Based on a satellite or airborne platform, threedimensional map of the detected area can be achieved, which provides a powerful tool for the planetary exploration, position, navigation, and disaster relief. As a passion for precision, researchers worldwide are seeking a way to realize higher range resolution. The development of mode locking and frequency stabilization technique in recent years has commercialized the femtosecond laser frequency combs [1–3]. It is natural to extend the light source to range measurement since the broad spectrum supports high range resolution. The lidar system with femtosecond laser frequency combs usually adopts a configuration similar to that of Michelson's interferometer. In terms of the detection method, Cui et al. [4] and Matsumoto et al. [5] adopt the crosscorrelation method; a fine scanning system must be equipped in the system, and the target distance is obtained through the maximum value of the cross correlation. This is the most traditional ranging method using the femtosecond laser, and an ambiguity range of pulse-to-pulse distance can be realized. Xia and Zhang [6,7] use a dispersive interferometer method; target
n
Corresponding author. E-mail address: [email protected] (Y. Zhan).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.11.043
distance can be calculated from the data recorded in the oscilloscope (OSC) or optical spectrum analyzer (OSA). The measurement principle is convenient to apply; however, the dynamic range is restricted by the bandwidth of the OSC, which is 8.7 mm in the experiment by using 7 GHz bandwidth OSC [7]. Swann and Newbury [8] employ the frequency resolved dispersion compensation method, which is an extremely attractive method; the coherent signal is spectrally resolved into N channels, phase compensation is implemented to each channel and then summed up to realize range detection. However, details of the principle are missed in the paper, and the reader is confused. Coddington et al. [9] apply a pair of stabilized femtosecond laser frequency combs of slightly different frequencies so the ambiguity range can be extended greatly. The ambiguity range is 30 km in the paper which is the longest value reported in the literature until now. As two phase locked femtosecond lasers are required, the system is both expensive and complicated. Lepetit et al. [10], Rovati et al. [11], Doloca et al. [12], and Pesatori et al. [13] measure the distance by the phase delay of the returning optical pulse train with respect to the outgoing one. Lepetit et al. [10] and Rovati et al. [11] use optical coherent detection methods and spectrally resolve the coherent signal to the line CCDs. Phase information induced by a signal range is obtained. The difference is that Rovati et al. [11] replace the femtosecond laser by a low cost multi-longitudinal mode laser. Doloca et al. [12] and Pesatori et al. [13] use independent photodetectors to measure the signal pulse and reference pulse;
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phase difference is judged in an electric domain. The difference between them is that Doloca et al. [12] evaluate the phase shift of two distinct comb frequencies. A coarse measurement is performed by a comb line of 100 MHz, and a fine measurement is performed by a comb line of 11.4 GHz; the two results are combined to obtain range information. Pesatori et al. [13] expand the time interval corresponding to the phase delay by downconverting beat notes from high harmonics of the pulse repetition rate (up to 10 GHz) to several kilohertz in the electric domain; this way both high range resolution (based on 10 GHz) and convenient phase measurement (based on the time period of several kilohertz) can be realized. As the target range is calculated by the phase difference, the target distance will be affected by any phase disturbance, such as changes in temperature, pressure, humidity and so on. It means that the phase detection method is exact strictly with the operation environment. In this paper, for the first time to our knowledge, the detailed theory of the frequency resolved dispersion compensation method used in the femtosecond ranging system is illustrated. And it is also compared with the traditional cross correlation method to show its advantages. This paper is arranged as follows. Section 1 presented the introduction part. In Section 2 the system adopting the traditional cross correlation method is described. The principle, ambiguity range and distance accuracy, influence of the optical path difference and the spectral width, and the requirement of the scanning system's step resolution are included. In Section 3 the system using the frequency resolved dispersion compensation method is described. The principle, ambiguity range and distance accuracy, the requirement of optical delay line's adjustment range, and the influence of the channel number, channel spacing, and channel bandwidth are given. Section 4 presents the conclusion.
2. Femtosecond laser range finder based on the traditional cross correlation method 2.1. Principle A schematic of the femtosecond laser range finder based on the traditional cross correlation method is shown in Fig. 1. The optical source adopts phase-stabilized mode-locked femtosecond laser. This source emits a broad spectrum of light that comprises a set of narrow individual optical lines in frequency space separated by the laser repetition rate. A beam splitter divides the light into two
parts. One part serves as a local oscillator (LO), and the other part serves as the main oscillator (MO). The LO is in the non-varying arm, and light is reflected by a retro-reflector. The optical path of the MO can be varied with a scanning system. The two beams combine with each other, and are then detected by a photodetector. The electric field of LO and MO is expressed, respectively, as ELO and ESIG in the following equations: N
ELO ðtÞ ¼ ∑ a1;n ðωn Þ exp ðiωn tÞ
ð1Þ
n¼1 N
ESIG ðtÞ ¼ ∑ a2;n ðωn Þ exp ðiωn tÞ
ð2Þ
n¼1
where a1;n ðωn Þ and a2;n ðωn Þ are the signal spectra of LO and MO respectively; ωn ¼ ω0 þnωrep , ωrep ¼ 2πf rep is the angular repetition rate, and ω0 is the offset frequency due to the difference of group velocity and phase velocity. After traversing different paths x1 and x2 in air, the two signals can be expressed as ELO ðt; x1 Þ and ESIG ðt; x2 Þ in the following equations: N
ELO ðt; x1 Þ ¼ ∑ a1;n ðωn ÞRLO exp ðiωn t kω x1 þ ϕ1;n Þ
ð3Þ
n¼1 N
ESIG ðt; x2 Þ ¼ ∑ a2;n ðωn ÞRSIG exp ðiωn t kω x2 þ ϕ2;n Þ
ð4Þ
n¼1
where kω ¼ nω ω=c is the wave vector; nω is the refractive index in air deduced by the Ciddor equation [14], which is closely related to the system environment, such as temperature, atmospheric pressure, humidity and so on; RLO and RSIG represent the transmission losses in the LO and MO paths; ϕ1;n and ϕ2;n are the phase offsets in the two arms. As the femtosecond pulses are too fast to respond, the output of the detector shown in Eq. (5) is proportional to the average value of the two beams cross correlation signal: I ω ¼ ℜ ½ELO ðt; x1 Þ þESIG ðt; x2 Þ½ELO ðt; x1 Þ þ ESIG ðt; x2 Þn ð5Þ where ℜ is the proportion factor. Assuming x ¼ x2 x1 , ϕ ¼ ϕ2;n ϕ1;n , and substituting Eqs. (3) and (4) into Eq. (5), we can get the following equation: N h i I ω ðxÞ ¼ ℜ ∑ j a1;n ðωn ÞRLO j 2 þ j a2;n ðωn ÞRSIG j 2 n¼1 N
þ 2 ∑ j a1;n ðωn ÞRLO :a2;n ðωn ÞRSIG j cos ðkω x þ ϕÞ n¼1
ð6Þ
As shown in Eq. (6), I ω varies with the optical path difference x. Eq. (6) consists of three items. The first two items are DC signals.
Fig. 1. Schematic of the femtosecond laser range finder based on the traditional cross correlation method.
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The influence of optical path difference x is reflected in the last item. For convenience, set ϕ as a constant. Assuming that the optical spectrum is Gauss shaped, the normalized spectrum and the dispersion situation of the optical source are shown in Fig. 2, using the system and environment parameters shown in Table 1. As shown in Fig. 2, the refractive index difference in the spectral range is 0.1 ppm. The optical path difference can be controlled accurately with a scanning system, like a piezo-electric system. The AC output of the detector is shown in Fig. 3 when the optical path difference is in the range of ð 100 μm; 100 μmÞ. It is observed that the closer to the zero optical path difference, the stronger the detector's output. The target distance is obtained through seeking the scanning range corresponding to the maximum value of the detector's output. 2.2. Ambiguity range and distance accuracy As the LO and MO come from the same light source, we can regard a1;n ðωn Þ ¼ a2;n ðωn Þ, when the system environment stays the same. The target distance is contained in the third term of Eq. (6). Expanding the third term in Eq. (6), the following equation can be obtained: h n ω x n ω x ω1 1 ω2 2 þ ϕ þ cos þϕ I ef f ¼ Q cos c c n ω x n ω x i ωn n ωN N þ ϕ þ⋯ þ cos þϕ ð7Þ þ ⋯ cos c c where Q ¼ 2ja1;n ðωn ÞRLO jja2;n ðωn ÞRSIG j. between two adjacent items is phadif f ¼
nωn ωn x nωðn 1Þ ωn 1 x c c
The
phase
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in refractive index, the dispersion should also be considered due to the wide spectrum. Setting the step resolution of the scanning system at 200 nm, 50 nm, and 25 nm, the difference between mlrep and the maximum value of the cross correlation after algorithm extraction [15] is shown in Fig. 4, when the system parameters are the same as in Table 1. As shown in Fig. 4, the maximum distance deviations from mlrep are 2.5 μm, 0.65 μm and 0.65 μm corresponding to the step resolutions of 200 nm, 50 nm, and 25 nm. When the step resolution is sparse, higher distance accuracy is obtained through Table 1 System and environment parameters. Parameter
Description
Value
T P h λc λBW f rep RLO RSIG
Environment temperature Atmosphere pressure Air humidity Center wavelength Spectral full width at half maximum Pulse repetition rate
20 1C 1 atm 50% 1550 nm 40 nm 1 GHz
Optical path attenuation Signal path attenuation
0.5 0.5
difference ð8Þ
As the refractive index differs a little in the spectrum, set nωn ¼ nωðn 1Þ nc ng , where nc is the refractive index of the center wavelength, and ng is the group refractive index. The pulse-to-pulse separation is lrep ¼ c=ng f rep . When the optical path difference x is an integral multiple of lrep ,phadif f ðng x=cÞðωn ωn 1 Þ ¼ 2πm, where m is an integral number, all frequency combs in the spectrum are coherent to enhance, and a local maximum value in the cross correlation is produced. So the ambiguity range based on the traditional cross correlation method is the pulse-topulse separation. Since the scanning system has a finite step size, it is impossible to position the optical path difference x at an exact value of an integral multiple of lrep . Besides, although there is a little difference
Fig. 3. Normalized cross correlation when the optical path difference is in the range of ð 100 μm; 100 μmÞ with a step resolution of 50 nm.
Fig. 2. Normalized optical spectrum (left), and dispersion of the light source (right).
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Fig. 4. Distance deviation versus variable optical path difference under the step resolutions of 200 nm, 50 nm and 25 nm.
smaller step resolution. However, the distance accuracy remains the same when the step resolution is small enough. It can be explained by the fact that there does exist an offset between the real maximum value of the cross correlation and mlrep owing to the dispersion problem in the wide spectrum, and this offset value is contained in the distance accuracy. So the distance accuracy of the femtosecond laser range finders based on the traditional cross correlation method is within 0.65 μm. 2.3. Influence of optical path difference Adopting the parameters in Table 1, the phase difference within the frequency combs is shown in Fig. 5, and the corresponding normalized cross correlation is shown in Fig. 6 when the optical path difference is 0, 1500lrep , 3000lrep , and 4500lrep . As illustrated in Fig. 5, when the optical path difference is zero, the phase of every comb is in strict conformance. With the increment of optical path difference, the phase difference of each frequency comb gradually becomes evident, so there is a reduction in the maximum value of the cross correlation as shown in Fig. 6.
Fig. 5. Phase difference between the frequency combs under the optical path difference of 0, 1500lrep , 3000lrep , and 4500lrep .
2.4. Influence of spectrum width In order to manifest the influence of the spectrum width, set the FWHM (full width at half maximum) of the spectrum at 40 nm and 4 nm respectively; other parameters are the same as in Table 1.
Normalized cross correlation graphs are shown in Fig. 7 when the optical path difference is in the range of ð 700 μm; 700 μmÞ, which is carried out by the scanning system.
F. Yang et al. / Optics Communications 316 (2014) 179–189
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Fig. 6. Cross correlation under the optical path difference of 0, 1500lrep , 3000lrep , and 4500lrep .
Fig. 7. Normalized cross correlation when the FWHM of the light spectrum is 40 nm and 4 nm.
It is observed that the cross correlation width of the 4 nm FWHM is 10 times that of the 40 nm FWHM. As illustrated in chapter 3 of reference [16], the permitted maximum optical path difference for interference is λc 2 =λBW , which is in inverse proportion to the spectral width. Similar to this, a narrower spectrum shows broader cross correlation width.
2.5. Requirement of the scanning system's step resolution The target distance, which is represented by the optical path difference, is contained in the third term of Eq. (6). For an authentic cross correlation graph, the step resolution of the scanning system Δx should be controlled, namely the phase
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Fig. 8. Schematic of femtosecond laser range finder based on the frequency resolved dispersion compensation method.
variance kω Δx in the third term of Eq. (6) should be kept low enough to ensure sufficient sample numbers in one cycle. Taking the optical center frequency as an example, the step resolution should be less than 155 nm to make sure that at least 10 points are sampled in one cycle. In order to extract a maximum value in the cross correlation, the scanning range needs to contain at least one pulse-to-pulse distance. The pulse-to-pulse distance lrep is nearly 300 mm using the 1 GHz repetition rate in Table 1. It means that the optical path difference should be varied 106 times with submicron step resolution to complete one time target distance measurement.
3. Femtosecond laser range finder based on the frequency resolved dispersion compensation method 3.1. Principle A schematic of the femtosecond laser range finder based on the frequency resolved dispersion compensation method is shown in Fig. 8. The laser is of phase-stabilized mode-locked femtosecond type. A coupler splits the light into two parts, namely LO and MO paths. The LO path has a variable delay line, whose delay time can be controlled accurately. Light in the MO path enters into a circulator, and is then transmitted into the target through a telescope. The return light from the target is collected by the same telescope, and mixed with the LO in a coupler. The monostatic setup simplifies the optical adjustment greatly. Then the mixed coherent signal is spectrally resolved into N channels by using a spectral filter, such as arrayed waveguide grating, dense wavelength division multiplexing and so on. The N channels signals are detected by photon detectors and sampled separately, phase compensation is implemented independently to each channel in software, and then summed together to produce a range image. The signal can also be expressed as Eq. (7), where x is the physical optical path difference between two arms. Assuming that the spacing between channels is Δf , and p frequency combs are contained in each channel, the spectrum width contained in each channel after filtering is FBW ¼ pf rep . If channel spacing is Δf ¼ 1000 GHz, spectrum width after filtering is FBW ¼ 50 GHz, and channel number is N ¼ 7, the normalized spectrum after spectrum filtering is shown in Fig. 9.
Fig. 9. Normalized spectrum when the channel spacing is 1000 GHz, individual channel bandwidth is 50 GHz, and channel number is 7.
Fig. 10. Final signal after dispersion compensation when the sum of physical optical path difference and compensated optical path is in the range of ð 10 μm; 610 μmÞ.
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After spectrum filtering, the signal of nth and n þ1th channel can be expressed as follows: I n ¼ Q cos ðkn1 x þ ϕÞ þ cos ðkn2 x þ ϕÞ þ ⋯ þ cos ðknp x þ ϕÞ ; 1 rn r ðN 1Þ I n þ 1 ¼ Q cos ðkðn þ 1Þ1 x þ ϕÞ þ cos ðkðn þ 1Þ2 x þ ϕÞ þ ⋯ ð10Þ þ cos ðkðn þ 1Þp x þ ϕÞ where 2πΔf ¼ ðωn þ 1 ωn Þ. Ifnωn ¼ nωðn 1Þ nc ng , the corresponding phase difference between the adjacent channels can be illustrated as follows: phaadj ¼ kðn þ 1Þi x kni x ¼
2πng Δf 2πx x ½n f nni f ni c c ðn þ 1Þi ðn þ 1Þi
ð11Þ
185
In essence, dispersion compensation is phase compensation. It is implemented by delaying a variable optical path xi through software. The phase compensated to the nth channel is nð2πng Δf =cÞxi . The signal of nth and n þ1th channels after dispersion compensation can be expressed as follows: I n;com ¼ Q cos ðkn1 x þ ϕ þnψÞ þ cos ðkn2 x þ ϕ þ nψÞ þ ⋯ þ cos ðknp x þ ϕ þ nψÞ (
I n þ 1;com ¼ Q
cos ½kðn þ 1Þ1 x þ ϕ þ ðn þ 1Þψ þ cos ½kðn þ 1Þ2 x þ ϕ þ ðn þ 1Þψ þ ⋯ þ cos ½kðn þ 1Þp x þ ϕ þ ðn þ 1Þψ
)
ð12Þ where ψ ¼ ð2πng Δf =cÞxi , and 1 r n r ðN 1Þ. The final signal is obtained after summing up the N channels compensated signals: N
I sig ¼ ∑ I n;com ; n¼1
1rnrN
ð13Þ
The corresponding phase difference between the adjacent channels after dispersion compensation can be illustrated as follows: phaadj;com ¼ kðn þ 1Þi x kni x þ ψ Fig. 11. Range resolutions in the frequency resolved dispersion compensation system.
2πng Δf ðx þ xi Þ c
ð14Þ
lΔf ¼ c=ng Δf is the range resolution corresponding to the filter spacing. When x þ xi ¼ mlΔf , the corresponding phase difference
Fig. 12. Distance deviation versus variable optical path difference for the step resolution of 200 nm, 50 nm and 25 nm when the filter spacing is 1000 GHz, channel number is 7, and individual filter bandwidth is 100 GHz.
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between the adjacent channels phaadj;com 2πm, the channels are coherent to each other, and a nearly maximum value in I sig is obtained. Taking the specifications in Table 1 and the parameters according to Fig. 9, the final signal after dispersion compensation is shown in Fig. 10 when the sum of physical optical path difference
and compensated optical path x þ xi is in the range of ð 10 μm; 610 μmÞ. As the channel spacing isΔf ¼ 1000 GHz, the range resolution corresponding to the channel spacing is about 300 μm. So there is a peak value in the position near 0 μm, 300 μm and 600 μm, which is consistent with the above principle.
Fig. 13. Final signal under the pulse repetition rate of 20 GHz (left) and 30 GHz (right) when the channel spacing is 1000 GHz, channel number is 7, and individual filter bandwidth is 200 GHz.
Fig. 14. Final signals for the channel number of 3, 7, 11 and 15 when the filter spacing is 500 GHz, and individual filter bandwidth is 200 GHz.
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3.2. Range resolutions, ambiguity range and range accuracy in the femtosecond laser range finder based on the frequency resolved dispersion compensation method Range resolution is proportional to the inverse bandwidth. The range resolution corresponding to the channel spacing Δf islΔf p ð1=Δf Þ, range resolution corresponding to the individual filter bandwidth FBW is lFBW p ð1=FBWÞ, and range resolution corresponding to the full optical spectrum BW is lBW p ð1=BWÞ. The pulse to pulse separation is proportional to the inverse value of laser repetition rate f rep , namely lrep p ð1=f rep Þ. The relation among them is shown in Fig. 11. As illustrated, according to the sizes of above bandwidths, lBW olΔf o lFBW olrep . By adjusting the compensated optical path xi in software, the response (narrow peaks in Fig. 11) is scanned over the range under the dashed envelope. A sharp response peak is repeated at every range increment lΔf ¼ c=ng Δf . However, the signal intensity corresponding to these peaks is modulated by the dashed envelope shown in Fig. 11. The original ambiguity range adopting this method is lΔf . It can be increased by improving the scanning range of compensated optical path xi to experience a complete pulse-to-pulse separation lrep . The target distance is obtained through seeking xi corresponding to the maximum value of the final signal. In this way, the range ambiguity can be increased to pulse-to-pulse separation lrep . In order to compare with the range accuracy adopting the traditional cross correlation method, the step resolution of xi is
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also set at 200 nm, 50 nm, and 25 nm. The difference between mlrep and the maximum value of the final signal after algorithm extraction [15] is shown in Fig. 12, when the system parameters are the same as in Table 1, the channel spacing is Δf ¼ 1000 GHz, channel number is N ¼ 7, and individual filter bandwidth is FBW ¼ 100 GHz. As shown in Fig. 12, the maximum distance deviations from mlrep are 1.1 μm, 0.65 μm and 0.65 μm corresponding to the step resolutions of 200 nm, 50 nm, and 25 nm. When the step resolution is sparse, higher distance accuracy is obtained through smaller step resolution. However, the distance accuracy remains the same when the step resolution is small enough. It can be explained by the fact that there does exist an offset between the real maximum value of the cross correlation and mlrep owing to the dispersion problem in the wide spectrum, and this offset value is already contained in the distance accuracy. So the distance accuracy adopting the frequency resolved dispersion compensation method is within 0.65 μm. In other words, the ambiguity range and distance accuracy of the femtosecond laser range finders based on the frequency resolved dispersion compensation method are the same as those of the system adopting traditional cross correlation. The optical path difference in the system adopting the traditional cross correlation method is varied by hardware. An electric engine and fine translation stage as well as control and feedback system should all be incorporated with the system. However, in the system adopting the frequency resolved dispersion compensation
Fig. 15. Final signals for the channel spacing of 500 GHz, 1000 GHz, 1500 GHz, and 2000 GHz when the channel number is 5 and the individual filter bandwidth is 100 GHz.
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method, the optical path difference is varied by software; no electric engine and fine translation stage as well as control and feedback system are needed, which simplifies the system structure and improves the system reliability. 3.3. Requirement of the adjustment range of the variable delay line As discussed in Section 2.2, when the optical path difference x ¼ mlrep , where m is an integer, all frequency combs in the spectrum are coherent to enhance, and a nearly maximum value in the cross correlation is produced. However, in the frequency resolved dispersion compensation system, if x ¼ mlrep , only the frequency combs in the individual filtered channel are coherent to enhance. Coherence between channels is decided by ð2πng Δf = cÞ ðx þ xi Þ, as shown in Eq. (14). If Δf ¼ mf rep , ð2πng Δf =cÞðx þ xi Þ is an integral multiple of 2π when x þ xi ¼ mlrep . It means that peaks are repeated at every range increment of lrep . Otherwise, only one peak can be obtained when x þ xi ¼ 0. To demonstrate intuitively, except the laser repetition rate, the system parameters are the same as in Table 1. The normalized final signals of f rep ¼ 20 GHzðΔf ¼ mf rep Þ and f rep ¼ 30 GHz ðΔf a mf rep Þ are given in Fig. 13 when filter channel number is N ¼ 7, individual filter bandwidth is FBW ¼ 200 GHz, channel spacing is Δf ¼ 1000 GHz, and x þ xi is in the range of ð 7:5 μm; 16:5 mmÞ, namely two times integral multiple of lrep is obtained. In the left figure, when x þ xi ¼ mlrep (circle marked), the maximum values appear. It means that when the filter spacing
Δf is an integral multiple of laser repetition rate f rep , measurement can be accomplished by delaying the optical path in software in a scope of pulse-to-pulse separation lrep ; the system has no requirement on the length of variable delay line. Thus, the optical variable delay line can be removed. In the right figure, there is only one maximum when x þxi ¼ 0 (circle marked); at other mlrep ðm a0Þ (circle marked), no maximum appears. It means that when the filter spacing Δf is not an integral multiple of laser repetition rate f rep , the length of the variable delay line should be adjusted to make sure that the physical optical path difference meets the demand jxj r lrep . As a result, the relation of the channel spacing and laser repetition rate should be controlled to facilitate the measurement. 3.4. Influence of channel number, channel spacing, and individual filter bandwidth 3.4.1. Influence of channel number Using the parameters in Table 1, and filter spacing Δf ¼ 500 GHz, individual filter bandwidth FBW ¼ 200 GHz, x þxi is in the range of ð 20 μm; 1220 μmÞ, namely three times integral multiple of lrep is obtained ðm ¼ 0; 1; 2Þ; the normalized final signals are shown in Fig. 14 when channel numbers are 3, 7, 11, and 15. As illustrated, the larger the channel number, the sharper the final normalized signal, namely the better the final signal. It can be explained by the fact that when the filter spacing and individual filter bandwidth are the same, more energies are combined with
Fig. 16. Final signals for the individual filter bandwidth of 100 GHz, 200 GHz, 400 GHz, and 800 GHz when the channel number is 7 and channel spacing is 1000 GHz.
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increased channel number, like the main peak of multiple-slits diffraction being much brighter than that of double slit interference. However, in the real system, the number of channels cannot be increased unrestrictedly. The complexity of the system should also be considered. 3.4.2. Influence of channel spacing Using parameters in Table 1, channel number N ¼ 5, and individual filter bandwidth FBW ¼ 100 GHz, the normalized final signals are shown in Fig. 15 when the channel spacing Δf is 500 GHz, 1000 GHz, 1500 GHz, and 2000 GHz. As shown in Fig. 15, the narrower the channel spacing, the sharper the final normalized signal, namely the better the final signal. It can be explained by the fact that the spectrum strength decreases with departure from the spectrum center; the narrower the channel spacing, the stronger the combined energy to form the final signal, so the better the final signal. However, compared with the effect of channel number, the influence of channel spacing is much weaker. 3.4.3. Influence of filter bandwidth Using parameters in Table 1, channel number is N ¼ 7, and filter spacing Δf ¼ 1000 GHz, the normalized final signals are shown in Fig. 16 when individual filter bandwidth FBW is 100 GHz, 200 GHz, 400 GHz, and 800 GHz. As shown, the modulation effect at integral multiples of lΔf , which is the range resolution corresponding to the filter spacing Δf , is much more obvious with the increase of individual filter bandwidth FBW because the larger the individual filter bandwidth FBW, the smaller the range resolution lFBW corresponding to FBW, and the intensity of mlΔf is modulated by the range window of lFBW as shown in Fig. 11. Except the modulation effect, no apparent difference can be illustrated in Fig. 16. 4. Conclusions Two laser range finder systems using broadband femtosecond lasers as optical sources are discussed and compared in detail. One system uses the traditional cross correlation method, and the other system uses the frequency resolved dispersion compensation method. Better than 0.65 μm range accuracy and ambiguity range of pulse-to-pulse distance can be realized in both systems. In the system using the traditional cross correlation method, the main peak value of the cross correlation is reduced with the increase of the optical path difference; the FWHM of the cross correlation's width is decreased with the increase of spectral
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width; submicron step resolution of the scanning system is required to yield an authentic cross correlation graph. In the system adopting frequency resolved dispersion compensation method, no scanning system is needed; all process is finished in software; the channel spacing should be a multiple integral of the laser repetition rate to facilitate the measurement; otherwise the optical path difference must be controlled to less than a pulse-to-pulse separation; the influence of channel number is much more significant than the channel spacing and individual filter bandwidth; of course, the channel number ought to be considered in combination with the system complexity. Using the wide spectrum of the femtosecond light source, both systems can be applied in the field of range detection with submicron accuracy, while the system stability can be guaranteed because of the adoption of mature communication band devices.
Acknowledgments The authors acknowledge the support of National Natural Science Foundation of China (61108067 and 11005017), the central project in Colleges and Universities, and the constructive comments from the reviewers and editors.
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