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We theoretically and experimentally examined the effects of the different positions between receiving lens (RL) and detector, namely, defocus, off-axis, and tilt, on the time-domain pulsed echo laser profile (TDPELP). Results show that distortions including saturation and broadening of TDPELP are obtained, regardless of the position between RL and detector. Thus, we adjust the focal length of RL to successfully obtain an optimal TDPELP using tunable lens under extreme situations, such as too strong or too weak intensity of the TDPELP.
© 2017 Optical Society of America
1. Introduction
With the development of techniques on pulsed laser and fast detection, time-domain pulsed echo laser profile (TDPELP) has been attracting much attention in recent years. This profile is widely used in various fields, such as laser ranging [1], three-dimensional (3D) imaging [2], surface profiles [3], target recognition [4], tracking [5], and ghost image [6]. For example, compared with traditional target recognition based on a 3D image, the efficiency of target recognition is dramatically improved in TDPELP-based technique because of the absence of scanning imaging process [4].
An avalanche photodiode (APD) is a typical device [7] for detecting laser echo power in the aforementioned applications (e.g. laser ranging and 3D imaging, and a TDPELP affects the performances of ranging and 3D sensors [4]. However, TDPELP-based applications are limited under extreme conditions including strong and weak intensity, because ranging error is resulted from broadening [8, 9], while low signal-noise-ratio (SNR) is resulted from weak intensity. Thus, electrical methods have been proposed to improve the detectable range [10, 11]. However, the main drawback of electrical methods is the simultaneous enhancement or suppression of noise and signal. Liu et al. [12] recently designed a new structure to compensate the temporal pulse distortion in a saturated amplifier. The method can be suitably used in a transmitting laser system but not for an echo laser receiving system, because various positions between the receiving lens (RL) and APD should be considered. A detector should be traditionally located on the focal position of the RL to obtain the highest echo power [13–15]. However, accurately maintaining an APD on the focal plane of the RL in practical use is difficult, and the position between the RL and APD may be changed by unknown forces during operation, especially when the system including APD and RL is used in moveable platforms (e.g. cars and planes). Therefore, extreme situations of strong or weak intensity will result in poor performances on measuring dynamic ranges [16]. Studying the position between the RL and APD is important and beneficial to improve the performances of TDPELP-based systems. Typical ranging system based on time-of-flight (TOF) employ time-domain information to determine start and stop time stamps. Therefore, analysis on time-domain feature is suitable for such systems. Although the spatial effects caused by the target and atmosphere have been previously investigated [17, 18], to the best of our knowledge, detailed study on the time-domain effects of the positions between the RL and APD has not yet been performed.
The main contribution of the paper is the theoretical and experimental evaluation of the effects of the position between the RL and APD. Accordingly, we provide a simple approach to adjust the TDPELP under extreme conditions using tunable lens, and the range error is decreased by the use of this method. The rest of the paper is organized as follows. Section 2 presents a theoretical analysis of TDPELP models with different positions between the RL and APD, that is, defocused, off-axis, and tilted RL with respect to APD. Combinations of the different positions were also considered. Then, we perform simulations and corresponding experiments to test the validity of the models. In section 3, we provide a simple approach to improve the performance of the device using tunable lens. In section 4, we discuss the relationship between focal length and the TDPELP under different transmitting powers. This relationship can be used to determine the optimal focal length in practical use. Finally, important conclusions are drawn in section 5.
2. Models
2.1 Defocus
The detection of echo power is shown in Fig. 1, in which pulsed laser beams illuminate the target after collimated by a transmitting lens (TL). The scattered and reflected pulsed laser beams from the target focus on the APD. The TDPELP is obtained using the read out circuit. In general, the APD is placed on the focal plane [13–15].
The intensity distribution of the pulsed laser is a combination of spatial and temporal functions of the Gaussian function [18, 19]. Here, we only discuss the time-domain feature Pt(t), as follows:
where Et is the pulse energy, and τ is the transmitting pulse width. After propagating with a distance R, the laser beam on target of area illuminated is determined as follows:where Pillu(t) is the transmitting laser power at the location of the target, Ta is the one-way atmospheric transmission, Tot is the transmission efficiency of TL, R is the distance between the optical system and target, and c is the light speed. The reflected pulse from the target is viewed as a secondary pulsed laser source Ps(t), as follows [14]:where ρr is the total hemispherical reflectivity of the target, and At is the area of the target. In general, the focal length of the RL is much smaller than the distance between the target and the system, and the APD is placed at the focal plane. Therefore, the intensity of the TDPELP is determined as follows:where Pr(t) is the intensity of the TDPELP when the APD is on the focal plane, Tot and Tor are the optical efficiencies of the TL and RL, respectively, Srec = π(d/2)2 is the area of the RL(d is the diameter of the RL), Sillu is the area illuminated, and ηD is the quantum efficiency. With D as the size of detector, and To = Tor·Tot as the total optical efficiency, the following equation can be obtained:where τr is the received pulse width, which is affected by the tilt angle of the target [20, 21]. The corrected Pr(t) is written as follows:where w0 is the waist radius of the laser, w(z) is the beam radius, and λ is the wavelength. We introduce a correction coefficient (k) to compensate for pulse broadening caused by the saturation of the APD. Ideally, k is 1 in the absence of pulse broadening. Typically, k is larger than 1 and a variant parameter because of the different APDs. The k value is determined experimentally when the APD and corresponding circuit are given. Eqution (6) shows the echo power of the TDPELP when the APD is on the focal plane. The size of the TDPELP varies with the different positions between the RL and APD when the APD is not on the focal plane (Fig. 2). The intensities of the TDPELP at positions I and VI [Fig. 2(b)] are weaker than those at positions II, III, V, and IV, because the active area of the APD is smaller than the area of the TDPELP, relatively decreasing the intensity. The smallest size of the TDPELP is obtained at position III, i.e., when the APD is on the focal plane. When APD is located at position II or IV, where the size of the TDPELP is larger than that at position III but smaller than that of the active area of the APD, the intensity is not changed despite the defocusing of APD. However, when APD is located on the defocused position, the intensity is inversely proportional to the square of the distance of the defocused site [22], i.e., Prd’(t) ∝1/△l2, where △l is the defocus offset. Therefore, the TDPELP power under the defocus offset (△l) is as follows:where Prd’(t) is the TDPELP power under defocus, r0 is the radius of the beam spot at the focal plane, i.e., the size of the TDPELP at position III [Fig. 2(b)]. Equation (7) theoretically shows that the echo power of TDPELP decreases with increasing defocus offset.
Fig. 2 Positions between receiving lens (RL) and avalanche photodiode (APD) under focused and defocused conditions. (a) Several typical positions between the RL and APD. (b) Corresponding cross-sections of APD and TDPELP.
2.2 Off-axis
Three positions between the active area of the APD and the TDPELP are studied when only the off-axis effects are considered (Fig. 3). Figure 3(a) shows the small off-axis, where the TDPELP is encircled by the active area of APD. The expression of the TDPELP power is identical to Pr(t) [Eq. (6)]. Figures 3(c) and 3(d) illustrate the large off-axis case, in which no response from the APD is obtained because of the weak intensity for detection. Therefore, the echo power of the case shown in Fig. 3(b) is determined by the overlapping area between the APD and TDPELP. According to the geometrical relationship in sector [Fig. 3(e)], the following expressions are obtained:
Fig. 3 Effects of different off-axis conditions on the TDPELP. (a)△d < (D/2)– r0. (b) Overlap between the active area of APD, i.e. (D/2)–r0<△d<(D/2) + r0. (c) TDPELP and the active area of APD are tangent, i.e., △d = (D/2) + r0. (d) The spot leaving from the active area of APD, i.e., △d>(D/2) + r0. (e) Details on the overlap of (b).
Similarly, in sector, the following expressions are obtained:
Meanwhile, based on cosine law, θ1 and θ2, can be written as follows:
The illuminated area of the active area of the APD, i.e., the overlapping area between the APD and TDPELP, is as follows:
Therefore, according to the relationship between Δd, r0 and D/2, the effective TDPELP power under the defocus offset (Δd) is as follows:
2.3 Tilt between RL and APD
Tilting between RL and APD (Fig. 4) results in an elliptical, rather than circular, TDPELP on the APD, which is contrary to the effect caused by the off-axis case. The distance between the ellipse and the circle is labeled as Δb, and the major and minor axes of the ellipse are labeled as a and b, respectively. The crossing point (x, y) between the ellipse and circle is determined as follows:
According to Eq. (13), we obtainThe length of O3G in Fig. 4(f) is calculated as follows:whereFig. 4 Relative positions between RL and APD under tilted APD or RL (a) Tilt angle between RL and APD. (b) △b≤(D/2)–a. (c) (D/2)–a <△b<(D/2) + a. (d) and (e) are situations when △b≥(D/2) + a. (f) Details on the overlap of (c).
Therefore, angles γ1 and γ2 are obtained as follows:
The overlapping area between the ellipse and the circle includes sectors and, which are determined as follows [23]:
Therefore, the overlapping area Slap is as follows:
According to the positions between the TDPELP and APD, three situations are obtained, as follows:
Equations (7), (12) and (20) show the effects resulting from the defocus, off-axis, and tilt between the RL and APD, respectively.
3. Experiments and results
3.1 Experimental setup
We performed experiments to test the results in section 2 using the experiment setup shown in Fig. 5. The pulse laser (PL) has a central wavelength of 905 nm, pulse width of 100 ns, peak power larger than 300 W, and range of repetition frequency from 1 kHz to 10 kHz. The PL is triggered by the function wave generator. Then, the pulse collimated by the TL goes through the attenuator and illuminates the diffused reflector (DR; the reflectance is 0.8, and the distance between the DR and the system is 8 m). The reflected or scattered laser pulse is received by the RL [including a fixed focal lens (Canon®, EF 100 mm, f/2.8)] and a tunable lens (Optotune®, ML-20-35-NIR, clear aperture of 20 mm, response range of 700–1100 nm) and focused on the APD (active area of 4 mm, 65A/W@905 nm). In the experiment, we use a relatively large active area of APD to analyze how the TDPELP is affected by the different positions between RL and APD. We use a tunable lens with a rotational stage to adjust the position with respect to APD. The vision system (Allied Vision®, Bigeye G-132B NIR Cool, resolution1280 × 1024) is used to quantitatively evaluate the TDPELP affected by the relative positions between RL and APD.
Fig. 5 Experimental setup and components. FWG-function wave generator, PL-pulsed laser, TL-transmitting lens, DR-diffused reflector, RL-receiving lens, and APD-avalanche photo diode.
3.2 Modeling verifications
3.2.1 Single effect on the TDPELP
We experimentally validate the models using the three positions between the RL and APD. We used normalized echo power in the following experiments because of the universality of theory and the distinct units of TDPELP used between theoretical and experimental studies. First, we study the TDPELP under focused and defocused conditions (Fig. 6). Three TDPELPs are saturated when △l are −10, 0, and 10 mm [Figs. 6(c)–6(e)]. These results illustrate that too strong echo power results in deep saturation of TDPELP, and this phenomenon is shown in Fig. 6(d) with a blue line. The width of the TDPELE is broadened, and the relative difference between simulations and experiment is 2.6% [calculated as (390 ns–380 ns)/380 ns) × 100; Fig. 6(d)]. This result illustrates that the simulation results agree well with experimental results. Similar results can be found when △l = ± 20 mm [Figs. 6(b) and 6(f)]. The APD on the focal plane is not the best position, because the intensity of the TDPELP surpasses the linear response range of the APD. For example, the TDPELP is deeply saturated when APD is located on the focal plane [Fig. 6(d)]. However, such strong intensity can be adjusted by placing the APD at the defocus position. When |△l | is 20 mm, the speckle size of the TDPELP is approximately 5 mm, which is larger than the APD. This result indicates that the echo power slightly decreases under the defocused position, so an unsaturated TDPELP is obtained. Thus, a potential approach to adjust TDPELP by manually defocusing under distinct echo power is provided.
Fig. 6 Comparison of simulation and experimental results of the TDPELP under diverse cases. (a) Experimental results under different △l values. (b) and (f) are the cases when △l is ± 20 mm. (c) and (e) are cases when △l is ± 10 mm. (d) is the case for the focal position.
The effects of off-axis and tilt between the RL and APD by experiments are shown in Fig. 7(a), which illustrates the different TDPELPs under various △d values. The amplitude of the TDPELP increases from 0.4 to 1.0 when △d decreases from 1.6 mm to 0.1 mm. The pulse width increases from 108 ns (unsaturated) to 483 ns (deep saturated), which illustrates the strong intensity in the saturated TDPELP. Actually, the TDPELP is saturated when △d is less than 1.0 mm. For example, the pulse width is 108 ns when △d is 1.6 mm [Fig. 7(b)]. The pulse widths are 317 and 483 ns when △d are 0.4 and 0.1 mm, respectively. Besides that, we find that the time stamps at peak under different Δd are variant. Compare with time stamps between Δd = 0.1mm and 1.6mm, the difference of time stamp is 12ns, which means that the ranging error is 1.8m (0.5 × 3 × 108m/s × 12ns × 10−9 = 1.8m). Therefore, distorted TDPELP results in range error under strong intensity.
Fig. 7 Comparison of simulations and experimental results under off-axis case. (a) △d increases from 0.1 mm to 1.6 mm. (b) Δd = 1.6 mm.
The effects of tilt between the RL and APD by experiments are shown Fig. 8. The amplitude of the TDPELP increases from 0.4 to 1 when the tilt angle α decreases from 1.75° to 0.5°. The experimental results agree well with the simulations. For example, the relative error of pulse width between these results is 3.5% [calculated as (147–142)/142] when α is 0.75°. The TDPELP is saturated when α is less than 1.5°. For example, when α is 0.75°, the width of the TDPELP is 147 ns [Fig. 8(b)]. When α is 1.75°, the width of TDPELP is 107 ns (Fig. 8a)]. Additionally, compared with Fig. 8(a) and Fig. 7(a), no deep saturation is found in tilted cases, illustrating that the loss of echo power resulting from tilt angle is larger than in the defocus and off-axis. Thus, the effects from off-axis are larger than those from tilting between the RL and APD.
Fig. 8 Comparison of simulations and experimental results with tilt angle between the RL and APD. (a) when α is increased from 0.5° to 1.75°. (d) when α is 0.75°.
3.2.2 Combined effects on the TDPELP
We further study the TDPELP under the combined defocus and varied off-axis distances [Fig. 9(a)]. △d changes from 0.1 mm to 1.6 mm, and the distance between the RL and APD is 144 mm, with APD on the defocus plane. The speckle size of TDPELP is approximately 3.5 mm. Therefore, when △d is not large, e.g., △d ranges from 0.1 mm to 0.7 mm, the TDPELP is saturated because of the strong intensity focusing on APD [Fig. 9(b)]. Increasing △d from 0.1 mm to 1.6 mm results in decreased TDPELP width from 209 ns to 104 ns, which shows that unsaturated TDPELP is still obtained under combined effects.
Fig. 9 TDPELPs under combined defocus and off-axis cases. (a) Experimental diagram. (b) TDPELs under different △d values when the distance between RL and APD is 144 mm, i.e., defocused case.
In practical use, the effects may result from the combined cases of defocus, off-axis, and tilt. We perform experiments by combining these factors, show in Fig. 10, where l is 144 mm, φ is 0.5°, and △d ranges from 0.1 mm to 1.6 mm. The results are similar to those in Fig. 9(b), and the intensity of the TDPELP decreases with increasing off-axis distance. The intensity of TDPELP decreases from 1 (saturated) to 0.42 (unsaturated) when △d increases from 0.1 mm to 1.6 mm, and the pulse width decreases from 187 ns to 103 ns. The width of TDPELP is unsaturated when the amplitude is less than 0.68 [Fig. 10(b)], in which the pulse width is 108 ns.
Fig. 10 Experiments under the combined effects of defocus, off-axis, and tilt angles between the RL and APD. (a) Experimental diagram. (b) Experimental results under combined effects with different △d values when the distance between the RL and APD is 144 mm, and tilt angle is 0.5°.
3.3 Improvements on the TDPELP via tunable lens
The theoretical analytical and experimental results show similar effects on TDPELP. These effects include saturation, broadening, and no response, regardless of the positions between the RL and APD. Manual defocusing of APD is reasonable under strong echo power. Therefore, such results provide a possible approach to adjust TDPELP using tunable lens under different situations. We conduct two experiments to verify that TDPELP can be adjusted using tunable lens under different conditions (Fig. 11). Two pairs of curves of TDPELP are obtained, i.e., A and A’, as well as B and B’. For saturated cases (Fig. 11 with red point line), the parameters are set to l = 144 mm, φ = 0°, and △d is 0.1 mm. The A’ curve is saturated and pulse is broadened because of too strong echo power. Such a TDPELP under these cases can be adjusted to the A curve using tunable lens by increasing l from 144 mm to 158 mm. Compare with time stamps at peaks between A and A’, the difference of time stamp is 28ns, which means that the ranging error is 4.2m (0.5 × 3 × 108 m/s × 28ns × 10−9 = 4.2m). The results show that the ranging error is decreased by the use of tunable lens. For cases with no response (l is 150 mm, d is 2.5 mm, and φ is 0.5°), although APD is on the focal plane, the TDPELP cannot be detected by APD, because the echo is not on the APD (B’ curve in Fig. 11). However, after adjusting the tunable lens by increasing l from 150 mm to 163 mm, an acceptable TDPELP can be obtained as shown in the B curve. The results show that TDPELP can be manually adjusted by the tunable lens. The focal length can be adjusted by a loop controlling system with the advantage of the high response speed of the tunable lens, until an acceptable TDPELP is obtained.
4. Discussions
The experimental results show that two features of the TDPELP, namely amplitude and pulse width, are useful under the different positions between RL and APD, and these features directly reflect the TDPELP. Therefore, we discuss these two features under different intensities of a transmitting laser, which is achieved by changing the attenuation rate, and the results are shown in Fig. 12.
Fig. 12 Two features of TDPELP are affected by the relative positions between the tunable RL and APD. (a) Amplitude of TDPELP versus displacement. (b) Width of TDPELP versus displacement. Comparison of the simulation and experimental results at attenuation rate of [(c) and (d)] 0.4 and [(e) and (f)] 0.7.
The amplitude of the TDPELP is approximately 4 V, and the TDPELP is saturated when the distance between the RL and APD is close to 150 mm. Under attenuation rate of 0.4, the amplitude of TDPELP increases from 0 V to 3.6 V, when the displacement increases from 128 mm to 150 mm. Then, the amplitude decreases from 3.6 V to 0 V, when the displacement increases from 150 mm to 170 mm. All curves are almost symmetrical with respect to the axis of l = 150 mm [Figs. 12(a) and 12(b)]. This characteristic indicates that the effects on the TDPELP from the front and back positions of the defocus are almost identical (points A and A’ in Figs. 12(a) and 12(b), respectively). Additionally, no response is obtained when the displacement is too large or too small. For example, when attenuation rate = 0.7, no response is obtained when the displacement is less than 130 mm or larger than 163 mm [red circles in Figs. 12(e) and 12(f)]. Therefore, the corresponding width of the TDPELP is zero [Fig. 12(d)]. The simulation and experimental results agree well with each other [Figs. 12(c)–12(f)]. In the simulation, a response is obtained when the displacement is less than 137 mm [Figs. 12(e) and 12(f)]. However, no response is obtained experimentally when the displacement is less than 137 mm. We manually set a threshold at an attenuation rate of 0.7 to maintain the universality of the theory. Therefore, we obtain consistent tendency between experiments and simulations. The width of each curve can be divided into two zones, shown in Fig. 12 (b) with blue and yellow shadows, respectively. The non-broadening zone indicates that the system can obtain better performance than the TDPELP broadening zone. Moreover, the non-broadening zone under stronger laser intensity is shorter than that under the small one, such as when high laser power is used, low attenuation rate is 0.4, and the range of constant pulse is 4 mm (142–138 mm). However, under low laser power, i.e., high attenuation rate (e.g., 0.7), the range of the non-broadening zone is 6 mm (138–132 mm). Therefore, lower laser power is helpful to increase the range of the constant pulse width.
5. Conclusions and future works
We theoretically study the effects of the different positions between the RL and APD, namely, defocus, off-axis, and tilt. Then, we conduct experiments under individual and combined effects. The results show good agreement between the simulations and experiments. Results show that distortions including saturation and broadening of TDPELP are obtained, regardless of the position between RL and detector. The amplitude and width of the TDPELP varied under the different positions, and the width is increased by intensity. Such properties provide the feasibility of adjusting intensity of TDPELP by tunable optical elements. We successfully performed experiments to adjust the TDPELP under extreme situations of too strong intensity and no response from the APD. The results show that an optimal TDPELP can be easily obtained through the proposed simple approach.
Funding
National Science Foundation (NSF) (51327005, 61605008, and 91420203); Jiangsu Province Natural Science Foundation of China (No. BK20160375); National Key Foundation for Exploring Scientific Instrument (2014YQ350461).
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