## Investigation of spherical aberration effects on coherent lidar performance |

Optics Express, Vol. 21, Issue 22, pp. 25670-25676 (2013)

http://dx.doi.org/10.1364/OE.21.025670

Acrobat PDF (1199 KB)

### Abstract

In this paper we demonstrate experimentally the performance of a monostatic coherent lidar system under the influence of phase aberrations, especially the typically predominant spherical aberration (SA). The performance is evaluated by probing the spatial weighting function of the lidar system with different telescope configurations using a hard target. It is experimentally and numerically proven that the SA has a significant impact on lidar antenna efficiency and optimal beam truncation ratio. Furthermore, we demonstrate that both effective probing range and spatial resolution of the system are substantially influenced by SA and beam truncation.

© 2013 Optical Society of America

## 1. Introduction

1. The final report of the EU FP6 project UPWIND, http://www.upwind.eu/

2. J. Mann, J. P. Cariou, M. Courtney, P. Parmentier, T. Mikkelsen, R. Wagner, P. J. P. Lindelöw, M. Sjöholm, and K. Enevoldsen, “Comparison of 3D turbulence measurements using three staring wind lidars and a sonic anemometer,” Meteorol. Z. **18**, 135–140(2009). [CrossRef]

7. J. Y. Wang, “Optimal truncation of a lidar transmitted beam,” Appl. Opt. **27**, 4470–4474(1988). [CrossRef] [PubMed]

8. B. J. Rye, “Primary aberration contribution to incoherent backscatter heterodyne lidar resturns,” Appl. Opt. **21**, 839–844(1982) [CrossRef] [PubMed]

8. B. J. Rye, “Primary aberration contribution to incoherent backscatter heterodyne lidar resturns,” Appl. Opt. **21**, 839–844(1982) [CrossRef] [PubMed]

8. B. J. Rye, “Primary aberration contribution to incoherent backscatter heterodyne lidar resturns,” Appl. Opt. **21**, 839–844(1982) [CrossRef] [PubMed]

## 2. Experimental setup

*L*

_{1}and

*L*

_{2}is adjusted such that the lidar signal is optimized with the hard target (rotating belt) placed at a range of 80 m. Around 0.5 mW of the diode laser output is tapped within the optical ciculator and is used as the local oscillator (LO) for the heterodyne detection. Both the signal from the rotating belt and the LO is focused onto the detector through the optical circulator. The ”virtual” back propagated local oscillator (BPLO) from the detector plane matches the transmit beam with a Gaussian field amplitude profile of radius,

*w*, at the plane of lens

*L*

_{2}. This configuration is commonly referred to as the Wang design [7

7. J. Y. Wang, “Optimal truncation of a lidar transmitted beam,” Appl. Opt. **27**, 4470–4474(1988). [CrossRef] [PubMed]

*ρ*=

*w/r*

_{L}_{2}where

*r*

_{L2}is the radius of

*L*

_{2}aperture, can be probed by changing the focal length (

*f*

_{1}) of lens

*L*

_{1}, since the imaging magnification of the beam is dependent on the focal length ratio between

*L*

_{1}and

*L*

_{2}. Two different

*L*

_{2}, both with exit aperture radius of

*r*

_{L2}= 35.65 mm (3 inch optics), are used in our experiments in order to evaluate the system under different degrees of SA. The lenses (

*L*

_{2}) are respectively a singlet lens (

*f*

_{2}= 200 mm) that is not corrected for SA and a doublet lens (

*f*

_{2}= 216 mm) designed for reduced SA. The correlation overview between the

*L*

_{1}focal lengths and

*ρ*can be found in Table 1. A quantitative illustration of the SA introduced by the

*L*

_{2}lenses is shown in Fig. 2(a), where the optical path difference (OPD) is measured in number of waves. The curves in Fig. 2(a)are generated using a Zemax simulation with monochromatic input, zero incident angle and assuming circular symmetry in the transverse plane, i.e. only the symmetrical components of wavefront errors are presented. The first six nonzero Zernike fringe coefficients, generated in Zemax, are listed in Table 2. Those coefficients are used to generate the OPD curves and they differ slightly from the standard Zernike coefficients, which is evident from the polynomials provided in the table. A detailed description of those coefficients can be found in the user’s manual of Zemax [9]. It is evident from the table values that the SA (

*Z*

_{9}) is the dominant source of the waverfront errors.

*L*

_{2}is possible using those OPD curves. The SA can be incorporated in the theoretical simulation by introducing an extra phase term,

*ϕ*to the truncated Gaussian field at

_{SA}*L*

_{2}according to Eq. (1)[8

**21**, 839–844(1982) [CrossRef] [PubMed]

*E*

_{0}is the peak amplitude,

*r*is the radial coordinate,

*ϕ*(

*r*) is the phase of the beam due to field curvature,

*ϕ*is the phase term induced by the SA and the OPD(

_{SA}*r*) is either curves shown in Fig. 2(a). Due to the circular symmetry, the field on the target plane can be calculated numerically by a simple Fourier-Bessel transformation of Eq. (1)with an appropriate field curvature [8

**21**, 839–844(1982) [CrossRef] [PubMed]

## 3. Results and discussions

*L*

_{2}case, the transverse irradiance profiles in different distances,

*z*after the singlet

*L*

_{2}are calculated and shown in Fig. 2(b). A side by side comparison between the numerical simulation and the experimental counterpart of the beam profiles is displayed in Fig. 3. The observed beam profiles are recorded with aid of an IR-detection card. An intensity clipping level was introduced in the presentation of the numerical results in order to simulate the saturation effect of the IR-detection card. In Fig. 3the consistency between the simulations and the measurements is quite evident, qualitatively validating the accuracy of our theoretical simulations.

*ρ*≈ 0.8 through the

*L*

_{2}is required [3]. However, in the presence of SA, optimal

*ρ*will differ from 0.8 and the overall antenna efficiency will decrease compared to the aberration-free case. It is easiest to calculate the antenna efficiency,

*η*using the target-plane formalism based on Siegman’s antenna theorem [3, 10

_{a}10. A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Appl. Opt. **5**, 1588–1594(1966). [CrossRef] [PubMed]

*λ*is the wavelength,

*R*is the intended imaging or probing range (80 m in our case),

*A*is the area of

_{r}*L*

_{2}and

*I*

_{target}is the irradiance distribution of the output beam at the target plane position,

*z*and is normalized by the total beam power before truncation.

*z*=

*R*as a function of

*ρ*is shown for both

*L*

_{2}lenses and compared with the aberration-free case (the dashed lines). The simulation is constructed such, that the algorithm is imitating the same distance optimization procedure as in the experiments. It is evident that not only the overall antenna efficiency has decreased as the consequence of SA, but the optimum

*ρ*also shifts with different degrees of aberrations. Even for the doublet case where the SA is minimal, there is still a clear shift of the optimal

*ρ*and a quite noticeable drop in the maximum antenna efficiency. It demonstrates the importance of considering the aberration effects in designing a lidar system. The result in Fig. 4(a)indicates that optimal

*ρ*decreases with increasing degree of SA.

*ρ*, normalized to the maximum data point of the doublet case, is also shown in Fig. 4(a). The experimental data shows the same tendencies as the numerical calculations. While the singlet data coincides quite well with the simulations, there is a slight deviation for the doublet case. Since the simulation only includes the SA effect, it is reasonable that the doublet data can deviate from the simulation due to other aberration effects in the rest of the system (we do observe small degree of astigmatism from the output of the optical circulator). The singlet data is, on the other hand, dominated by the SA, which explains the good agreement with the simulation.

*L*

_{1}and

*L*

_{2}are presented in Fig. 5along with their numerical counterparts. In general the numerical results coincide quite well with the experimental data, but the singlet case gives a much better match than the doublet case. As we discussed earlier the simulations only include the SA effect, since the measurement with singlet

*L*

_{2}is dominated by SA while the doublet

*L*

_{2}case is not, it is expected that singlet lens case will provide a better fit. The simulation results do suggest that the full system (not only

*L*

_{2}) potentially suffers from other aberration effects like astigmatism and coma, which are comparable with the SA introduced by the doublet

*L*

_{2}, since the experimental data shows a visible broadening of the weighting function compared with their numerical counterparts. Comparing the simulation with the experimental data there is a broadening of 21% for the green dash line (

*f*

_{1}= 8.1 mm) while the broadening is 70% for the blue dash line case (

*f*

_{1}= 18.8 mm). The residual broadening is likely due to the astigmatism of the beam from the optical circulator, which also explains the observed increase in the degree of broadening with the focal length of

*L*

_{1}.

*L*

_{2}is minor in terms of signal strength enhancement but more on improved spatial resolution. For the more general turbulent air flow in the probing volume, higher signal strength enhancement due to tighter spatial confinement is of course expected.

**21**, 839–844(1982) [CrossRef] [PubMed]

*ρ*the weighting function will suffer from both peak shift and broadening effects under the influence of SA. Recalling Eq. (2)and the transverse irradiance profiles in Fig. 2(b)it is expected that the weighting function for the singlet

*L*

_{1}case will have a peak around 60 m, since the effective beam confinement is tightest there and not at the intended imaging range, 80 m. The data shown in Fig. 5provides, to our knowledge, the first experimental confirmation of these tendencies. But our numerical and experimental results also show that both the peak shift and the broadening effect can be compensated to certain degree by selecting a

*ρ*appropriate for a particular degree of SA, which is not described in the theoretical work by Rye [8

**21**, 839–844(1982) [CrossRef] [PubMed]

*L*

_{2}only suffers from minor SA, nevertheless, we still observe a weighting function peak shift of 2.4 m (for

*L*

_{1}focal length of 8.1 mm), indicating the sensitivity of the lidar system to the SA effect. We also note in Fig. 5that the peak shift is larger for lower

*ρ*. Thus, in applications where delicate probing range control is necessary, it is advisable to calibrate the lidar system with a hard target by mapping the weighting function.

## 4. Conclusion

*L*

_{2}. This corrective measure results from the novel finding in this work that the optimal truncation ratio depends on the degree of SA. Furthermore we have shown that both SA and truncation ratio influence the peak shift and width of the weighting function. In applications where precise probing range and spatial resolution are essential, a weighting function calibration of the lidar system using a hard target might be necessary. It is worth to stress that this study can also be applied to accurately model the weighting function of pulsed coherent lidar systems [2

2. J. Mann, J. P. Cariou, M. Courtney, P. Parmentier, T. Mikkelsen, R. Wagner, P. J. P. Lindelöw, M. Sjöholm, and K. Enevoldsen, “Comparison of 3D turbulence measurements using three staring wind lidars and a sonic anemometer,” Meteorol. Z. **18**, 135–140(2009). [CrossRef]

## Acknowledgments

## References and links

1. | The final report of the EU FP6 project UPWIND, http://www.upwind.eu/ |

2. | J. Mann, J. P. Cariou, M. Courtney, P. Parmentier, T. Mikkelsen, R. Wagner, P. J. P. Lindelöw, M. Sjöholm, and K. Enevoldsen, “Comparison of 3D turbulence measurements using three staring wind lidars and a sonic anemometer,” Meteorol. Z. |

3. | T. Fujii and T. Fukuchi, eds. |

4. | Y. Zhao, M. J. Post, and R. M. Hardesty, “Receiving efficiency of pulsed coherent lidars. 1: theory,” Appl. Opt. |

5. | R. G. Frehlich and M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. |

6. | B. J. Rye and R. G. Frehlich, “Optimal truncation and optical efficiency of an apertured coherent lidar focused on an incoherent backscatter target,” Appl. Opt. |

7. | J. Y. Wang, “Optimal truncation of a lidar transmitted beam,” Appl. Opt. |

8. | B. J. Rye, “Primary aberration contribution to incoherent backscatter heterodyne lidar resturns,” Appl. Opt. |

9. | ZEMAX Optical Design Program User’s Manual (July8, 2011) pp. 196–199. |

10. | A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Appl. Opt. |

11. | J. Mann, A. Peña, F. Bingöl, R. Wagner, and M. S. Courtney, “Lidar scanning of momentum flux in and above the atmospheric surface layer,” J. Atmos. Oceanic Technol. |

**OCIS Codes**

(280.0280) Remote sensing and sensors : Remote sensing and sensors

(280.3340) Remote sensing and sensors : Laser Doppler velocimetry

(280.3640) Remote sensing and sensors : Lidar

**ToC Category:**

Remote Sensing

**History**

Original Manuscript: July 24, 2013

Revised Manuscript: September 3, 2013

Manuscript Accepted: October 10, 2013

Published: October 21, 2013

**Citation**

Qi Hu, Peter John Rodrigo, Theis F. Q. Iversen, and Christian Pedersen, "Investigation of spherical aberration effects on coherent lidar performance," Opt. Express **21**, 25670-25676 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-25670

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### References

- The final report of the EU FP6 project UPWIND, http://www.upwind.eu/
- J. Mann, J. P. Cariou, M. Courtney, P. Parmentier, T. Mikkelsen, R. Wagner, P. J. P. Lindelöw, M. Sjöholm, and K. Enevoldsen, “Comparison of 3D turbulence measurements using three staring wind lidars and a sonic anemometer,” Meteorol. Z.18, 135–140(2009). [CrossRef]
- T. Fujii and T. Fukuchi, eds. Laser Remote Sensing(CRC Press, 2005).
- Y. Zhao, M. J. Post, and R. M. Hardesty, “Receiving efficiency of pulsed coherent lidars. 1: theory,” Appl. Opt.29, 4111–4119(1990). [CrossRef] [PubMed]
- R. G. Frehlich and M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt.30, 5325–5352(1991). [CrossRef] [PubMed]
- B. J. Rye and R. G. Frehlich, “Optimal truncation and optical efficiency of an apertured coherent lidar focused on an incoherent backscatter target,” Appl. Opt.31, 2891–2899(1992). [CrossRef] [PubMed]
- J. Y. Wang, “Optimal truncation of a lidar transmitted beam,” Appl. Opt.27, 4470–4474(1988). [CrossRef] [PubMed]
- B. J. Rye, “Primary aberration contribution to incoherent backscatter heterodyne lidar resturns,” Appl. Opt.21, 839–844(1982) [CrossRef] [PubMed]
- ZEMAX Optical Design Program User’s Manual (July8, 2011) pp. 196–199.
- A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Appl. Opt.5, 1588–1594(1966). [CrossRef] [PubMed]
- J. Mann, A. Peña, F. Bingöl, R. Wagner, and M. S. Courtney, “Lidar scanning of momentum flux in and above the atmospheric surface layer,” J. Atmos. Oceanic Technol.27, 959–976(2010). [CrossRef]

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