## A unified approach to the analysis of incoherent Doppler lidars

Optics Express, Vol. 14, Issue 17, pp. 7670-7677 (2006)

http://dx.doi.org/10.1364/OE.14.007670

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### Abstract

This work describes the use of a generalized modal scattering matrix theory as a fast, efficient approach to the analysis of incoherent Doppler lidars. The new technique uses Bessel beams, a type of optical vortices, as the basic modal expansion characterizing optical signals. The tactic allows solving both multilayered reflections problems and spatial diffraction phenomena using scattering parameters associated with the transmitted and reflected spectrum of vortices. Here, we will show the capabilities of the technique by considering realistic incoherent Doppler systems based on Fabry-Perot etalons.

© 2006 Optical Society of America

## 1. Introduction

5. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**, 651–654 (1986). [CrossRef]

6. J. Durnin, J.J. Miceli Jr., and J.H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. **58**, 1499–1501 (1987). [CrossRef] [PubMed]

## 2. A generalized scattering matrix representation

5. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**, 651–654 (1986). [CrossRef]

6. J. Durnin, J.J. Miceli Jr., and J.H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. **58**, 1499–1501 (1987). [CrossRef] [PubMed]

*U*as:

*J*is the

_{m}*m-th*order Bessel function of the first kind, and

*r*and

*φ*denotes the cylindrical coordinates. The product of these Bessel functions with the harmonic functions in Eq. (1) defines the so-called Bessel beams. The spatial frequencies

*v*

_{mn}distinguish any of the optical modes used to describe the complex amplitude

*U*. A number of (

*2M+1)×(N+1)*modes have been considered. Once the modal basis has been set, the wave

*U*can be explicitly described as a vector with components

*f*. The number of modes needed depend on the illumination beam studied. For example, for a typical gaussian beam we need to consider just a few dozen modes. For more complex illuminations, such as those considering optical aberrations, a higher number of modes may be required.

_{mn}*l*at port

*i*,

*a*, produces reflected Bessel modes at port

_{i,l}*i, b*and transmitted Bessel modes at port

_{i,l}*j*,

*b*. As the device is diffractive, each input Bessel beam generates a complete set of output modes. The generalized scattering parameters can be defined at input port

_{j,l}*i*with input mode

*m*and output port

*j*and mode

*n*:

*and reflection*

**T***matrices makes explicit the physical meaning of the four submatrices of*

**R***. Arbitrary lidar systems can be analyzed using this generalized scattering parameters. Each subsystem is connected to other subsystems in the same way as microwave circuits. With this modal method, reflection between subsystems are taken in count with the S-parameter formulism. This formalism allows to consider diffractive optics by defining non-diagonal S-matrix. The computationally expensive 2-D convolution of plane waves in classical beam propagation methods is here a simple product of full modal S-matrix. If the optical device is not diffractive, the S-matrix is diagonal. Non-diffractive optical elements have diagonal S-matrix, as it has the non-diffractive propagation of our Bessel beams. To obtain the S parameter of a lidar system from the S parameters of the composed subsystem the same analysis techniques than in microwave networks can be used [9*

**S**_{j}9. V.A. Monaco and P. Tiberio, “Computer-Aided Analysis of Microwave Circuits,” IEEE Trans. Microwave Theory Tech. **22**, 249–263 (1974). [CrossRef]

## 3. Modal parameters for Fabry-Perot etalons

*d*. In our technique framework, we describe the etalon as a three linear two-port elements, i.e., two reflective layers and one free-space propagation. Basically, any mirror can be represented by a field scattering matrix

*with*

**S***R*is the complex reflectivity of the mirror surfaces and

*π/2*phase change introduced to the transmitted waves by the reflective layer. Obviously

*R*and, consequently, the scattering parameters

*S*and

_{11}, S_{21}, S_{12},*S*, have no dependency with the spatial frequencies

_{22}*v*

_{mn}of the illumination wave.

*v*

_{mn}used in Eq. (1) to describe the waves traveling the etalon cavity. Since each Bessel beam component has a different associated spatial frequency

*v*

_{mn}, each travels a differently the distance between the two etalon parallel planes, and relative phase delays are thus introduced. The well-known direction cosine

*(1-(λv*along the propagation axis must be regarded as the factor describing the phase delay associated to the Bessel-wave component with spatial frequency

_{mn})^{2})^{1/2}*v*

_{mn}. Now, the required free-space propagation can be analyzed through a scattering matrix

*with*

**S***S*and

_{11}= S_{22}= 0*S*, where

_{21}= S_{12}= exp [-j k d (1-(λv_{mn})^{2})^{1/2}]*λ*. is the illumination wavelength and

*k= 2π/λ*. is the wavenumber. Certainly, these results consider the foundations of scalar diffraction theory. In fact, as with the angular spectrum of plane waves, we are formulating the scalar diffraction in a linear framework where the basic optical disturbances are our Bessel beams: If a complex field distribution is analyzed across any plane, the various spatial components can be identified as Bessel waves traveling away from that plane so that the field amplitude across any other plane can be calculated by adding the contributions of these Bessel waves taking due account of the phase shifts they undergone during propagation. Using these propagation terms, we make sure that our estimations collect any diffraction effect due to diffracting structures (per example, apertures limiting the incoming light) or finite illumination sources (such as Gaussian field illuminations). Also, we use scalar diffraction in its most general forms, i.e., we do not need to consider Fresnel and Fraunhofer approximations used to reduce the mathematical manipulations. Our estimations are valid for any possible range making possible to consider both near and far fields of the illumination.

_{21}, S

_{12}and reflectivities S

_{11}, S

_{22}we have defined for them. It is easy to apply Mason’s rule to obtain the S-matrix components of a Fabry-Perot resonator:

*l*(element of the S matrix diagonal as this network is not dispersive) is obtained:

*T*and

_{1}*T*are the two etalon transmittances,

_{2}*R*and

_{1}*R*are the etalon reflectances, and

_{2}*τ*is the transmittance (attenuation) between the two etalons and

_{21}*τ*is the transmittance (isolation) in the reflected signal between the two etalons.

_{12}12. J.A. McKay, “Single and tandem Fabry-Perot Etalons as solar background filters for lidar,” Appl. Opt. **38**, 5851–5857 (1999). [CrossRef]

*l*, and they depend on the the radial angular spectrum frequency associated to this wave (

*ρ*). Thus, expressions (4) and (5) takes into account the angular dispersion of the light at the input. As a simple example, Fig.6 shows the transmittances function of the two-etalon combination (reflectivity R=90%, separations of 0.24 mm and 2.12 mm). This figure shows the ideal case without interreflections simulated sing an isolator between the two etalons (

_{l}=v_{mn}*τ*=0,

_{12}*τ*=1), and without attenuator (

_{21}*τ*=

_{12}*τ*=1), and with attenuator of 5% (

_{21}*τ*=

_{12}*τ*=0.95). This figure shows that interreflections can be reduced using an attenuator, but the cost is a reduction in the peak transmittance. This fact is well known in microwave area, where attenuators are common connected between mismatched devices to improve the return loss of the second device. Multiple etalon systems improve the background rejection with a modest cost to signal transmittance due the attenuation losses [12

_{21}12. J.A. McKay, “Single and tandem Fabry-Perot Etalons as solar background filters for lidar,” Appl. Opt. **38**, 5851–5857 (1999). [CrossRef]

## 4. Conclusion

*VBS*(Vortical Beam Spectra), a self-contained software package implementing our new, generalized modal scattering matrix theory and including a library of blocks to model the behavior of optical elements and systems.

*VBS*tools are helping us to implement commercial and research optical interferometric systems and their components. The software serves a broad range of tasks across a variety of optical problems from analysis and design to optimization and modeling.

## References and links

1. | J. A. Dobrowolski, |

2. | R. E. Collin, |

3. | K. Kurokawa, “Power waves and the scattering matrix,” IEEE Trans.Microwave Theory Tech. |

4. | J. W. Goodman, |

5. | J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

6. | J. Durnin, J.J. Miceli Jr., and J.H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. |

7. | A. Yariv, |

8. | M. Nieto-Vesperinas, |

9. | V.A. Monaco and P. Tiberio, “Computer-Aided Analysis of Microwave Circuits,” IEEE Trans. Microwave Theory Tech. |

10. | D. M. Pozart, |

11. | G. Hernandez, |

12. | J.A. McKay, “Single and tandem Fabry-Perot Etalons as solar background filters for lidar,” Appl. Opt. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.6200) Instrumentation, measurement, and metrology : Spectrometers and spectroscopic instrumentation

(280.3640) Remote sensing and sensors : Lidar

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 19, 2006

Revised Manuscript: August 2, 2006

Manuscript Accepted: August 3, 2006

Published: August 21, 2006

**Citation**

Antonio Lázaro and Aniceto Belmonte, "A unified approach to the analysis of incoherent Doppler lidars," Opt. Express **14**, 7670-7677 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7670

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### References

- J. A. Dobrowolski, Introduction to Computer Methods for Microwave Circuit Analysis and Design (Artech House, Boston, 1991).
- R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, New York, 1966).
- K. Kurokawa, "Power waves and the scattering matrix," IEEE Trans.Microwave Theory Tech. 13, 194-202 (1965). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, Boston, 1996).
- J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4,651-654 (1986). [CrossRef]
- J. Durnin, J.J. Miceli, Jr. and J.H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
- A. Yariv, Introduction to Optical Electronics (Holt, Reinhart and Winston, New York, 1985).
- M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (John Wiley & Sons, Inc., New York, 1991).
- V.A. Monaco, P. Tiberio, "Computer-Aided Analysis of Microwave Circuits," IEEE Trans. Microwave Theory Tech. 22, 249-263 (1974). [CrossRef]
- D. M. Pozart, Microwave Engineering (Wiley, New York, 2004).
- G. Hernandez, Fabry-Perot Interferometers (Cambridge U.Press, Cambridge, UK, 1988).
- J.A. McKay, "Single and tandem Fabry-Perot Etalons as solar background filters for lidar," Appl. Opt. 38, 5851-5857 (1999). [CrossRef]

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