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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 17 — Aug. 21, 2006
  • pp: 7670–7677
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A unified approach to the analysis of incoherent Doppler lidars

Antonio Lázaro and Aniceto Belmonte  »View Author Affiliations


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

The scattering matrix is the most common used technique to study propagation waves in transmission lines and microwave systems [1–3

1. J. A. Dobrowolski, Introduction to Computer Methods for Microwave Circuit Analysis and Design (Artech House, Boston, 1991).

]. However, the application of scattering theory in optics has usually been limited to multilayer propagation of plane waves, and the analysis of spatial diffraction, the most characteristic effect of any optical element, has had to be considered through techniques much less competent. One of the most used approaches, the angular spectrum propagation method, analyzes the propagation of optical perturbations through systems consisting of optical elements, apertures, and free space regions using two-dimensional fast Fourier transforms [4

4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, Boston, 1996).

]. By using Fourier techniques, the incident beams are decomposed in plane waves that propagate through the optical system. Unfortunately, although this method requires a larger number of plane waves to represent typical optical perturbances, it deals very poorly with optical interferometric systems (e.g. Fabry-Perot and Fizeau filters) where multiple reflections need to be accurately considered to describe correctly their behavior. Consequently, methods other than those based on angular spectra are required to solve this set of problems.

In Section 2, we describe the principles of the technique and define the main parameters needed to model both basic optical elements and more complex optical systems. Section 3 describe the scattering matrix associated to a Fabry-Perot etalon and consider its use in the modeling of a complex, realistic incoherent interferometric system based on Fabry-Perot etalons. Section 3 recapitulates the main conclusions of this research.

2. A generalized scattering matrix representation

In this work we suggest a novel approach to fast and accurate analysis of large-scale incoherent optical lidar systems based on interferometric devices. We show that such systems may be described by scattering matrices (S matrices) and are connected by free-space propagations with corresponding S matrices. The properties of the entire optical interferoemeter are then accurately described by recursive combination of the individual S matrices of the functional optical components and the free-space propagations into a total S matrix.

To be specifics, using this new base we decompose any arbitrary wave U as:

Urφ=m=MMn=0NfmnJm(vmnr)ejmφ,
(1)

To illustrate our approach, we consider a interferometric system that consists of linear two-port components, including free-space propagations. In the most general case, each component may be a diffractive optical device. For the arbitrary diffractive optical device shown in Fig.1, each incident Bessel beam with order l at port i, ai,l, produces reflected Bessel modes at port i, bi,l and transmitted Bessel modes at port j, bj,l. 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 i with input mode m and output port j and mode n:

Sji,mn=bj,nai,m|ak,v=0,ki,vm
(2)

For a simple two-port component, this matrix can be written as:

(b1b2)=S(a1a2)=(R11T12T21R22)(a1a2)
(3)

Fig. 1 Outgoing and incoming waves in two-port optical network. We denote the incoming and outgoing waves at ports i by the L=(2M+1)×(N+1)-component vectors ai, and bi, respectively

3. Modal parameters for Fabry-Perot etalons

It is straightforward to apply our analysis method to the study of Fabry-Perot interferometers. Figure 2 shows a simple, schematic Fabry-Perot etalon composed by two mirrors separated a distance 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 S with S11=S22=R and S21=S12=j1R, where R is the complex reflectivity of the mirror surfaces and j=1 takes into consideration the π/2 phase change introduced to the transmitted waves by the reflective layer. Obviously R and, consequently, the scattering parameters S11, S21, S12, and S22, have no dependency with the spatial frequencies v mn of the illumination wave.

On the other hand, the free-space propagation scattering description needs to consider an implicit dependency on the spatial frequencies 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-(λvmn)2)1/2 along the propagation axis must be regarded as the factor describing the phase delay associated to the Bessel-wave component with spatial frequency v mn. Now, the required free-space propagation can be analyzed through a scattering matrix S with S11 = S22 = 0 and S21 = S12 = exp [-j k d (1-(λvmn)2)1/2], where 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.

Fig. 2. A Fabry-Perot interferometer can be analyzed as a multiport optical system composed of three different two-port elements. Waves b2 transmitted by the first reflective layer propagated a distance d to reach the second reflective layer as incoming waves a5. By properly defining the scattering matrix of any of these three elements, our modal approach allows to describes any interference and diffractive problem characterizing the Fabry-Perot etalon behavior.

S11(vmn)=S22(vmn)=R+Rexp[jkd1(λvmn)2]1Rexp[jkd1(λvmn)2]
S21(vmn)=S12(vmn)=1R2exp[jkd1(λvmn)2]1Rexp[jkd1(λvmn)2]
(4)

Obviously, this synthetic approach must lead to the same results that the genuine one considering a matrix for any component on every etalon. We have tested and confirmed this result. Also, it is worth to note that results in (4) are similar –but not identical– to the well known Airy formulas for the transmission of an ideal Fabry-Perot resonator [11

11. G. Hernandez, Fabry-Perot Interferometers (Cambridge U.Press, Cambridge, UK, 1988).

]. Here, we consider not just the interference of multiple resonator modes traveling back and forth between the two mirrors, but any diffraction effect and illumination divergence not contemplated in the basic Airy expressions.

Fig. 3. Flow chart for a Fabry-Perot interferometer. In bold line, the signal path for an input wave that propagates to the output of the etalon. To find the transfer function of the etalon system represented by this block diagram we use Mason’s gain rule.

Next, we show the result of applying our techniques to a tandem of two Fabry-Perot resonators acting as an incoherent Doppler interferometric system. Direct detection Doppler lidars probing atmospheric winds need to consider both Mie scattering from aerosols in the lower troposphere and Rayleigh scattering form the lower stratosphere. As the spectral widths of aerosol (MHz) and molecular (GHz) backscattered signals are orders of magnitude different, two different interferometric channels must be defined in any reliable working lidar system. In an effort to increase the overall system efficiency, usual instrument configurations capture the light reflected by a high-spectral-resolution aerosol Fabry-Perot etalon into another wider passband molecular etalon. Certainly, the analysis and optimization of such complex instruments has to consider in a realistic way the limitations caused by any cross-talk between light channels and any interference and diffractive effect.

Fig.6 (left) shows the flowchart representing two etalons in cascade. Using Mason rules, the two-etalon transmittance for a given Bessel mode l (element of the S matrix diagonal as this network is not dispersive) is obtained:

T(ρl)=S21(ρl)=T1τ21T21τ21R2τ12R1
(5)

where T1 and T2 are the two etalon transmittances, R1 and R2 are the etalon reflectances, and τ21 is the transmittance (attenuation) between the two etalons and τ12 is the transmittance (isolation) in the reflected signal between the two etalons.

Note, that expression (5) has the same form of the well-known Fabry-Perot Airy function. Alternatively, this result an be obtained by summing the multiple wave reflections between 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]

]. However, using the proposed flow chart method, compact analytical expressions for triple or other optical circuits can be easily obtained. Distinctively, expressions (4) and (5) give the transmittance for each Bessel mode l, and they depend on the the radial angular spectrum frequency associated to this wave (ρl=vmn). 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 (τ12=0, τ21=1), and without attenuator (τ12= τ21=1), and with attenuator of 5% (τ12= τ21=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

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

].

Fig. 4. Flow chart (up) for two etalons with a attenuator used to reduce reflections between etalons. In bold line, it is market the signal path for an input wave that propagate to the output of the second etalon. Transmittance for the dual-etalon solar filter without absorber, with 5% absorber, and isolated etalons omitting reflections (down).

4. Conclusion

Interferometers used for atmospheric wind studies need to produce narrow and smooth interference fringes. By using our generalized scattering matrix formalism, we have analyzed the details of mode formation in multiple Fabry-Perot incoherent Doppler lidar systems and we have extended these results to the apparently more complex situation of incoherent Doppler lidars based on Fizeau wedge interferometers. The unique capabilities of the technique have been used to address the optimization of the interferometric device parameters, those producing the sharpest fringes in the detection plane. We have chosen to show the sensitivity of the method by studying the wind measurement uncertainty inherent to the lidar instrumentation as an indication of the performance of systems based on multiple Fabry-Perot etalons and Fizeau wedges. We intend to present these results in a companion paper.

References and links

1.

J. A. Dobrowolski, Introduction to Computer Methods for Microwave Circuit Analysis and Design (Artech House, Boston, 1991).

2.

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, New York, 1966).

3.

K. Kurokawa, “Power waves and the scattering matrix,” IEEE Trans.Microwave Theory Tech. 13, 194–202 (1965). [CrossRef]

4.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, Boston, 1996).

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]

7.

A. Yariv, Introduction to Optical Electronics (Holt, Reinhart and Winston, New York, 1985).

8.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (John Wiley & Sons, Inc., New York, 1991).

9.

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

10.

D. M. Pozart, Microwave Engineering (Wiley, New York, 2004).

11.

G. Hernandez, Fabry-Perot Interferometers (Cambridge U.Press, Cambridge, UK, 1988).

12.

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

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

  1. J. A. Dobrowolski, Introduction to Computer Methods for Microwave Circuit Analysis and Design (Artech House, Boston, 1991).
  2. R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, New York, 1966).
  3. K. Kurokawa, "Power waves and the scattering matrix," IEEE Trans.Microwave Theory Tech. 13, 194-202 (1965). [CrossRef]
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, Boston, 1996).
  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]
  7. A. Yariv, Introduction to Optical Electronics (Holt, Reinhart and Winston, New York, 1985).
  8. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (John Wiley & Sons, Inc., New York, 1991).
  9. V.A. Monaco, P. Tiberio, "Computer-Aided Analysis of Microwave Circuits," IEEE Trans. Microwave Theory Tech. 22, 249-263 (1974). [CrossRef]
  10. D. M. Pozart, Microwave Engineering (Wiley, New York, 2004).
  11. G. Hernandez, Fabry-Perot Interferometers (Cambridge U.Press, Cambridge, UK, 1988).
  12. 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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