## Abstract

Heterodyne efficiency of a coherent lidar system reflects the matching of phase and amplitude between a local oscillator (LO) beam and received signal beam and is, therefore, an indicator of system performance. One aspect of a lidar system that affects heterodyne efficiency is aberrations present in optical components. A method for including aberrations in the determination of heterodyne efficiency is presented. The effect of aberrations on heterodyne efficiency is demonstrated by including Seidel aberrations in the mixing of two perfectly matched gaussian beams. Results for this case are presented as animations that illustrate the behavior of the mixing as a function of time. Extension of this method to propagation through lidar optical systems is discussed.

© Optical Society of America

## 1. Introduction

An indicator of the performance of a coherent lidar system is its heterodyne efficiency. This parameter reflects the matching between local oscillator (LO) and received signal and is optimized when they are perfectly matched in both amplitude and phase distribution. Evaluation of heterodyne efficiency in coherent systems has been thoroughly examined from a theoretical perspective.[1–8] An exhaustive bibliography may be found in Ref 6. Aberrations present in the lidar optical system have been studied by considering the receiver as an integral unit or as a single telescope mirror.[5] Calculation of heterodyne efficiency for a specific coherent lidar receiver system, including propagation of measured beam distributions through individual components, has also been presented.[8,9] However, aberrations of the components were not considered in the calculations.

In this paper a method is proposed for including aberrations of individual components in the evaluation of heterodyne efficiency of coherent optical systems, with specific application to lidar receivers. Aberrations imparted to a beam by the components will be quantified by incorporating optical testing data in the model. Both the signal and local oscillator (LO) beams will be propagated through the optical system so that inherent phase contributions of elements will be considered. Examples of intensity distributions for the mixing of deterministic signal and LO fields under the simplification of no propagation will be shown as animations to illustrate the effect of aberration. A simple example of propagation through an afocal telescope will be used to illustrate extension of the technique to lidar optical systems.

## 2. Formulation for heterodyne efficiency with aberrations

Heterodyne efficiency is a the ratio of averaged effective coherent power to total incoherent power in the system.[8] This is optimized when the signal and LO beams are well matched in both amplitude and phase distribution. Following the procedure given in Ref 8, an analytic expression for heterodyne efficiency is developed in the presence of aberrations. Specific aberrations will then be discussed and the Seidel terms will be used to demonstrate the effect aberrations have on heterodyne efficiency. Uniform quantum efficiency over the area of the detector is assumed.

#### 2.1 Analytic expression for heterodyne efficiency

For a monostatic coherent lidar system with a short transmitted pulse, the heterodyne efficiency may be expressed

where the brackets indicate statistical average, **P** is a detector
coordinate, z is the range to the target, D is the area of the detector, u is
the complex amplitude of a received field, P is the power in the received field
as measured by the detector and the subscripts S and LO indicate signal and
local oscillator, respectively.[8] Heterodyne efficiency directly affects system
performance through the signal-to-noise relation

where *h* is Planck’s constant, *v* is
the optical frequency, and B is the detector bandwidth. Introducing an
aberration to a beam contributes an additional term to its phase. This may be
represented by

where u_{0} represents the electric field without aberrations and
W(r,θ) is the aberration function.[10] The form of W will be discussed in more detail
below.

#### 2.2 Seidel aberrations and their effect on heterodyne efficiency

Wavefront aberrations are a deviation of a beam’s wavefront from an
expected, or reference surface, which is typically a sphere. A common method for
expressing aberrations in optical systems is as a power series^{11}

where λ is the system transmitted wavelength, W_{1,m,n} is the
aberration coefficient expressed as a number of wavelengths, η is a
normalized object field coordinate, ρ is a normalized pupil radius
and θ is pupil azimuth. Arguments based on rotational symmetry
indicate that only specific combinations of the indices l, m and n may occur.
The five second order terms are the Seidel, or primary, aberrations denoted as
spherical, coma, astigmatism, field curvature and distortion. Of the five Seidel
aberrations, spherical, coma and astigmatism will be used in this paper for
demonstration of the effect aberrations have on heterodyne efficiency. The
effect of tilt, a first order term of the power series, will also be shown since
it can be interpreted as an angular misalignment of two planar wavefronts at the
detector surface. Examples of the shape of a wavefront containing these
aberrations may be found in Ref 12.

For purposes of illustrating the effect that aberrations have on heterodyne
efficiency, consider mixing two ideal gaussian beams which are initially matched
in both amplitude and phase. One beam represents signal that has traversed an
optical system and, as a result, has aberration added to its wavefront. The
other beam will represent the local oscillator, which maintains a planar
wavefront. One-way transmission will be used for simplicity in this
demonstration, although two-way transmission is typical of lidar systems under
consideration. A truncation aperture representative of the optimum matched
gaussian profile (OMGP) for an atmospheric lidar system, where the 1/e field
radius is 0.87 times the aperture radius, will be used.[5] The result of aberrations in the signal beam can be seen
by examining the integrated intensity distribution on a detector as a function
of time and comparing that to the integrated intensity which would be detected
without aberrations. Mismatching between the signal and LO beams due to the
aberrations causes a redistribution of the intensity in the detected beam as a
function of time. Consequently, the magnitude of modulation of the integrated
intensity is reduced. The effective power measured by the detector is
proportional to the square of the modulation depth, so it is apparent that
aberrations will cause a reduction in system performance as measured by
heterodyne efficiency. This behavior is shown in the four animations of Figure 1, corresponding to the addition of aberrations
tilt (W_{011}), spherical(W_{040}), coma(W_{031}), and
astigmatism(W_{022}), respectively. Each animation contains a
representation of the intensity distribution on the detector as a function of
time when aberrations are present and a corresponding intensity distribution for
the case of no aberrations. Integrated intensity is also shown as a function of
time; two periods of the intermediate frequency are shown. It is seen that tilt
is the aberration which has the greatest effect on heterodyne efficiency. Figure 2 illustrates the reduction of heterodyne
efficiency as a function of the number of waves of aberration added to one of
the beams.

## 3. System modeling

#### 3.1 Theory

Understanding and optimizing the performance of a coherent lidar system is
improved by developing a model of the optical layout and applying properties
peculiar to that system in the heterodyne detection calculation. The two
approaches commonly used to evaluate a coherent lidar system are the
back-propagated local oscillator (BPLO) method and the forward propagation
method. The BPLO approach was proposed by Siegman^{1} and involves
propagating the LO distribution backward through the optical train beginning at
the detector surface and continuing through the system exit aperture to the
scattering plane where the resulting field is mixed with the transmitted field.
This method has been applied in the analysis of heterodyne efficiency of general
lidar systems for various target characteristics and for inclusion of aberrations.[5,8,9] In these studies the lidar receiver was modeled as
either a theoretical receiver response function or as a simplified response
function addressing a single receiver component.

Detailed analysis of a specific lidar system by Zhao, *et. al*.
develops an expression for the receiver response function using forward
propagation of a received signal beam.[8,9] Beginning at the scattering plane, the Huygens-Fresnel
principle is applied at each propagation step through the optical train, ending
at the detector surface.[10] The resulting expression takes the form

where K(S,P) is the receiver response function, P is a point in the plane of the optical element, t is the transfer function of an element and h is the propagation function which is defined using the Fresnel approximation as

where **r _{1}** and

**r**are coordinates in the two planes defining the boundaries of a propagation step. The transfer function, t, includes transmission or phase factors, such as the quadratic phase associated with focusing elements. The development in Ref. 8 continues by translating the expression in Eq. (5) to a comparable expression for the BPLO approach and demonstrates how the ordering of integration permits simplification of the calculation to reduce computation time and complexity.

_{2}Aberrations can be accommodated by the expression in Eq. (5) through the transfer function, t. As noted in Eq. (3), aberrations contribute to the phase of a beam and are included using the relation

where t_{0} represents the transfer function of the optical element
without aberrations. This form does not require W(**w**) to be expanded
as a power series; for example, the Zernike polynomials or rms wavefront
designations typically generated during optical testing may be used as an
alternative.

#### 3.2 Implementation

An objective in developing a method for evaluating the effect of aberrations on lidar systems was to implement it in a manner that could be extended to optimize design parameters and evaluate system performance under nonideal conditions such as misalignment. A commercial software package developed for physical optics modeling was chosen to implement the propagation in order to minimize the software development time required to establish system performance and provide flexibility in system modeling. Alternative propagation algorithms implemented in the package reduce the computation requirements from forward propagation with the Huygens-Fresnel approach but produce comparable beam distributions. Optical system specifications are similar to those used by optics design software packages where components are defined in terms of physical dimensions, positions and rotations. Aberrations can be applied in numerous forms, including Seidel terms, Zernike polynomials and rms definitions. Forward or backward propagation through a system can be chosen by proper orientation of components and beams.

As an example of the modeling approach, a two-mirror, off-axis beam expanding telescope was considered. The telescope has a 16:1 beam expansion ratio at f/1 using two parabolic mirrors.[13] Again, matched gaussian beams were assumed. Two beams were propagated through the telescope, one without aberrations and one with 0.1 wave rms aberration included at the primary mirror. The phase distribution of the beam propagated without aberrations is shown in Figure 3a. It is seen to remain planar over the area that would be intercepted by a detector, indicating that the telescope contributes negligible aberration. Figure 3b shows the corresponding phase distribution with aberrations. Reduction of heterodyne efficiency due to mixing this beam with a LO beam having no aberrations was calculated to be 31%.

## 4. Summary

Heterodyne efficiency is an important indicator of the performance of a coherent lidar system. The effect of aberrations on heterodyne mixing of two ideally matched gaussian beams has been illustrated by visualization of the intensity and integrated intensity on a detector surface. In addition, a method for systematically addressing aberrations of components in a complex lidar optical system has been developed and an example has been shown for a single component.

## Acknowledgments

This work was performed under contract NAS8-40836 in support of NASA/Marshall Space Flight Center and the Integrated Program Office, M.J. Kavaya and S.C. Johnson task initiators. I am grateful to F. Amzajerdian and G.D. Spiers for many helpful discussions. I am also grateful to B. J. Rye for his review of this paper and valuable suggestions.

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