1. Antenna Efficiency Optimization in Coherent Lidar Systems   Sammy Henderson, Pat Kratovil, and Charley Hale Beyond Photonics beyondphotonics.com

2. Overview
Achieving & Maintaining high Lidar Efficiency reduces Size, Weight, and Power of Coherent Lidar Systems
especially important for space-based lidar systems where prime power is limited
see paper We4 “Space-based Coherent Lidar for Wind Measurements with High Percentage Tropospheric Coverage”
Electro-optic efficiency must be maximized
transmit and receive optical path transmission efficiencies
detector quantum and shot noise efficiencies
Coherent Antenna Efficiency must be maximized
high transmit laser beam quality
low aberrations of optics
near-diffraction-limited alignment of transmit and BPLO beam paths
Improved Analytic Equations that accurately approximate the Antenna Efficiency of coherent lidar systems employing Truncated and Aberrated Gaussian beams have been developed
Allows better insight into important design trades and faster convergence to optimal designs
A Coherent Auto-alignment System can be implemented maintaining high Coherent Antenna Efficiency
Correction for Lag Angle and thermal or shock induced misalignments
Achieving and Maintaining Total Coherent Lidar Efficiency of 15-20% is Practical with Careful Design

3. Coherent Lidar Efficiency Definition Review
Short Pulse Lidar CNR
Electro-Optic Efficiency
includes optical, detector quantum, and shot noise efficiencies
Coherent Lidar Antenna Efficiency, is product of transmitter truncation and heterodyne efficiencies
Truncation efficiency is fraction of transmitted light incident on aperture that exits lidar system
Heterodyne efficiency is fraction of total power collected by the aperture that is coherently detected
𝐶𝑁𝑅 𝑧 = 𝜼 𝒂 𝒛 𝜼 𝒆𝒐 𝑇 2 𝐸 ℎ𝜐𝐵 𝑐𝛽 2 𝐴 𝑟 𝑧 2
𝜂 𝑒𝑜 = 𝜂 𝑡𝑜 𝜂 𝑟𝑜 𝜂 𝑞 𝜂 𝑠𝑛
𝜂 𝑎 𝑧 = 𝜂 𝑇𝑡 𝜂 ℎ 𝑧 = 𝜂 𝑇𝑡 𝜂 𝑇𝑏 𝜆 2 𝑧 2 𝐴 𝑟 𝐼 𝑛𝑡 𝑥,𝑦,𝑧 𝐼 𝑛𝑏 𝑥,𝑦,𝑧 𝑑𝑥𝑑𝑦
Pulse short wrt changes in other terms over pulse distance
Point out Antenna efficiency and electro-optic efficiency
Truncation Ratio of the Transmit and BPLO beams are 𝜂 𝑇𝑡 and 𝜂 𝑇𝑏 respectively.
Intensities are normalized wrt the total power in the target plane
RSS of Geometric and Diffraction Terms
Effective UGB Receiver Area
Point out Transmit and BPLO beams
This is for arbitrary Gaussian Beam sizes with Arbitrary overlap at target plane
Coherent detection effectively only received power in diffraction limited spot size
Note that power collected at 1000 meters is 100x down from power collected at 100 meters yet CNR essentally the same
Coherent Lidar only Detects Portion of Received Signal that is Matched to the Local Oscillator Field

4. Electro-optic Efficiency
Dual-Bal Lidar Det
Signal Port
T/R Polarizer
>098% T & R
/4
From Local Oscillator
Transmitter
R >99.8%
Matching Optics
T>99%
T >99.5%
50 /50
Fiber Coupler
T > 095%
Detector
𝜼 𝒒 >𝟎.𝟖𝟓
𝜼 𝒔𝒏 >𝟎.𝟗𝟓
Fiber Port
T > 099%
Telescope
Scanner
Optics
T >99%
To / From
Atmosphere
T >98%
𝜂 𝑒𝑜 = 𝜂 𝑡𝑜 𝜂 𝑟𝑜 𝜂 𝑞 𝜂 𝑠𝑛
Realistic Electro-optic Efficiency
𝜂 𝑒𝑜 >66.5%
𝜂 𝑜𝑡 >92.9%
𝜂 𝑜𝑟 >88.6%
𝜂 𝑞 >85%
𝜂 𝑠𝑛 >95%
Does not include Antenna Efficiency
or depolarization Losses
Coherent Lidar System with Electro-optic Efficiency of 65-70% is Realistic.

5. Approximate Analytic Equation for Antenna Efficiency
Developed Analytic Expressions that describe Antenna Efficiency for Truncated Gaussian beam Lidar Systems
Strehl ratio based Formulation
Significant Improvement over Gaussian Beam Equations typically used for Analytic Calculations
Initial Version Reported in CLRC 2016 Paper
Now have Developed More General Improved Equations
Arbitrary Transmit and BPLO beams
Includes Impact of, Truncation, Aberrations, and Misalignment
Eliminated Ringing at high Fresnel numbers
Improved Fit under Misalignment
Equations developed for all ranges (near and far field) but focus in this presentation due to time constraint is far field
𝜂 𝑎 (𝑧)≈2 𝜂 𝑐𝑙𝑡 𝜂 𝑇𝑡 𝜂 𝑐𝑙𝑏 𝜂 𝑇𝑏 𝜌 𝑇𝑏 2 𝜓(𝑧)∙𝑒𝑥𝑝 −2𝜓(𝑧) 𝛿𝜃 𝜃 𝑑𝑏 2
𝜌 𝑇𝑖 = 𝜔 𝑜𝑖 𝑎
𝜃 𝑑𝑏 = 𝜆 𝜋 𝜔 𝑜𝑏
𝜂 𝑐𝑙𝑖 ≈ 1−0.162 exp − 𝜌 𝑇𝑖 − 𝜌 𝑇𝑖 2 − 𝜌 𝑇𝑖 3
𝜂 𝑇𝑖 =1−𝑒𝑥𝑝 − 2 𝜌 𝑇𝑖 2
𝜓 𝑧 = 𝜔 𝑑 2 𝑧 𝜔 𝑒𝑡 2 𝑧 + 𝜔 𝑒𝑏 2 𝑧 = 𝜔 𝑒𝑡 2 𝑧 𝜔 𝑑 2 𝑧 + 𝜔 𝑒𝑏 2 𝑧 𝜔 𝑑 2 𝑧 −1
𝜔 𝑒𝑖 2 (𝑧)= 𝜔 𝑒𝑖,𝑛 𝑧 1− 𝑧 𝐹 𝜆𝑧 𝜋 𝜔 𝑒𝑖,𝑓 2

6. Far Field Analytic Equation
𝜔 𝑒𝑖,𝐹 = 𝑀 𝑖 2 𝜆𝑧 𝜋 𝜔 𝑜𝑖
For typical moderate truncation ratios and weak to moderate aberration levels, the total far field beam spreading, 𝑀 𝑖 2 , can be written to good approximation as the product of
Truncation - 𝑀 𝑖𝑇 2
Laser - 𝑀 𝑖𝐿 2
Optical aberrations - 𝑀 𝑖𝑂 2
So, in the far field
with
𝑀 𝑖 2 ≈ 𝑀 𝑖𝐿 2 𝑀 𝑖𝑂 2 𝑀 𝑖𝑇 2
𝑀 𝑖𝑇 2 ≈ 𝜂 𝑐𝑙𝑖 𝜂 𝑇𝑖 𝑓 𝑐2𝑖 𝑆 𝑖,𝐹,𝑇
𝑆 𝑖,𝐹,𝑇 = 1−ex p − 1 𝜌 𝑇
𝑀 𝑖𝐿 2 is laser beam quality factor
𝑀 𝑖𝑂 2 ≈ 𝑒 0.5 𝛾𝜋 𝜎 𝑖𝑂 2
with 𝜎 𝑖𝑂 = rms aberration (waves) over aperture
and 𝛾 varying between 1.8 and 2.8 depending on the aberration
type – spherical aberration is worst with 𝛾≈2.8
𝜂 𝑎 (𝑧)≈2 𝜂 𝑐𝑙𝑡 𝜂 𝑇𝑡 𝜂 𝑐𝑙𝑏 𝜂 𝑇𝑏 𝜌 𝑇𝑏 2 𝜓(𝑧)∙𝑒𝑥𝑝 −2𝜓(𝑧) 𝛿𝜃 𝜃 𝑑𝑏 2
𝑓 𝑐2𝑖 is a simple correction
factor depending only on the
truncation ratio, can also be
expressed as an analytic eqn.
with 𝜓 𝐹 ≈ 𝑀 𝑏𝐿 𝑀 𝑏𝑂 𝑀 𝑏𝑇 𝜌 𝑇𝑏 𝜌 𝑇𝑡 𝑀 𝑡𝐿 𝑀 𝑡𝑂 𝑀 𝑡𝑇 −1
If the beam sizes at the transmit aperture are matched
𝜂 𝑎,𝐹 ≈ 2𝜂 𝑎𝑚𝑎𝑥,𝐹 𝑀 𝑏𝐿 𝑀 𝑏𝑂 𝑀 𝑡𝐿 𝑀 𝑡𝑂 −1 𝑒𝑥𝑝 −2 𝑀 𝑇 𝑀 𝑏𝐿 𝑀 𝑏𝑂 𝑀 𝑡𝐿 𝑀 𝑡𝑂 −1 𝛿𝜃 𝜃 𝑑𝑏 2

7. Comparison with Numerical Calculations Far Field Antenna Efficiency
Analytic equation far field antenna efficiency predictions compared with published results for three coherent lidar antenna designs
Wang – matched transmit and BPLO with 𝜌 𝑇 =0.801
Rye Matched – matched transmit and BPLO with 𝜌 𝑇 = and pre-trucated LO
LO pre-truncation means must divide antenna eff. eqn by 𝜂 𝑇𝑏
Rye Unmatched – 𝜌 𝑇𝑡 =0.815 and 𝜌 𝑇𝑏 =1.186 and pre- truncated LO
Excellent Agreement of Analytic Theory to Numerical Far Field Antenna Efficiency Calculation

8. Impact of Aberrations
Achieving high antenna efficiency requires optical aberrations to be minimized
Worst aberration is spherical which is well approximated by 𝑀 𝑖𝑂 2 ≈ 𝑒 0.5 𝛾𝜋 𝜎 𝑖𝑂 𝑤𝑖𝑡ℎ 𝛾=2.8
Figures below show that this is independent of truncation ratio over practical ranges
Over practical truncation ranges of 0.75< 𝜌 𝑇 <1.2 aberrations well bounded by 1.8<𝛾<2.8
If dominated by spherical the aberrations must be held below ≈ 𝜆 20 rms to maintain >80% of peak
Easier accomplished at longer wavelengths – e.g., 𝜆 20 𝑎𝑡 2 𝜇𝑚 𝑖𝑠 𝜆 6.3 𝑎𝑡 𝜇𝑚
Maintaining >80% of peak antenna efficiency is achieved by minimizing optical aberrations
𝜸=𝟏.𝟖
𝜸=𝟐
𝜸=𝟐.𝟖
𝜸=𝟏.𝟖
𝜸=𝟐
𝜸=𝟐.𝟖
𝜸=𝟏.𝟖
𝜸=𝟐
𝜸=𝟐.𝟖

9. Example Antenna Efficiency with Aberrations
Antenna Efficiency is Approximately
Assume imperfect Wang design lidar with beam qualities and optical aberrations as noted in Table
Peak antenna efficiency for this example is reduced to ~73% of the perfect Wang design lidar
𝜂 𝑎 (𝑧)≈2 𝜂 𝑐𝑙𝑡 𝜂 𝑇𝑡 𝜂 𝑐𝑙𝑏 𝜂 𝑇𝑏 𝜌 𝑇𝑏 2 𝜓(𝑧)∙𝑒𝑥𝑝 −2𝜓(𝑧) 𝛿𝜃 𝜃 𝑑𝑏 2
with 𝜓 𝐹 ≈ 𝑀 𝑏𝐿 𝑀 𝑏𝑂 𝑀 𝑏𝑇 𝜌 𝑇𝑏 𝜌 𝑇𝑡 𝑀 𝑡𝐿 𝑀 𝑡𝑂 𝑀 𝑡𝑇 −1
Antenna Efficiency of 25-30% in practical in well-designed High-Power Coherent Lidar Systems. Even Higher is possible in Lower-Power Systems.

10. Impact of Misalignment on Far Field Antenna Efficiency
Analytic Antenna Efficiency Model Predicts Misalignment Loss very Accurately
Much better that scaled Untruncated Gaussian Beam Model
Better than previous CSAE Analytic Model (see 2016 CLRC paper)
𝜂 𝑎 (𝑧)≈2 𝜂 𝑐𝑙𝑡 𝜂 𝑇𝑡 𝜂 𝑐𝑙𝑏 𝜂 𝑇𝑏 𝜌 𝑇𝑏 2 𝜓(𝑧)∙𝑒𝑥𝑝 −2𝜓(𝑧) 𝛿𝜃 𝜃 𝑑𝑏 2
with 𝜓 𝐹 ≈ 𝑀 𝑏𝐿 𝑀 𝑏𝑂 𝑀 𝑏𝑇 𝜌 𝑇𝑏 𝜌 𝑇𝑡 𝑀 𝑡𝐿 𝑀 𝑡𝑂 𝑀 𝑡𝑇 −1
Matched Design with Various Truncation Ratios
Rye Unmatched Design
New Analytic Model

11. Comparison of Antenna Designs and Alignment Requirement
Rye Unmatched Design Compares Favorably to Wang Design
Higher efficiency
Reduced misalignment loss
Wang allows Simple Engineering Implementation with Single Mode Fiber in receiver so it is most commonly utilized
Maintaining 95% of peak Antenna Efficiency in the perfect Wang design requires 𝛿𝜃<0.27 𝜃 𝑑𝑏
For example for a large 1.5 meter diameter telescope this implies 𝛿𝜃 𝑙𝑏 <0.29 𝜇𝑟𝑎𝑑 in large beam space
In small beam space behind beam expander where optical alignment occurs angles increased by magnification 𝑀, so 𝛿𝜃 𝑠𝑏 <0.29∙𝑀 𝜇𝑟𝑎𝑑
If small beam diameter is held at 2𝜔 𝑜 =6 𝑚𝑚 then small beam alignment must be held to 𝛿𝜃 𝑠𝑏 <58 𝜇𝑟𝑎𝑑
Co-Alignment Requirement of Transmit and BPLO Beams is Achieved with Good Opto-Mechanical Design

12. Lag Angle Induced Misalignment
Rotation of lidar line of sight between transmit and receive time results in well know lag angle misalignment of received signal field wrt local oscillator field
This results in efficiency loss given by
At 320 km altitude orbital period is minutes
Ω 𝑜𝑟𝑏 =1.15 𝑚𝑟𝑎𝑑/𝑠 and 𝜃 𝐿 =2.81 𝜇𝑟𝑎𝑑
Assuming imperfect lidar described on Slide 9, with 𝜓 𝐹 = lag angle loss is shown in figure
𝜃 𝐿 =Ω 𝑡 𝑟𝑡
𝑡 𝑟𝑡 =2 𝑅 𝑐
𝜂 𝑎, 𝜃 𝐿 𝑧 ≈𝑒𝑥𝑝 −2𝜓 𝑧 𝜃 𝐿 𝜃 𝑑𝑏 2
If lidar utilizes matched beams
𝜂 𝑎, 𝜃 𝐿 𝑧 ≈𝑒𝑥𝑝 −2 𝑀 𝑇 𝑀 𝑏𝐿 𝑀 𝑏𝑂 𝑀 𝑡𝐿 𝑀 𝑡𝑂 −1 𝜃 𝐿 𝜃 𝑑𝑏 2
Lag Angle Losses in a Coherent Lidar
can be very Large and Must be Removed

13. Auto-alignment and Lag Angle Compensation
Beyond Photonics is Developing a Coherent Lidar Auto-alignment and Lag Angle Compensation System
Angle rate sensor attached to lidar provides information required to compensate for platform rotations.
Back-propagated local oscillator is compared to transmit beam position in far field
Alignment errors corrected with PZT actuated Servo Mirror
Will be demonstrated in 2019

14. Summary
Maintaining High Lidar Efficiency in Coherent Lidar is Feasible with Good Design Practices – Efficiency of 15-20% is Realistic
65-70% electro-optic efficiency
25-30% antenna efficiency without misalignment
Alignment efficiency >95% by opto-mechanical design and/ or auto-alignment
Analytic Expressions that Approximate Well the Antenna Efficiency of Coherent Lidar Systems have been Developed
Good fit to Numerically Calculated Efficiencies
Allows for Faster Design Trades aimed at Design Optimization
Lag Angle Must be Corrected in Space-based Coherent Lidar Systems Requiring Large Apertures
Beyond Photonics has Designed and is Developing an Automated Lag Angle Compensation System that will allow Alignment Efficiency of Coherent Lidar Systems to be Maintained at >95% of Peak
Will be demonstrated in late 2019