# Presentation: 532329

## Optomechanical Design and Analysis of Adaptive Optical Systems  using FEA and Optical Design Software

**Victor Genberg, Keith Doyle, Gregory Michels**
**Sigmadyne, Inc. **
**803 West Avenue, Rochester, NY**
**Phone:  585-235-7460**
**email: genberg@sigmadyne.com**

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated Optomechanical Analysis

**Optical**
**Analysis**

**Surface**
**Deformations**

**Interpolated**
**Temperatures**

**      ****static & dynamic**
**      ****linear / nonlinear **
**      ****stress**
**      ****displacement**
** ****shock & vibration**
**      ****thermo-elastic**
**      ****inertial **
**      ****buckling**

**Steady-State & Transient**
**conduction **
**convection**
**radiation**

**Stress**
**Birefringence**

**Thermo-Optic**
**Effects**

**Optical**
** ****Performance **
** ****Metrics     **

**Structual**
**Optimization**

**Thermal**
**Analysis**

**Structural**
**Analysis**

**Interface Programs**
**     ****Zernike Fitting **
**      ****Interpolation**** **

**Wavefront Analysis**
**Point Spread Function**
**Modulation Transfer Function**
**Encircled Energy**

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated OptoMechanical Analysis

- Example Telescope:  Must pass structural distortions to optical model for analysis

- Finite Element Model

- Optical Model

- Copyright Sigmadyne, Inc.

- 5/8/03

## Zernike Polynomials  

- Standard Zernike polynomials  (See Born & Wolf, *Principles of Optics)*
- use as many terms as required to represent the data
- Fringe Zernike polynomials are a subset of the Standard Zernikes
- include higher-order symmetrical terms (r10 & r12) that are more important to wavefront propagation; eliminates the higher-order azimuthal terms

**Polynomial series with two real variables, r and ******

- r - dimensionless normalized radius
- polar angle
- Anm & Bnm - polynomial coefficients

- Copyright Sigmadyne, Inc.

- 5/8/03

## Zernike Surfaces

**Bias/Piston: **1** **

**Tilt:  **rcos() / rsin()

**Power/Defocus:  **2r2-1

**Pri-Astigmatism**
- 2r2cos(2) / 2r2sin(2)** **

- -1.0

- +1.0

**Zernike Surfaces**

- Copyright Sigmadyne, Inc.

- 5/8/03

## Zernike Surfaces

**Pri-Coma: **
- (3r3-2r)cos() / (3r3-2r)sin()

**Pri-Trefoil:**
**  **r3cos(3) / r3sin(3)** **

**Pri-Spherical: **
- 6r4-6r2+1

**Sec-Astigmatism:**
- (4r4-3r2)cos(2) / (4r4-3r2)sin(2)
** **

- +1.0

- -1.0

**Zernike Surfaces**

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated OptoMechanical Analysis - Current Technology

- FEA code (Nastran) => surface deformations
- SigFit => Fit Zernikes to FEA data, output in Optics format
- Optics code (CodeV) => read Zernikes, calculate system optical response
- Disadvantages
- requires optical engineer in the loop
- analysis process turnaround is slow
- can not use in FEA optimization loop

- Copyright Sigmadyne, Inc.

- 5/8/03

## Why Adaptive Optical System

- Optical surfaces are deformed and moved based on measured or anticipated information to compensate for unwanted disturbances
- Uses
- Fabrication & assembly errors in deployable systems
- Thermoelastic & humidity distortion
- Atmospheric disturbance in ground based telescopes
- Vibrations & dynamic disturbances

- Copyright Sigmadyne, Inc.

- 5/8/03

## Adaptive Performance Can Be Simulated With Finite Element Analysis
Generate two sets of deformation predictions
Uncorrected disturbances
Actuator influences
Solve for actuator inputs, x1,x2,x3...xn, to minimize surface error, E

If focus compensation exists elsewhere, terms like 22-1 or R can be added as augment actuators

- Generate two sets of deformation predictions
- Uncorrected disturbances
- Actuator influences
- Solve for actuator inputs, x1,x2,x3...xn, to minimize surface error, E
- If focus compensation exists elsewhere, terms like 22-1 or R can be added as *augment* actuators

**Adaptive Simulation Method - Conceptual **

- x1

- x2

- xn

- Input Disturbance

- Actuator 1

- Actuator n

- Actuator 2

- Surface Error

- Copyright Sigmadyne, Inc.

- 5/8/03

## Adaptive Analysis - Current Technology

- FEA code => surface distortions
- FEA code => actuator influence functions
- SigFit => read FEA data, calculate actuator force to correct that surface
- Optics code => read SigFit data, calculate system response
- Disadvantages
- Error correction for that single surface, _not_ system response
- Not correcting other optical surfaces effects
- Can not combine multiple adaptive surfaces
- To correct system level effects, the system wavefront error must be related back to the adaptive optic as an equivalent surface distortion.

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated System Analysis - New Technology

- Optics code => system response sensitivity due to unit Zernikes at each surface
- FEA code => surface distortions of all surfaces
- FEA code => influence functions for all actuators (if adaptive)
- SigFit => calculate system response
- fit Zernikes to FEA distortions of each surface
- multiply by system sensitivities to get system response
- SigFit => calculate corrected system response (if adaptive)
- fit Zernikes to FEA influence functions
- calculate actuator forces to minimize system error
- Advantage
- speeds up analysis turn around
- using system level performance generates superior designs

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated System Analysis - New Technology

- Optical surfaces:  n = 1 to S		Number adaptive surfaces: t = 1 to T
- Zernike in/surface: j = 1 to Z 		Zernike out/system: k = 1 to Z
- Load case number: i = 1 to L		Actuator number: m = 1 to M
- Sensitivity matrix = Zernike out (k)  for Zernike in (j) at surface (n)  = Skjn
- Disturbance fit = fit each load case (i) with Zernike (j) at surface (n) = Cjin
- Actuator influence = fit with Zernike (j) at surface (t) = Bjmt
- System response = Zernike (k) at output location (0) for load case (i) = Zki0

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated System Analysis - New Technology

- System level response = Zernikes at output (ie Exit Pupil)

- Where S is the Zernike sensitivities from Code V
- 		Skjn = matrix of size (Z x Z x N)
- and C is the Zernike fit to FEA deformations for each load case
- 		Cjin = matrix of size (Z x L x N)
- Resulting Zo is reported along with Surface RMS and Peak-Valley
- Output a visualization file showing net response at output location

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated System Analysis - Adaptive - New Technology

- System level response at Output location due to Actuators

- Where B is the Zernike fit to Actuator influence functions
- 		Bjmt = matrix of size (Z x M x T)
- Define system level error E as

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated System Analysis - Adaptive - New Technology

- Minimize System Error with respect to Actuator forces

- Solve resulting linear system for A

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated System Analysis - Example

- Example: Telescope

- Finite Element Model

- Optical Model

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated System Analysis - Example

- Adaptive PM (9 force actuators in red, 3 displacement actuators in blue)

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated System Analysis - Example

- Load Case:  1g along optical axis
- Added 5 of astigmatism on SM (represents a thermal distortion)
- PM sits on 3 points (displacement actuators)
- 1g distortion = 4.62 RMS
- mostly trefoil = 12.5
- SM sits on 3 edge points (with 5 astigmatism added)
- 1g distortion = 2.18 RMS
- trefoil = 2.0
- added astigmatism = 5.0
- Note:  Surface distortions have a doubling effect on reflected wavefront error

- =

- +

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated System Analysis - Example

-        No Correction                    PM Correction                    Sys Correction

- PM

- SM

- Sys

- RMS=
- 4.62

- RMS=
- 0.11

- RMS=
- 2.18

- RMS=
- 2.18

- RMS=
- 2.18

- RMS=
- 2.18

- RMS=
- 11.25

- RMS=
- 4.31

- RMS=
- 0.23

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated System Analysis - Example

- Correcting PM disturbance only
- Adaptive PM reduced PM error
- Did _not_ correct SM error, so SM effects still in System error
- Correcting System response
- Adaptive PM corrected PM error _and_ the SM error
- Resulting System error greatly reduced

- Copyright Sigmadyne, Inc.

- 5/8/03

## Integrated System Analysis -Example: Compare Sys Response with CodeV

- Copyright Sigmadyne, Inc.

- 5/8/03

## Summary

- SigFit’s new System Level Analysis allows more rapid turn around of analyses
- Optics engineer needed up front to get sensitivities
- Design and analysis under control of structural engineer
- Can optimize on system level response
- Reduces the need to budget each optic separately
- Improves and simplifies system level analyses
- Can correct multiple surfaces’ effects with single adaptive optic
- Can combine multiple adaptive optics to correct system response
- More accurate & useful than correcting a single surface’s effect
- User features
- Visualization plots of System Level Response
- Future development
- Add System Level Response to SigFit dynamics
- Add System Level Response to SigFit optimization equations for Nastran

- Copyright Sigmadyne, Inc.

- 5/8/03

## References

**Genberg, V., **_**Sigfit Version 2003-r1 Reference Manual**_**, Sigmadyne, Inc., January, 2003**
**Doyle, K., Genberg, V., Michels, G., **_**Integrated Optomechanical Analysis**_**,**
**SPIE Press, TT58, October, 2002.**
**Genberg, V., Michels, G.,  "OptoMechanical Analysis of Segmented/Adaptive Optics", **
**SPIE Paper 4444-10, August 2001.**
**Michels, G.,  Genberg, V.,  "Design Optimization of Actively Controlled Optics", **
**SPIE Paper 4198-17, November 2000.**

- Copyright Sigmadyne, Inc.

- 5/8/03