## Fizeau interferometric cophasing of segmented mirrors |

Optics Express, Vol. 20, Issue 28, pp. 29457-29471 (2012)

http://dx.doi.org/10.1364/OE.20.029457

Acrobat PDF (1044 KB)

### Abstract

Segmented mirror telescope designs address issues of mechanical rigidity, but introduce the problem of aligning, or cophasing, the separate segments to conform to the optimum mirror shape. While several solutions have been widely adopted, a few difficulties persist — the introduction of non-common path errors and an artificial division of the problem into coarse and fine regimes involving separate dedicated hardware solutions. Here we propose a novel method that addresses many of these issues. Fizeau Interferometric Cophasing of Segmented Mirrors (FICSM) uses non-redundant sparse aperture interferometry to phase mirror segments to interferometric precision using unexceptional science hardware. To show the potential of this technique we numerically simulate conditions on NASA’s James Webb Space Telescope (JWST), showing that the FICSM method has the potential to phase the primary mirror from an initial state with segment-to-segment pistons as large as 150 microns and tilts as large as 0.5 arcseconds, to produce a final state with 0.75 nm rms segment-to-segment pistons and 3.7 mas rms segment tilts. The image undergoes monotonic improvement during this process. This results in a rms wavefront error of 3.65 nm, well below the 100 nm requirement of JWST’s coarse phasing algorithm.

© 2012 OSA

## 1. Introduction

1. D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE **8442** (2012), 84422H.

2. G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, and D. Kirkman, “Phasing the mirror segments of the Keck telescopes: the broadband phasing algorithm,” Appl. Opt. **37**, 140–155 (1998). [CrossRef]

3. G. Chanan, C. Ohara, and M. Troy, “Phasing the mirror segments of the Keck telescopes II: the narrow-band phasing algorithm,” Appl. Opt. **39**, 4706–4714 (2000). [CrossRef]

4. J. Rodríguez-Ramos and J. Fuensalida, “Phasing of segmented mirrors: a new algorithm for piston detection,” MNRAS **328**, 167–173 (2001). [CrossRef]

5. G. Chanan, M. Troy, and E. Sirko, “Phase discontinuity sensing: a method for phasing segmented mirrors in the infrared,” Appl. Opt. **38**, 704–713 (1999). [CrossRef]

7. P. Tuthill, J. Monnier, W. Danchi, E. Wishnow, and C. Haniff, “Michelson interferometry with the Keck I telescope,” Pub. Ast. Soc. Pac. **112**, 555–565 (2000). [CrossRef]

8. J. Monnier, P. Tuthill, M. Ireland, R. Cohen, A. Tannirkulam, and M. Perrin, “Mid-infrared size survey of young stellar objects: Description of Keck segment-tilting experiment and basic results,” ApJ **700**, 491 (2009). [CrossRef]

*N*hole aperture mask exhibits

*n*=

_{b}*N*(

*N*− 1)/2 baselines (or hole pairs). For the case of a non-redundant array geometry, each baseline produces a fringe pattern at a unique spatial frequency, hence the number of baselines is greater than the number of holes when

*N*> 3. The distinguishable fringe patterns comprising such an interferogram are most readily studied in the Fourier plane (or image spectrum), where the amplitude and phase at any given point for which there is power can be termed the complex visibility on that specific baseline. The square of the absolute value of the image spectrum is referred to as the power spectrum, and we note that NRM often does not fill the Fourier plane completely, in which case only a subset of the available spatial frequencies passed by the unocculted primary mirror are measured by a single NRM image.

*μ*m segment pistons to a state with a final root-mean-square (rms) wavefront error of 100 nm to maximize telescope performance [9]. To comfortably test this range, we opt for a maximum initial segment-to-segment piston error of 150

*μ*m.

10. R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE **8442** (2012), 84422R.

11. A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE **8442** (2012), 84422S.

### 1.1. Phasing segmented mirrors with NRM

## 2. Mathematical basis

*f*(

*λ*) (energy per wavelength interval per square meter). The combined intensity distribution on the detector expressed in terms of the field (

*E*) at each wavelength will be given by Eq. (1) below. We define (

*ξ*,

*η*) to be the coordinates in the image plane (expressed as angles on the sky), and (

*x*,

*y*) to be the coordinates in the pupil plane (in meters). where

*λ*

_{1}and

*λ*

_{2}are the extremal wavelengths of the filter’s bandpass.

*E*can be expressed as a sum of the fields

*E*from each subaperture. We write the shape of each subaperture as

_{i}*C*(

_{i}*x*−

*x*,

_{i}*y*−

*y*), where the hole center is (

_{i}*x*,

_{i}*y*), and write its piston as

_{i}*p*and tip/tilt (expressed as mirror gradients) as (

_{i}*m*,

_{i}*n*) in the (

_{i}*x*,

*y*) directions. For circular holes,

*C*(

_{i}*x*−

*x*,

_{i}*y*−

*y*) would be a uniform disk with radius

_{i}*r*. We can then write each

_{i}*E*as the Fourier transform (ℱ) of the subaperture function, which becomes: Here we measure piston as a physical height of a segment, so the wavefront error (or optical path difference) is twice this value.

_{i}*E*shown in Eq. (3).

_{i}*A*(

_{i}*ξ*−

*m*,

_{i}*η*−

*n*) is the Fourier Transform of the subaperture function

_{i}*C*, shifted in the image plane (relative to the centroid of the image intensity distribution) due to the tilt of mirror

_{i}*i*. The phase term due to piston is unaffected by the Fourier Transform.

14. E. Sabatke, J. Burge, and D. Sabatke, “Analytic diffraction analysis of a 32-m telescope with hexagonal segments for high-contrast imaging,” Appl. Opt. **44**, 1360–1365 (2005). [CrossRef] [PubMed]

^{*}denoting a complex conjugate): where the

*E*are given by Eq. (3).

_{i}*I*terms then correspond to the fringes generated from interference between subapertures

_{i,j}*i*and

*j*. In the special case

*i*=

*j*this reduces to a form of the familiar Airy pattern shifted due to the tilt on the mirror associated with that subaperture. In the case

*i*≠

*j*, this gives interference fringes etched into the overlap of the two shifted Airy patterns from subapertures

*i*and

*j*, with an added phase due to the difference in piston.

*I*, using En. 6: Clearly, the phase terms disappear when

_{i,j}*i*=

*j*, giving the familiar Airy pattern.

*λ*

_{1}and

*λ*

_{2}is small compared to the wavelengths themselves, the difference in phase between the fringes changes over a much larger scale, characterized by the coherence length of the filter, Here Δ

*λ*is the full width at half maximum (FWHM) of

*f*(

*λ*) and

*λ*= (

_{c}*λ*

_{1}+

*λ*

_{2})/2. Since the fringes are affected by relative piston only, the zero point of the pistons is unspecified. It can be chosen to be convenient for the descriptions of segment positioning.

## 3. The basis of the FICSM algorithm

### 3.1. Measuring tilts

### 3.2. Measuring piston

15. R. J. Allen, “Station-keeping requirements for constellations of free-flying collectors used for astronomical imaging in space,” Pub. Ast. Soc. Pac. **119**, 914–922 (2007). [CrossRef]

*V*) of the splodge is compared to each theoretically generated one from the lookup table using the goodness-of-fit parameter (

*u*,

*v*) coordinates refer to the frequencies in the (

*ξ*,

*η*) directions of the image.

*χ*

^{2}space having local minima centered on the correct piston and spaced by

*λ*. In order for this step to converge to the correct solution (so that the correct fringes line up), the coarse measurement must be accurate to within half a wavelength. Occasionally this condition is not met, leading to a final piston measurement that is an integral number of wavelengths (usually 1) from the actual value. While this is a rare occurrence (observed in one out of 100 complete simulations below), they are easily recognized and rectified.

*cλ*

_{2}−

*dλ*

_{1}, where

*c*and

*d*are unknown integers (usually −1, 0 or 1). This equation has a unique solution when assuming small

*c*and

*d*, and these can then be multiplied by the wavelength and subtracted from the measured piston to reconstruct the actual piston.

### 3.3. Turning baseline data into estimates for mirrors

2. G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, and D. Kirkman, “Phasing the mirror segments of the Keck telescopes: the broadband phasing algorithm,” Appl. Opt. **37**, 140–155 (1998). [CrossRef]

*p*to the measured piston difference

_{i}*δ*between mirrors

_{i,j}*j*and

*i*: Since all measurements are of relative piston, we also impose a constraint that one of the mirrors be defined to have zero piston. This allows all of the mirror pistons to be expressed relative to a single reference point. In practice it may be more useful to require the total piston be zero, to minimize mirror motion.

*M*for each baseline in terms of that on each mirror

_{i,j}*t*: Since we measure tilts in two directions, we have a separate set of equations for each. This method is described for one, and is repeated for the other direction.

_{i}*δ*with

*M*and replacing

*p*with

*t*.

*N*unknowns, and can be expressed in matrix form as:

*A*to be expressed in the form

*A*=

*Uwv*, where

^{T}*w*is a diagonal matrix with diagonal elements (

*w*). This leads to the least squares solutions for piston, given by [16]: where diag(1/

_{j}*w*) is the diagonal matrix with diagonal elements 1/

_{j}*w*.

_{j}## 4. Cophasing JWST: a numerical case-study

- Using the specifications of the telescope, optics and detector, the coarse piston lookup table is computed and loaded.
- The initial state of the mirror is prepared by applying a random tip/tilt and piston to each segment.
- Cophasing begins by taking a narrow bandwidth image using the current mirror state.
- The image is processed with the tip/tilt fitting program, and the best-estimate tilts corrected in the present pupil.
- Images in two different broad bandwidth filters are taken.
- The two broad band images are processed with the piston fitting program, and the results compared.
- If they agree, the present pupil is corrected for the mean of the two measurements.
- If they disagree, the two measurements are used to reconstruct the true piston, which is then used to correct the pupil.
- Steps 3 – 8 are repeated once.
- The final fit residual pistons are computed by comparison with Step 2.

### 4.1. Numerical simulations: setup and configuration

18. A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE **7440** (2009), 74400Y. [CrossRef]

*μ*m in piston, and a final wavefront error of less than 100 nm. To comfortably test the viability of this method, a maximum piston capture range of 150

*μ*m was adopted, measured between any two segments. Using the coherence length of 96

*μ*m from the first broadband filter, this corresponds to more than 3 coherence lengths at the wavefront (6 coherence lengths for the second filter). We adopted a reasonable error budget of a maximum 0.5 arcseconds of tilt, chosen through consideration of the plate scale, 65 mas per pixel. This corresponds to more than 7.5 pixels, and is an estimate of the residual tilts from initial alignment steps rather than an exploration of the capture range of FICSM.

### 4.2. Numerical simulations: results

^{9}photons. Errors due to inaccuracy in segment actuator motion were not considered. The cophasing algorithm produced a final rms residual piston of 0.75 nm, and an rms residual tilt of 3.7 mas, showing that we can expect the method to work to well within the accuracy required from the JWST coarse phasing system. By comparison with the 65 mas plate scale of the detector, a tilt of this size would result in a misalignment on the detector of less than 6% of a pixel.

*μ*m using the common approximation

*S*=

*e*

^{−(2πσ/λ)2}, where

*σ*is the rms wavefront error and

*λ*is the operating wavelength. This is more than an order of magnitude lower than the 100 nm requirement for coarse phasing the JWST primary mirror, and appears to be achievable with existing hardware and most of its science cameras.

## 5. Extending FICSM to other configurations

8. J. Monnier, P. Tuthill, M. Ireland, R. Cohen, A. Tannirkulam, and M. Perrin, “Mid-infrared size survey of young stellar objects: Description of Keck segment-tilting experiment and basic results,” ApJ **700**, 491 (2009). [CrossRef]

## 6. Conclusions

*μ*m and 0.5 arcsecond respectively) and incorporated realistic noise sources, yet delivered final residuals of a few nanometers in piston and 10 milliarcseconds of segment tilt – an improvement of more than 5 orders of magnitude. The method can be carried out using any of JWST’s scientific imaging cameras. Results were achieved after two iterations through a procedure requiring three exposures and two mirror adjustments each pass. Operational details of the technique follow existing procedures. We conclude that the FICSM technique has the potential to cophase the JWST primary mirror to more than an order of magnitude better than its coarse phasing requirements, and may also be useful to future segmented-mirror telescopes.

## Acknowledgments

## References and links

1. | D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE |

2. | G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, and D. Kirkman, “Phasing the mirror segments of the Keck telescopes: the broadband phasing algorithm,” Appl. Opt. |

3. | G. Chanan, C. Ohara, and M. Troy, “Phasing the mirror segments of the Keck telescopes II: the narrow-band phasing algorithm,” Appl. Opt. |

4. | J. Rodríguez-Ramos and J. Fuensalida, “Phasing of segmented mirrors: a new algorithm for piston detection,” MNRAS |

5. | G. Chanan, M. Troy, and E. Sirko, “Phase discontinuity sensing: a method for phasing segmented mirrors in the infrared,” Appl. Opt. |

6. | A. Cheetham, Cophasing JWST’s segmented mirror using sparse aperture interferometry University of Sydney Hons. Thesis, (2011). |

7. | P. Tuthill, J. Monnier, W. Danchi, E. Wishnow, and C. Haniff, “Michelson interferometry with the Keck I telescope,” Pub. Ast. Soc. Pac. |

8. | J. Monnier, P. Tuthill, M. Ireland, R. Cohen, A. Tannirkulam, and M. Perrin, “Mid-infrared size survey of young stellar objects: Description of Keck segment-tilting experiment and basic results,” ApJ |

9. | JWST Mission Operations Concept Document, Revision C (2008). |

10. | R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE |

11. | A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE |

12. | J. Lloyd, F. Martinache, M. Ireland, J. Monnier, S. Pravdo, S. Shaklan, and P. Tuthill, “Direct detection of the brown dwarf GJ 802B with adaptive optics masking interferometry,” ApJ Lett. |

13. | R. Bracewell, |

14. | E. Sabatke, J. Burge, and D. Sabatke, “Analytic diffraction analysis of a 32-m telescope with hexagonal segments for high-contrast imaging,” Appl. Opt. |

15. | R. J. Allen, “Station-keeping requirements for constellations of free-flying collectors used for astronomical imaging in space,” Pub. Ast. Soc. Pac. |

16. | W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, |

17. | M. Perrin, R. Soummer, E. Elliott, M. Lallo, and A. Sivaramakrishnan, “Simulating point-spread functions for the James Webb Space Telescope with WebbPSF,” Proc. SPIE |

18. | A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.6085) Instrumentation, measurement, and metrology : Space instrumentation

(110.1080) Imaging systems : Active or adaptive optics

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: September 13, 2012

Revised Manuscript: November 9, 2012

Manuscript Accepted: November 24, 2012

Published: December 19, 2012

**Citation**

Anthony C. Cheetham, Peter G. Tuthill, Anand Sivaramakrishnan, and James P. Lloyd, "Fizeau interferometric cophasing of segmented mirrors," Opt. Express **20**, 29457-29471 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-28-29457

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### References

- D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.
- G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, and D. Kirkman, “Phasing the mirror segments of the Keck telescopes: the broadband phasing algorithm,” Appl. Opt.37, 140–155 (1998). [CrossRef]
- G. Chanan, C. Ohara, and M. Troy, “Phasing the mirror segments of the Keck telescopes II: the narrow-band phasing algorithm,” Appl. Opt.39, 4706–4714 (2000). [CrossRef]
- J. Rodríguez-Ramos and J. Fuensalida, “Phasing of segmented mirrors: a new algorithm for piston detection,” MNRAS328, 167–173 (2001). [CrossRef]
- G. Chanan, M. Troy, and E. Sirko, “Phase discontinuity sensing: a method for phasing segmented mirrors in the infrared,” Appl. Opt.38, 704–713 (1999). [CrossRef]
- A. Cheetham, Cophasing JWST’s segmented mirror using sparse aperture interferometry University of Sydney Hons. Thesis, (2011).
- P. Tuthill, J. Monnier, W. Danchi, E. Wishnow, and C. Haniff, “Michelson interferometry with the Keck I telescope,” Pub. Ast. Soc. Pac.112, 555–565 (2000). [CrossRef]
- J. Monnier, P. Tuthill, M. Ireland, R. Cohen, A. Tannirkulam, and M. Perrin, “Mid-infrared size survey of young stellar objects: Description of Keck segment-tilting experiment and basic results,” ApJ700, 491 (2009). [CrossRef]
- JWST Mission Operations Concept Document, Revision C (2008).
- R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.
- A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.
- J. Lloyd, F. Martinache, M. Ireland, J. Monnier, S. Pravdo, S. Shaklan, and P. Tuthill, “Direct detection of the brown dwarf GJ 802B with adaptive optics masking interferometry,” ApJ Lett.650, L131 (2006). [CrossRef]
- R. Bracewell, The Fourier Transform & its Applications, 3rd ed. (McGraw-Hill Science/Engineering/Math, 2000).
- E. Sabatke, J. Burge, and D. Sabatke, “Analytic diffraction analysis of a 32-m telescope with hexagonal segments for high-contrast imaging,” Appl. Opt.44, 1360–1365 (2005). [CrossRef] [PubMed]
- R. J. Allen, “Station-keeping requirements for constellations of free-flying collectors used for astronomical imaging in space,” Pub. Ast. Soc. Pac.119, 914–922 (2007). [CrossRef]
- W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge Univ Press, 1986).
- M. Perrin, R. Soummer, E. Elliott, M. Lallo, and A. Sivaramakrishnan, “Simulating point-spread functions for the James Webb Space Telescope with WebbPSF,” Proc. SPIE8442 (2012), 84423D.
- A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE7440 (2009), 74400Y. [CrossRef]

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