Polarization phase shifting dispersed fringe sensor

Gene Olczak

doi:10.1364/OE.20.003703

1. Introduction

Future astronomical systems will call for higher performance with ever-larger observatories whether in space or on the ground. These complex systems typically include segmented optics or multiple apertures.

A critical component of the co-phasing of segmented aperture optics is the unambiguous measurement and removal of piston errors between segments. The dispersed fringe sensor (DFS) has been demonstrated as an effective means of measuring mirror segment piston error for telescopes with primary mirror apertures below 10 meters [1]. With larger proposed telescopes such as The Thirty Meter Telescope (TMT) and The European Large Telescope (ELT) including ever more segments, there is a need for improvement in the co-phasing capability for segmented primary mirrors [2].

Phase shifting technology has been applied to interferometry for several decades with great success [3]. More recently polarization phase shifting (PPS) has been demonstrated effectively in interferometry for highly dynamic systems [4,5]. In this paper a novel DFS that employs polarization phase shifting technology [6] is introduced. This novel technology provides system designers and engineers with a new tool to extend the dynamic range of a DFS (including the dispersed Hartmann sensor variant).

2. Theory of polarization phase shifting dispersed fringe sensing

The DFS can be thought of as an extension of Young’s dual slit experiment. In the classic experiment two symmetrically disposed rectangular slits are illuminated with mutually coherent monochromatic light. This arrangement results in a far field diffraction pattern where the fringe spacing is proportional to the illuminant wavelength. In the context of a telescope, the slits are analogous to a pupil conjugate and the far field pattern is observed at a plane equivalent to the telescope image surface.

If a uniform optical path difference (OPD) is introduced at one of the two slits then the positions of the fringe intensity minima and maxima are displaced in a direction perpendicular to the axis of symmetry of the slits. The fringe displacement is proportional to the OPD between the two slits. The fringe pattern versus OPD is ambiguous, as the intensity pattern is equivalent for integer wavelength offsets of the OPD.

The magnitude of the fringe displacement at a shorter wavelength is larger than that of the fringe displacement at a longer wavelength for a fixed OPD. The DFS takes advantage of this by employing broadband illumination and an optically dispersive element. The direction of dispersion is set perpendicular to the direction of slit symmetry. In this configuration a unique two dimensional fringe pattern is observed as a function of OPD within the capture range of the DFS. The intensity pattern observed is a straight but tapered fringe in the case of zero OPD, but morphs into a barber-pole-like tilted fringe pattern where the degree and direction of tilt is a function of the OPD (the observed tilt is due to the differential wavelength dependent fringe displacement). The fringe pattern washes out outside the capture range of the DFS.

It is important to keep in mind that the tilted fringe pattern has a transform relationship with the system wavefront. The tilt observed in DFS fringes is not equivalent to wavefront tilt i.e. as measured with an interferometer.

Consider the case of a single split-aperture two-segment mirror system with a DFS. Given a grism (combination grating and prism) with linear dispersion rate $C_{0}$ , the wavelength $λ$ in the dispersion direction x at the detector is given by [7]

λ (x) = λ_{0} + \frac{\partial λ}{\partial x} x = λ_{0} + C_{0} x,

where the central wavelength

λ_{0}

and

\frac{\partial λ}{\partial x}

are characteristics of the grism dispersion. Equation (1) can be solved for the dispersive characteristic function at the detector such that

x_{λ} (λ) = \frac{λ - λ_{0}}{C_{0}} .

The shape of the split aperture used for a DFS can be varied. For simplicity, a double-slit aperture geometry is assumed. The sensor to be discussed is depicted in Fig. 1 . Illumination from a segmented system to the left of the figure impinges a set of pupil conjugate components. The first component is a grism. Following this is a double slit aperture mask. Each slit of width w and length l is located such that the segment boundary bisects the mid-point on the aperture mask in the y direction. The effective focal length of the system is given by f. Ignoring polarization components, the intensity profile at the detector is given by [8]

I_{0} (x, y) = 2 {[\frac{l w}{f}]}^{2} \int_{λ_{m i n}}^{λ_{m a x}} S (λ) \frac{1}{λ^{2}} sinc {[\frac{l (x - x_{λ} (λ))}{λ f}]}^{2} sinc {[\frac{w y}{λ f}]}^{2} {1 + \cos (\frac{2 π}{λ} [\frac{η_{0} y}{f} + δ])} d λ,

where S(

λ

) is the spectral intensity of the source,

η_{0}

is the center to center spacing between the slits in the y direction and sinc denotes the normalized sinc function

\frac{\sin (π x)}{π x}

. The piston error is a uniform OPD between segments, denoted as

δ

. As discussed in Ref. 7 and 8, the resultant intensity pattern forms a fringe pattern that can be used to determine the relative piston error between mirror segments by direct interrogation of the fringe intensity.

Fig. 1 Depiction of a polarization phase shifting dispersed fringe sensor.

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Note the polarization components in Fig. 1. To the right of the upper slit aperture is a linear polarizing filter with a transmission axis that is parallel to x. To the right of the lower slit aperture is a linear polarizing filter with a transmission axis that is parallel to y. Following the aperture is an achromatic quarter wave plate (QWP) with a fast axis that bisects the transmission axes of the preceding polarizers. Given this arrangement, the polarization state of the light exiting the QWP is circular with the polarization state from each slit having opposing spin directions (this configuration has some similarity to a quantum eraser experiment [9]).

In-plane rotation of the linear polarizing filter produces a relative phase shift between the two (no longer orthogonal) transmitted beams [10–13]. The result is a relative phase shift between the two slits as a function of the orientation of the transmission axis of the filter $θ_{n}$ such that the focused intensity pattern at the detector is given by

I_{n} (x, y) = \frac{1}{2} {[\frac{l w}{f}]}^{2} \int_{λ_{m i n}}^{λ_{m a x}} S (λ) \frac{1}{λ^{2}} sinc {[\frac{l (x - x_{λ} (λ))}{λ f}]}^{2} sinc {[\frac{w y}{λ f}]}^{2} {1 + \cos (\frac{2 π}{λ} [\frac{η_{0} y}{f} + δ] + 2 θ_{n})} d λ

where the n subscript denotes the index of a plurality of possible filter orientations. The use of multiple filter orientations provides for polarization phase shifting (PPS) between the two sub apertures (slits) and modulation of the resultant fringe pattern. In an ideal system, the intensity at the detector is reduced by a factor of four compared to an ideal system without PPS for each I_n. From Eq. (3) and Eq. (4) one can observe that the morphology of the intensity pattern for the PPS DFS is the same as a conventional DFS when

θ_{n} = 0

.

Note: The rotating filter is shown on the right of the lens in Fig. 1. In practice it can be located in the pupil at left as well (this may be preferable if the lens has a low f-number).

The addition of PPS provides the opportunity to employ analysis methods that have not previously been considered for a DFS. As discussed in Ref. 3, phase shifting techniques are generally more robust than direct inspection methods (such as fringe tracing). As a result, the majority of interferometers used in modern optical testing employ phase shifting techniques (The method of PPS used here provides a novel mechanism for generating generalized pupil piston diversity, a technology that has applications beyond DFS [14].).

One simple method that is employed in instantaneous phase capture systems uses four intensity images [15] {I_1, I_2, I_3, I₄}. This can be accomplished with a set of cameras and polarization splitting optics or a single camera with a micropolarizer array (see also Ref. 4 and 5). Simultaneous capture can be beneficial in a high vibration test environment.

Using rotation angles {0, $\frac{π}{4}$ , $\frac{π}{2}$ , $\frac{3 π}{4}$ } radians for $θ_{1 - 4}$ results in a relative phase shift of {0, $\frac{π}{2}$ , $π$ , $\frac{3 π}{2}$ } radians for each respective intensity image. A back focus phase surface can be constructed from the four intensity images using the arctangent function such that (Ref. 3)

ϕ (x, y) = \tan^{- 1} [\frac{I_{1} (x, y) - I_{3} (x, y)}{I_{2} (x, y) - I_{4} (x, y)}] .

The piston error is determined from a linear fit to $ϕ (x, y)$ . In practice, interpretation of $ϕ (x, y)$ requires two-dimensional phase unwrapping in the presence of large piston errors. There are many choices of algorithms for two-dimensional phase unwrapping. In the example that follows, a Goldstein unwrapping algorithm [16] is employed. As is demonstrated below, the measured piston error ( $\hat{δ}$ ) is found to be linearly proportional to a fit of the Zernike polynomial coefficient to $ϕ (x, y)$ . So

\hat{δ} = M^{- 1} \hat{Z} .

where M⁻¹ is an inverse sensitivity matrix and

\hat{Z}

is a fit to the Zernike fringe polynomial terms for tilt, power and astigmatism over

ϕ (x, y)

. Each individual fringe pattern forms a set of diverging lines, the shape of which depends on the value of

\hat{δ}

[17]. Changes in

ϕ (x, y)

as a function of

\hat{δ}

are dominated by tilt and astigmatism.

An added benefit of PPS is that the fringe modulation $γ (x, y)$ can be calculated directly from the phase shifted images; $γ (x, y)$ provides a data quality metric that is useful for system diagnostics and data pruning. $γ (x, y)$ is given by (Ref. 3)

γ (x, y) = 2 {(\frac{{[I_{1} (x, y) - I_{3} (x, y)]}^{2} + {[I_{2} (x, y) - I_{4} (x, y)]}^{2}}{I_{1} (x, y) + I_{2} (x, y) + I_{3} (x, y) + I_{4} (x, y)})}^{1 / 2} .

The numerator in Eq. (7) provides an estimate of what the fringe image would look like if the slit apertures were illuminated with mutually incoherent light. The incoherent image can be useful for data pruning and estimating the orientation of the null fringe ( $\hat{δ}$ = 0). It is defined as

I_{i n c o h} = I_{1} (x, y) + I_{2} (x, y) + I_{3} (x, y) + I_{4} (x, y) .

3. Model and simulations

A PPS DFS system is modeled with the parameters in Table 1 . The simulations that follow are performed using double precision numerical calculations in MATLAB® software. The integral over $λ$ is approximated as a Riemann sum with unity weighting for 128 bins over the defined bandwidth. The sample distance in x and y is set to 3.175 microns (an arbitrary but reasonable value). A sampled uniform pseudo-random noise generator is used to analyze dynamic noise effects (As discussed in Ref. 3, phase shifting is known to be insensitive to fixed pattern noise.). The nominal value of signal to noise (SNR) used in the figure below is 1000 (intensity maximum minus minimum divided by standard deviation).

Table 1. Simulation Parameters

View Table

Given the geometry from Table 1 the boundary between the segments forms a zero clearance gap. The size of each fringe image is 267 microns in y (plotted vertically) and 800 microns in x (plotted horizontally). Each of four phase shifted image for $δ$ = 0 microns and $δ = - 250$ microns is depicted in Fig. 2 . The impact of phase shifting on each of the fringe images is easily visible in the figure (the vertical walking of the dark fringe in (a)). One can also see a change in the fringe rotation when changing the state of $δ .$ The range is chosen such that the mean modulation over the fringe drops to 20% for the maximum value of $δ$ .

Fig. 2 (a) Phase shifted dispersed fringe images for $δ$ = 0 microns, $θ_{n} =$ {0, $\frac{π}{4}$ , $\frac{π}{2}$ , $\frac{3 π}{4}$ } from left to right. (b) Phase shifted dispersed fringe images for $δ$ = −250 microns, $θ_{n} =$ {0, $\frac{π}{4}$ , $\frac{π}{2}$ , $\frac{3 π}{4}$ } from left to right.

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Given the fringe images, the phase and modulation are calculated according to Eq. (5) and Eq. (7). The resulting phase and modulation maps are depicted in Fig. 3 . The difference in I_incoh for the two cases is shown in Fig. 4 . By the symmetry of the difference one can see that the orientation of I_incoh is unchanged for the two piston conditions. This is beneficial for calibration of the PSS DFS since the grism dispersion direction can be determined directly from the data. Conventional DFS can suffer from piston measurement errors due to uncertainty in the dispersion direction [18].

Fig. 3 (a) Phase maps in radians for $δ$ = 0 microns (left) and $δ$ = −250 microns (right). (b) Modulation maps for $δ$ = 0 microns (left) and $δ$ = −250 microns (right). (c) Incoherent images for $δ$ = 0 microns (left) and $δ$ = −250 microns (right).

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Fig. 4 Difference map of I_incoh for $δ$ = 0 microns minus I_incoh for $δ$ = −250 microns.

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The modulation and incoherent images are used to trim the data before applying the two dimensional phase unwrapping algorithm. The nominal threshold values are chosen as 10% for modulation and 20% for intensity. The unwrapped phase maps are depicted in Fig. 5 .

Fig. 5 Unwrapped phase maps in radians for $δ$ = 0 microns (left) and $δ$ = −250 microns (right).

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To show that relationship between $δ$ and $\hat{δ}$ , simulations with modest SNR values {50, 100, 1000} are performed. The value of $δ$ is varied from −250 microns to + 250 microns with a 0.1 micron increment for $| δ | < 2$ microns and a 2 micron increment elsewhere. The finer increment is used around the zero crossing to capture the small non linearity in that region. The values for M⁻¹ are found from a least squared fit to the noise free model (using singular value decomposition). The differences between $δ$ and $\hat{δ}$ are plotted in Fig. 6 . The R² fit value in all cases is 1 for practical purposes and no variation in data quality is visibly discernable on a linear plot of $\hat{δ}$ versus $δ$ over the simulated sensing range.

Fig. 6 Piston estimation error for SNR {50, 100, 100} (left to right).

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For the high noise case (SNR = 50 or 16.7 dB) the root-mean-square (RMS) error over the entire range is 83.6 nanometers with a noticeable degradation toward the extremes of the range. For $| δ | < 200$ microns the RMS error is 17.7 nanometers. For the SNR = 100 case the RMS error is 10.4 nanometers and one can see the zero crossing feature beginning to dominate the error plot. This feature is systematic and may be handled with a look-up table but practically the 30 nanometer peak to valley error provides for co phasing ability such that the mirrors can be aligned well within the range of a mature fine phasing algorithm (though the magnitude of the non linearity is a function of the PPS DFS design range).

For the SNR = 1000 case the zero crossing error is clearly visible as well as a low-amplitude odd-symmetry non-linearity. Here the RMS error is 8.5 nanometers. Compared to existing piston sensing devices, the PPS DFS is expected have higher precision (See Table l in Ref. [7]) especially when the devices are tuned for an equivalent sensing range.

3. Concluding remarks

A novel PPS DFS has been introduced. The addition of phase shifting to a dispersed fringe sensor allows for the design of higher performance sub systems to support more complex segmented aperture systems. Because the hardware and software implementations are simple, it may also be straight-forward to upgrade existing systems that employ a static DFS to a PPS DFS if needed. Elsewhere, a PPS DFS has been demonstrated as effective even in the presence of very substantial figure error in a multi-segmented aperture system [19].

Beyond the scope of this paper are the details of a PPS DFS implementation for a multi segment complex aperture telescope e.g., tradeoffs between edge sensing, full segment aperture sensing, the threshold for handing off to final phasing. Errors may include grism diffraction order leakage, polarization leakage, effects of mirror figure error, phase errors in the achromatic waveplate, electronic channel noise, phase analysis centering and phase coordinate normalization.

Crosstalk between grism channels is also a concern; for a large telescope with many segments the piston detection process will likely be designed with several grisms operating in parallel (See Ref. 1 and 2). It is expected that the PPS DFS will be more robust than a conventional DFS with respect to cross talk and spurious illumination allowing a higher density of fringe patterns to be captured simultaneously. Future studies will focus on the hardware implementation of a mult-channel PPS DFS.

Acknowledgments

Thanks to William Castle (ITT Exelis), Alex Ceres (ITT Exelis) and Cormic Merle (ITT Exelis) for their helpful discussions and review of this paper.

References and links

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4. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004). [CrossRef]

5. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005). [CrossRef] [PubMed]

6. G. Olczak and C. Merle, “Polarization modulated image conjugate piston sensing and phase retrieval system,” ITT Manufacturing Enterprises, Inc., US Patent 7,864,333 (2011)

7. F. Shi, D. C. Redding, A. E. Lowman, C. W. Bowers, L. A. Burns, P. Petrone III, C. M. Ohara, and S. A. Basinger, “Segmented mirror coarse phasing with a dispersed fringe sensor: experiment on NGST’s wavefront control testbed,” Proc. SPIE 4850, 318–328 (2003). [CrossRef]

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16. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory Algorithms, and Software (John Wiley and Sons, Inc. 1998)

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19. G. Olczak, “Recent advances in wavefront sensing at ITT,” ITT Internal Report (2009).

Parameter	Value
F	1 meter
$λ_{m a x}$	780 nanometers
$λ_{m i n}$	720 nanometers
$λ_{0}$	750 nanometers
C₀	6 microns/meter
L	25 millimeters
W	12.5 millimeters
$η_{0}$	12.5 millimeters

Polarization phase shifting dispersed fringe sensor

Abstract

1. Introduction

2. Theory of polarization phase shifting dispersed fringe sensing

3. Model and simulations

3. Concluding remarks

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (8)

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