## Optical phase unwrapping in the presence of branch points

Optics Express, Vol. 16, Issue 10, pp. 6985-6998 (2008)

http://dx.doi.org/10.1364/OE.16.006985

Acrobat PDF (206 KB)

### Abstract

Strong turbulence causes phase discontinuities known as branch points in an optical field. These discontinuities complicate the phase unwrapping necessary to apply phase corrections onto a deformable mirror in an adaptive optics (AO) system. This paper proposes a non-optimal but effective and implementable phase unwrapping method for optical fields containing branch points. This method first applies a least-squares (LS) unwrapper to the field which isolates and unwraps the LS component of the field. Four modulo-2*π*-equivalent non-LS components are created by subtracting the LS component from the original field and then restricting the result to differing ranges. 2*π* phase jumps known as branch cuts are isolated to the non-LS components and the different non-LS realizations have different branch cut placements. The best placement of branch cuts is determined by finding the non-LS realization with the lowest normalized cut length and adding it to the LS component. The result is an unwrapped field which is modulo-2*π*-equivalent to the original field while minimizing the effect of phase cuts on system performance. This variable-range ‘*ϕ _{LS}
*+

*ϕ*’ unwrapper, is found to outperform other unwrappers designed to work in the presence of branch points at a reasonable computational burden. The effect of improved unwrapping is demonstrated by comparing the performance of a system using a fixed-range ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ realization unwrapper against the variable-range ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ unwrapper in a closed-loop simulation. For the 0.5 log-amplitude variance turbulence tested, the system Strehl performance is improved by as much as 41.6 percent at points where fixed-range ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ unwrappers result in particularly poor branch cut placement. This significant improvement in previously poorly performing regions is particularly important for systems such as laser communications which require minimum Strehl ratios to operate successfully.

_{non-LS}© 2008 Optical Society of America

## 1. Introduction/Background

*branch points*where intensity is zero and phase is undefined. [3

3. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A **15** (1998). [CrossRef]

*π*phase discontinuity known as

*branch cuts*in the field leading to the branch points.

### 1.1. Wavefront Sensors

5. J. Barchers, “The performance of wavefront sensors in strong turbulence,” Proc. SPIE **4839** (2003). [CrossRef]

### 1.2. Irrotational and Rotational Fields

*ϕ*(

_{Tot}*x*,

*y*) can be divided into irrotational

*ϕ*(

_{Irr}*x*,

*y*) and rotational

*ϕ*(

_{Rot}*x*,

*y*) components. [2] Under weak turbulence conditions, only the irrotational component of the phase is present because the total field is irrotational. [2] In the absence of detector noise, any rotational component is the result of

*branch points*(Sec. 1.3) which are caused by strong turbulence.

### 1.3. Branch Points

**F**(

**r**)≠0 where

*ϕ*(

*x*,

*y*). Thus starting at a point

*A*and integrating phase differential along a closed path around a branch point back to

*A*yields a different phase than what was started with. Since the phase at point

*A*has not changed, the integration along the closed path will necessarily be an integer multiple of 2

*π*. A clockwise integration around a single positive branch point yields +2

*π*while a negative branch point yields -2

*π*. If multiple branch points are contained within an integration path, their effects sum with positive branch points cancelling out the effects of negative branch points. In principle, integrating phase gradients along a path around a point is the way the presence of branch points is determined. [3

3. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A **15** (1998). [CrossRef]

### 1.4. Phase Cuts

*π*phase discontinuities. In a sampled phase map, a phase cut is defined as wherever there is more than a

*π*difference in phase between adjacent samples.

*π*compensation is applied and cuts are essentially ignored. The phase is not required to be unwrapped prior to implementation on a segmented DM.

### 1.5. Wrapping Cuts

*π*to the field on one side of the wrapping cut. Cuts which form a closed path within the field are also unwrapping cuts and can be eliminated similarly by either adding or subtracting an integer multiple of 2

*π*to the interior or exterior of the cut path. As an example, Fig. 1 depicts a wrapped and unwrapped phase. Note that the wrapped phase is limited in range to [-

*π*,

*π*) while the unwrapped phase is not.

### 1.6. Branch Cuts

6. D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. **31** (1992). [CrossRef] [PubMed]

*π*. As the line integral crosses the branch cut, however, ∓2

*π*is added so that the closed line integral sums to zero as it would if there were not a branch point within the closed path. Branch cuts can be placed in a variety of ways, all of which will still compensate for the non-zero curl of branch points in the phase. Two examples of phase cut placement are shown in Fig. 3. The poor unwrap is created by simply unwrapping the field from left to right. The minimum cut distance unwrap was manually created to minimize the length of the branch cuts.

### 1.7. Least-Squares Unwrappers

#### 1.7.1. Unweighted LS Unwrappers

*N*×

*N*array of phases, an unweighted LS unwrapper is developed as

**G**is a 2

*N*(

*N*-1)×

*N*

^{2}transformation matrix that converts the

*N*

^{2}vector of phases

*ϕ*into a 2

*N*(

*N*-1) vector of phase differentials in the

*x*and

*y*directions

**s**and the inverse notation is taken to be the pseudo-inverse. In weak turbulence,

**s**is most commonly the phase gradients provided by a Shack-Hartmann WFS. If actual phases

*ϕ*

_{Tot}are available, the phase differentials

**s**are developed as

**s**=𝒲 (

**G**

*ϕ*

_{Tot}) where 𝒲() indicates the wrapping operation of limiting the differentials s to some 2

*π*interval. An important point is that while creating an

*N*

^{2}×

*N*

^{2}pseudo inverse is computationally daunting, the problem is alleviated somewhat by

**G**being sparse and fixed for a given AO system. Much of the work can be pre-computed a single time rather than having to be determined in real time during execution.

#### 1.7.2. Weighted LS Unwrappers

**W**is an 2

*N*(

*N*-1)×2

*N*(

*N*-1) diagonal array of weights. It works essentially the same as an unweighted LS unwrapper, but the pseudo inverse cannot be pre-computed because the weighting matrix is not typically constant. This makes a weighted LS unwrapper difficult to implement in real-time systems.

#### 1.7.3. LS Unwrappers and the Hidden Phase

### 1.8. Non-LS Component of the Field

*π*-equivalent to the original field, and the non-LS component will be non-existent. If the original field has branch points and is rotational, the effects of those branch points will be isolated in the non-LS component. As such it is sometimes referred to as the rotational component. [2] Strictly speaking, the rotational component is not unique[10

10. Wikipedia (2007). Website, URL http://en.wikipedia.org/wiki/Vector-potential.

*π*range.

## 2. Improved Unwrapper

*LS*() indicates applying an LS unwrapper operation to the vector of wrapped phases

*ϕ*and 𝒲() indicates wrapping the phase to some 2

_{Tot}*π*range.

*ϕ*. Thus total phase

_{non-LS}*ϕ*adjusted to remove wrapping cuts while still retaining branch cuts can be determined as

_{Tot}*𝒰*() indicates an unwrapping process which removes wrapping cuts (but not branch cuts). While removing any wrapping cuts, this unwrapped result is modulo-2

*π*-equivalent to

*ϕ*, maintaining both the irrotational and rotational components of the field. This has been covered in several texts [2] and is a common way of including rotational phase effects in the AO systems being developed to operate under strong turbulence conditions. [11]

_{Tot}*ϕ*is free of phase cuts, the non-LS portion

_{LS}*ϕ*must be examined in order to reduce the impact of phase cuts. Being wrapped,

_{Non-LS}*ϕ*is restricted to some 2

_{Non-LS}*π*range, say [0,2

*π*). If the range is changed to [-

*π*/2,3

*π*/2) then all the points whose phase is in [3

*π*/2,2

*π*) would have 2

*π*subtracted from them. The resulting field would be modulo-2

*π*equivalent to the original field, but would have branch cuts in different positions. The field depicted in Fig. 5 is re-depicted in Fig. 6 alongside unwraps for the same field with

*ϕ*having differing range restrictions. The unwrap with

_{Non-LS}*ϕ*restricted to [0,2

_{Non-LS}*π*) has terrible branch cut placement. The remaining three realizations depicted are much more reasonable, with the realization created by limiting

*ϕ*to [-

_{Non-LS}*π*,

*π*) having the lowest normalized cut length. It should be noted that the creation of four realizations is reasonable because the majority of the computational load is in executing the LS unwrapper which only has to be done once.

### 2.1. Unwrapping Metric - Normalized Cut Length

*π*-equivalent phase realizations, it is necessary to compare different branch cut placements so that the best one can be chosen. Short cuts through regions of minimal illumination have the least impact on system performance. [6

6. D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. **31** (1992). [CrossRef] [PubMed]

*F*is the ‘Fine’ field,

*E*is the estimated DM field and * is the conjugation operator. [12

12. T. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE **5553** (2004). [CrossRef]

## 3. Simulation and Results

*r*

_{0}. The log-amplitude variance of intensity is a measure of the scintillation of the field and a reasonable indication of the turbulence strength. [13] Both variances reflect strong turbulence which would create branch points.

*ϕ*ranges. The average and maximum score for all realizations is then determined for each of the four ranges. These data show how the unwrapper would perform if the range was fixed to a particular range. The integrated cut intensity is also recorded for the

_{non-LS}*ϕ*range which gives the lowest score. The average and maximum is determined for this best of four

_{non-LS}*ϕ*ranges and compared against the average and maximum scores from the fixed ranges.

_{non-LS}*ϕ*+

_{LS}*ϕ*’ mean normalized cut length is reduced in both cases. Perhaps more importantly, the worst realizations are avoided in a variable-range ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ unwrapper so that the maximum normalized cut length is dramatically reduced. The weighted variable-range ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ unwrapper has the effect of influencing the LS portion of the field towards the areas of higher intensity. The non-LS portion of the field is then influenced towards the areas of lower intensity and branch cuts are forced into darker portions of the field. While a weighted LS unwrapper has the best performance, the computational cost of a weighted unwrapper is significant (see section 4).

_{non-LS}## 4. Comparison to Other Unwrappers

*ϕ*+

_{LS}*ϕ*’ method, it was compared to other unwrappers designed to work with branch points. The other unwrappers are the fixed-range ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ unwrap, Goldstein’s branch cut placement unwrap method[2], Wave-prop’s xphase[14], and Fried’s smoothphase.[4

_{non-LS}4. D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Optics Communications **200** (2001). [CrossRef]

*ϕ*+

_{LS}*ϕ*’ unwrapper is the same as the variable-range ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ but only develops a single unwrap realization instead of choosing the best of four realizations. Goldstein’s branch cut placement method attempts to determine minimum length branch cuts that connect branch points. [2] Xphase is a Matlab unwrapping function from the AOTools Matlab toolbox. It is designed to work with fields containing branch points and attempt to place branch cuts in low intensity regions of the field. [14] It should be noted that xphase required the 32×32 field to be zero-padded to 64×64 in order to work properly. Fried’s smoothphase unwrapper separates the field into rotational and irrotational components by first determining the rotational component (after balancing the number of branch points by adding additional branch points along the edge of the field as necessary). Once separated, the irrotational component can be unwrapped and then recombined with the rotational component of the field. [4

_{non-LS}4. D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Optics Communications **200** (2001). [CrossRef]

*ϕ*+

_{LS}*ϕ*’ unwrapper using an unweighted LS gives the best performance at a reasonable computation burden. The variable-range ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ unwrapper using a weighted LS improves performance still more, but at an unreasonable computational burden. AOTools xphase unwrapper gave slightly improved results compared to a variable-range ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ using an unweighted LS unwrapper, but at over six times the computational burden.

_{non-LS}## 5. Impact on System Performance

*r*

_{0}=4

*D*where

_{SA}*D*is the diameter of a sub-aperture, sample rate=223

_{SA}*f*where

_{G}*f*is the Greenwood frequency of the atmosphere, average SNR 200. The simulation used a leak-free integrator controlled feedback with a error signal gain of 0.4. The control law was applied immediately before the unwrapper, whose output then went to the DM.

_{G}*ϕ*range ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ unwrappers was compared against using a variable

_{non-LS}*ϕ*range ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ unwrapper. Strehl ratio performance of the various simulations are plotted in Figs. 8 and shows how different fixed ranges have different periods of reduced performance. Average results are tabulated in table 5 as well as average and maximum improvements when using a ‘best of four’ unwrapper. The new unwrapper improved the average Strehl ratio performance between 3.3% and 7.6% against the four fixed

_{non-LS}*ϕ*range unwrappers with considerably less variability. The maximum improvement of the new unwrapper against the four fixed

_{non-LS}*ϕ*range unwrappers was more dramatic, ranging from 23.0% to 41.6%.

_{non-LS}*ϕ*+

_{LS}*ϕ*’ unwrapper is inconsistent, the average improvement of the variable-range ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ unwrapper over a fixed-range ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ unwrapper is difficult to determine. In order to develop an average improvement, the simulations were extended to 10,000 frames to provide each fixed-range of the ‘

_{non-LS}*ϕ*+

_{LS}*ϕ*’ unwrappers with areas of both good and bad performance. The results of the simulation were put into histograms and then summed to form cumulative distribution functions (CDFs) shown in Fig. 9. The CDFs show how the variable range

_{non-LS}*ϕ*unwrapper improves performance. The CDF of the variable range

_{non-LS}*ϕ*unwrapper is shifted to the right when compared to the CDF of the fixed range

_{non-LS}*ϕ*unwrapper. Not only does this indicate improved average performance, but indicates more significant improvement for systems such as laser communication where performance thresholds which inhibit operation below a certain Strehl ratio.

_{non-LS}## 6. Conclusion

## Disclaimer

## References and links

1. | J. W. Hardy, |

2. | D. C. Ghiglia and M. D. Pritt, |

3. | D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A |

4. | D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Optics Communications |

5. | J. Barchers, “The performance of wavefront sensors in strong turbulence,” Proc. SPIE |

6. | D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. |

7. | D. L. Fried, “The Nature of the Branch Point Problem in Adaptive Optics,” Proc. SPIE |

8. | C. A. Primmerman, “Atmospheric-compensation experiments in strong-scintillation conditions,” Appl. Opt. |

9. | M. C. Roggemann, “Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wavefront correction,” J. Opt. Soc. Am. A |

10. | Wikipedia (2007). Website, URL http://en.wikipedia.org/wiki/Vector-potential. |

11. | J. Barchers, “Personal Communication,” (2007). |

12. | T. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE |

13. | L. C. Andrews and R. C. Philips, |

14. | T. J. Brennan and P. H. Roberts, |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: January 30, 2008

Revised Manuscript: April 23, 2008

Manuscript Accepted: April 23, 2008

Published: May 1, 2008

**Citation**

Todd M. Venema and Jason D. Schmidt, "Optical phase unwrapping in the presence of branch points.," Opt. Express **16**, 6985-6998 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-6985

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### References

- J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).
- D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping Theory, Algorithms, and Software (John Wiley & Sons, Inc., 1998).
- D. L. Fried, "Branch point problem in adaptive optics," J. Opt. Soc. Am. A 15, 2759-2768 (1998). [CrossRef]
- D. L. Fried, "Adaptive optics wave function reconstruction and phase unwrapping when branch points are present," Opt. Commun. 200, 43-72 (2001). [CrossRef]
- J. Barchers, "The performance of wavefront sensors in strong turbulence," Proc. SPIE 4839, 217 (2003). [CrossRef]
- D. L. Fried and J. L. Vaughn, "Branch cuts in the phase function," Appl. Opt. 31, 2865-2882 (1992). [CrossRef] [PubMed]
- D. L. Fried, "The nature of the branch point problem in Adaptive Optics," Proc. SPIE 3381, 38-46 (1998). [CrossRef]
- C. A. Primmerman, "Atmospheric-compensation experiments in strong-scintillation conditions," Appl. Opt. 34, 2081-2088 (1995). [CrossRef] [PubMed]
- M. C. Roggemann, "Branch-point reconstruction in laser beam projection through turbulence with finite-degreeof-freedom phase-only wavefront correction," J. Opt. Soc. Am. A 17, 53-62 (2000). [CrossRef]
- Wikipedia (2007). Website, URL http://en.wikipedia.org/wiki/Vector-potential.
- J. Barchers, "Personal Communication," (2007).
- T. Rhoadarmer, "Development of a self-referencing interferometer wavefront sensor," Proc. SPIE 5553, 112-126 (2004). [CrossRef]
- L. C. Andrews and R. C. Philips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 1998).
- T. J. Brennan and P. H. Roberts, AO Tools The Adaptive Optics Toolbox-Users Guide version 1.2 (Optical Sciences Company, 2006).

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