Simple model for image-plane polychromatic speckle contrast

Speckle mitigation for wavefront sensing in the presence of weak turbulence

When measuring atmospheric turbulence along the propagation path to an extended non-cooperative target, a wavefront sensor normally suffers from severe noise due to speckle. In this work, we quantify the benefits of speckle mitigation via polychromatic illumination for a Shack-Hartmann wavefront sensor. We obtain results over a wide range of conditions by using the spectral-slicing approach to polychromatic wave-optics simulations. To quantify speckle noise, even when turbulence is present, we introduce a metric involving racetrack-mode strength in slope-discrepancy space. The results show that polychromatic illumination greatly reduces speckle noise under realistic conditions. Even with near worst-case conditions, 15 coherence lengths per resolution cell reduce the wavefront-measurement error by 56%.

Polychromatic wave-optics models for image-plane speckle. 1. Well-resolved objects

Polychromatic laser light can reduce speckle noise in wavefront-sensing and imaging applications that use direct-detection schemes. To help quantify the achievable reduction in speckle, this paper investigates the accuracy and numerical efficiency of three separate wave-optics methods. Each method simulates the active illumination of extended objects with polychromatic laser light. In turn, this paper uses the Monte Carlo method, the depth-slicing method, and the spectral-slicing method, respectively, to simulate the laser-object interaction. The limitations and sampling requirements of all three methods are discussed. Further, the numerical efficiencies of the methods are compared over a range of conditions. The Monte Carlo method is found to be the most efficient, while spectral slicing is more efficient than depth slicing for well-resolved objects. Finally, Hu's theory is used to quantify method accuracy when possible (i.e., for well-resolved objects). In general, the theory compares favorably to the simulation methods.

Polychromatic wave-optics models for image-plane speckle. 2. Unresolved objects

Polychromatic laser light can reduce speckle noise in many wavefront-sensing and imaging applications. To help quantify the achievable reduction in speckle noise, this study investigates the accuracy of three polychromatic wave-optics models under the specific conditions of an unresolved object. Because existing theory assumes a well-resolved object, laboratory experiments are used to evaluate model accuracy. The three models use Monte-Carlo averaging, depth slicing, and spectral slicing, respectively, to simulate the laser-object interaction. The experiments involve spoiling the temporal coherence of laser light via a fiber-based, electro-optic modulator. After the light scatters off of the rough object, speckle statistics are measured. The Monte-Carlo method is found to be highly inaccurate, while depth-slicing error peaks at 7.8% but is generally much lower in comparison. The spectral-slicing method is the most accurate, always producing results within the error bounds of the experiment.

Influence of geometry on polychromatic speckle contrast

Understanding speckle behavior is very important in speckle metrology application. The contrast of a polychromatic speckle depends not only on surface roughness and the coherence length of a light source, as shown in previous works, but also on optical geometry. We applied the Fresnel approach of diffraction theory for the free-space geometry and derived a simple analytical relationship between contrast, coherence length, size of illuminated spot, and distances between source, object, and observation plane. The effect of contrast reduction is found to be significant for low-coherence light sources.

Robustness of reduced temporal phase unwrapping in the measurement of shape

The predictions of success rate and depth uncertainty for the negative exponential sequence used for temporal phase unwrapping of shape data are generalized to include the effect of a reduced sequence and speckle noise in single-channel and multichannel systems, respectively. To cope with the reduction of the sequence, a scaling factor is introduced. A thorough investigation is made of the performance of this algorithm, called the reduced temporal phase-unwrapping algorithm. Two different approaches are considered: a single-channel approach in which all the necessary images are acquired sequentially in time and a multichannel approach in which the three channels of a color CCD camera are used to carry the phase-stepped images for each fringe density in parallel. The performance of these two approaches are investigated by numerical simulations. The simulations are based on a physical model in which the speckle contrast, the fringe modulation, and random noise are considered the sources of phase errors. Expressions are found that relate the physical quantities to phase errors for the single-channel and the multichannel approaches. In these simulations the single-channel approach was found to be the most robust. Expressions that relate the measurement accuracy and the unwrapping reliability, respectively, with the reduction of the fringe sequence were also found. As expected, the measurement accuracy is not affected by a shorter fringe sequence, whereas a significant reduction in the unwrapping reliability is found as compared with the complete negative exponential sequence.