Optics: measuring and testing – By light interference
Reexamination Certificate
1998-10-16
2001-04-17
Kim, Robert (Department: 2877)
Optics: measuring and testing
By light interference
Reexamination Certificate
active
06219142
ABSTRACT:
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims the benefit of the filing of U.S. Provisional Patent Application Serial No. 60/063,745, entitled “Method and Apparatus for the Phase Retrieval of Multidimensional Spectrograms and Sonograms,” filed on Oct. 17, 1997, and the specification thereof is incorporated herein by reference.
BACKGROUND OF THE INVENTION
1. Field of the Invention (Technical Field)
The present invention relates to phase retrieval in wave phenomena. The technical field of this invention is multidimensional phase retrieval. The preferred embodiment of this invention applies to ultrafast laser diagnostics.
2. Background Art
Interference phenomena are produced through the interaction of at least two spatially distinguishable waves. Sometimes diffraction and interference are not clearly distinguished. Interference occurs when two or more wavefronts interact while diffraction occurs naturally when a single wave is limited in some way. The nature of an interference pattern depends on several factors including the amplitudes and phases of the incoming waves. If the incoming waves are in phase, then the amplitude of the waves may add whereas, if the phase of the incoming waves are separated by 180°, then cancellation will result. These phenomena, the adding and canceling of amplitude, are typically referred to as constructive and destructive interference. For example, if light from a single laser is split such that it produces two separate beams, and these beams travel two different paths, then it is likely that the beams are no longer in phase or phase coherent. If the two beams are then recombined, the resulting image will be an interference pattern. In some instances, it is helpful to detect the interference pattern as an image in a plane or on a planar surface. However, the interference pattern can also be captured as intensity (absolute value of amplitude squared) versus time. The intensity versus time curve, resulting from the interference between the two beams, can be reconstructed if the individual frequency components are known i.e., amplitudes and phases.
Phase differences in the interference waves can be represented as shifts in origin. For example, the Fourier transform of a function f(x) (which converts a wave from amplitude as a function of time or space to amplitude as a function of frequency) is as follows:
F(
k
)=∫
f
(
x
)exp(−
ikx
)dx (1)
where the limits on the integral are from x=−∞ to x=+∞. When the origin, or phase, is shifted, the Fourier transform is represented as,
F
1
(
k
)=∫f(
x−x
o
)exp−
ikx
)dx (2)
or as,
F
1
(
k
)=F(k)exp(—ikx
o
) (3).
Thus, the form of the Fourier transform differs only by the phase factor exp(−ikx
o
), remembering that the amplitudes |F
1
(k)| and |F(k)| and the intensity, the amplitudes squared, are equal. In many wave problems the function f(x) is complex, i.e., f(x)=Re[f(x)]+ilm[f(x)], and an analysis of transforms of complex functions applies. A plane wave has a wavefront in a plane of constant phase normal to the direction of propagation. Often a plane wave may be written as
E=E
o
exp(−i&ohgr;t) (4)
In Equation 4, the wave is represented as E, as a function of time, where the frequency characteristics are captured in the exponential term. In Equation 4, the amplitude is at a maximum at t=0.
Unfortunately, if phase information for the two pulses is not known ahead of time, then there is no unique combination of frequencies and amplitudes that may be combined to produce an identical plot of intensity versus time. Essentially, there is not enough information in the intensity versus time plot alone to be able to reconstruct the phase of the original pulses. The method to solve for the phase in these types of problems is referred to as phase retrieval. In essence, the phase retrieval problem is similar to solving a single equation for two unknowns—many solutions exist. To overcome this problem, a constraint must be imposed, i.e., an additional equation. Additionally, the constraint must be physically reasonable. Most phase retrieval problems are solved through imposition of a reasonable constraint that leads to a unique solution. The type of constraint depends on the application. In crystallography, symmetry conditions are typically imposed. In most instances for crystallographic application, the symmetry conditlon is applied to what is known as the outer bounds of the region. Such constraints benefit from prior knowledge of the way atoms are arranged in a crystalline structure. For other applications, other constraints must be found in order to retrieve phase.
To begin solution of such problems, an initial estimate of the phase is necessary. However, the guess is not so critical and any reasonable starting point can be used to obtain a solution. Of course, knowing that many of the solution techniques use iterative processes, a more reasonable guess typically results in fewer iterations in arriving at a unique solution.
In the field of phase retrieval, there is a class of problems known as Reconstruction from Multiple Fourier Intensities (RMI). This problem involves the reconstruction of two functions from multiple Fourier intensities of the product of relative displacements of the two functions. The solution to this important problem has significant applications to any situation where the intensity of the Fourier transform of the product of two functions is recorded for multiple relative displacements of the functions including transmission microscopy (optical, electron, and x-ray) and ultrafast laser diagnostics. For example, in the field of ultrafast laser diagnostics, the solution to the RMI problem is known as a frequency-resolved optical gatng (FROG) trace inversion, spectrogram inversion, or sonogram inversion and is used to find the intensity and phase of an ultrashort laser pulse. In the case of FROG, for example (see FIG.
1
), a gate pulse which is one function is scanned, in time, across a pulse (the other function) to be measured. For each time delay, the spectrum of the pulse that results from a well-defined nonlinear interaction of these two pulses is recorded. (The nonlinear interaction produces the product of the gate and pulse.) The resulting spectrogram, or FROG trace, is a plot of intensity versus time and frequency of the pulse. Unfortunately, it is only possible to obtain intensity information of the spectrum. Consequently, the key parameters of the pulse, the intensity and phase, cannot be obtained directly from this plot. An iterative two-dimensional phase retrieval method must be used to find the phase in order to extract the functions, and hence, the pulse characteristics from its spectrogram. Methods currently exist, but they require a priori knowledge of the gate function or are slow and cumbersome, requiring large amounts of computational power to arrive at the result. There is a need for fast inversion methods for ultrashort pulse measurement devices, and the same method will be generally applicable to other fields.
Ultrafast laser systems have a large number of applications in biochemistry, chemistry, physics, and electrical engineering. These systems generate laser pulses with durations of 10 picoseconds or less and such systems are used to explore kinetics in proteins, examine carrier relaxation in semiconductors, or image through turbid media. They are also used as an ultrafast probe in electronic circuits. By using ultrafast diagnostic systems, highly advanced semiconductors, electronic circuitry, and even biomedical products can be developed and tested for commercial applications. Furthermore, new applications requiring shaped ultrashort pulses in both intensity and phase such as coherent control of chemical reactions are beginning to be developed. The continued development of these applications will require, fast, high quality, and easy-to-use ultrafast las
Kim Robert
Lee Andrew H.
Myers Jeffrey D.
Pangrle Brian J.
Southwest Sciences Incorporated
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