2. Introduction to Artificial Neural Network
2.3 Summary
From the case studies shown in this chapter, we found that ANN is useful for numerous applications. Most of the applications of ANN use the back-propagation structure. Function approximation is useful to simulate the behavior of a physical system, which the underlying processes inside the system are unclear. We can build an ANN to simulate the relationship between the input and output of a physical system. The most sensitive issue of ANN relates to the training process. The training process requires much CPU time and may yield poor performances if an inappropriate ANN structure is implemented and is trained with inappropriate data sets.
Chapter 3 Complete Characterization of Ultrashort Coherent Optical Pulses with SHG Spectral
Measurement
3.1 Introduction
As explained in Chapter 1, the complete field characterization of coherent optical pulses is the first step to invoke these optical pulses for optical metrology. Several techniques had been developed to offer the complete characterization of coherent optical phases, such as frequency-resolved optical gating (FROG) first reported by D. Kane and R. Trebino [2], and spectral-phase interferometry for direct electric field reconstruction (SPIDER) developed by T. Tanabe, et al. [3].
The basic concept of FROG is quite similar to the autocorrelation measurement but FROG measures the spectrums at different time delays instead of optical intensity only.
Retrieving the spectral phases and then yielding a complete-field information of the coherent pulse under study is via an iteration algorithm. SPIDER can directly measure the spectral phase of a coherent pulse with a spectral-shearing interferometer, which separates the incoming coherent pulse into two parts and sent one part through a linear spectral phase modulator, and the other through a linear temporal phase modulator. And then by superpose these two parts together to yield a spectral-shearing interferogram.
The spectral phase and therefore the complete field information of the coherent pulse can be deduced directly from the interferogram without involving any further iterative calculation.
Along the development of complete coherent pulse characterization, Dorrer, et al.
had invoked a self-referencing device based on the concept of shearing interferometry in the space and frequency domains to perform the spatio-temporal characterization of ultrashort light pulses [22]. Weiner et al. [23] had demonstrated the spreading of femtosecond optical pulses into picosecond-duration pseudo-noise bursts. In this case, pulse spreading was accomplished by encoding pseudorandom binary phase codes onto the optical frequency spectrum. Subsequently, decoding of the spectral phases restores the original pulse. Shelton et al. have generated a coherently synthesized optical pulse from two independent mode-locked femtosecond lasers, providing a route to extend the coherent bandwidth available for ultrafast science [24]. Applications of coherent light pulse characterization techniques in femto-chemistry had been well reviewed in [25].
Another attractive approach to characterize coherent laser pulse is to use an adaptive feedback-controlled apparatus to tailor the spectral phase of a coherent pulse to achieve the maximum second harmonic generation output from a nonlinear optical crystal [26]. In this way, the compensating spectral phases carry the spectral phase information about the coherent pulse under study.
Control the quantum evolution of a complex system is an important advance in optical metrology. The technique has now been coined as coherent or quantum control.
Adaptive coherent pulse control [27-30] is the most successful scheme to be used for quantum control. Several algorithms have been developed to tailor a coherent optical field for a specific target on the basis of fitness information [31-36]. In this regard, a freezing phase concept had been proposed for adaptive coherent control with a femtosecond pulse shaper [26].
Our main goal of this study is to develop an artificial neuron network (ANN) model which can be used to retrieve the spectral phase of a coherent pulse directly from the spectrum of the second harmonic generation (SHG) with a nonlinear optical crystal. The SHG spectrum is affected by both the SHG process and the spectrum of the incident light pulse. In this chapter, we will develop an ANN to help us retrieving the spectral phase and therefore the complete-field information of a coherent pulse (phase and spectrum) with the measured spectrum of second harmonic generation.
Assuming the temporal profile of a coherent pulse is known, therefore we only need to adjust the spectral phase of the input pulse to generate the maximum SHG output from a nonlinear crystal. From the measured SHG spectrum, we retrieve the spectral phase of the input coherent pulse with an artificial neuron network. If the approach is successful, we can simply retrieve the complete field information of a
coherent pulse in real time directly from a measured SHG spectrum without time-consuming computation. The apparatus needs only NLO (Nonlinear Optics) crystal and a spectroscope.
3.2 Theory
Considering an incident coherent optical pulse E( )
A( )
ei ( ) with a spectrum of A w( ) and spectral phase distribution of j w( ). The second harmonicgeneration spectrum can be expressed as (2 ) ( )* ( ) 2
ISHG
E
E
d . (3-1) Assuming the spectrum of the coherent pulse to be Gaussian, and the spectral phase profile can be properly depicted with a polynomial of order 6, usually factor the phase ofa high order is much small than the low order we cut it off at the order six.
2 2
In general, the phase terms of order zero and one do not have any effect on SHG. The spectral phase profile can be further simplified by including terms from two to six only.
Note that from the point of view of theory, it shall be impossible to retrieve the spectral phase of a coherent pulse directly from the SHG spectrum of a coherent pulse.
Therefore, in the following we will conduct some simulations to test the feasibility of
3.3 Simulation 1
3.3.1 Preparation of the training data set
To prepare the training process of ANN, we sampled the spectrum and phase of a coherent Gaussian pulse to generate 64 data points. The second harmonic spectrum is presented with a data array of 127 data points because the second harmonic spectrum was calculated via a convolution operation.
The schematic of the training process is detailed in Fig 3.1. The input into a back-propagation artificial neuron network is the data of the second harmonic generation pulse comprising a spectral profile array and a spectral phase array.
Fig. 3.1. The schematic showing the training process of a back-propagation artificial neuron network. The input data to the ANN is prepared from the SHG Spectrum generated by a coherent pulse with a Gaussian amplitude profile and a desired phase profile.
BPANN Error
Desired Phase SHG Spectrum
3.3.2 Creation of a Backward Propagation Artificial Neuron Network
The input layer of the BP ANN was designed to accommodate the 127 inputs of the second harmonic generation spectrum. The output layer generates the retrieved spectral phase profile for the coherent pulse under study. A typical training data for the coherent pulse under study is shown in Fig. 3.2. The resulting SHG spectrum obtained from the training data is presented in Figure 3.3. We had investigated BP ANN with different numbers of hidden nodes and different learning parameters to find out the best learning performance of the artificial neuron network. The results will be discussed in the following section.
0 10 20 30 40 50 60 70 -5
0 5 10 15 20 25
Ph ase (rad )
Pixel
Amp Phase
Fig. 3.2. Typical training data prepared for the BP ANN. The data comprises the spectrum and the spectral phase of the coherent pulse under study.
0 20 40 60 80 100 120 140 0.00
0.01 0.02 0.03
I (a.u.)
pixel
SHG
3.3.3 Results and Discussion of Simulation 1
In this simulation, we used a two-layer BP ANN to find out how many hidden nodes are needed to yield a satisfactory learning performance. The active function of each node was chosen to be the sigmoid function and the rate constant of learning rate was set to be 0.1 for the first and the second weighting layers. In the learning phase, we trained the network by 1000 epochs. To evaluate the performance of the training, we used a correlation coefficient r, which is define as
x y
x y
r S
S S . (3-4)
Fig. 3.3. The resulting SHG spectrum obtained from the training data shown in Fig. 3.2.
Here
The correlation coefficient can reveal the underlying relation between two sets of data. It value lies between -1 and 1 with value one implies a prefect linear dependence. In our case, r=1 means the output of the ANN is same as the desired output.
0 10 20 30 40 50 60 70
Figure 3.4 presents a plot of the population number with r > 0.9 in 1000 data points as a function of the number of hidden nodes used in the BP ANN. We can find that by increasing the number of hidden nodes the number of data points with the correlation coefficient higher than 0.9 increases, implying that the predicted values with ANN can
Fig. 3.4.The curves showing the relation of the population number with r > 0.9 in 1000 data points with the number of hidden nodes in the BP ANN.
increasing tendency becomes stagnated. Therefore, the best choice is a BP ANN with each hidden layer containing about 20 hidden nodes.
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
From the distribution of r presented in Fig 3.5, we found that only about 35% of the predicted values of spectral phase has a correlation coefficient higher than 0.9.
Apparently, the learning performance of this SP ANN is not satisfactory. In view that the spectral phases used are expressed in terms of polynomials, we may be able to solve the problem with an increase of the information content in the training data by including more orthogonal phase profiles. Therefore, in the next section, we will try to express the spectral phase profile in terms of Legendre polynomials of order 2 to 6, which are shown
Fig. 3.5. The resulting distribution of r with 1000 test data points , and 1000 test data samples. The number of hidden nodes used is 32.
in Figure 3.6.
3.4.1 Preparation of the Training Data Set
Legendre polynomials form a complete set of orthogonal basis for a continuous function. By expanding the spectral phase profile of a coherent pulse into Legendre polynomials, we can significantly increase the information content with a minimum number of Legendre polynomials. Indeed as shown in Fig. 3.7, by including Legendre polynomials of order 2 to 6 in the spectral phase profilej w( ), more complicated SHG spectrum can be synthesized. The method significantly increases the degrees of freedom
Fig. 3.6. L2, L3, L4, L5, L6: the Legendre polynomials of order 2 to 6.
in the phase retrieval procedure. Six SHG spectra are prepared for the training of BP ANN by including more Legendre polynomials in the spectral phase are shown in Figure 3.7.
3.4.2 Creation of the Backward Propagation Artificial Neuron Network
For this study, we built another BP ANN which contains 762 input nodes and 64 output nodes. The 762 (6x127=762) input nodes are designated for the six SHG spectrums and 64 output nodes are for the retrieval spectral phase of the coherent pulse under study. The learning rate is set to 0.1 for both the first weighting layer and the second weighting layer. The data-flow schematic for the training is shown as Fig 3.8.
Fig 3.7. Six SHG spectra are prepared by including more Legendre polynomials in the spectral phase.
3.4.3 Results and Discussion of Simulation 2
The simulation results showing the relation of the population number with r > 0.9 in 1000 data points with respect to the number of hidden nodes in the BP ANN used are presented in Fig. 3.9. The results are quite encouraging in view that the number of the predicted phases with a correlation coefficient r> 0.9 can reach more than 90% of the test samples.
Fig 3.8. The data-flow schematic for the BP ANN training. Six SHG Spectra as shown in Fig. 3.7 were used.
Desire Phase Six SHG spectrums
Error BPANN
0 10 20 30 40 50 60 70 300
400 500 600 700 800 900
Num. of s am p le r > 0.9
Hidden Node Num.
Test Set Data Set
We presented in Fig. 3.10 some representative profiles with r>0.9 to give some hints of how well the BP ANN performs. From this Figure, we can see that the phase profile retrieved by our ANN agrees very well with the target profile.
Fig 3.9. The curves showing the plot of the population number with r >
0.9 in 1000 data points as a function of the number of hidden nodes in the BP ANN.
0 10 20 30 40 50 60 70
3.5 The Proposed Experimental Setup
For an experimental realization of the technique, one may concern how to conduct the training of BP ANN and then how use the trained ANN to perform the complete field characterization of a coherent pulse experimentally. We proposed an advanced apparatus with a pulse shaper to yield an adaptive feedback control loop as depicted in Fig. 3.11.
By using this apparatus, we can first measure the spectrum of the laser pulse under study.
We can produce many possible phase distorted versions of the coherent pulse by combining the measured spectrum with a variety of spectral phases expressed as a series
Fig 3.10. A representative phase profile retrieved from the BP ANN with r>0.9 is plotted to reveal the close agreement with the target phase profile.
ANN. After training, the trained ANN can be invoked to retrieve the spectral phase of the coherent pulse with the measured SHG spectrum. The apparatus offers a possibility to perform a complete-field characterization of a coherent excitation and quantum control of a physical system in a single setup.
3.6 Conclusions
We developed a BP ANN which can be invoked to retrieve the spectral phase of a coherent pulse from the measured SHG spectrum. We proposed a setup to be used for the experimental realization of the concept. By using this apparatus, we only need to measure the spectrum of the coherent pulse under study and then combine the spectrum with a variety of spectral phase profiles to prepare the SHG spectra for training the BP ANN. The trained BP ANN can be invoked to retrieve the spectral phase profile for the
SHG crystal
Spectrometer Phase
Modulator Plus
ANN System Phase information
Phase retrieve
Fig. 3.11. Proposed Experimental setup.
the predicted phases can achieve the target profile with more than 90% confidence.
Thanks to the computation efficiency of BP ANN, the technique developed in this study offers a possibility to perform a complete-field characterization of a coherent excitation to a physical system and quantum control of the system in a single setup.
Chapter 4 Real-Time Localization of Nano Objects at the Nanometer Scales
4.1 Introduction
An isolated fluorescent molecule or nano object will be observed like a light spot under an optical microscope. The spatial profile of the light spot simply reveals the point spread function of the optical microscope used. The peak position of the light spot can be determined with an accuracy of 1 nm if the signal-to-noise ratio of the detection is high enough. This impressive feature of nanometer localization with optical microscopy had recently inspired many applications including Fluorescence Imaging with One Nanometer Accuracy (FIONA) [37], sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM) [38], and fluorescence photoactivation localization microscopy (FPALM) [39], etc. Important biophysical mechanisms at the subcelluar scales had been discovered [40].
In the historical point of view, we noticed that a modified Hough transformation had been developed to detect a circular object [41] for recognizing and classifying interesting features present in a phase-contrast (PC) cytological image. Fillard, et al. had invoked the frequency dependence of the argument of Fourier transform to analyze an in-focus two-dimensional Airy disk [42]. Alexander, et al. proposed a method to
eliminate the systematic error in centroid estimation and achieved a subpixel accuracy [43]. It is interesting to know that a diffractive optical element (DOE) [44] had been applied to effectively locate a laser spot on a projection screen. The method could be invoked to achieve nanometer localization for optical microscopy. Anderson had presented an algebraic solution to the problem of localizing single fluorescent particle with sub-diffraction-limit accuracy [45]. Qu, et al. [46] had demonstrated nanometer localization of multiple single-molecules (NALMS) by using fluorescent microscopy and photobleaching properties of fluorophores. Cui, et al. [47] had devised an optimized algorithm useful for localizing light spots in high noise background. A method [48]
combining the radial basis network with anisotropic Gaussian basis function had been used to detect the position of a fluorescent protein. Fillard [49] relied on the Fourier phase frequency dependence to achieve sub-pixel localization accuracy of a light spot.
Enderlein [50] had proposed a method useful for tracking single fluorescent molecules diffusing in a two-dimensional membrane by invoking a rotating laser focus to track the position of the molecule. More information about single-molecular imaging and spectroscopy can be found in [51].
Based on the technical review, we found the major issue in the localization and tracking of nano objects is how to localize these objects accurately and rapidly with minimum invasiveness. To achieve the goal, many algorithms had been developed.
Fitting the light spot to a 2D Gaussian function is the most popular technique in this field. Another useful technique is to retrieve the peak position of a light spot via a center-of-mass approach [4]. However, to invoke these two techniques to localize a light spot with large size often fails to yield an accurate result. Therefore, in this chapter we will develop an ANN model to rapidly localize multiple light spots with high accuracy.
We invoked the feature of function approximation of ANN. We expect that the localization accuracy of ANN can be further improved when more data are accumulated.
Comparing to the 2D Gaussian fitting method, our ANN localization method is also less sensitive to noise influence.
4.2 Data Preparation for Training the Artificial Neuron Network
Localization Model
To train and test the performance of an ANN localization model, we prepared an image of 10 bright spots with 256x256 pixels as shown in Figure 4.1. The brightness profile of the light spots is Gaussian. The main target of this study is to construct a trained ANN model which can be invoked to yield the peak positions with a localization error less than one pixel.
4.3 First Test Run of the Artificial Neuron Network Localization Model
In this study, we constructed a BP ANN for localizing bright spots in an observing region. To serve this purpose, we began at an image of one spot with 25x25=625 pixels shown in Fig. 4.2. We input this image into the BP ANN. The output is the coordinates (x, y) of the spot. Therefore, the BP ANN possesses a total of 625 input nodes, 2 output nodes, and a hidden layer of 30 nodes. The activation function of the hidden layer is chosen to be hyperbolic tangent (tanh), while the activation function of the output layer is sigmoid function. We set the rate constant of learning to be 1.0 and 0.1 for the first and the second weighting layer, respectively.
Fig 4.1. An image of 10 bright spots prepared for training the artificial neuron network localization model.
The performance of the BP ANN trained by 10000 epochs with 1000 samples is shown in Figure 4.3. The distribution of the localization error deduced from the results with a training set of 1000 images or 1000 test images reveals that the localization error can be smaller than one pixel. However, in this example, we did not take into account the noise influence and only single bright spot is included.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0
50 100 150 200
Num. of sample
Localization Error (Pixel)
Data Test
Fig. 4.3. The distribution of the localization error deduced from a run with either
Fig. 4.2. One of the images with 25x25=625 pixels and containing single bright spot, used for training the artificial neuron network localization model.
4.3.1 Difficulty Encountered by the Artificial Neuron Network to
Localize Multiple Light Spots
In the previous section, we found that the BP ANN can be used to successfully localize a single bright spot with a localization error less than one pixel. However, it is difficult to use single BP ANN to searching an area with multiple light spots. In fact, in a real situation, we do not know how many spots are involved in the beginning. Therefore, we try to combine two BP ANNs with the first ANN to determine how many spots are involved in a large region, and divide the region into a series of smaller domains with
In the previous section, we found that the BP ANN can be used to successfully localize a single bright spot with a localization error less than one pixel. However, it is difficult to use single BP ANN to searching an area with multiple light spots. In fact, in a real situation, we do not know how many spots are involved in the beginning. Therefore, we try to combine two BP ANNs with the first ANN to determine how many spots are involved in a large region, and divide the region into a series of smaller domains with