"
114 | )
115 | );
116 | },
117 |
118 | /**
119 | * helper function to hide the search marks again
120 | */
121 | hideSearchWords: () => {
122 | document
123 | .querySelectorAll("#searchbox .highlight-link")
124 | .forEach((el) => el.remove());
125 | document
126 | .querySelectorAll("span.highlighted")
127 | .forEach((el) => el.classList.remove("highlighted"));
128 | localStorage.removeItem("sphinx_highlight_terms")
129 | },
130 |
131 | initEscapeListener: () => {
132 | // only install a listener if it is really needed
133 | if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) return;
134 |
135 | document.addEventListener("keydown", (event) => {
136 | // bail for input elements
137 | if (BLACKLISTED_KEY_CONTROL_ELEMENTS.has(document.activeElement.tagName)) return;
138 | // bail with special keys
139 | if (event.shiftKey || event.altKey || event.ctrlKey || event.metaKey) return;
140 | if (DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS && (event.key === "Escape")) {
141 | SphinxHighlight.hideSearchWords();
142 | event.preventDefault();
143 | }
144 | });
145 | },
146 | };
147 |
148 | _ready(() => {
149 | /* Do not call highlightSearchWords() when we are on the search page.
150 | * It will highlight words from the *previous* search query.
151 | */
152 | if (typeof Search === "undefined") SphinxHighlight.highlightSearchWords();
153 | SphinxHighlight.initEscapeListener();
154 | });
155 |
--------------------------------------------------------------------------------
/docs/build/html/_static/language_data.js:
--------------------------------------------------------------------------------
1 | /*
2 | * language_data.js
3 | * ~~~~~~~~~~~~~~~~
4 | *
5 | * This script contains the language-specific data used by searchtools.js,
6 | * namely the list of stopwords, stemmer, scorer and splitter.
7 | *
8 | * :copyright: Copyright 2007-2024 by the Sphinx team, see AUTHORS.
9 | * :license: BSD, see LICENSE for details.
10 | *
11 | */
12 |
13 | var stopwords = ["a", "and", "are", "as", "at", "be", "but", "by", "for", "if", "in", "into", "is", "it", "near", "no", "not", "of", "on", "or", "such", "that", "the", "their", "then", "there", "these", "they", "this", "to", "was", "will", "with"];
14 |
15 |
16 | /* Non-minified version is copied as a separate JS file, if available */
17 |
18 | /**
19 | * Porter Stemmer
20 | */
21 | var Stemmer = function() {
22 |
23 | var step2list = {
24 | ational: 'ate',
25 | tional: 'tion',
26 | enci: 'ence',
27 | anci: 'ance',
28 | izer: 'ize',
29 | bli: 'ble',
30 | alli: 'al',
31 | entli: 'ent',
32 | eli: 'e',
33 | ousli: 'ous',
34 | ization: 'ize',
35 | ation: 'ate',
36 | ator: 'ate',
37 | alism: 'al',
38 | iveness: 'ive',
39 | fulness: 'ful',
40 | ousness: 'ous',
41 | aliti: 'al',
42 | iviti: 'ive',
43 | biliti: 'ble',
44 | logi: 'log'
45 | };
46 |
47 | var step3list = {
48 | icate: 'ic',
49 | ative: '',
50 | alize: 'al',
51 | iciti: 'ic',
52 | ical: 'ic',
53 | ful: '',
54 | ness: ''
55 | };
56 |
57 | var c = "[^aeiou]"; // consonant
58 | var v = "[aeiouy]"; // vowel
59 | var C = c + "[^aeiouy]*"; // consonant sequence
60 | var V = v + "[aeiou]*"; // vowel sequence
61 |
62 | var mgr0 = "^(" + C + ")?" + V + C; // [C]VC... is m>0
63 | var meq1 = "^(" + C + ")?" + V + C + "(" + V + ")?$"; // [C]VC[V] is m=1
64 | var mgr1 = "^(" + C + ")?" + V + C + V + C; // [C]VCVC... is m>1
65 | var s_v = "^(" + C + ")?" + v; // vowel in stem
66 |
67 | this.stemWord = function (w) {
68 | var stem;
69 | var suffix;
70 | var firstch;
71 | var origword = w;
72 |
73 | if (w.length < 3)
74 | return w;
75 |
76 | var re;
77 | var re2;
78 | var re3;
79 | var re4;
80 |
81 | firstch = w.substr(0,1);
82 | if (firstch == "y")
83 | w = firstch.toUpperCase() + w.substr(1);
84 |
85 | // Step 1a
86 | re = /^(.+?)(ss|i)es$/;
87 | re2 = /^(.+?)([^s])s$/;
88 |
89 | if (re.test(w))
90 | w = w.replace(re,"$1$2");
91 | else if (re2.test(w))
92 | w = w.replace(re2,"$1$2");
93 |
94 | // Step 1b
95 | re = /^(.+?)eed$/;
96 | re2 = /^(.+?)(ed|ing)$/;
97 | if (re.test(w)) {
98 | var fp = re.exec(w);
99 | re = new RegExp(mgr0);
100 | if (re.test(fp[1])) {
101 | re = /.$/;
102 | w = w.replace(re,"");
103 | }
104 | }
105 | else if (re2.test(w)) {
106 | var fp = re2.exec(w);
107 | stem = fp[1];
108 | re2 = new RegExp(s_v);
109 | if (re2.test(stem)) {
110 | w = stem;
111 | re2 = /(at|bl|iz)$/;
112 | re3 = new RegExp("([^aeiouylsz])\\1$");
113 | re4 = new RegExp("^" + C + v + "[^aeiouwxy]$");
114 | if (re2.test(w))
115 | w = w + "e";
116 | else if (re3.test(w)) {
117 | re = /.$/;
118 | w = w.replace(re,"");
119 | }
120 | else if (re4.test(w))
121 | w = w + "e";
122 | }
123 | }
124 |
125 | // Step 1c
126 | re = /^(.+?)y$/;
127 | if (re.test(w)) {
128 | var fp = re.exec(w);
129 | stem = fp[1];
130 | re = new RegExp(s_v);
131 | if (re.test(stem))
132 | w = stem + "i";
133 | }
134 |
135 | // Step 2
136 | re = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/;
137 | if (re.test(w)) {
138 | var fp = re.exec(w);
139 | stem = fp[1];
140 | suffix = fp[2];
141 | re = new RegExp(mgr0);
142 | if (re.test(stem))
143 | w = stem + step2list[suffix];
144 | }
145 |
146 | // Step 3
147 | re = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/;
148 | if (re.test(w)) {
149 | var fp = re.exec(w);
150 | stem = fp[1];
151 | suffix = fp[2];
152 | re = new RegExp(mgr0);
153 | if (re.test(stem))
154 | w = stem + step3list[suffix];
155 | }
156 |
157 | // Step 4
158 | re = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/;
159 | re2 = /^(.+?)(s|t)(ion)$/;
160 | if (re.test(w)) {
161 | var fp = re.exec(w);
162 | stem = fp[1];
163 | re = new RegExp(mgr1);
164 | if (re.test(stem))
165 | w = stem;
166 | }
167 | else if (re2.test(w)) {
168 | var fp = re2.exec(w);
169 | stem = fp[1] + fp[2];
170 | re2 = new RegExp(mgr1);
171 | if (re2.test(stem))
172 | w = stem;
173 | }
174 |
175 | // Step 5
176 | re = /^(.+?)e$/;
177 | if (re.test(w)) {
178 | var fp = re.exec(w);
179 | stem = fp[1];
180 | re = new RegExp(mgr1);
181 | re2 = new RegExp(meq1);
182 | re3 = new RegExp("^" + C + v + "[^aeiouwxy]$");
183 | if (re.test(stem) || (re2.test(stem) && !(re3.test(stem))))
184 | w = stem;
185 | }
186 | re = /ll$/;
187 | re2 = new RegExp(mgr1);
188 | if (re.test(w) && re2.test(w)) {
189 | re = /.$/;
190 | w = w.replace(re,"");
191 | }
192 |
193 | // and turn initial Y back to y
194 | if (firstch == "y")
195 | w = firstch.toLowerCase() + w.substr(1);
196 | return w;
197 | }
198 | }
199 |
200 |
--------------------------------------------------------------------------------
/pynlo/utility/fft.py:
--------------------------------------------------------------------------------
1 | # -*- coding: utf-8 -*-
2 | """
3 | Aliases to fast FFT implementations and associated helper functions.
4 |
5 | """
6 |
7 | __all__ = ["fft", "ifft", "rfft", "irfft",
8 | "fftshift", "ifftshift"]
9 |
10 |
11 | # %% Imports
12 |
13 | from scipy.fft import next_fast_len, fftshift as _fftshift, ifftshift as _ifftshift
14 | import mkl_fft
15 |
16 |
17 | # %% Helper Functions
18 |
19 | #---- FFT Shifts
20 | def fftshift(x, axis=-1):
21 | """
22 | Shift the origin from the beginning to the center of the array.
23 |
24 | This function is used after an `fft` operation to shift from fft to monotic
25 | ordering.
26 |
27 | Parameters
28 | ----------
29 | x : array_like
30 | Input array
31 | axis : int, optional
32 | The axis over which to shift. The default is the last axis.
33 |
34 | Returns
35 | -------
36 | ndarray
37 | The shifted array.
38 |
39 | """
40 | return _fftshift(x, axes=axis)
41 |
42 | def ifftshift(x, axis=-1):
43 | """
44 | Shift the origin from the center to the beginning of the array.
45 |
46 | The inverse of fftshift. This function is used before an `fft` operation to
47 | shift from monotonic to fft ordering. Although identical for even-length
48 | `x`, `ifftshift` differs from `fftshift` by one sample for odd-length `x`.
49 |
50 | Parameters
51 | ----------
52 | x : array_like
53 | Input array.
54 | axis : int, optional
55 | The axis over which to shift. The default is the last axis.
56 |
57 | Returns
58 | -------
59 | ndarray
60 | The shifted array.
61 |
62 | """
63 | return _ifftshift(x, axes=axis)
64 |
65 |
66 | # %% Transforms
67 |
68 | #---- FFTs
69 | def fft(x, fsc=1.0, n=None, axis=-1, overwrite_x=False):
70 | """
71 | Use MKL to perform a 1D FFT of the input array along the given axis.
72 |
73 | Parameters
74 | ----------
75 | x : array_like
76 | Input array, can be complex.
77 | fsc : float, optional
78 | The forward transform scale factor. The default is 1.0.
79 | n : int, optional
80 | Length of the transformed axis of the output. If `n` is smaller than
81 | the length of the input, the input is cropped. If it is larger, the
82 | input is padded with zeros.
83 | axis : int, optional
84 | Axis over which to compute the FFT. The default is the last axis.
85 | overwrite_x : bool, optional
86 | If True, the contents of x may be overwritten during the computation.
87 | The default is False.
88 |
89 | Returns
90 | -------
91 | complex ndarray
92 | The transformed array.
93 |
94 | """
95 | return mkl_fft.fft(x, n=n, axis=axis, overwrite_x=overwrite_x, forward_scale=fsc)
96 |
97 | def ifft(x, fsc=1.0, n=None, axis=-1, overwrite_x=False):
98 | """
99 | Use MKL to perform a 1D IFFT of the input array along the given axis.
100 |
101 | Parameters
102 | ----------
103 | x : array_like
104 | Input array, can be complex.
105 | fsc : float, optional
106 | The forward transform scale factor. Internally, this function sets the
107 | reverse transform scale factor as ``1/(n*fsc)``. The default is 1.0.
108 | n : int, optional
109 | Length of the transformed axis of the output. If `n` is smaller than
110 | the length of the input, the input is cropped. If it is larger, the
111 | input is padded with zeros.
112 | axis : int, optional
113 | Axis over which to compute the inverse FFT. The default is the last
114 | axis.
115 | overwrite_x : bool, optional
116 | If True, the contents of x may be overwritten during the computation.
117 | The default is False.
118 |
119 | Returns
120 | -------
121 | complex ndarray
122 | The transformed array.
123 |
124 | """
125 | return mkl_fft.ifft(x, n=n, axis=axis, overwrite_x=overwrite_x, forward_scale=fsc)
126 |
127 | #---- Real FFTs
128 | def rfft(x, fsc=1.0, n=None, axis=-1):
129 | """
130 | Use MKL to perform a 1D FFT of the real input array along the given axis.
131 | The output array is complex and only contains positive frequencies.
132 |
133 | The length of the transformed axis is ``n//2 + 1``.
134 |
135 | Parameters
136 | ----------
137 | x : array_like
138 | Input array, must be real.
139 | fsc : float, optional
140 | The forward transform scale factor. The default is 1.0.
141 | n : int, optional
142 | Number of points to use along the transformed axis of the input. If
143 | `n` is smaller than the length of the input, the input is cropped. If
144 | it is larger, the input is padded with zeros.
145 | axis : int, optional
146 | Axis over which to compute the FFT. The default is the last axis.
147 |
148 | Returns
149 | -------
150 | complex ndarray
151 | The transformed array.
152 |
153 | """
154 | return mkl_fft.rfft_numpy(x, n=n, axis=axis, forward_scale=fsc)
155 |
156 | def irfft(x, fsc=1.0, n=None, axis=-1):
157 | """
158 | Use MKL to perform a 1D IFFT of the input array along the given axis. The
159 | input is assumed to contain only positive frequencies, and the output is
160 | always real.
161 |
162 | If `n` is not given the length of the transformed axis is ``2*(m-1)``,
163 | where `m` is the length of the transformed axis of the input. To get an odd
164 | number of output points, `n` must be specified.
165 |
166 | Parameters
167 | ----------
168 | x : array_like
169 | Input array, can be complex.
170 | fsc : float, optional
171 | The forward transform scale factor. Internally, this function sets the
172 | reverse transform scale factor as ``1/(n*fsc)``. The default is 1.0.
173 | n : int, optional
174 | Length of the transformed axis of the output. For `n` output points,
175 | ``n//2+1`` input points are necessary. If the input is longer than
176 | this, it is cropped. If it is shorter than this, it is padded with
177 | zeros.
178 | axis : int, optional
179 | Axis over which to compute the inverse FFT. The default is the last
180 | axis.
181 |
182 | Returns
183 | -------
184 | ndarray
185 | The transformed array.
186 |
187 | """
188 | return mkl_fft.irfft_numpy(x, n=n, axis=axis, forward_scale=fsc)
189 |
--------------------------------------------------------------------------------
/examples/silica-pcf_supercontinuum.py:
--------------------------------------------------------------------------------
1 | # -*- coding: utf-8 -*-
2 | """
3 | This example demonstrates supercontinuum generation due to soliton effects in
4 | silica-based photonic crystal fiber. The simulation uses the nonlinear
5 | Schrödinger equation (NLSE) and is based on the parameters given in part V-A of
6 | Dudley et al. (2006).
7 |
8 | References
9 | ----------
10 | Dudley JM, Genty G, Coen S. Supercontinuum generation in photonic crystal
11 | fiber. Reviews of modern physics. 2006;78(4):1135-84.
12 | https://doi.org/10.1103/RevModPhys.78.1135
13 |
14 | """
15 |
16 |
17 | # %% Imports
18 | import numpy as np
19 | from scipy.constants import pi, c
20 | from matplotlib import pyplot as plt
21 |
22 | import pynlo
23 | from pynlo import utility as ut
24 |
25 |
26 | # %% Pulse Properties
27 | """
28 | We start by initializing a hyperbolic secant pulse using one of the built-in
29 | pulse shapes of the `Pulse` class. The first few parameters constrain the
30 | frequency grid, the number of points and the frequency range, while the last
31 | three set the initial pulse properties, its center frequency, pulse energy, and
32 | pulse width.
33 |
34 | """
35 | n_points = 2**12
36 | v_min = c/1500e-9 # 200 THz
37 | v_max = c/500e-9 # 600 THz
38 |
39 | v0 = c/835e-9 # 360 THz
40 | e_p = 285e-12 # 285 pJ
41 | t_fwhm = 50e-15 # 50 fs
42 |
43 | pulse = pynlo.light.Pulse.Sech(n_points, v_min, v_max, v0, e_p, t_fwhm)
44 |
45 |
46 | # %% Mode Properties
47 | """
48 | We next need to define both the linear and nonlinear properties of the
49 | waveguide. In this example, we are only given the waveguide properties at a
50 | single frequency so we must extrapolate to the rest of the frequency grid. For
51 | the beta parameter this is accomplished using a Taylor series, but for the
52 | nonlinearity we use the `gamma_to_g3` conversion function from the
53 | `utility.chi3` submodule. This function calculates the generalized 3rd-order
54 | nonlinear parameter (required by the PyNLO propagation models) from the gamma
55 | parameter and the optical shock time scale. If available, the nonlinear
56 | parameter can also be generated from the refractive index, effective area, and
57 | nonlinear susceptibility, see `utility.chi3` for more details. The Raman effect
58 | is implemented using the Raman response function given in section 2.3.3 of
59 | Agrawal.
60 |
61 | References
62 | ----------
63 | Agrawal GP. Nonlinear Fiber Optics. Sixth ed. London; San Diego,
64 | CA;: Academic Press; 2019.
65 | https://doi.org/10.1016/B978-0-12-817042-7.00009-9
66 |
67 | """
68 | length = 15e-2 # 15 cm
69 |
70 | #---- Phase Coefficient
71 | beta_n = 11*[0]
72 | beta_n[2] = -11.830 * 1e-12**2/1e3 # -11.830 ps**2 / km
73 | beta_n[3] = 8.1038e-2 * 1e-12**3/1e3 # 8.1038e-2 ps**3 / km
74 | beta_n[4] = -9.5205e-5 * 1e-12**4/1e3 # -9.5205e-5 ps**4 / km
75 | beta_n[5] = 2.0737e-7 * 1e-12**5/1e3 # 2.0737e-7 ps**5 / km
76 | beta_n[6] = -5.3943e-10 * 1e-12**6/1e3 # -5.3943e-10 ps**6 / km
77 | beta_n[7] = 1.3486e-12 * 1e-12**7/1e3 # 1.3486e-12 ps**7 / km
78 | beta_n[8] = -2.5495e-15 * 1e-12**8/1e3 # -2.5495e-15 ps**8 / km
79 | beta_n[9] = 3.0524e-18 * 1e-12**9/1e3 # 3.0524e-18 ps**9 / km
80 | beta_n[10] = -1.7140e-21 * 1e-12**10/1e3 # -1.7140e-21 ps**10 / km
81 |
82 | beta = ut.taylor_series(2*pi*pulse.v0, beta_n)(2*pi*pulse.v_grid)
83 |
84 | #---- 3rd-Order Nonlinearity
85 | gamma = 0.11 # 0.11 / W * m
86 | t_shock = 0.56e-15 # 0.56 fs
87 | g3 = ut.chi3.gamma_to_g3(pulse.v_grid, gamma, t_shock)
88 |
89 | # Raman effect
90 | r_weights = [0.245*(1-0.21), 12.2e-15, 32e-15] # resonant contribution
91 | b_weights = [0.245*0.21, 96e-15] # boson contribution
92 | rv_grid, raman = ut.chi3.raman(pulse.n, pulse.dt, r_weights, b_weights)
93 |
94 | mode = pynlo.medium.Mode(pulse.v_grid, beta, g3=g3, rv_grid=rv_grid, r3=raman)
95 |
96 |
97 | # %% Model
98 | """
99 | The NLSE model is initialized with the pulse and mode objects defined above. At
100 | this stage we also estimate the optimal initial step size given a target local
101 | error.
102 |
103 | """
104 | model = pynlo.model.NLSE(pulse, mode)
105 |
106 | #---- Estimate step size
107 | local_error = 1e-6
108 | dz = model.estimate_step_size(local_error=local_error)
109 |
110 |
111 | # %% Simulate
112 | """
113 | The model's `simulate` method runs the simulation. We input the total
114 | propagation length, the initial step size, local error, and the number of
115 | simulation steps we wish to record. We receive the output pulse and the
116 | propagations distance, pulse spectrum, and complex envelope at each record
117 | point. To view real-time simulation results (updated whenever the simulation
118 | reaches a record point), set the `plot` keyword to "time", "frq", or "wvl".
119 |
120 | """
121 | pulse_out, z, a_t, a_v = model.simulate(
122 | length, dz=dz, local_error=local_error, n_records=100, plot=None)
123 |
124 |
125 | # %% Plot Results
126 | """
127 | For comparison with Dudley, we plot the evolution in the time and wavelength
128 | domains. For accurate representation of the density, plotting over wavelength
129 | requires converting the power from a per Hz basis to a per m basis. This is
130 | accomplished by multiplying the frequency domain power spectral density by the
131 | ratio of the frequency and wavelength differentials. The power spectral density
132 | is then converted to decibel (dB) scale to increase the visible dynamic range.
133 |
134 | """
135 | fig = plt.figure("Simulation Results", clear=True)
136 | ax0 = plt.subplot2grid((3,2), (0, 0), rowspan=1)
137 | ax1 = plt.subplot2grid((3,2), (0, 1), rowspan=1)
138 | ax2 = plt.subplot2grid((3,2), (1, 0), rowspan=2, sharex=ax0)
139 | ax3 = plt.subplot2grid((3,2), (1, 1), rowspan=2, sharex=ax1)
140 |
141 | p_l_dB = 10*np.log10(np.abs(a_v)**2 * model.dv_dl)
142 | p_l_dB -= p_l_dB.max()
143 | ax0.plot(1e9*c/pulse.v_grid, p_l_dB[0], color="b")
144 | ax0.plot(1e9*c/pulse.v_grid, p_l_dB[-1], color="g")
145 | ax2.pcolormesh(1e9*c/pulse.v_grid, 1e3*z, p_l_dB,
146 | vmin=-40.0, vmax=0, shading="auto")
147 | ax0.set_ylim(bottom=-50, top=10)
148 | ax2.set_xlabel('Wavelength (nm)')
149 |
150 | p_t_dB = 10*np.log10(np.abs(a_t)**2)
151 | p_t_dB -= p_t_dB.max()
152 | ax1.plot(1e12*pulse.t_grid, p_t_dB[0], color="b")
153 | ax1.plot(1e12*pulse.t_grid, p_t_dB[-1], color="g")
154 | ax3.pcolormesh(1e12*pulse.t_grid, 1e3*z, p_t_dB,
155 | vmin=-40.0, vmax=0, shading="auto")
156 | ax1.set_ylim(bottom=-50, top=10)
157 | ax3.set_xlabel('Time (ps)')
158 |
159 | ax0.set_ylabel('Power (dB)')
160 | ax2.set_ylabel('Propagation Distance (mm)')
161 | fig.tight_layout()
162 | fig.show()
163 |
--------------------------------------------------------------------------------
/examples/ppln_cascaded-chi2.py:
--------------------------------------------------------------------------------
1 | # -*- coding: utf-8 -*-
2 | """
3 | This example demonstrates supercontinuum generation due to phase-mismatched
4 | second harmonic generation (SHG) in periodically poled lithium niobate (PPLN).
5 | The simulation uses the unidirectional propagation equation (UPE) and is based
6 | on the parameters given for the second example in part III of Conforti et al.
7 | (2010).
8 |
9 | References
10 | ----------
11 | Conforti M, Baronio F, De Angelis C. Nonlinear envelope equation for broadband
12 | optical pulses in quadratic media. Physical review. A, Atomic, molecular, and
13 | optical physics. 2010;81(5).
14 | https://doi.org/10.1103/PhysRevA.81.053841
15 |
16 | """
17 |
18 |
19 | # %% Imports
20 | import numpy as np
21 | from scipy.constants import pi, c
22 | from matplotlib import pyplot as plt
23 |
24 | import pynlo
25 | from pynlo import utility as ut
26 |
27 |
28 | # %% Pulse Properties
29 | """
30 | We first initialize a Gaussian pulse using one of the built-in pulse shapes of
31 | the `Pulse` class. The first few parameters constrain the frequency grid, the
32 | number of points and the frequency range, while the last three set the initial
33 | pulse properties, its center frequency, pulse energy, and pulse width. Since we
34 | are going to be simulating 2nd-order effects, the `alias` parameter has been
35 | increased to support two alias-free Nyquist zones.
36 |
37 | """
38 | n_points = 2**13
39 | v_min = c/3500e-9 # c / 3500 nm
40 | v_max = c/450e-9 # c / 450 nm
41 |
42 | v0 = c/1580e-9 # c / 1580 nm
43 | e_p = 1e-9 # 1 nJ
44 | t_fwhm = 50e-15 # 50 fs
45 |
46 | pulse = pynlo.light.Pulse.Gaussian(n_points, v_min, v_max, v0, e_p, t_fwhm,
47 | alias=2) # anti-aliasing
48 |
49 |
50 | # %% Mode Properties
51 | """
52 | In this example we use the refractive index, effective area, and nonlinear
53 | susceptibility to calculate the linear and nonlinear properties of the mode.
54 | The material used in this case is congruent lithium niobate, and we use the
55 | Sellmeier equations from Jundt (1997) for the refractive index. The generalized
56 | 2nd-order nonlinear parameter is weighted for second-harmonic generation using
57 | the `g2_shg` function from the `utility.chi2` submodule. Poling is implemented
58 | by indicating the location of each domain inversion. This is calculated using
59 | the `domain_inversion` function of the `utility.chi2` submodule. This example
60 | only uses a constant poling period, but the function also supports the
61 | generation of arbitrarily chirped poling periods.
62 |
63 | """
64 | length = 7e-3 # 7 mm
65 | a_eff = 15e-6 * 15e-6 # 15 um * 15 um
66 |
67 | #---- Phase Coefficient
68 | def n_cLN(v, T=24.5):
69 | """
70 | Refractive index of congruent lithium niobate.
71 |
72 | References
73 | ----------
74 | Dieter H. Jundt, "Temperature-dependent Sellmeier equation for the index of
75 | refraction, ne, in congruent lithium niobate," Opt. Lett. 22, 1553-1555
76 | (1997). https://doi.org/10.1364/OL.22.001553
77 |
78 | """
79 | a1 = 5.35583
80 | a2 = 0.100473
81 | a3 = 0.20692
82 | a4 = 100.
83 | a5 = 11.34927
84 | a6 = 1.5334e-2
85 | b1 = 4.629e-7
86 | b2 = 3.862e-8
87 | b3 = -0.89e-8
88 | b4 = 2.657e-5
89 |
90 | wvl = c/v * 1e6 # um
91 | f = (T-24.5)*(T+570.82)
92 | n2 = (a1 + b1*f + (a2 + b2*f)/(wvl**2 - (a3 + b3*f)**2)
93 | + (a4 + b4*f)/(wvl**2 - a5**2) - a6*wvl**2)
94 | return n2**0.5
95 | n_eff = n_cLN(pulse.v_grid)
96 |
97 | beta = n_eff * 2*pi*pulse.v_grid/c
98 |
99 | #---- 2nd-order nonlinearity
100 | d_eff = 27e-12 # 27 pm / V
101 | chi2_eff = 2 * d_eff
102 | g2 = ut.chi2.g2_shg(v0, pulse.v_grid, n_eff, a_eff, chi2_eff)
103 |
104 | # Poling
105 | p0 = 30e-6 # 30 um poling period
106 | z_invs, domains, poled = ut.chi2.domain_inversions(length, 2*pi/p0)
107 |
108 | mode = pynlo.medium.Mode(pulse.v_grid, beta, g2=g2, g2_inv=z_invs)
109 |
110 |
111 | # %% Model
112 | """
113 | The UPE model is initialized with the pulse and mode objects defined above. At
114 | this stage we also use the target local error to estimate the optimal initial
115 | step size.
116 |
117 | """
118 | model = pynlo.model.UPE(pulse, mode)
119 |
120 | #---- Estimate step size
121 | local_error = 1e-6
122 | dz = model.estimate_step_size(local_error=local_error)
123 |
124 |
125 | # %% Simulate
126 | """
127 | The model's `simulate` method runs the simulation. We input the total
128 | propagation length, the initial step size, local error, and the number of
129 | simulation steps we wish to record. We receive the output pulse and the
130 | propagations distance, pulse spectrum, and complex envelope at each record
131 | point. To view real-time simulation results (updated whenever the simulation
132 | reaches a record point), set the `plot` keyword to "time", "frq", or "wvl".
133 |
134 | """
135 | pulse_out, z, a_t, a_v = model.simulate(
136 | length, dz=dz, local_error=local_error, n_records=100, plot=None)
137 |
138 |
139 | # %% Plot Results
140 | """
141 | The results are plotted in the time and frequency domains. Although the
142 | 3rd-order nonlinearity is not directly included in the propagation model we see
143 | several 3rd-order phenomena due to cascaded 2nd-order effects. Phase-mismatched
144 | SHG yields an effective Kerr effect (see DeSalvo et al. (1992)), which leads to
145 | soliton and dispersive wave generation. Additionally, the high frequency spike
146 | is third-harmonic generation due to cascaded second-harmonic and sum-frequency
147 | generation from the pump frequency. The other spike is due to SHG quasi-phase
148 | matched off of the 3rd-order spatial harmonic of the poling structure.
149 |
150 | References
151 | ----------
152 | R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and
153 | H. Vanherzeele, "Self-focusing and self-defocusing by cascaded second-order
154 | effects in KTP," Opt. Lett. 17, 28-30 (1992)
155 |
156 | """
157 | fig = plt.figure("Simulation Results", clear=True)
158 | ax0 = plt.subplot2grid((3,2), (0, 0), rowspan=1)
159 | ax1 = plt.subplot2grid((3,2), (0, 1), rowspan=1)
160 | ax2 = plt.subplot2grid((3,2), (1, 0), rowspan=2, sharex=ax0)
161 | ax3 = plt.subplot2grid((3,2), (1, 1), rowspan=2, sharex=ax1)
162 |
163 | p_v_dB = 10*np.log10(np.abs(a_v)**2)
164 | p_v_dB -= p_v_dB.max()
165 | ax0.plot(1e-12*pulse.v_grid, p_v_dB[0], color="b")
166 | ax0.plot(1e-12*pulse.v_grid, p_v_dB[-1], color="g")
167 | ax2.pcolormesh(1e-12*pulse.v_grid, 1e3*z, p_v_dB,
168 | vmin=-40.0, vmax=0, shading="auto")
169 | ax0.set_ylim(bottom=-50, top=10)
170 | ax2.set_xlabel('Frequency (THz)')
171 |
172 | p_t_dB = 10*np.log10(np.abs(a_t)**2)
173 | p_t_dB -= p_t_dB.max()
174 | ax1.plot(1e12*pulse.t_grid, p_t_dB[0], color="b")
175 | ax1.plot(1e12*pulse.t_grid, p_t_dB[-1], color="g")
176 | ax3.pcolormesh(1e12*pulse.t_grid, 1e3*z, p_t_dB,
177 | vmin=-40.0, vmax=0, shading="auto")
178 | ax1.set_ylim(bottom=-50, top=10)
179 | ax3.set_xlabel('Time (ps)')
180 |
181 | ax0.set_ylabel('Power (dB)')
182 | ax2.set_ylabel('Propagation Distance (mm)')
183 | fig.tight_layout()
184 | fig.show()
185 |
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