gaussian_beams.py 15.1 KB
Newer Older
1
2
3
4
5
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
from __future__ import unicode_literals

6
import pykat.exceptions as pkex
7
import numpy as np
8
import math
9
import copy
10
11
import warnings
import cmath
12
from math import factorial
13
from scipy.special import hermite
14
from pykat.math.jacobi import jacobi
15
from pykat.SIfloat import SIfloat
16
17
18

class gauss_param(object):
    """
19
    Use beam_param instead, will be the future name of this object.
20
    
21
22
    Gaussian beam complex parameter
    
Daniel Brown's avatar
Daniel Brown committed
23
    beam_param is effectively a complex number with extra
24
25
    functionality to determine beam parameters.
    
Daniel Brown's avatar
Daniel Brown committed
26
    Defaults to 1064e-9m for wavelength and refractive index 1
27
28
29
    usage:
        q = gauss_param(w0=w0, z=z)
        q = gauss_param(z=z, zr=zr)
30
        q = gauss_param(w=w, rc=rc)
31
        q = gauss_param(q=a) # where a is a complex number
32
33
34
35
36
37
38
        
        or change default wavelength and refractive index with:
        
        q = gauss_param(wavelength, nr, w0=w0, zr=zr)
    """
    
    def __init__(self, wavelength=1064e-9, nr=1, *args, **kwargs):
39
40
41
        if self.__class__ != beam_param:
            warnings.warn("Name changed. Use beam_param instead of gauss_param.")
            
42
        self.__q = None
43
44
        self.__lambda = SIfloat(wavelength)
        self.__nr = SIfloat(nr)
45
46
        
        if len(args) == 1:
Daniel Brown's avatar
Daniel Brown committed
47
            self.__q = complex(args[0])
48
49
50
51
52
        
        elif len(kwargs) == 1:
            if "q" in kwargs:
                self.__q = complex(kwargs["q"])        
            else:
53
                raise pkex.BasePyKatException("Must specify: z and w0 or z and zr or rc and w or q, to define the beam parameter")
54
                
55
56
57
        elif len(kwargs) == 2:        
            
            if "w0" in kwargs and "z" in kwargs:
58
                q = SIfloat(kwargs["z"]) + 1j * math.pi*SIfloat(kwargs["w0"])**2/(self.__lambda/self.__nr)
59
            elif "z" in kwargs and "zr" in kwargs:
60
                q = SIfloat(kwargs["z"]) + 1j * SIfloat(kwargs["zr"]) 
61
            elif "rc" in kwargs and "w" in kwargs:
62
                one_q = 1 / SIfloat(kwargs["rc"]) - 1j * SIfloat(wavelength) / (math.pi * SIfloat(nr) * SIfloat(kwargs["w"])**2)
63
64
                q = 1/one_q
            else:
65
                raise pkex.BasePyKatException("Must specify: z and w0 or z and zr or rc and w or q, to define the beam parameter")
66
67
68
69
70
71
72
                
            self.__q = q
        else:
            raise pkex.BasePyKatException("Incorrect usage for gauss_param constructor")
    
    @property
    def wavelength(self): return self.__lambda
73
74
    @wavelength.setter
    def wavelength(self,value): self.__lambda = SIfloat(value)
75
76
77
78
79
80
81
82
83
84
85
86
87
88
    
    @property
    def nr(self): return self.__nr
    
    @property
    def q(self): return self.__q
    
    @property
    def z(self): return self.__q.real
    
    @property
    def zr(self): return self.__q.imag
    
    @property
89
    def w(self):
90
        return np.abs(self.__q)* np.sqrt(self.__lambda / (self.__nr * math.pi * self.__q.imag))
91
    
92
    def beamsize(self, z=None, wavelength=None, nr=None, w0=None):
93

94
        if z is None:
95
96
97
98
            z = self.z
        else:
            z = np.array(z)
                
99
        if wavelength is None:
100
101
102
103
            wavelength = self.wavelength
        else:
            wavelength = np.array(wavelength)
            
104
        if nr is None:
105
106
107
108
            nr = self.nr
        else:
            nr = np.array(nr)
            
109
        if w0 is None:
110
111
112
113
114
115
116
117
118
            w0 = self.w0
        else:
            w0 = np.array(w0)
        
        q = z + 1j * math.pi * w0 **2 / wavelength
        
        return np.abs(q)*np.sqrt(wavelength / (nr * math.pi * q.imag))
    
    def gouy(self, z=None, wavelength=None, nr=None, w0=None):
119
        if z is None:
120
121
122
123
            z = self.z
        else:
            z = np.array(z)
                
124
        if wavelength is None:
125
126
127
128
            wavelength = self.wavelength
        else:
            wavelength = np.array(wavelength)
            
129
        if nr is None:
130
131
132
133
            nr = self.nr
        else:
            nr = np.array(nr)
            
134
        if w0 is None:
135
136
137
138
139
140
141
142
            w0 = self.w0
        else:
            w0 = np.array(w0)
        
        q = z + 1j * math.pi * w0 **2 / wavelength
        
        return np.arctan2(q.real, q.imag)
        
143
144
    @property
    def w0(self):
145
        return np.sqrt(self.__q.imag * self.__lambda / (self.__nr * math.pi))    
146
147
148

    @property
    def Rc(self):
149
150
151
152
153
154
155
156
157
        def __rc(z, zr):
            if z != 0:
                return z * (1 + (zr/z)**2)
            else:
                return float("inf")
                
        v = np.vectorize(__rc)
        
        return v(self.z, self.zr)
158
    
159
    def curvature(self, z=None, wavelength=None, nr=None, w0=None):
160
        if z is None:
161
162
163
164
            z = self.z
        else:
            z = np.array(z)
                
165
        if wavelength is None:
166
167
168
169
            wavelength = self.wavelength
        else:
            wavelength = np.array(wavelength)
            
170
        if nr is None:
171
172
173
174
            nr = self.nr
        else:
            nr = np.array(nr)
            
175
        if w0 is None:
176
177
178
179
180
181
182
183
            w0 = self.w0
        else:
            w0 = np.array(w0)
        
        q = z + 1j * math.pi * w0 **2 / wavelength
        
        return q.real * (1+ (q.imag/q.real)**2)
        
184
185
186
187
188
189
190
191
192
193
194
195
196
    @staticmethod
    def overlap(q1, q2):
        """
        Computes the projection from one beam parameter to another to give a measure of the
        overlap between the two beam parameters.
        
        This function was provided by Paul Fulda and Antonio Perreca, which came originally
        from Chris Mueller.
        
        Added on 20/4/2015
        """
        return abs(4*q1.imag * q2.imag)/abs(q1.conjugate()-q2)**2
        
197
    def conjugate(self):
Daniel Brown's avatar
Daniel Brown committed
198
        return beam_param(self.__lambda, self.__nr, self.__q.conjugate())
199
    
200
201
202
    def __abs__(self):
        return abs(complex(self.__q))
        
203
204
205
206
207
208
209
    def __complex__(self):
        return self.__q
    
    def __str__(self):
        return str(self.__q)
    
    def __mul__(self, a):
Daniel Brown's avatar
Daniel Brown committed
210
        return beam_param(self.__lambda, self.__nr, self.__q * complex(a))
211
212
    
    def __imul__(self, a):
213
        self.__q *= complex(a)
214
215
216
217
218
        return self
        
    __rmul__ = __mul__
    
    def __add__(self, a):
Daniel Brown's avatar
Daniel Brown committed
219
        return beam_param(self.__lambda, self.__nr, self.__q + complex(a))
220
221
222
223
224
225
226
227
    
    def __iadd__(self, a):
        self.__q += complex(a)
        return self
        
    __radd__ = __add__
    
    def __sub__(self, a):
Daniel Brown's avatar
Daniel Brown committed
228
        return beam_param(self.__lambda, self.__nr, self.__q - complex(a))
229
230
231
232
233
    
    def __isub__(self, a):
        self.__q -= complex(a)
        return self
        
234
    def __rsub__(self, a):
Daniel Brown's avatar
Daniel Brown committed
235
        return beam_param(self.__lambda, self.__nr, complex(a) - self.__q)
236
237
    
    def __div__(self, a):
Daniel Brown's avatar
Daniel Brown committed
238
        return beam_param(self.__lambda, self.__nr, self.__q / complex(a))
239
240
241
242
243
244
    
    def __idiv__(self, a):
        self.__q /= complex(a)
        return self
    
    def __pow__(self, q):
Daniel Brown's avatar
Daniel Brown committed
245
        return beam_param(self.__lambda, self.__nr, self.__q**q)
246
247

    def __neg__(self, q):
Daniel Brown's avatar
Daniel Brown committed
248
        return beam_param(self.__lambda, self.__nr, -self.__q)
249
250
        
    def __eq__(self, q):
251
        if q is None:
252
253
            return False
            
254
255
256
257
258
259
260
261
262
263
        return complex(q) == self.__q
        
    @property
    def real(self): return self.__q.real
    @real.setter
    def real(self, value): self.__q.real = SIfloat(value)
    
    @property
    def imag(self): return self.__q.imag
    @imag.setter
264
    def imag(self, value): self.__q.imag = SIfloat(value)
265
266
267
268

    # reverse beam direction 
    def reverse(self):
        self.__q = -1.0 * self.__q.real + 1j * self.__q.imag
269

270

271
272
class beam_param(gauss_param):
    pass
273

274
class HG_mode(object):
275
    """ Hermite-Gauss mode profile. Example usage:
276
277
    import pykat.optics.gaussian_beams as gb
    qx=gb.beam_param(w0=1e-3,z=0)
278
    beam=gb.HG_mode(qx,n=2,m=0)
279
280
    beam.plot()
    """    
281
282
283
284
    def __init__(self, qx, qy=None, n=0, m=0):
        self._qx = copy.deepcopy(qx)
        self._2pi_qrt = math.pow(2.0/math.pi, 0.25)
        
285
        if qy is None:
286
            self._qy = copy.deepcopy(qx)
287
        else:
288
            self._qy = copy.deepcopy(qy)
289
    
290
291
292
293
        self._n = int(n)
        self._m = int(m)
        self._hn = hermite(self._n)
        self._hm = hermite(self._m)
294
295
296
297
298
299
        self._calc_constants()
        
    @property
    def n(self): return self._n
    @n.setter
    def n(self,value): 
300
        self._n = int(value)
301
        self._calc_constants()
302
        self._hn = hermite(self._n)
303
304
305
306
307

    @property
    def m(self): return self._m
    @m.setter
    def m(self,value): 
308
        self._m = int(value)
309
        self._calc_constants()
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
        self._hm = hermite(self._m)
            
    @property
    def q(self):
        if self._qx.q == self._qy.q:
            return self._qx.q
        else:
            return (self._qx.q, self._qy.q)
    @q.setter
    def q(self, value):
        if value.__class__ == beam_param:
            self._qx = copy.deepcopy(value)
            self._qy = copy.deepcopy(value)
        else:
            self._qx = beam_param(q=complex(value))
            self._qy = beam_param(q=complex(value))
    
    @property
    def qx(self):
        return self._qx.q
        
    @qx.setter
    def qx(self, value):
        if value.__class__ == beam_param:
            self._qx = copy.deepcopy(value)
        else:
            self._qx = beam_param(q=complex(value))
    
    @property
    def qy(self):
        return self._qy.q
341
        
342
343
344
345
346
347
    @qy.setter
    def qy(self, value):
        if value.__class__ == beam_param:
            self._qy = copy.deepcopy(value)
        else:
            self._qy = beam_param(q=complex(value))
348
349
350
351
352
353
354
355
356
    
    @property
    def constant_x(self):
        return self.__xpre_const
        
    @property
    def constant_y(self):
        return self.__ypre_const
        
357
358
    def _calc_constants(self):
        self.__xpre_const = math.pow(2.0/math.pi, 0.25)
359
        self.__xpre_const *= np.sqrt(1.0/(self._qx.w0 * 2**(self._n) * np.math.factorial(self._n)))
360
        self.__xpre_const *= np.sqrt(1j*self._qx.imag / self._qx.q)
361
        self.__xpre_const *= ((1j*self._qx.imag * self._qx.q.conjugate())/(-1j*self._qx.imag * self._qx.q)) ** ( self._n/2.0)
362
363
        
        self.__ypre_const = math.pow(2.0/math.pi, 0.25)
364
        self.__ypre_const *= np.sqrt(1.0/(self._qy.w0 * 2**(self._m) * np.math.factorial(self._m)))
365
        self.__ypre_const *= np.sqrt(1j*self._qy.imag / self._qy.q)
Daniel Brown's avatar
Daniel Brown committed
366
        self.__ypre_const *= ((1j*self._qy.imag * self._qy.q.conjugate())/(-1j*self._qy.imag * self._qy.q)) **(self._m/2.0)
367
368
369
370
371
372
373
374
375
    
        self.__sqrt2_wxz = math.sqrt(2) / self._qx.w
        self.__sqrt2_wyz = math.sqrt(2) / self._qy.w
        
        self.__kx =  2*math.pi / self._qx.wavelength
        self.__ky =  2*math.pi / self._qy.wavelength
        
        self.__invqx = 1/ self._qx.q
        self.__invqy = 1/ self._qy.q
376
    
Daniel Brown's avatar
Daniel Brown committed
377
    def Un(self, x):
378
379
        return self.__xpre_const * self._hn(self.__sqrt2_wxz * x) * np.exp(-0.5j * self.__kx * x*x * self.__invqx)
    
Daniel Brown's avatar
Daniel Brown committed
380
381
    def Um(self, y):
        return self.__ypre_const * self._hm(self.__sqrt2_wyz * y) * np.exp(-0.5j * self.__ky * y*y * self.__invqy)
382
        
383
384
385
386
    def Unm(self, x, y):
        _un = self.Un(x)  
        _um = self.Um(y)
        return np.outer(_un, _um)
387
388
        
    def plot(self, ndx=100, ndy=100, xscale=4, yscale=4):
389
        """ Make a simple plot the HG_mode """
390
391
        import pykat.plotting 
        import matplotlib.pyplot as plt
392
393
394
395
396
397
398
        
        xrange = xscale * np.linspace(-self._qx.w, self._qx.w, ndx)
        yrange = yscale * np.linspace(-self._qy.w, self._qy.w, ndy)

        dx = xrange[1]-xrange[0]
        dy = yrange[1]-yrange[0]

399
        data = self.Unm(xrange,yrange)
400

401
402
403
404
        fig = pykat.plotting.figure()
        axes = plt.imshow(np.abs(data.T), aspect=dx/dy, extent=[min(xrange),max(xrange),min(yrange),max(yrange)])
        plt.xlabel('x [m]')
        plt.ylabel('y [m]')
405
        cbar = fig.colorbar(axes)
406
        plt.show()
407
        
408

409
def HG2LG(n,m):
410
411
    """A function for Matlab which returns the coefficients and mode indices of
    the LG modes required to create a particular HG mode.
412
    Usage: coefficients,ps,ls = HG2LG(n,m)
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
    
    n,m:          Indces of the HG mode to re-create.
    coeffcients:  Complex coefficients for each order=n+m LG mode required to
                  re-create HG_n,m.
    ps,ls:        LG mode indices corresponding to coefficients.
    """
    # Mode order
    N = n+m;
    
    # Create empty vectors for LG coefficients/ indices
    coefficients = np.linspace(0,0,N+1,dtype=np.complex_)
    ps = np.linspace(0,0,N+1)
    ls = np.linspace(0,0,N+1)
    
    # Calculate coefficients
    for j in np.arange(0,N+1):
        
        # Indices for coefficients
        l = 2*j-N
        p = int((N-np.abs(l))/2)
        
        ps[j] = p
        ls[j] = l
        
        signl = np.sign(l)
        if (l==0):
            signl = 1.0

        # Coefficient
        c = (signl*1j)**m * np.sqrt(factorial(N-m)*factorial(m)/(2**N * factorial(np.abs(l)+p)*factorial(p)))
443
        c = c * (-1.0)**p * (-2.0)**m * jacobi(m,np.abs(l)+p-m,p-m,0.0)
444
445
446
447
448
449

        coefficients[j] = c
        
    return coefficients, ps, ls 
        

450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
    
def LG2HG(p,l):
    """ Function to compute the amplitude coefficients
    of Hermite-Gauss modes whose sum yields a Laguerre Gauss mode
    of order n,m.
    Usage: coefficients, ns, ms = LG2HG(p,l)
    p:    Radial LG index
    l:    Azimuthal LG index
    The LG mode is written as u_pl with 0<=|l|<=p.
    The output is a series of coefficients for u_nm modes,
    given as complex numbers and respective n,m indices
    coefficients (complex array): field amplitude for mode u_nm
    ns (int array): n-index of u_nm
    ms (int array): m-index of u_nm

    
    The formula is adpated from M.W. Beijersbergen et al 'Astigmatic
    laser mode converters and transfer of orbital angular momentum',
    Opt. Comm. 96 123-132 (1993)
    We adapted coefficients to be compatible with our
    definition of an LG mode, it differs from
    Beijersbergen by a (-1)^p factor and has exp(il\phi) rather
    than exp(-il\phi).  Also adapted for allowing -l.
    Andreas Freise, Charlotte Bond    25.03.2007"""

    # Mode order
    N=2*p+np.abs(l)

    # Indices for coefficients
    n = np.abs(l)+p
    m = p

    # create empty vectors
    coefficients = np.linspace(0,0,N+1,dtype=np.complex_)
    ns = np.linspace(0,0,N+1)
    ms = np.linspace(0,0,N+1)

    # l positive or negative
    signl = np.sign(l)
    if (l==0):
        signl = 1.0
    
    # Beijersbergen coefficients
    for j in np.arange(0,N+1):
        ns[j]=N-j
        ms[j]=j

        c=(-signl*1j)**j * math.sqrt(factorial(N-j)*factorial(j)/(2**N * factorial(n)*factorial(m)))
        coefficients[j] = c * (-1.0)**p * (-2)**j * jacobi(j,n-j,m-j,0.0)
    
    return coefficients, ns, ms