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
19
20
21
22
23
class gauss_param(BeamParam):
    pass

class beam_param(BeamParam):
    pass
        
class BeamParam(object):
24
    """
25
    Gaussian beam complex parameter.
26
    
27
    BeamParam is effectively a complex number with extra
28
29
    functionality to determine beam parameters.
    
Daniel Brown's avatar
Daniel Brown committed
30
    Defaults to 1064e-9m for wavelength and refractive index 1
31
    usage:
32
33
34
35
        q = BeamParam(w0=w0, z=z)
        q = BeamParam(z=z, zr=zr)
        q = BeamParam(w=w, rc=rc)
        q = BeamParam(q=a) # where a is a complex number
36
37
38
        
        or change default wavelength and refractive index with:
        
39
        q = BeamParam(wavelength, nr, w0=w0, zr=zr)
40
41
42
    """
    
    def __init__(self, wavelength=1064e-9, nr=1, *args, **kwargs):
43
44
        if self.__class__ != BeamParam:
            warnings.warn("Name changed. Use BeamParam instead of gauss_param or beam_param.")
45
            
46
        self.__q = None
47
48
        self.__lambda = SIfloat(wavelength)
        self.__nr = SIfloat(nr)
49
50
        
        if len(args) == 1:
Daniel Brown's avatar
Daniel Brown committed
51
            self.__q = complex(args[0])
52
53
54
55
56
        
        elif len(kwargs) == 1:
            if "q" in kwargs:
                self.__q = complex(kwargs["q"])        
            else:
57
                raise pkex.BasePyKatException("Must specify: z and w0 or z and zr or rc and w or q, to define the beam parameter")
58
                
59
60
61
        elif len(kwargs) == 2:        
            
            if "w0" in kwargs and "z" in kwargs:
62
                q = SIfloat(kwargs["z"]) + 1j * math.pi*SIfloat(kwargs["w0"])**2/(self.__lambda/self.__nr)
63
            elif "z" in kwargs and "zr" in kwargs:
64
                q = SIfloat(kwargs["z"]) + 1j * SIfloat(kwargs["zr"]) 
65
            elif "rc" in kwargs and "w" in kwargs:
66
                one_q = 1 / SIfloat(kwargs["rc"]) - 1j * SIfloat(wavelength) / (math.pi * SIfloat(nr) * SIfloat(kwargs["w"])**2)
67
68
                q = 1/one_q
            else:
69
                raise pkex.BasePyKatException("Must specify: z and w0 or z and zr or rc and w or q, to define the beam parameter")
70
71
72
73
74
75
76
                
            self.__q = q
        else:
            raise pkex.BasePyKatException("Incorrect usage for gauss_param constructor")
    
    @property
    def wavelength(self): return self.__lambda
77
78
    @wavelength.setter
    def wavelength(self,value): self.__lambda = SIfloat(value)
79
80
81
82
83
84
85
86
87
88
89
90
91
92
    
    @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
93
    def w(self):
94
        return np.abs(self.__q)* np.sqrt(self.__lambda / (self.__nr * math.pi * self.__q.imag))
95
    
96
    def beamsize(self, z=None, wavelength=None, nr=None, w0=None):
97

98
        if z is None:
99
100
101
102
            z = self.z
        else:
            z = np.array(z)
                
103
        if wavelength is None:
104
105
106
107
            wavelength = self.wavelength
        else:
            wavelength = np.array(wavelength)
            
108
        if nr is None:
109
110
111
112
            nr = self.nr
        else:
            nr = np.array(nr)
            
113
        if w0 is None:
114
115
116
117
118
119
120
121
122
            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):
123
        if z is None:
124
125
126
127
            z = self.z
        else:
            z = np.array(z)
                
128
        if wavelength is None:
129
130
131
132
            wavelength = self.wavelength
        else:
            wavelength = np.array(wavelength)
            
133
        if nr is None:
134
135
136
137
            nr = self.nr
        else:
            nr = np.array(nr)
            
138
        if w0 is None:
139
140
141
142
143
144
145
146
            w0 = self.w0
        else:
            w0 = np.array(w0)
        
        q = z + 1j * math.pi * w0 **2 / wavelength
        
        return np.arctan2(q.real, q.imag)
        
147
148
    @property
    def w0(self):
149
        return np.sqrt(self.__q.imag * self.__lambda / (self.__nr * math.pi))    
150
151
152

    @property
    def Rc(self):
153
154
155
156
157
158
159
160
161
        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)
162
    
163
    def curvature(self, z=None, wavelength=None, nr=None, w0=None):
164
        if z is None:
165
166
167
168
            z = self.z
        else:
            z = np.array(z)
                
169
        if wavelength is None:
170
171
172
173
            wavelength = self.wavelength
        else:
            wavelength = np.array(wavelength)
            
174
        if nr is None:
175
176
177
178
            nr = self.nr
        else:
            nr = np.array(nr)
            
179
        if w0 is None:
180
181
182
183
184
185
186
187
            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)
        
188
189
190
191
192
193
194
195
196
197
198
199
200
    @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
        
201
    def conjugate(self):
202
        return BeamParam(self.__lambda, self.__nr, self.__q.conjugate())
203
    
204
205
206
    def __abs__(self):
        return abs(complex(self.__q))
        
207
208
209
210
211
212
213
    def __complex__(self):
        return self.__q
    
    def __str__(self):
        return str(self.__q)
    
    def __mul__(self, a):
214
        return BeamParam(self.__lambda, self.__nr, self.__q * complex(a))
215
216
    
    def __imul__(self, a):
217
        self.__q *= complex(a)
218
219
220
221
222
        return self
        
    __rmul__ = __mul__
    
    def __add__(self, a):
223
        return BeamParam(self.__lambda, self.__nr, self.__q + complex(a))
224
225
226
227
228
229
230
231
    
    def __iadd__(self, a):
        self.__q += complex(a)
        return self
        
    __radd__ = __add__
    
    def __sub__(self, a):
232
        return BeamParam(self.__lambda, self.__nr, self.__q - complex(a))
233
234
235
236
237
    
    def __isub__(self, a):
        self.__q -= complex(a)
        return self
        
238
    def __rsub__(self, a):
239
        return BeamParam(self.__lambda, self.__nr, complex(a) - self.__q)
240
241
    
    def __div__(self, a):
242
        return BeamParam(self.__lambda, self.__nr, self.__q / complex(a))
243
244
245
246
247
248
    
    def __idiv__(self, a):
        self.__q /= complex(a)
        return self
    
    def __pow__(self, q):
249
        return BeamParam(self.__lambda, self.__nr, self.__q**q)
250
251

    def __neg__(self, q):
252
        return BeamParam(self.__lambda, self.__nr, -self.__q)
253
254
        
    def __eq__(self, q):
255
        if q is None:
256
257
            return False
            
258
259
260
261
262
263
264
265
266
267
        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
268
    def imag(self, value): self.__q.imag = SIfloat(value)
269
270
271
272

    # reverse beam direction 
    def reverse(self):
        self.__q = -1.0 * self.__q.real + 1j * self.__q.imag
273
274
        
        
275
class HG_mode(object):
276
    """ Hermite-Gauss mode profile. Example usage:
277
    import pykat.optics.gaussian_beams as gb
278
    qx=gb.BeamParam(w0=1e-3,z=0)
279
    beam=gb.HG_mode(qx,n=2,m=0)
280
281
    beam.plot()
    """    
282
283
284
285
    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)
        
286
        if qy is None:
287
            self._qy = copy.deepcopy(qx)
288
        else:
289
            self._qy = copy.deepcopy(qy)
290
    
291
292
293
294
        self._n = int(n)
        self._m = int(m)
        self._hn = hermite(self._n)
        self._hm = hermite(self._m)
295
296
297
298
299
300
        self._calc_constants()
        
    @property
    def n(self): return self._n
    @n.setter
    def n(self,value): 
301
        self._n = int(value)
302
        self._calc_constants()
303
        self._hn = hermite(self._n)
304
305
306
307
308

    @property
    def m(self): return self._m
    @m.setter
    def m(self,value): 
309
        self._m = int(value)
310
        self._calc_constants()
311
312
313
314
315
316
317
318
319
320
        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):
321
        if value.__class__ == BeamParam:
322
323
324
            self._qx = copy.deepcopy(value)
            self._qy = copy.deepcopy(value)
        else:
325
326
            self._qx = BeamParam(q=complex(value))
            self._qy = BeamParam(q=complex(value))
327
328
329
330
331
332
333
    
    @property
    def qx(self):
        return self._qx.q
        
    @qx.setter
    def qx(self, value):
334
        if value.__class__ == BeamParam:
335
336
            self._qx = copy.deepcopy(value)
        else:
337
            self._qx = BeamParam(q=complex(value))
338
339
340
341
    
    @property
    def qy(self):
        return self._qy.q
342
        
343
344
    @qy.setter
    def qy(self, value):
345
        if value.__class__ == BeamParam:
346
347
            self._qy = copy.deepcopy(value)
        else:
348
            self._qy = BeamParam(q=complex(value))
349
350
351
352
353
354
355
356
357
    
    @property
    def constant_x(self):
        return self.__xpre_const
        
    @property
    def constant_y(self):
        return self.__ypre_const
        
358
359
    def _calc_constants(self):
        self.__xpre_const = math.pow(2.0/math.pi, 0.25)
360
        self.__xpre_const *= np.sqrt(1.0/(self._qx.w0 * 2**(self._n) * np.math.factorial(self._n)))
361
        self.__xpre_const *= np.sqrt(1j*self._qx.imag / self._qx.q)
362
        self.__xpre_const *= ((1j*self._qx.imag * self._qx.q.conjugate())/(-1j*self._qx.imag * self._qx.q)) ** ( self._n/2.0)
363
364
        
        self.__ypre_const = math.pow(2.0/math.pi, 0.25)
365
        self.__ypre_const *= np.sqrt(1.0/(self._qy.w0 * 2**(self._m) * np.math.factorial(self._m)))
366
        self.__ypre_const *= np.sqrt(1j*self._qy.imag / self._qy.q)
Daniel Brown's avatar
Daniel Brown committed
367
        self.__ypre_const *= ((1j*self._qy.imag * self._qy.q.conjugate())/(-1j*self._qy.imag * self._qy.q)) **(self._m/2.0)
368
369
370
371
372
373
374
375
376
    
        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
377
    
Daniel Brown's avatar
Daniel Brown committed
378
    def Un(self, x):
379
380
        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
381
382
    def Um(self, y):
        return self.__ypre_const * self._hm(self.__sqrt2_wyz * y) * np.exp(-0.5j * self.__ky * y*y * self.__invqy)
383
        
384
385
386
387
    def Unm(self, x, y):
        _un = self.Un(x)  
        _um = self.Um(y)
        return np.outer(_un, _um)
388
389
        
    def plot(self, ndx=100, ndy=100, xscale=4, yscale=4):
390
        """ Make a simple plot the HG_mode """
391
392
        import pykat.plotting 
        import matplotlib.pyplot as plt
393
394
395
396
397
398
399
        
        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]

400
        data = self.Unm(xrange,yrange)
401

402
403
404
405
        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]')
406
        cbar = fig.colorbar(axes)
407
        plt.show()
408
        
409

410
def HG2LG(n,m):
411
412
    """A function for Matlab which returns the coefficients and mode indices of
    the LG modes required to create a particular HG mode.
413
    Usage: coefficients,ps,ls = HG2LG(n,m)
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
443
    
    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)))
444
        c = c * (-1.0)**p * (-2.0)**m * jacobi(m,np.abs(l)+p-m,p-m,0.0)
445
446
447
448
449
450

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

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
501
    
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