ó ~9­\c@ s´dZddlmZmZddddddd d d d d dddddgZddlmZmZmZm Z m Z m Z m Z m Z mZmZmZddlmZde fd„ƒYZdefd„ƒYZdefd„ƒYZdefd„ƒYZdefd„ƒYZdefd„ƒYZd efd„ƒYZd e fd„ƒYZd efd„ƒYZd„Zd„Zd „Zd!„ZeZ d"„Z!d#„Z"d$S(%sÜ Gaussian optics. The module implements: - Ray transfer matrices for geometrical and gaussian optics. See RayTransferMatrix, GeometricRay and BeamParameter - Conjugation relations for geometrical and gaussian optics. See geometric_conj*, gauss_conj and conjugate_gauss_beams The conventions for the distances are as follows: focal distance positive for convergent lenses object distance positive for real objects image distance positive for real images iÿÿÿÿ(tprint_functiontdivisiontRayTransferMatrixt FreeSpacetFlatRefractiontCurvedRefractiont FlatMirrort CurvedMirrortThinLenst GeometricRayt BeamParametertwaist2rayleightrayleigh2waisttgeometric_conj_abtgeometric_conj_aftgeometric_conj_bft gaussian_conjtconjugate_gauss_beams( tatan2tExprtItimtMatrixtootpitretsqrttsympifyttogether(t filldedentcB s\eZdZd„Zd„Zed„ƒZed„ƒZed„ƒZed„ƒZ RS(sÕ Base class for a Ray Transfer Matrix. It should be used if there isn't already a more specific subclass mentioned in See Also. Parameters ========== parameters : A, B, C and D or 2x2 matrix (Matrix(2, 2, [A, B, C, D])) Examples ======== >>> from sympy.physics.optics import RayTransferMatrix, ThinLens >>> from sympy import Symbol, Matrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat Matrix([ [1, 2], [3, 4]]) >>> RayTransferMatrix(Matrix([[1, 2], [3, 4]])) Matrix([ [1, 2], [3, 4]]) >>> mat.A 1 >>> f = Symbol('f') >>> lens = ThinLens(f) >>> lens Matrix([ [ 1, 0], [-1/f, 1]]) >>> lens.C -1/f See Also ======== GeometricRay, BeamParameter, FreeSpace, FlatRefraction, CurvedRefraction, FlatMirror, CurvedMirror, ThinLens References ========== .. [1] https://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis cG s®t|ƒdkr=|d|df|d|dff}nat|ƒdkr‚t|dtƒr‚|djdkr‚|d}nttdt|ƒƒƒ‚tj||ƒS(Niiiiis` Expecting 2x2 Matrix or the 4 elements of the Matrix but got %s(ii(tlent isinstanceRtshapet ValueErrorRtstrt__new__(tclstargsttemp((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR#os+ cC sÕt|tƒr%ttj||ƒƒSt|tƒrJttj||ƒƒSt|tƒrÁ|t|jfdfƒ}|d|djdtƒ}t|j t t |ƒƒdt t |ƒƒƒStj||ƒSdS(Niitcomplextz_r(i( RRRt__mul__R R tqtexpandtTruetwavelenRRR(tselftotherR&R*((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR)}s cC s|dS(sß The A parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.A 1 i(ii((R.((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pytA‹s cC s|dS(sß The B parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.B 2 ii(ii((R.((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pytBšs cC s|dS(sß The C parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.C 3 ii(ii((R.((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pytC©s cC s|dS(sß The D parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.D 4 i(ii((R.((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pytD¸s ( t__name__t __module__t__doc__R#R)tpropertyR0R1R2R3(((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR8s5  cB seZdZd„ZRS(sQ Ray Transfer Matrix for free space. Parameters ========== distance See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import FreeSpace >>> from sympy import symbols >>> d = symbols('d') >>> FreeSpace(d) Matrix([ [1, d], [0, 1]]) cC stj|d|ddƒS(Nii(RR#(R$td((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR#ás(R4R5R6R#(((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyRÈscB seZdZd„ZRS(s¶ Ray Transfer Matrix for refraction. Parameters ========== n1 : refractive index of one medium n2 : refractive index of other medium See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import FlatRefraction >>> from sympy import symbols >>> n1, n2 = symbols('n1 n2') >>> FlatRefraction(n1, n2) Matrix([ [1, 0], [0, n1/n2]]) cC s8tt||fƒ\}}tj|ddd||ƒS(Nii(tmapRRR#(R$tn1tn2((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR#ÿs(R4R5R6R#(((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyRåscB seZdZd„ZRS(s' Ray Transfer Matrix for refraction on curved interface. Parameters ========== R : radius of curvature (positive for concave) n1 : refractive index of one medium n2 : refractive index of other medium See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import CurvedRefraction >>> from sympy import symbols >>> R, n1, n2 = symbols('R n1 n2') >>> CurvedRefraction(R, n1, n2) Matrix([ [ 1, 0], [(n1 - n2)/(R*n2), n1/n2]]) cC sJtt|||fƒ\}}}tj|dd||||||ƒS(Nii(R9RRR#(R$tRR:R;((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR#s!(R4R5R6R#(((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyRscB seZdZd„ZRS(sê Ray Transfer Matrix for reflection. See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import FlatMirror >>> FlatMirror() Matrix([ [1, 0], [0, 1]]) cC stj|ddddƒS(Nii(RR#(R$((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR#6s(R4R5R6R#(((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR$scB seZdZd„ZRS(s— Ray Transfer Matrix for reflection from curved surface. Parameters ========== R : radius of curvature (positive for concave) See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import CurvedMirror >>> from sympy import symbols >>> R = symbols('R') >>> CurvedMirror(R) Matrix([ [ 1, 0], [-2/R, 1]]) cC s)t|ƒ}tj|ddd|dƒS(Niiiþÿÿÿ(RRR#(R$R<((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR#Ss (R4R5R6R#(((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR:scB seZdZd„ZRS(sd Ray Transfer Matrix for a thin lens. Parameters ========== f : the focal distance See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import ThinLens >>> from sympy import symbols >>> f = symbols('f') >>> ThinLens(f) Matrix([ [ 1, 0], [-1/f, 1]]) cC s)t|ƒ}tj|ddd|dƒS(Niiiÿÿÿÿ(RRR#(R$tf((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR#qs (R4R5R6R#(((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyRXscB s5eZdZd„Zed„ƒZed„ƒZRS(s Representation for a geometric ray in the Ray Transfer Matrix formalism. Parameters ========== h : height, and angle : angle, or matrix : a 2x1 matrix (Matrix(2, 1, [height, angle])) Examples ======== >>> from sympy.physics.optics import GeometricRay, FreeSpace >>> from sympy import symbols, Matrix >>> d, h, angle = symbols('d, h, angle') >>> GeometricRay(h, angle) Matrix([ [ h], [angle]]) >>> FreeSpace(d)*GeometricRay(h, angle) Matrix([ [angle*d + h], [ angle]]) >>> GeometricRay( Matrix( ((h,), (angle,)) ) ) Matrix([ [ h], [angle]]) See Also ======== RayTransferMatrix cG s t|ƒdkrEt|dtƒrE|djdkrE|d}nKt|ƒdkrt|df|dff}nttdt|ƒƒƒ‚tj||ƒS(Niiis` Expecting 2x1 Matrix or the 2 elements of the Matrix but got %s(ii(RRRR R!RR"R#(R$R%R&((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR#¢s% cC s|dS(s0 The distance from the optical axis. Examples ======== >>> from sympy.physics.optics import GeometricRay >>> from sympy import symbols >>> h, angle = symbols('h, angle') >>> gRay = GeometricRay(h, angle) >>> gRay.height h i((R.((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pytheight®scC s|dS(s0 The angle with the optical axis. Examples ======== >>> from sympy.physics.optics import GeometricRay >>> from sympy import symbols >>> h, angle = symbols('h, angle') >>> gRay = GeometricRay(h, angle) >>> gRay.angle angle i((R.((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pytangle¿s(R4R5R6R#R7R>R?(((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR zs& cB seZdZdddgZd„Zed„ƒZed„ƒZed„ƒZed„ƒZ ed „ƒZ ed „ƒZ ed „ƒZ RS( s‰ Representation for a gaussian ray in the Ray Transfer Matrix formalism. Parameters ========== wavelen : the wavelength, z : the distance to waist, and w : the waist, or z_r : the rayleigh range Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.q 1 + 1.88679245283019*I*pi >>> p.q.n() 1.0 + 5.92753330865999*I >>> p.w_0.n() 0.00100000000000000 >>> p.z_r.n() 5.92753330865999 >>> from sympy.physics.optics import FreeSpace >>> fs = FreeSpace(10) >>> p1 = fs*p >>> p.w.n() 0.00101413072159615 >>> p1.w.n() 0.00210803120913829 See Also ======== RayTransferMatrix References ========== .. [1] https://en.wikipedia.org/wiki/Complex_beam_parameter .. [2] https://en.wikipedia.org/wiki/Gaussian_beam tzR(R-cK sÀtt||fƒ\}}tj|||ƒ}||_||_t|ƒdkrctdƒ‚nYd|kr…t|dƒ|_n7d|kr°t t|dƒ|ƒ|_n tdƒ‚|S(Nis/Constructor expects exactly one named argument.R(tws-The constructor needs named argument w or z_r( R9RRR#R-R@RR!R(R (R$R-R@tkwargstinst((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR# s     cC s|jt|jS(s The complex parameter representing the beam. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.q 1 + 1.88679245283019*I*pi (R@RR((R.((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR*s cC s|jd|j|jdS(s The radius of curvature of the phase front. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.radius 1 + 3.55998576005696*pi**2 ii(R@R((R.((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pytradius's cC s#|jtd|j|jdƒS(sI The beam radius at `1/e^2` intensity. See Also ======== w_0 : the minimal radius of beam Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.w 0.001*sqrt(0.2809/pi**2 + 1) ii(tw_0RR@R((R.((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyRA6scC st|jt|jƒS(sE The beam waist (minimal radius). See Also ======== w : the beam radius at `1/e^2` intensity Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.w_0 0.00100000000000000 (RR(RR-(R.((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyREJscC s|jt|jS(sï Half of the total angular spread. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.divergence 0.00053/pi (R-RRE(R.((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyt divergence^s cC st|j|jƒS(sÚ The Gouy phase. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.gouy atan(0.53/pi) (RR@R((R.((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pytgouyms cC sd|jtS(sª The minimal waist for which the gauss beam approximation is valid. The gauss beam is a solution to the paraxial equation. For curvatures that are too great it is not a valid approximation. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.waist_approximation_limit 1.06e-6/pi i(R-R(R.((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pytwaist_approximation_limit|s( R4R5R6t __slots__R#R7R*RDRARERFRGRH(((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR Õs- cC s+tt||fƒ\}}|dt|S(s^ Calculate the rayleigh range from the waist of a gaussian beam. See Also ======== rayleigh2waist, BeamParameter Examples ======== >>> from sympy.physics.optics import waist2rayleigh >>> from sympy import symbols >>> w, wavelen = symbols('w wavelen') >>> waist2rayleigh(w, wavelen) pi*w**2/wavelen i(R9RR(RAR-((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR “scC s-tt||fƒ\}}t|t|ƒS(sjCalculate the waist from the rayleigh range of a gaussian beam. See Also ======== waist2rayleigh, BeamParameter Examples ======== >>> from sympy.physics.optics import rayleigh2waist >>> from sympy import symbols >>> z_r, wavelen = symbols('z_r wavelen') >>> rayleigh2waist(z_r, wavelen) sqrt(wavelen*z_r)/sqrt(pi) (R9RRR(R(R-((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR ©scC smtt||fƒ\}}t|ƒtks?t|ƒtkrYt|ƒtkrU|S|S||||SdS(s¶ Conjugation relation for geometrical beams under paraxial conditions. Takes the distances to the optical element and returns the needed focal distance. See Also ======== geometric_conj_af, geometric_conj_bf Examples ======== >>> from sympy.physics.optics import geometric_conj_ab >>> from sympy import symbols >>> a, b = symbols('a b') >>> geometric_conj_ab(a, b) a*b/(a + b) N(R9RtabsR(tatb((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR ¾s$cC s*tt||fƒ\}}t|| ƒ S(sp Conjugation relation for geometrical beams under paraxial conditions. Takes the object distance (for geometric_conj_af) or the image distance (for geometric_conj_bf) to the optical element and the focal distance. Then it returns the other distance needed for conjugation. See Also ======== geometric_conj_ab Examples ======== >>> from sympy.physics.optics.gaussopt import geometric_conj_af, geometric_conj_bf >>> from sympy import symbols >>> a, b, f = symbols('a b f') >>> geometric_conj_af(a, f) a*f/(a - f) >>> geometric_conj_bf(b, f) b*f/(b - f) (R9RR (RKR=((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyRÚscC sžtt|||fƒ\}}}dd||d||d|}dtd||d||dƒ}|d||d||d}|||fS(sŸ Conjugation relation for gaussian beams. Parameters ========== s_in : the distance to optical element from the waist z_r_in : the rayleigh range of the incident beam f : the focal length of the optical element Returns ======= a tuple containing (s_out, z_r_out, m) s_out : the distance between the new waist and the optical element z_r_out : the rayleigh range of the emergent beam m : the ration between the new and the old waists Examples ======== >>> from sympy.physics.optics import gaussian_conj >>> from sympy import symbols >>> s_in, z_r_in, f = symbols('s_in z_r_in f') >>> gaussian_conj(s_in, z_r_in, f)[0] 1/(-1/(s_in + z_r_in**2/(-f + s_in)) + 1/f) >>> gaussian_conj(s_in, z_r_in, f)[1] z_r_in/(1 - s_in**2/f**2 + z_r_in**2/f**2) >>> gaussian_conj(s_in, z_r_in, f)[2] 1/sqrt(1 - s_in**2/f**2 + z_r_in**2/f**2) iiÿÿÿÿi(R9RR(ts_intz_r_inR=ts_outtmtz_r_out((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyRøs #!&("c K stt|||fƒ\}}}||}t||ƒ}t|ƒdkr[tdƒ‚nµd|kr|ttdƒƒ‚n”d|krÝt|dƒ}|dtd|d|d|dƒ}t|||ƒd}n3d|krþttdƒƒ‚nttd ƒƒ‚|||fS( sà Find the optical setup conjugating the object/image waists. Parameters ========== wavelen : the wavelength of the beam waist_in and waist_out : the waists to be conjugated f : the focal distance of the element used in the conjugation Returns ======= a tuple containing (s_in, s_out, f) s_in : the distance before the optical element s_out : the distance after the optical element f : the focal distance of the optical element Examples ======== >>> from sympy.physics.optics import conjugate_gauss_beams >>> from sympy import symbols, factor >>> l, w_i, w_o, f = symbols('l w_i w_o f') >>> conjugate_gauss_beams(l, w_i, w_o, f=f)[0] f*(1 - sqrt(w_i**2/w_o**2 - pi**2*w_i**4/(f**2*l**2))) >>> factor(conjugate_gauss_beams(l, w_i, w_o, f=f)[1]) f*w_o**2*(w_i**2/w_o**2 - sqrt(w_i**2/w_o**2 - pi**2*w_i**4/(f**2*l**2)))/w_i**2 >>> conjugate_gauss_beams(l, w_i, w_o, f=f)[2] f is,The function expects only one named argumenttdistsD Currently only focal length is supported as a parameterR=iiRMsG The functions expects the focal length as a named argument( R9RR RR!tNotImplementedErrorRRR( R-twaist_int waist_outRBRPR@R=RMRO((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pyR"s$%!   ,  N(#R6t __future__RRt__all__tsympyRRRRRRRRRRRtsympy.utilities.miscRRRRRRRRR R R R R RRRR(((s<lib/python2.7/site-packages/sympy/physics/optics/gaussopt.pytsF L "[¾     *