B ˜‘[[Nã@s@dZddlmZmZddddddd d d d d dddddgZddlmZmZmZm Z m Z m Z m Z m Z mZmZmZddlmZGdd„de ƒZGdd„deƒZGdd„deƒZGdd„deƒZGdd„deƒZGdd„deƒZGdd „d eƒZGdd „d e ƒZGdd „d eƒZdd „Zdd „Zd d„Zd!d„ZeZ d"d„Z!d#d„Z"d$S)%aÜ 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 é)Úprint_functionÚdivisionÚRayTransferMatrixÚ FreeSpaceÚFlatRefractionÚCurvedRefractionÚ FlatMirrorÚ CurvedMirrorÚThinLensÚ GeometricRayÚ BeamParameterÚwaist2rayleighÚrayleigh2waistÚgeometric_conj_abÚgeometric_conj_afÚgeometric_conj_bfÚ gaussian_conjÚconjugate_gauss_beams) Úatan2ÚExprÚIÚimÚMatrixÚooÚpiÚreÚsqrtÚsympifyÚtogether)Ú filldedentc@sPeZdZdZdd„Zdd„Zedd„ƒZedd „ƒZed d „ƒZ ed d „ƒZ dS)raÔ 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] http://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis cGs€t|ƒdkr.|d|df|d|dff}nFt|ƒdkr`t|dtƒr`|djdkr`|d}nttdt|ƒƒƒ‚t ||¡S)Nérééé)r"r"z` Expecting 2x2 Matrix or the 4 elements of the Matrix but got %s)ÚlenÚ isinstancerÚshapeÚ ValueErrorrÚstrÚ__new__)ÚclsÚargsÚtemp©r-ú>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.A 1 )rrr-)r5r-r-r.ÚA‹s zRayTransferMatrix.AcCs|dS)zß The B parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.B 2 )rr!r-)r5r-r-r.ÚBšs zRayTransferMatrix.BcCs|dS)zß The C parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.C 3 )r!rr-)r5r-r-r.ÚC©s zRayTransferMatrix.CcCs|dS)zß The D parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.D 4 )r!r!r-)r5r-r-r.ÚD¸s zRayTransferMatrix.DN) Ú__name__Ú __module__Ú __qualname__Ú__doc__r)r1Úpropertyr7r8r9r:r-r-r-r.r8s5   c@seZdZdZdd„ZdS)raQ 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]]) cCst |d|dd¡S)Nr!r)rr))r*Údr-r-r.r)ászFreeSpace.__new__N)r;r<r=r>r)r-r-r-r.rÈsc@seZdZdZdd„ZdS)ra¶ 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]]) cCs(tt||fƒ\}}t |ddd||¡S)Nr!r)Úmaprrr))r*Ún1Ún2r-r-r.r)ÿszFlatRefraction.__new__N)r;r<r=r>r)r-r-r-r.råsc@seZdZdZdd„ZdS)ra' 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]]) cCs8tt|||fƒ\}}}t |dd||||||¡S)Nr!r)rArrr))r*ÚRrBrCr-r-r.r)szCurvedRefraction.__new__N)r;r<r=r>r)r-r-r-r.rsc@seZdZdZdd„ZdS)rzê Ray Transfer Matrix for reflection. See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import FlatMirror >>> FlatMirror() Matrix([ [1, 0], [0, 1]]) cCst |dddd¡S)Nr!r)rr))r*r-r-r.r)6szFlatMirror.__new__N)r;r<r=r>r)r-r-r-r.r$sc@seZdZdZdd„ZdS)r a— 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]]) cCst|ƒ}t |ddd|d¡S)Nr!réþÿÿÿ)rrr))r*rDr-r-r.r)SszCurvedMirror.__new__N)r;r<r=r>r)r-r-r-r.r :sc@seZdZdZdd„ZdS)r ad 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]]) cCst|ƒ}t |ddd|d¡S)Nr!réÿÿÿÿ)rrr))r*Úfr-r-r.r)qszThinLens.__new__N)r;r<r=r>r)r-r-r-r.r Xsc@s0eZdZdZdd„Zedd„ƒZedd„ƒZdS) r a 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 cGstt|ƒdkr2t|dtƒr2|djdkr2|d}n6t|ƒdkrT|df|dff}nttdt|ƒƒƒ‚t ||¡S)Nr!r)r"r!r"z` Expecting 2x1 Matrix or the 2 elements of the Matrix but got %s)r$r%rr&r'rr(r))r*r+r,r-r-r.r)¢s  zGeometricRay.__new__cCs|dS)a0 The distance from the optical axis. Examples ======== >>> from sympy.physics.optics import GeometricRay >>> from sympy import symbols >>> h, angle = symbols('h, angle') >>> gRay = GeometricRay(h, angle) >>> gRay.height h rr-)r5r-r-r.Úheight®szGeometricRay.heightcCs|dS)a0 The angle with the optical axis. Examples ======== >>> from sympy.physics.optics import GeometricRay >>> from sympy import symbols >>> h, angle = symbols('h, angle') >>> gRay = GeometricRay(h, angle) >>> gRay.angle angle r!r-)r5r-r-r.Úangle¿szGeometricRay.angleN)r;r<r=r>r)r?rHrIr-r-r-r.r zs& c@sveZdZdZdddgZdd„Zedd„ƒZed d „ƒZed d „ƒZ ed d„ƒZ edd„ƒZ edd„ƒZ edd„ƒZ dS)r a‡ Representation for a gaussian ray in the Ray Transfer Matrix formalism. Parameters ========== wavelen : the wavelength, z : the distance to waist, and w : the waist, or z_r : the rayleigh range Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.q 1 + 1.88679245283019*I*pi >>> p.q.n() 1.0 + 5.92753330865999*I >>> p.w_0.n() 0.00100000000000000 >>> p.z_r.n() 5.92753330865999 >>> from sympy.physics.optics import FreeSpace >>> fs = FreeSpace(10) >>> p1 = fs*p >>> p.w.n() 0.00101413072159615 >>> p1.w.n() 0.00210803120913829 See Also ======== RayTransferMatrix References ========== .. [1] http://en.wikipedia.org/wiki/Complex_beam_parameter .. [2] http://en.wikipedia.org/wiki/Gaussian_beam Úzr0r4cKs„tt||fƒ\}}t |||¡}||_||_t|ƒdkrBtdƒ‚n>d|krZt|dƒ|_n&d|krxt t|dƒ|ƒ|_ntdƒ‚|S)Nr!z/Constructor expects exactly one named argument.r0Úwz-The constructor needs named argument w or z_r) rArrr)r4rJr$r'r0r )r*r4rJÚkwargsZinstr-r-r.r) s  zBeamParameter.__new__cCs|jt|jS)a 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 )rJrr0)r5r-r-r.r2s zBeamParameter.qcCs|jd|j|jdS)a 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 r!r")rJr0)r5r-r-r.Úradius's zBeamParameter.radiuscCs|jtd|j|jdƒS)aI 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) r!r")Úw_0rrJr0)r5r-r-r.rK6szBeamParameter.wcCst|jt|jƒS)aE 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 )rr0rr4)r5r-r-r.rNJszBeamParameter.w_0cCs|jt|jS)zï 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 )r4rrN)r5r-r-r.Ú divergence^s zBeamParameter.divergencecCst|j|jƒS)zÚ The Gouy phase. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.gouy atan(0.53/pi) )rrJr0)r5r-r-r.Úgouyms zBeamParameter.gouycCsd|jtS)aª 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 r")r4r)r5r-r-r.Úwaist_approximation_limit|sz'BeamParameter.waist_approximation_limitN)r;r<r=r>Ú __slots__r)r?r2rMrKrNrOrPrQr-r-r-r.r Õs-       cCs"tt||fƒ\}}|dt|S)a^ 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 r")rArr)rKr4r-r-r.r “scCs"tt||fƒ\}}t|t|ƒS)ajCalculate 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) )rArrr)r0r4r-r-r.r©scCsRtt||fƒ\}}t|ƒtks*t|ƒtkr>t|ƒtkr:|S|S||||SdS)a¶ 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)rArÚabsr)ÚaÚbr-r-r.r¾scCs tt||fƒ\}}t|| ƒ S)ap 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) )rArr)rTrGr-r-r.rÚscCsˆtt|||fƒ\}}}dd||d||d|}dtd||d||dƒ}|d||d||d}|||fS)aŸ 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) r!rFr")rArr)Ús_inZz_r_inrGÚs_outÚmZz_r_outr-r-r.røs #$$ c KsÎtt|||fƒ\}}}||}t||ƒ}t|ƒdkr>tdƒ‚n†d|krTttdƒƒ‚npd|kr¢t|dƒ}|dtd|d|d|dƒ}t|||ƒd}n"d|kr¸ttdƒƒ‚n ttd ƒƒ‚|||fS) aá 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*(-sqrt(w_i**2/w_o**2 - pi**2*w_i**4/(f**2*l**2)) + 1) >>> 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 r!z,The function expects only one named argumentZdistzD Currently only focal length is supported as a parameterrGr"rrVzG The functions expects the focal length as a named argument) rArr r$r'ÚNotImplementedErrorrrr) r4Zwaist_inZ waist_outrLrXrJrGrVrWr-r-r.r"s$%     ( N)#r>Z __future__rrÚ__all__ZsympyrrrrrrrrrrrZsympy.utilities.miscrrrrrrr r r r r rrrrrrr-r-r-r.ÚsJ4  "[?*