\hQzSr/SQrSSKJr SSKJrJr SSKJr SSK J r J r SSK J r SSKJr SS KJrJr SS KJr SS KJr "S S \5r"SS\5r"SS\5r"SS\5r"SS\5r"SS\5r"SS\5r"SS\5r"SS\5rS%Sjr Sr!S r"S!r#\#r$S"r%S#r&g$)&a 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 )RayTransferMatrix FreeSpaceFlatRefractionCurvedRefraction FlatMirror CurvedMirrorThinLens GeometricRay BeamParameterwaist2rayleighrayleigh2waistgeometric_conj_abgeometric_conj_afgeometric_conj_bf gaussian_conjconjugate_gauss_beams)Expr)Ipi)sympify)imre)sqrt)atan2)MatrixMutableDenseMatrix)together) filldedentcd\rSrSrSrSrSr\S5r\S5r \S5r \S5r S r g ) r;aB Base class for a Ray Transfer Matrix. It should be used if there is not 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 c6[U5S:XaUSUS4USUS44nO`[U5S:Xa1[US[5(aUSRS:XaUSnO [ [ S[ U5-55e[R"X5$)Nr)r$r$z` Expecting 2x2 Matrix or the 4 elements of the Matrix but got %slen isinstancershape ValueErrorrstr__new__clsargstemps U/var/www/auris/envauris/lib/python3.13/site-packages/sympy/physics/optics/gaussopt.pyr,RayTransferMatrix.__new__ss t9>!Wd1g&a$q'(:;D Y!^47F++GMMV+7DZ))+.t9)567 7~~c((c [U[5(a [[U5[U5-5$[U[5(a [[U5[U5-5$[U[5(au[U5[UR 4S45-nUSUS- R SS9n[ UR[[U55[[U55S9$[R"X5$)Nr#rr#T)complex)z_r) r(rrr r qexpandwavelenrrr__mul__)selfotherr0r8s r1r;RayTransferMatrix.__mul__s e. / /$VD\&-%?@ @ | , ,t VE] :; ; } - -$< D'9 ::Daa(((6A !)"Q%%-be_6 6>>$. .r3c US$)z The A parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.A 1 )rrr<s r1ARayTransferMatrix.ADzr3c US$)z The B parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.B 2 )rr#r@rAs r1BRayTransferMatrix.BrDr3c US$)z The C parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.C 3 )r#rr@rAs r1CRayTransferMatrix.CrDr3c US$)z The D parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.D 4 )r#r#r@rAs r1DRayTransferMatrix.DrDr3r@N) __name__ __module__ __qualname____firstlineno____doc__r,r;propertyrBrFrIrL__static_attributes__r@r3r1rr;sb5n ) /        r3rc\rSrSrSrSrSrg)ra  Ray Transfer Matrix for free space. Parameters ========== distance See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import FreeSpace >>> from sympy import symbols >>> d = symbols('d') >>> FreeSpace(d) Matrix([ [1, d], [0, 1]]) c4[RUSUSS5$Nr#rrr,)r.ds r1r,FreeSpace.__new__ ((aAq99r3r@NrNrOrPrQrRr,rTr@r3r1rrs 0:r3rc\rSrSrSrSrSrg)rax 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]]) c^[[X45up[RUSSSX- 5$rXmaprrr,)r.n1n2s r1r,FlatRefraction.__new__s-Wrh' ((aAru==r3r@Nr]r@r3r1rrs 6>r3rc\rSrSrSrSrSrg)ri a Ray Transfer Matrix for refraction on curved interface. Parameters ========== R : Radius of curvature (positive for concave). n1 : Refractive index of one medium. n2 : Refractive index of other medium. See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import CurvedRefraction >>> from sympy import symbols >>> R, n1, n2 = symbols('R n1 n2') >>> CurvedRefraction(R, n1, n2) Matrix([ [ 1, 0], [(n1 - n2)/(R*n2), n1/n2]]) cr[[XU45upn[RUSSX#- U- U- X#- 5$rXra)r.Rrcrds r1r,CurvedRefraction.__new__(s;!- r ((aRWaKNBEJJr3r@Nr]r@r3r1rr s :Kr3rc\rSrSrSrSrSrg)ri-z Ray Transfer Matrix for reflection. See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import FlatMirror >>> FlatMirror() Matrix([ [1, 0], [0, 1]]) c4[RUSSSS5$rXrY)r.s r1r,FlatMirror.__new__?r\r3r@Nr]r@r3r1rr-s ":r3rc\rSrSrSrSrSrg)riCaS Ray Transfer Matrix for reflection from curved surface. Parameters ========== R : radius of curvature (positive for concave) See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import CurvedMirror >>> from sympy import symbols >>> R = symbols('R') >>> CurvedMirror(R) Matrix([ [ 1, 0], [-2/R, 1]]) cP[U5n[RUSSSU- S5$)Nr#rrrr,)r.rhs r1r,CurvedMirror.__new__\( AJ ((aBqD!<>> from sympy.physics.optics import ThinLens >>> from sympy import symbols >>> f = symbols('f') >>> ThinLens(f) Matrix([ [ 1, 0], [-1/f, 1]]) cP[U5n[RUSSSU- S5$)Nr#rrp)r.fs r1r,ThinLens.__new__{rrr3r@Nr]r@r3r1rras 2=r3rc>\rSrSrSrSr\S5r\S5rSr g)r iaV Representation for a geometric ray in the Ray Transfer Matrix formalism. Parameters ========== h : height, and angle : angle, or matrix : a 2x1 matrix (Matrix(2, 1, [height, angle])) Examples ======== >>> from sympy.physics.optics import GeometricRay, FreeSpace >>> from sympy import symbols, Matrix >>> d, h, angle = symbols('d, h, angle') >>> GeometricRay(h, angle) Matrix([ [ h], [angle]]) >>> FreeSpace(d)*GeometricRay(h, angle) Matrix([ [angle*d + h], [ angle]]) >>> GeometricRay( Matrix( ((h,), (angle,)) ) ) Matrix([ [ h], [angle]]) See Also ======== RayTransferMatrix c&[U5S:Xa1[US[5(aUSRS:XaUSnO<[U5S:Xa US4US44nO [ [ S[ U5-55e[R"X5$)Nr#r)r$r#r$z` Expecting 2x1 Matrix or the 2 elements of the Matrix but got %sr&r-s r1r,GeometricRay.__new__s t9>ja&99GMMV+7D Y!^!WJa +DZ))+.t9)567 7~~c((r3c US$)z The distance from the optical axis. Examples ======== >>> from sympy.physics.optics import GeometricRay >>> from sympy import symbols >>> h, angle = symbols('h, angle') >>> gRay = GeometricRay(h, angle) >>> gRay.height h rr@rAs r1heightGeometricRay.height Awr3c US$)z 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@rAs r1angleGeometricRay.angler~r3r@N) rNrOrPrQrRr,rSr|rrTr@r3r1r r s5%N ) r3r c\rSrSrSrSSjr\S5r\S5r\S5r \S5r \S 5r \S 5r \S 5r \S 5r\S 5r\S5r\S5rSrg)r ia& 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. n : the refractive index of medium. 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 Nc[U5n[U5n[U5nUbUc [U5nO-UbUc[[U5X5nOUcUc [S5e[R"XX#U5$)NzMust specify one of w and z_r.)rr r*rr,)r.r:zr7wns r1r,BeamParameter.__new__sq'" AJ AJ ?qy#,C ]s{ W8C [QY=> >||C!!44r3c URS$)Nrr/rAs r1r:BeamParameter.wavelen yy|r3c URS$)Nr#rrAs r1rBeamParameter.z$rr3c URS$)Nr$rrAs r1r7BeamParameter.z_r(rr3c URS$)Nr%rrAs r1rBeamParameter.n,rr3cBUR[UR--$)z 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 )rrr7rAs r1r8BeamParameter.q0svv$(( ""r3cZURSURUR- S---$)z 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$)rr7rAs r1radiusBeamParameter.radius?s)vvqDHHTVVOa//00r3clUR[SURUR- S--5-$)aC The radius of the beam w(z), at any position z along the beam. The beam radius at `1/e^2` intensity (axial value). 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$)w_0rrr7rAs r1rBeamParameter.wNs.(xxQ$&&/A!55666r3cn[UR[UR-- UR-5$)a The minimal radius of beam at `1/e^2` intensity (peak value). See Also ======== w : the beam radius at `1/e^2` intensity (axial value). Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.w_0 0.00100000000000000 )rr7rrr:rAs r1rBeamParameter.w_0ds)$DHHbi(566r3cBUR[- UR- $)z Half of the total angular spread. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.divergence 0.00053/pi )r:rrrAs r1 divergenceBeamParameter.divergencexs||Btxx''r3cB[URUR5$)z The Gouy phase. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.gouy atan(0.53/pi) )rrr7rAs r1gouyBeamParameter.gouysTVVTXX&&r3c.SUR-[- $)as The minimal waist for which the gauss beam approximation is valid. Explanation =========== The gauss beam is a solution to the paraxial equation. For curvatures that are too great it is not a valid approximation. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.waist_approximation_limit 1.06e-6/pi r$)r:rrAs r1waist_approximation_limit'BeamParameter.waist_approximation_limits&~b  r3r@)NNr#)rNrOrPrQrRr,rSr:rr7rr8rrrrrrrTr@r3r1r r s-d 5 # # 1 177*77& ( ( ' '!!r3r cL[[X45upUS-U-[-U- $)a. Calculate the rayleigh range from the waist of a gaussian beam. See Also ======== rayleigh2waist, BeamParameter Examples ======== >>> from sympy.physics.optics import waist2rayleigh >>> from sympy import symbols >>> w, wavelen = symbols('w wavelen') >>> waist2rayleigh(w, wavelen) pi*w**2/wavelen r$)rbrr)rr:rs r1r r s+$Wql+JA a46"9W r3cR[[X45up[U[- U-5$)a>Calculate 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) )rbrrr)r7r:s r1r r s'"w/LC Bw r3c[[X45upUR(dUR(aUR(aU$U$X-X-- $)a Conjugation relation for geometrical beams under paraxial conditions. Explanation =========== Takes the distances to the optical element and returns the needed focal distance. See Also ======== geometric_conj_af, geometric_conj_bf Examples ======== >>> from sympy.physics.optics import geometric_conj_ab >>> from sympy import symbols >>> a, b = symbols('a b') >>> geometric_conj_ab(a, b) a*b/(a + b) )rbr is_infinite)abs r1r r s@0 w DA}} MMq(q(sAE{r3cB[[X45up[X*5*$)aE Conjugation relation for geometrical beams under paraxial conditions. Explanation =========== 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) )rbrr )rrvs r1rrs$6 w DA a $ $$r3c[[XU45upnSSXS-X- - -- SU- -- nS[SX- S-- X- S--5- nUSX- S-- X- S--- nX5U4$)a] Conjugation relation for gaussian beams. Parameters ========== s_in : The distance to optical element from the waist. z_r_in : The rayleigh range of the incident beam. f : The focal length of the optical element. Returns ======= a tuple containing (s_out, z_r_out, m) s_out : The distance between the new waist and the optical element. z_r_out : The rayleigh range of the emergent beam. m : The ration between the new and the old waists. Examples ======== >>> from sympy.physics.optics import gaussian_conj >>> from sympy import symbols >>> s_in, z_r_in, f = symbols('s_in z_r_in f') >>> gaussian_conj(s_in, z_r_in, f)[0] 1/(-1/(s_in + z_r_in**2/(-f + s_in)) + 1/f) >>> gaussian_conj(s_in, z_r_in, f)[1] z_r_in/(1 - s_in**2/f**2 + z_r_in**2/f**2) >>> gaussian_conj(s_in, z_r_in, f)[2] 1/sqrt(1 - s_in**2/f**2 + z_r_in**2/f**2) r#rur$)rbrr)s_inz_r_inrvs_outmz_r_outs r1rrsR'D!#45OD! "dQY112QqS8 :E $TVaKFHq=0 11Adfq[VXM9:G A r3c [[XU45upnX!- n[X5n[U5S:wa [ S5eSU;a[ [ S55eSU;aA[US5nUS[SUS-- US-US-- - 5- -n[XuU5SnO.SU;a[ [ S55e[ [ S 55eXxU4$) a 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 r#z,The function expects only one named argumentdistzD Currently only focal length is supported as a parameterrvr$rrzG The functions expects the focal length as a named argument) rbrr r'r*NotImplementedErrorrrr) r:waist_in waist_outkwargsrrrvrrs r1rrKs V$'wI0N#O GyAx)A 6{aGHH 6 !*.G#HI I  F3K AQq!tVad1a4i/001dq)!, 6 !*.G#HI I%JKL L  r3Nr5)'rR__all__sympy.core.exprrsympy.core.numbersrrsympy.core.sympifyr$sympy.functions.elementary.complexesrr(sympy.functions.elementary.miscellaneousr(sympy.functions.elementary.trigonometricrsympy.matrices.denserrsympy.polys.rationaltoolsrsympy.utilities.miscrrrrrrrrr r r r r rrrrr@r3r1rs. (!&&99:;.+N*Nb:!::>&>B K( KF:":,=$=<= =FT%TvJ!DJ!b, *>%<&-`=r3