+ iQ|Rt.R#Ot^R IHt^R IHtHt^R IHt^R I H t H t ^RI H t ^RIHt^RIHtHt^RIHt^RIHt!RR]4t!RR]4t!RR]4t!RR]4t!RR]4t!RR]4t!RR]4t!RR]4t!RR ]4tR$Rlt Rt!Rt"Rt#]#t$R t%R!t&R"#)%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 BeamParameter)Expr)Ipi)sympify)imre)sqrt)atan2)MatrixMutableDenseMatrix)together) filldedentcpa]tRt^;toRtRtRt]R4t]R4t ]R4t ]R4t Rt Vt R #) raB 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 c~\V4^8Xd&V^,V^,3V^,V^,33pMs\V4^8Xd@\V^,\4'd#V^,PR8Xd V^,pM$\ \ R\ V4,44h\P!W4#)z` Expecting 2x2 Matrix or the 4 elements of the Matrix but got %s)rlen isinstancershape ValueErrorrstr__new__clsargstemps&* }/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/physics/optics/gaussopt.pyr!RayTransferMatrix.__new__ss t9>!Wd1g&a$q'(:;D Y!^47F++GMMV+7DZ))+.t9)567 7~~c((c X\V\4'd%\\V4\V4,4#\V\4'd%\\V4\V4,4#\V\4'd\V4\VP 3R34,pV^,V^,, P RR7p\ VP\\V44\\V44R7#\P!W4#)T)complex)z_rr*) rrrr r qexpandwavelenrrr__mul__)selfotherr%r.s&& r&r1RayTransferMatrix.__mul__s e. / /$VD\&-%?@ @ | , ,t VE] :; ; } - -$< D'9 ::Daa(((6A !)"Q%%-be_6 6>>$. .r(c VR,#)z The A parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.A 1 )r6r2s&r&ARayTransferMatrix.ADzr(c VR,#)z The B parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.B 2 )r6r*r7r8s&r&BRayTransferMatrix.Br;r(c VR,#)z The C parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.C 3 )r*r6r7r8s&r&CRayTransferMatrix.Cr;r(c VR,#)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*r7r8s&r&DRayTransferMatrix.Dr;r(r7N)__name__ __module__ __qualname____firstlineno____doc__r!r1propertyr9r=r@rC__static_attributes____classdictcell__ __classdict__s@r&rr;se5n ) /        r(c*a]tRt^toRtRtRtVtR#)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\PV^V^^4#r-rr!)r#ds&&r&r!FreeSpace.__new__ ((aAq99r(r7NrErFrGrHrIr!rKrLrMs@r&rrs0::r(c*a]tRt^toRtRtRtVtR#)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]]) cf\\W34wr\PV^^^W, 4#r-maprrr!)r#n1n2s&&&r&r!FlatRefraction.__new__s-Wrh' ((aAru==r(r7NrUrMs@r&rrs6>>r(c*a]tRtRtoRtRtRtVtR#)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]]) c\\WV34wrp\PV^^W#, V, V, W#, 4#r-rX)r#RrZr[s&&&&r&r!CurvedRefraction.__new__(s;!- r ((aRWaKNBEJJr(r7NrUrMs@r&rr s:KKr(c*a]tRtRtoRtRtRtVtR#)ri-z Ray Transfer Matrix for reflection. See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import FlatMirror >>> FlatMirror() Matrix([ [1, 0], [0, 1]]) c4\PV^^^^4#r-rQ)r#s&r&r!FlatMirror.__new__?rTr(r7NrUrMs@r&rr-s"::r(c*a]tRtRtoRtRtRtVtR#)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]]) cX\V4p\PV^^RV, ^4#)r*rrr!)r#r_s&&r&r!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]]) cX\V4p\PV^^RV, ^4#)r*rg)r#fs&&r&r!ThinLens.__new__{rir(r7NrUrMs@r&rras2==r(cJa]tRtRtoRtRt]R4t]R4tRt Vt R#)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^\V4^8Xd@\V^,\4'd#V^,PR8Xd V^,pMI\V4^8XdV^,3V^,33pM$\ \ R\ V4,44h\P!W4#)r*z` Expecting 2x1 Matrix or the 2 elements of the Matrix but got %s)rr*rr"s&* r&r!GeometricRay.__new__s t9>ja&99GMMV+7D Y!^!WJa +DZ))+.t9)567 7~~c((r(c V^,#)z The distance from the optical axis. Examples ======== >>> from sympy.physics.optics import GeometricRay >>> from sympy import symbols >>> h, angle = symbols('h, angle') >>> gRay = GeometricRay(h, angle) >>> gRay.height h r7r8s&r&heightGeometricRay.height Awr(c V^,#)z The angle with the optical axis. Examples ======== >>> from sympy.physics.optics import GeometricRay >>> from sympy import symbols >>> h, angle = symbols('h, angle') >>> gRay = GeometricRay(h, angle) >>> gRay.angle angle r7r8s&r&angleGeometricRay.anglerur(r7N) rErFrGrHrIr!rJrsrwrKrLrMs@r&r r s8%N ) r(ca]tRtRtoRtRRlt]R4t]R4t]R4t ]R4t ]R 4t ]R 4t ]R 4t ]R 4t]R 4t]R4t]R4tRtVtR#)r ia& Representation for a gaussian ray in the Ray Transfer Matrix formalism. Parameters ========== wavelen : the wavelength, z : the distance to waist, and w : the waist, or z_r : the rayleigh range. n : the refractive index of medium. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.q 1 + 1.88679245283019*I*pi >>> p.q.n() 1.0 + 5.92753330865999*I >>> p.w_0.n() 0.00100000000000000 >>> p.z_r.n() 5.92753330865999 >>> from sympy.physics.optics import FreeSpace >>> fs = FreeSpace(10) >>> p1 = fs*p >>> p.w.n() 0.00101413072159615 >>> p1.w.n() 0.00210803120913829 See Also ======== RayTransferMatrix References ========== .. [1] https://en.wikipedia.org/wiki/Complex_beam_parameter .. [2] https://en.wikipedia.org/wiki/Gaussian_beam Nc\V4p\V4p\V4pVeVf \V4pM1VeVf\\V4W4pMVfVf \R4h\P!WW#V4#)NzMust specify one of w and z_r.)rwaist2rayleighrr r!)r#r0zr,wns&&&&&&r&r!BeamParameter.__new__sq'" AJ AJ ?qy#,C ]s{ W8C [QY=> >||C!!44r(c(VP^,#)r6r$r8s&r&r0BeamParameter.wavelen yy|r(c(VP^,#r-rr8s&r&r|BeamParameter.z$rr(c(VP^,#)rrr8s&r&r,BeamParameter.z_r(rr(c(VP^,#)rr8s&r&r~BeamParameter.n,rr(c RVP\VP,,#)z The complex parameter representing the beam. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.q 1 + 1.88679245283019*I*pi )r|r r,r8s&r&r.BeamParameter.q0svv$(( ""r(c zVP^VPVP, ^,,,#)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,r8s&r&radiusBeamParameter.radius?s)vvqDHHTVVOa//00r(c VP\^VPVP, ^,,4,#)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) )w_0rr|r,r8s&r&r}BeamParameter.wNs.(xxQ$&&/A!55666r(c \VP\VP,, VP,4#)a The minimal radius of beam at `1/e^2` intensity (peak value). See Also ======== w : the beam radius at `1/e^2` intensity (axial value). Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.w_0 0.00100000000000000 )rr,r r~r0r8s&r&rBeamParameter.w_0ds)$DHHbi(566r(c RVP\, VP, #)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 )r0r rr8s&r& divergenceBeamParameter.divergencexs||Btxx''r(c B\VPVP4#)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) )rr|r,r8s&r&gouyBeamParameter.gouysTVVTXX&&r(c >^VP,\, #)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 )r0r r8s&r&waist_approximation_limit'BeamParameter.waist_approximation_limits&~b  r(r7)NNr*)rErFrGrHrIr!rJr0r|r,r~r.rr}rrrrrKrLrMs@r&r r s-d 5 # # 1 177*77& ( ( ' '!!r(cl\\W34wrV^,V,\,V, #)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 )rYrr )r}r0r~s&&&r&r{r{s+$Wql+JA a46"9W r(cb\\W34wr\V\, V,4#)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) )rYrrr )r,r0s&&r&rayleigh2waistrs'"w/LC Bw r(c\\W34wrVP'gVP'dVP'dV#T#W,W,, #)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) )rYr is_infinite)abs&&r&geometric_conj_abrsF0 w DA}}} MMMq(q(sAE{r(cB\\W34wr\W)4)#)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) )rYrr)rrms&&r&geometric_conj_afrs$6 w DA a $ $$r(cz\\WV34wrp^RW^,W, , ,, ^V, ,, p^\^W, ^,, W, ^,,4, pV^W, ^,, W, ^,,, pW5V3#)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) rl)rYrr)s_inz_r_inrms_outmz_r_outs&&& r& gaussian_conjrsR'D!#45OD! "dQY112QqS8 :E $TVaKFHq=0 11Adfq[VXM9:G A r(c \\WV34wrpW!, p\W4p\V4^8wd \ R4hRV9d\ \ R44hRV9dj\VR,4pV^\^V^,, V^,V^,, , 4, ,p\WuV4^,pM/RV9d\ \ R44h\ \ R44hWxV3#)a Find the optical setup conjugating the object/image waists. Parameters ========== wavelen : The wavelength of the beam. waist_in and waist_out : The waists to be conjugated. f : The focal distance of the element used in the conjugation. Returns ======= a tuple containing (s_in, s_out, f) s_in : The distance before the optical element. s_out : The distance after the optical element. f : The focal distance of the optical element. Examples ======== >>> from sympy.physics.optics import conjugate_gauss_beams >>> from sympy import symbols, factor >>> l, w_i, w_o, f = symbols('l w_i w_o f') >>> conjugate_gauss_beams(l, w_i, w_o, f=f)[0] f*(1 - sqrt(w_i**2/w_o**2 - pi**2*w_i**4/(f**2*l**2))) >>> factor(conjugate_gauss_beams(l, w_i, w_o, f=f)[1]) f*w_o**2*(w_i**2/w_o**2 - sqrt(w_i**2/w_o**2 - pi**2*w_i**4/(f**2*l**2)))/w_i**2 >>> conjugate_gauss_beams(l, w_i, w_o, f=f)[2] f z,The function expects only one named argumentdistzD Currently only focal length is supported as a parameterrmrzG The functions expects the focal length as a named argument) rYrr{rrNotImplementedErrorrrr) r0waist_in waist_outkwargsrr|rmrrs &&&, r&conjugate_gauss_beamsrKs V$'wI0N#O GyAx)A 6{aGHH 6 !*.G#HI I  F3K AQq!tVad1a4i/001dq)!, 6 !*.G#HI I%JKL L  r(N)rrrrrrrr r r{rrrgeometric_conj_bfrrr-)'rI__all__sympy.core.exprr sympy.core.numbersr r sympy.core.sympifyr$sympy.functions.elementary.complexesrr(sympy.functions.elementary.miscellaneousr(sympy.functions.elementary.trigonometricrsympy.matrices.denserrsympy.polys.rationaltoolsrsympy.utilities.miscrrrrrrrrr r r{rrrrrrr7r(r&rs. (!&&99:;.+N*Nb:!::>&>B K( KF:":,=$=<= =FT%TvJ!DJ!b, *>%<&-`=r(