繞射

研究歷史

「光不僅會沿直線傳播、折射和反射，還能夠以第四種方式傳播，即通過繞射的形式傳播。」（"Propositio I. Lumen propagatur seu diffunditur non solum directe, refracte, ac reflexe, sed etiam alio quodam quarto modo, diffracte."[6][8]:149[9]:95

「這樣，我就展示了人們能夠通過何種方式來構想光以球面波連續不斷地傳播出去……」（ "J'ai donc montré de quelle façon l'on peut concevoir que la lumière s'étend successivement par des ondes sphériques, ..."[13]:章1,p18

物理機制

${\displaystyle F={\frac {a^{2}}{L\lambda }}\geq 1}$

${\displaystyle F={\frac {a^{2}}{L\lambda }}\ll 1}$

光的繞射

單縫繞射

${\displaystyle (d/2)\,\sin \theta =\lambda /2}$

${\displaystyle d\,\sin \theta _{min,n}=n\lambda }$

${\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}(d\pi \sin \theta /\lambda )}$

• ${\displaystyle I(\theta )}$為給定角度位置處的輻照度
• ${\displaystyle I_{0}}$初始輻照度
• ${\displaystyle x\neq 0}$時，${\displaystyle \operatorname {sinc} (x)=\sin(\pi x)/(\pi x)}$，在原點處${\displaystyle \operatorname {sinc} (0)=1}$

雙縫繞射

${\displaystyle I(\theta )=I_{m}(\cos ^{2}\beta )\left({\frac {\sin \alpha }{\alpha }}\right)^{2}}$

繞射光柵

${\displaystyle d\left(\sin {\theta _{m}}+\sin {\theta _{i}}\right)=m\lambda .}$

• ${\displaystyle \theta _{i}}$i為光波入射到光柵的角度，如果是垂直入射到平面光柵，則${\displaystyle \sin {\theta _{i}}=0}$
• ${\displaystyle d}$為光柵刻線的間距，也成為光柵常數
• ${\displaystyle m}$為非零整數

繞射光柵強度分佈

${\displaystyle P=D(\theta )*I(\theta )}$

${\displaystyle D={\frac {\sin(\pi *d*\sin(\theta )/\lambda )^{2}*\lambda ^{2}}{(\pi ^{2}*d^{2}*\sin(\theta )^{2})}}}$

I 是干涉因子：

${\displaystyle I={\frac {\sin(\pi *a*\sin(\theta )*N/\lambda )^{2}}{(N^{2}*\sin(\pi *a*\sin(\theta )/\lambda )^{2})}}}$

圓孔繞射

${\displaystyle I(\theta )=I_{0}\left({\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right)^{2}}$

一般孔隙的情況

${\displaystyle \nabla ^{2}\mathbf {E} -\ {\frac {1}{c^{2}}}\ {\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}=-{\frac {1}{\epsilon _{0}}}\left(-\nabla \rho -\ {\frac {1}{c^{2}}}\ {\frac {\partial ^{2}\mathbf {J} }{\partial t^{2}}}\right)}$
${\displaystyle \nabla ^{2}\mathbf {B} -\ {\frac {1}{c^{2}}}\ {\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}=-\mu _{0}\nabla \times \mathbf {J} }$

${\displaystyle \nabla ^{2}\Psi -\ {\frac {1}{c^{2}}}\ {\frac {\partial ^{2}\Psi }{\partial t^{2}}}=-F(\mathbf {r} ,t)}$

${\displaystyle F(\mathbf {r} ,t)=f(\mathbf {r} )e^{-i\omega t}}$

${\displaystyle \nabla ^{2}\psi +k^{2}\psi =-f(\mathbf {r} ),}$（1）

${\displaystyle \nabla ^{2}G(\mathbf {r} ,\mathbf {r} ')+k^{2}G(\mathbf {r} ,\mathbf {r} ')=-\delta (\mathbf {r} -\mathbf {r} ')}$

${\displaystyle \psi (\mathbf {r} )=\int _{\mathbb {V} }f(\mathbf {r} ')G(\mathbf {r} ,\mathbf {r} ')\mathrm {d} ^{3}\mathbf {r} '}$

${\displaystyle \nabla ^{2}G(\mathbf {R} ,\mathbf {O} )+k^{2}G(\mathbf {R} ,\mathbf {O} )=-\delta (\mathbf {R} )}$

${\displaystyle {\frac {1}{R}}{\frac {\mathrm {d} ^{2}}{\mathrm {d} R^{2}}}(RG)+k^{2}G=-\delta (\mathbf {R} )}$

${\displaystyle G(\mathbf {R} ,\mathbf {O} )={\frac {e^{ikR}}{4\pi R}}}$

${\displaystyle G(\mathbf {r} ,\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}}$

{\displaystyle {\begin{aligned}\psi &=\int _{\mathbb {V} }f(\mathbf {r} ')G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{3}\mathbf {r} '=\int _{\mathbb {V} }f_{s}({\boldsymbol {\rho }}')\delta (z')G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{3}\mathbf {r} '\\&=\int _{\mathbb {S} }f_{s}({\boldsymbol {\rho }}')G(\mathbf {r} ,{\boldsymbol {\rho }}')\,\mathrm {d} \sigma '=\int _{\mathbb {S} }f_{s}({\boldsymbol {\rho }}'){\frac {e^{ik|\mathbf {r} -{\boldsymbol {\rho }}'|}}{4\pi |\mathbf {r} -{\boldsymbol {\rho }}'|}}\,\mathrm {d} \sigma '\\\end{aligned}}}

${\displaystyle \psi =\int _{\mathbb {V} }f(\mathbf {r} ')G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{3}\mathbf {r} '=\int _{\mathbb {S} }f_{s}(\mathbf {r} '){\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} \sigma '}$

${\displaystyle \psi (\mathbf {r} )=C\int _{\mathbb {S} }\psi _{inc}(\mathbf {r} '){\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} \sigma '}$

${\displaystyle \psi (\mathbf {r} )=CE_{0}\int _{\mathbb {S} }{\frac {e^{ik|\mathbf {r} -{\boldsymbol {\rho }}'|}}{|\mathbf {r} -{\boldsymbol {\rho }}'|}}\,\mathrm {d} \sigma '}$

{\displaystyle {\begin{aligned}\psi (\mathbf {r} )&\approx CE_{0}{\frac {e^{ikr}}{r}}\int _{\mathbb {S} }e^{-ik{\hat {\mathbf {r} }}\cdot {\boldsymbol {\rho }}'}\,\mathrm {d} \sigma '\\&=CE_{0}{\frac {e^{ikr}}{r}}\int _{0}^{a}\int _{0}^{2\pi }e^{-ik\rho '\sin {\theta }\cos {(\phi -\phi ')}}\,\rho '\mathrm {d} \phi '\mathrm {d} \rho '\\&=2\pi a^{2}CE_{0}{\frac {e^{ikr}}{r}}\ {\frac {J_{1}(ka\sin {\theta })}{ka\sin {\theta }}}\\\end{aligned}}}

${\displaystyle I(\theta )=\psi ^{*}\psi /2=I_{0}\left[{\frac {2J_{1}(ka\sin {\theta })}{ka\sin {\theta }}}\right]^{2}}$

{\displaystyle {\begin{aligned}\psi (\mathbf {r} )&={\frac {1}{4\pi }}\int _{\mathbb {S} }\left[\psi (\mathbf {r} ')\nabla \left({\frac {e^{ikR}}{R}}\right)-{\frac {e^{ikR}}{R}}\nabla \psi (\mathbf {r} ')\right]\cdot \,\mathrm {d} {\boldsymbol {\sigma }}'\\&=-\ {\frac {1}{4\pi }}\int _{\mathbb {S} }\left({\frac {e^{ikR}}{R}}\right)\left[\nabla \psi (\mathbf {r} ')+ik\left(1+{\frac {i}{kR}}\right){\hat {\mathbf {R} }}\psi (\mathbf {r} ')\right]\cdot \,\mathrm {d} {\boldsymbol {\sigma }}'\qquad \qquad \qquad \qquad (2)\\\end{aligned}}}

激光的繞射

${\displaystyle \mathbf {E_{s}} =\mathbf {E_{0}} \exp \left(-{\frac {|\mathbf {r} |^{2}}{\mathbf {\omega _{0}} ^{2}}}\right)}$

${\displaystyle d=f_{1}+f_{2}}$

繞射對於光學系統解像度的制約

${\displaystyle d=1.22\lambda N,\,}$

${\displaystyle \sin \theta =1.22{\frac {\lambda }{D}}}$

繞射波的普遍性質

• 繞射波的角間距與造成繞射的物體的尺寸負相關。也就是說，造成繞射的物體的尺寸越小，它所形成的繞射條紋越寬，反之亦然。例如，在單縫繞射裏，根據公式${\displaystyle d\,\sin \theta =\lambda }$，當入射波的波長${\displaystyle \lambda }$一定時，狹縫寬度${\displaystyle d}$越小，第一極小值對應的${\displaystyle \theta }$就越大，從而造成中間的亮紋寬度增加；
• 某一級繞射角的大小，只取決於入射波的波長與繞射物體尺寸的相對比值；
• 當造成繞射的物體結構具有周期性（例如繞射光柵），則繞射後的圖樣會變得更窄。例如，對比2條狹縫產生的繞射與5條狹縫產生的繞射，兩種情況的狹縫間距相等，不過5條狹縫產生的繞射圖樣更細（參見繞射光柵一節的第三幅插圖）。

粒子繞射

${\displaystyle \lambda ={\frac {h}{p}}}$

布拉格繞射

${\displaystyle m\lambda =2d\sin \theta \,}$

• ${\displaystyle \lambda }$為入射波（如常用的X射線等）波長
• ${\displaystyle d}$為晶面間距
• ${\displaystyle \theta }$為繞射角度
• ${\displaystyle m}$為繞射級數

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