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    Journals >Journal of Applied Optics >Volume 43 >Issue 3 >Page 510 > Article
    • Journal of Applied Optics
    • Vol. 43, Issue 3, 510 (2022)
    Derivation and application of expression methods based on law of reflection and refraction of light
    Xuhuan ZHOU1, Yunhui GONG1, Shaohua WU1,2,*, Yuankang WANG1..., Yafang HUANG1, Hongwei YE3, Guangwei WANG1 and Yicheng WANG1|Show fewer author(s)
    Author Affiliations
  • 1Yunnan KIRO Photonics Co.,Ltd., Kunming 650217, China
  • 2Kunming Institute of Physics, Kunming 650223, China
  • 3The Second Military Representative Department Garrisoned in Kunming of Chongqing Military Representative Bureau, Army Armaments Department of PLA, Kunming 650032, China
    • show less
    DOI: 10.5768/JAO202243.0305001 Cite this Article
    Xuhuan ZHOU, Yunhui GONG, Shaohua WU, Yuankang WANG, Yafang HUANG, Hongwei YE, Guangwei WANG, Yicheng WANG. Derivation and application of expression methods based on law of reflection and refraction of light[J]. Journal of Applied Optics, 2022, 43(3): 510 Copy Citation Text show less

    Abstract

    The reflection and refraction of light are the most basic methods for the application of geometric optics. In the processing of modern precision optical elements, the different expression methods can provide different solutions for light tracing, prism error analysis as well as prism assembly and adjustment. The traditional form expression methods of the law of reflection and the law of refraction of light were introduced, and the expression methods of vector, matrix and quaternion were derived. Through the simulation calculation assisted by Matlab, the reflected rays of incident light in two different areas of the Schmidt prism inspection optical path was symmetrically distributed on both sides of the reflected rays on the front surface in the horizontal direction, which was consistent with the practical application. The application of expression methods of vector, matrix and quaternion in Schmidt prism inspection optical path was realized. The three expression methods can provide scientific and practical solutions for light tracing, prism error analysis as well as prism assembly and adjustment.

    Keywords

    $ \frac{{{\text{sin}}I}}{{{\text{sin}}{i}}} = \frac{{n'}}{n} $(1)

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    $ {{\overrightarrow {\boldsymbol{B}}}} = \overrightarrow {{{{{\boldsymbol{A}}}}'_0}} + \left( { - \overrightarrow {{{{{\boldsymbol{A}}}}_0}} } \right) $(2)

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    $ \overrightarrow{{\boldsymbol{B}}}=-2\overrightarrow{{{\boldsymbol{N}}}_{0}}(\overrightarrow{{{\boldsymbol{A}}}_{0}} \cdot \overrightarrow{{{\boldsymbol{N}}}_{0}}) $(3)

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    $ \overrightarrow{{{\boldsymbol{A}}}'_{0}}=\overrightarrow{{{\boldsymbol{A}}}_{0}}-2\overrightarrow{{{\boldsymbol{N}}}_{0}}(\overrightarrow{{{\boldsymbol{A}}}_{0}} \cdot \overrightarrow{{{\boldsymbol{N}}}_{0}}) $(4)

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    $ {{n}}\sin I = n'\sin i $(5)

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    $ n\overrightarrow{{{\boldsymbol{A}}}_{0}}\times \overrightarrow{{{\boldsymbol{N}}}_{0}}=n'\overrightarrow{{{\boldsymbol{A}}}_{0}''}\times \overrightarrow{{{\boldsymbol{N}}}_{0}} $(6)

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    $ \overrightarrow{{\boldsymbol{A}}}\times \overrightarrow{{{\boldsymbol{N}}}_{0}}=\overrightarrow{{\boldsymbol{A}}''}\times \overrightarrow{{{\boldsymbol{N}}}_{0}} $(7)

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    $ \left(\overrightarrow{{{\boldsymbol{A}}}{''}}-{\overrightarrow{\boldsymbol{A}}}\right)\times \overrightarrow{{{{\boldsymbol{N}}}}_{{0}}}={0} $(8)

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    $ \overrightarrow{{\boldsymbol{C}}}=\overrightarrow{{\boldsymbol{A}}''}+\overrightarrow{13}=\overrightarrow{{\boldsymbol{A}}}+\overrightarrow{23} $(9)

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    $ \overrightarrow{13}=-\overrightarrow{{{\boldsymbol{N}}}_{0}}(\overrightarrow{{\boldsymbol{A}}''} \cdot \overrightarrow{{{\boldsymbol{N}}}_{0}}) $(10)

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    $ \overrightarrow{23}=-\overrightarrow{{{\boldsymbol{N}}}_{0}}(\overrightarrow{{\boldsymbol{A}}} \cdot \overrightarrow{{{\boldsymbol{N}}}_{0}}) $(11)

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    $ \overrightarrow{{\boldsymbol{A}}''}-\overrightarrow{{{\boldsymbol{N}}}_{0}}(\overrightarrow{{\boldsymbol{A}}''} \cdot \overrightarrow{{{\boldsymbol{N}}}_{0}})=\overrightarrow{{\boldsymbol{A}}}-\overrightarrow{{{\boldsymbol{N}}}_{0}}(\overrightarrow{{\boldsymbol{A}}} \cdot \overrightarrow{{{\boldsymbol{N}}}_{0}}) $(12)

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    $ \boldsymbolA″→⋅\boldsymbolN0→=n′cos(π−i)=−n′cosi=−n′2−n′2sin2i=−n′2−n2sin2I=−n′2−n2+n2cos2I=−n′2−n2+n2cos2I=−n′2−n2+(\boldsymbolA→⋅\boldsymbolN0→)2$()

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    $ \overrightarrow{{\boldsymbol{A}}''}=\overrightarrow{{\boldsymbol{A}}}-\overrightarrow{{{\boldsymbol{N}}}_{0}}\left[\overrightarrow{{\boldsymbol{A}}} \cdot \overrightarrow{{{\boldsymbol{N}}}_{0}}+\sqrt{{{n'}}^{2}-{{n}}^{2}+{\left(\overrightarrow{{\boldsymbol{A}}} \cdot \overrightarrow{{{\boldsymbol{N}}}_{0}}\right)}^{2}}\right] $(13)

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    $ \overrightarrow {{{\boldsymbol{A}}_0}'} = {{A}}_{0{{x}}}'\overrightarrow {\boldsymbol{i}} + {{A}}_{0{{y}}}'\overrightarrow {\boldsymbol{j}} + {{A}}_{0{{{\textit{z}}}}}'\overrightarrow {\boldsymbol{k}} $()

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    $ \overrightarrow {{{\boldsymbol{A}}_0}} = {{{A}}_{0{{x}}}}\overrightarrow {\boldsymbol{i}} + {{{A}}_{0{{y}}}}\overrightarrow {\boldsymbol{j}} + {{{A}}_{0{{{\textit{z}}}}}}\overrightarrow {\boldsymbol{k}} $()

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    $ \overrightarrow {{{\boldsymbol{N}}_0}} = {{{N}}_{0{{x}}}}\overrightarrow {\boldsymbol{i}} + {{{N}}_{0{{y}}}}\overrightarrow {\boldsymbol{j}} + {{{N}}_{0{{{\textit{z}}}}}}\overrightarrow {\boldsymbol{k}} $()

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    $ A_{0x}' = \left( {1 - 2N_{0x}^2} \right){A_{0x}} - 2{N_{0x}}{N_{0y}}{A_{0y}} - 2{N_{0x}}{N_{0{\textit{z}}}}{A_{0{\textit{z}}}} $()

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    $ A_{0y}' = - 2{N_{0x}}{N_{0y}}{A_{0x}} + \left( {1 - 2N_{0y}^2} \right){A_{0y}} - 2{N_{0y}}{N_{0{\textit{z}}}}{A_{0{\textit{z}}}} $()

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    $ A_{0{\textit{z}}}' = - 2{N_{0x}}{N_{0{\textit{z}}}}{A_{0x}} - 2{N_{0y}}{N_{0{\textit{z}}}}{A_{0y}} + \left( {1 - 2N_{0{\textit{z}}}^2} \right){A_{0{\textit{z}}}} $()

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    $ \left[ {A0x′A0y′A0z′} \right] = \left[ {1−2N0x2−2N0xN0y−2N0xN0z−2N0xN0y1−2N0y2−2N0yN0z−2N0xN0z−2N0yN0z1−2N0z2} \right]\left[ {A0xA0yA0z} \right] $(14)

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    $ {\boldsymbol{V}} = \left[ {1−2N0x2−2N0xN0y−2N0xN0z−2N0xN0y1−2N0y2−2N0yN0z−2N0xN0z−2N0yN0z1−2N0z2} \right] $(15)

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    $ {{\boldsymbol{A}}_0}' = {\boldsymbol{V}}{{\boldsymbol{A}}_0} $(16)

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    $ I=(−φ)+α−φ=(−β)+isin⁡φ=−yR $(17)

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    $ n\sin \alpha = n\sin \left( {I + \varphi } \right)= n\sin I\cos \varphi + n\cos I\sin \varphi $(18)

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    $ n'\sin \beta = n'\sin \left( {i + \varphi } \right)= n'\sin i\cos \varphi + n'\cos i\sin \varphi $(19)

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    $ n\sin I = n'\sin i $(20)

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    $ \left\{ y′=yn′sin⁡β=nsin⁡α−n′cos⁡i−ncos⁡IRy \right. $()

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    $ \left[ {y′n′sin⁡β} \right] = \left[ {10−n′cos⁡i−ncos⁡IR1} \right]\left[ {ynsin⁡α} \right] $(21)

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    $ \left[ {y′n′β} \right] = \left[ {10−n′−nR1} \right]\left[ {ynα} \right] $(22)

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    $ {\boldsymbol{\lambda}} ={{\lambda}}_{0} \cdot 1+{\lambda }_{1}{\boldsymbol{i}}+{\lambda }_{2}{\boldsymbol{j}}+{\lambda }_{3}{\boldsymbol{k}} $(23)

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    $ \left\{ \boldsymbolij=\boldsymbolk=−\boldsymbolji\boldsymboljk=\boldsymboli=−\boldsymbolk\boldsymbolki=\boldsymbolj=−\boldsymbolik\boldsymboli2=\boldsymbolj2=\boldsymbolk2=\boldsymbolijk=−1 \right. $(24)

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    $ {{\boldsymbol{\lambda }}^*} = {\lambda _0} - {\lambda _1}{\boldsymbol{i}} - {\lambda _2}{\boldsymbol{j}} - {\lambda _3}{\boldsymbol{k}} $(25)

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    $ \left| {\boldsymbol{\lambda }} \right| = \sqrt {\lambda _0^2 + \lambda _1^2 + \lambda _2^2 + \lambda _3^2} $(26)

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    $ {{\boldsymbol{\lambda }}^{ - 1}} = {{\boldsymbol{\lambda }}^*}/{\left| {\boldsymbol{\lambda }} \right|^2} $(27)

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    $ {\boldsymbol{P}} = {{{p}}_0} + {{{p}}_1}{\boldsymbol{i}} + {{{p}}_2}{\boldsymbol{j}} + {{p}_3}{\boldsymbol{k}} $()

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    $ {\boldsymbol{Q}} = {{{q}}_0} + {{{q}}_1}{\boldsymbol{i}} + {{{q}}_2}{\boldsymbol{j}} + {{{q}}_3}{\boldsymbol{k}} $()

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    $ {\boldsymbol{P}} + {\boldsymbol{Q}} = \left( {{{{p}}_0} + {{{q}}_0}} \right) + \left( {{{{p}}_1} + {{{q}}_1}} \right){\boldsymbol{i}} + \left( {{{{p}}_2} + {{{q}}_2}} \right){\boldsymbol{j}} + \left( {{{{p}}_3} + {{{q}}_3}} \right){\boldsymbol{k}} $(28)

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    $ {\boldsymbol{PQ}}={p}_{0}{q}_{0}-{\boldsymbol{p}} \cdot {\boldsymbol{q}}+{p}_{0}{\boldsymbol{q}}+{q}_{0}{\boldsymbol{p}}+{\boldsymbol{p}}\times {\boldsymbol{q}} $(29)

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    $ {\boldsymbol{PQ}}=-{\boldsymbol{p}} \cdot {\boldsymbol{q}}+{\boldsymbol{p}}\times {\boldsymbol{q}} $(30)

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    $ {\boldsymbol{p}} = {{{p}}_1}{\boldsymbol{i}} + {{{p}}_2}{\boldsymbol{j}} + {{{p}}_3}{\boldsymbol{k}} $()

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    $ {\boldsymbol{q}} = {{{q}}_1}{\boldsymbol{i}} + {{{q}}_2}{\boldsymbol{j}} + {{{q}}_3}{\boldsymbol{k}} $()

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    $ \boldsymbolq\boldsymbolp−1=\boldsymbolq\boldsymbolp∗|\boldsymbolp|2=−\boldsymbolqp|\boldsymbolp|2=−−\boldsymbolq⋅\boldsymbolp+\boldsymbolq×\boldsymbolp|\boldsymbolp|2=\boldsymbolp⋅\boldsymbolq+\boldsymbolp×\boldsymbolq|\boldsymbolp|2=pλqλcosθ+\boldsymbolζpλqλsinθpλ2=cos⁡θ+\boldsymbolζsin⁡θ $(31)

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    $ {\boldsymbol{q}} = \left( {\cos \theta + {\boldsymbol{\zeta}} \sin \theta } \right){\boldsymbol{p}} $(32)

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    $ {\boldsymbol{Q}} = \left( {\cos \dfrac{\theta }{2} + {\boldsymbol{\zeta}} \sin \dfrac{\theta }{2}} \right){\boldsymbol{P}}\left( {\cos \dfrac{\theta }{2} - {\boldsymbol{\zeta}} \sin \dfrac{\theta }{2}} \right)={\boldsymbol{uP}}{{\boldsymbol{u}}}^{*} $(33)

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    $ {{\boldsymbol{A}}}'_{0}=\left(\cos\dfrac\pi{2}+{{\boldsymbol{N}}}_{0}\sin\dfrac\pi{2}\right)\left(-{{\boldsymbol{A}}}_{0}\right)\left(\cos\dfrac\pi{2}-{{\boldsymbol{N}}}_{0}\sin\dfrac\pi{2}\right) = {{\boldsymbol{N}}_0}{{\boldsymbol{A}}_0}{{\boldsymbol{N}}_0} $(34)

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    $ {{\boldsymbol{E}}_0} = \frac{1}{{\sin {{I}}}}{{\boldsymbol{N}}_0} \times {{\boldsymbol{A}}_0} $(35)

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    $ {\boldsymbol{A}}_0' = {\boldsymbol{h}}{{\boldsymbol{A}}_0}{{\boldsymbol{h}}^*} $(36)

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    $ {\boldsymbol{A}}_0' = \left[ {\cos \left( {{{I - i}}} \right) + {{\boldsymbol{E}}_0}\sin \left( {{{I - i}}} \right)} \right]{{\boldsymbol{A}}_0} $(37)

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    $ {\boldsymbol{N}}'=\left(\mathrm{cos}\theta +{\boldsymbol{\zeta}} \mathrm{sin}\theta \right){\boldsymbol{N}} $(38)

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    $ \boldsymbolN′=(1+θ\boldsymbolζ)\boldsymbolN=\boldsymbolN+θ\boldsymbolζ\boldsymbolN=\boldsymbolN+θ(−\boldsymbolζ⋅\boldsymbolN+\boldsymbolζ×\boldsymbolN)=\boldsymbolN+\boldsymbolθ\boldsymbolζ×\boldsymbolN=\boldsymbolN+\boldsymbolθ\boldsymbolγ$(39)

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    $ {\gamma _2} = \left[ { - \cos \frac{{\text{π }}}{4}{\text{ }} - \sin \frac{{\text{π }}}{4}{\text{ }}0} \right] $()

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    $ {\gamma _3} = \left[ {\cos \frac{{3{\rm{\pi }}}}{8}{\text{ }} - \sin \frac{{3{\rm{\pi }}}}{8}{\text{ }}0} \right]$()

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    $ {\boldsymbol{N}}_2' = {{\boldsymbol{N}}_2} + {{\boldsymbol{\theta}} _2}{{\boldsymbol{\gamma}} _2} $()

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    $ {\boldsymbol{N}}_3' = {{\boldsymbol{N}}_3} + {{\boldsymbol{\theta}} _3}{{\boldsymbol{\gamma}} _3} $()

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    $ A1p3=[0,4θ22θ33−4θ22θ3+2θ2θ34−12θ2θ32+2θ2−4θ33+4θ3,−θ22θ34+6θ22θ32−θ22+8θ2θ33−8θ2θ3+θ34−6θ32+1,0] $()

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    $ A1p7=[0, −4θ26θ33+4θ26θ3−2θ25θ34+12θ25θ32−2θ25−4θ24θ33+4θ24θ3−4θ23θ34+24θ23θ32−4θ23+4θ22θ33−4θ22θ3−2θ2θ34+12θ2θ32−2θ2+4θ33−4θ3,−θ26θ34+6θ26θ32−θ26+8θ25θ33−8θ25θ3−θ24θ34+6θ24θ32−θ24+16θ23θ33−16θ23θ3+θ22θ34−6θ22θ32+θ22+8θ2θ33−8θ2θ3+θ34−6θ32+1, 0] $()

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    $ {{\boldsymbol{A}}_{{\text{1p}}7}} = \left[ {0{\text{ }}} \right. - 2{\theta _2} - 4{\theta _3}\left. {{\text{ }}1{\text{ }}0} \right] $(40)

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    $ {{\boldsymbol{A}}_{{\text{1p3}}}} = \left[ {0{\text{ }}2{\theta _2} + 4{\theta _3}} \right.\left. {{\text{ }}1{\text{ }}0} \right] $(41)

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    $ θI=−(2θ2+4θ3)/2=−θ2−2θ3=−ΔA−2ΔB=ΔC−ΔB=δ 67.5∘(因为ΔA+ΔB+ΔC=0) $()

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    Xuhuan ZHOU, Yunhui GONG, Shaohua WU, Yuankang WANG, Yafang HUANG, Hongwei YE, Guangwei WANG, Yicheng WANG. Derivation and application of expression methods based on law of reflection and refraction of light[J]. Journal of Applied Optics, 2022, 43(3): 510
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