-
光栅的衍射效率[11]是指第m级的衍射光能量与入射光总能量之比。计算光栅衍射效率的方法一般分为两大类:分别是标量衍射理论[12]和矢量衍射理论[13]。在标量衍射理论中,把光波当作标量来处理,只考虑电场或磁场的某一个横向分量,而假设其它有关分量也可以用同样方法来独立处理,完全忽略了电磁场矢量分量之间的耦合特性。基尔霍夫衍射理论[14]和瑞利-索末菲理论[15]都属于标量衍射理论。光栅的标量衍射理论是计算光栅衍射效率的一种近似方法,相较于矢量衍射理论其具有数学计算简单、物理概念清晰等特点[16-19]。下面将利用标量衍射理论对矩形槽光栅衍射光强度分布的一般公式进行推导。
矩形槽衍射光栅工作原理如图 1所示。图中,a为光栅缝宽,d为光栅周期(光栅常数)。本文中将分别讨论因加工造成光栅局部缝宽a和周期d出现的误差对光栅各级衍射效率产生的影响。
首先,对光栅衍射强度分布的一般公式进行推导。假设一束单位振幅的单色平行光垂直入射矩形光栅,用积分法求得通过该第n个单缝的衍射光在焦平面上的合成振动公式[20]为:
$ \mathit{\boldsymbol{y}} = {\mathit{\boldsymbol{E}}_0}{a_n}\frac{{\sin \left( {\frac{{{\rm{ \mathsf{ π} }}{a_n}\sin \theta }}{\lambda }} \right)}}{{\frac{{{\rm{ \mathsf{ π} }}{a_n}\sin \theta }}{\lambda }}} $
(1) 式中,y为通过第n个单缝的衍射光在焦平面上的合成振幅;E0为光波振幅;an为第n条狭缝宽度(n=0, 1, 2, …, N-1);θ为衍射角;λ为单色入射光波长。
令${\beta _n} = {a_n}\frac{{\sin \left({\frac{{{\rm{ \mathsf{ π} }}{a_n}\sin \theta }}{\lambda }} \right)}}{{\frac{{{\rm{ \mathsf{ π} }}{a_n}\sin \theta }}{\lambda }}}$,则有:
$ \mathit{\boldsymbol{y}} = {\mathit{\boldsymbol{E}}_0}{\mathit{\beta }_n} $
(2) 式中,由d0sinθ=λ可得θ=arcsin(λ/d0);d0为起始栅距, βn为第n条狭缝衍射光合成振幅系数。
通过N条光缝的N束衍射平行光,其相位均不相同,则通过第n条光缝的衍射光,其光波的向量表达式为:
$ \begin{array}{l} {\mathit{\boldsymbol{E}}_n} = {\mathit{\boldsymbol{E}}_0}{\mathit{\beta }_n}\exp \left[ {{\rm{i}}\left( {\mathit{\omega }t - kr + {\mathit{\delta }_n}} \right)} \right], \\ \;\;\;\;\;\;\;(n = 0, 1, 2, \cdots , N - 1) \end{array} $
(3) $ {\delta _n} = \frac{{2{\rm{ \mathsf{ π} }}\sin \theta }}{\lambda }\sum\limits_{n = 0}^{N - 1} {{d_n}} , (n = 0, 1, 2, \cdots , N - 1) $
(4) 式中,En为通过第n条光缝的光波振幅, ω为圆频率, t为时间, r为初相位, k为波数, δn为第n条狭缝衍射光相位角, dn为第n个栅距。
N束相干光合成的向量表达式为:
$ \begin{array}{l} \mathit{\boldsymbol{E}} = \sum\limits_{n = 0}^{N - 1} {{\mathit{\boldsymbol{E}}_n}} = \sum\limits_{n = 0}^{N - 1} {{\mathit{\boldsymbol{E}}_0}} {\mathit{\beta }_n}\exp \left[ {{\rm{i}}\left( {\mathit{\omega }t - kr + {\delta _n}} \right)} \right] = \\ \;\;\;\;\;\;\;\;{\mathit{\boldsymbol{E}}_0}\exp [{\rm{i}}(\mathit{\omega }t - kr)]\sum\limits_{n = 0}^{N - 1} {{\beta _n}} \exp \left( {{\rm{i}}{\delta _n}} \right) \end{array} $
(5) 则(5)式相对应的共轭波表达式为:
$ \begin{array}{l} {\mathit{\boldsymbol{E}}^*} = \sum\limits_{n = 0}^{N - 1} {\mathit{\boldsymbol{E}}_n^*} = \sum\limits_{n = 0}^{N - 1} {{\mathit{\boldsymbol{E}}_0}} {\mathit{\beta }_n}\exp \left[ { - {\rm{i}}\left( {\mathit{\omega }t - kr + {\mathit{\delta }_n}} \right)} \right] = \\ \;\;\;\;\;\;\;\;{\mathit{\boldsymbol{E}}_0}\exp [ - {\rm{i}}(\omega t - kr)]\sum\limits_{n = 0}^{N - 1} {{\beta _n}} \exp \left( { - {\rm{i}}{\delta _n}} \right) \end{array} $
(6) 则可得N束相干光合成光强度I, 表示为:
$ \begin{array}{l} I = \mathit{\boldsymbol{E}} \cdot {\mathit{\boldsymbol{E}}^*} = \mathit{\boldsymbol{E}}_0^2\left[ {{{\left( {\sum\limits_{n = 0}^{N - 1} {{\beta _n}} \cos {\delta _n}} \right)}^2} + } \right.\\ \;\;\;\;\;\;\;\;\left. {{{\left( {\sum\limits_{n = 0}^{N - 1} {{\beta _n}} \sin {\delta _n}} \right)}^2}} \right] = {I_0}F \end{array} $
(7) 式中, I0=E02为入射光强度, F为衍射-干涉复合因子。(7)式为光栅衍射强度分布的一般公式。
下面将分别对以上加工误差进行理论分析,这里主要讨论了两种情况。
(1) 改变光栅的局部周期d,其它周期d保持相同不变,以此来分析这种加工的周期变化对矩形光栅衍射效率能产生的影响。
由(7)式可知,当光栅狭缝不变时,即a为常数,则有:
$ \begin{array}{l} I = \mathit{\boldsymbol{E}}_0^2\left[ {{{\left( {\sum\limits_{n = 0}^{N - 1} {{\beta _n}} \cos {\delta _n}} \right)}^2} + {{\left( {\sum\limits_{n = 0}^{N - 1} {{\beta _n}} \sin {\delta _n}} \right)}^2}} \right] = \\ \;\;\;\;\;\;\;\mathit{\boldsymbol{E}}_0^2{\beta ^2}\left[ {{{\left( {\sum\limits_{n = 0}^{N - 1} {\cos } {\delta _n}} \right)}^2} + {{\left( {\sum\limits_{n = 0}^{N - 1} {\sin } {\delta _n}} \right)}^2}} \right] \end{array} $
(8) 式中, $\beta = a\frac{{\sin \left({\frac{{{\rm{ \mathsf{ π} }}a\sin \theta }}{\lambda }} \right)}}{{\frac{{{\rm{ \mathsf{ π} }}a\sin \theta }}{\lambda }}}$。
(2) 改变矩形光栅的部分缝宽,但各周期保持不变(即加工周期不存在误差),以此来分析这种加工光栅时产生的缝宽误差对矩形光栅衍射效率的影响。
由(7)式可知,当光栅周期d(光栅常数)不变时,即d为常数,则有:
$ I = \mathit{\boldsymbol{E}}_0^2\left[ {{{\left( {\sum\limits_{n = 0}^{N - 1} {{\beta _n}} \cos {\delta _n}} \right)}^2} + {{\left( {\sum\limits_{n = 0}^{N - 1} {{\beta _n}} \sin {\delta _n}} \right)}^2}} \right] $
(9) 当d为常数时,由(4)式可得:
$ {\delta _n} = \frac{{2{\rm{ \mathsf{ π} }}\sin \theta }}{\lambda }(n + 1)d $
(10) 将(10)式代入(9)式中,可得:
$ \begin{array}{l} I = \mathit{\boldsymbol{E}}_0^2\left[ {{{\left\{ {\sum\limits_{n = 0}^{N - 1} {{\beta _n}} \cos \left[ {\frac{{2{\rm{ \mathsf{ π} }}\sin \theta }}{\lambda }(n + 1)d} \right]} \right\}}^2} + } \right.\\ \left. {\;\;\;\;\;\;{{\left\{ {\sum\limits_{n = 0}^{N - 1} {{\beta _n}} \sin \left[ {\frac{{2{\rm{ \mathsf{ π} }}\sin \theta }}{\lambda }(n + 1)d} \right]} \right\}}^2}} \right] \end{array} $
(11) -
为分析方便,假设在讨论加工周期误差时,不讨论其它加工误差的存在。当N=5,入射光强为λ=632.8nm,理想狭缝宽a=0.0020mm,理想周期(光栅常数)d=0.0100mm时,如果有3个周期是理想的,而有两个周期因加工出现误差0.5μm或1.0μm的误差。针对这一情况,则利用(8)式进行计算,得到衍射光分布结果如图 2所示。
Figure 2. Comparison of diffraction light intensity distribution between ideal grating and error grating with the increase of cylce by formula (8)
图 2中曲线的变化显示了存在周期误差时, ±1, ±2, ±3和±4级次的衍射光强均会发生变化,随着周期误差的增加,各衍射主极大将朝着零级方向靠近。另外衍射光强会有相应降低,具体降低的值如表 1所示。
Table 1. Variations of all order diffraction light intensity under different error rates with the increase of cylce by formula (8)
cycle error rate ±1 ±2 ±3 ±4 5.00% -0.66% -2.41% -4.19% -0.90% 10.00% -3.69% -13.86% -27.63% -38.29% 表 1中给出了光栅周期存在5%和10%的误差时,各级次衍射光强的变化率。相较于理想光栅,随着误差增大,各级次衍射光强会有不同程度的减小,级次越高的衍射光强下降得越快。对所得数据进行归一化处理,如图 3所示。
Figure 3. Relative diffraction light intensity at ±1 order, ±2 order, ±3 order and ±4 order of each group with the increase of cycle by formula (8)
从图 3中可知,在加工过程中造成的矩形光栅周期误差。对比理想光栅可知,当加工误差使得光栅部分周期变大时,将会造成光栅(除零级外)各级次主极大衍射光强度都变小,但随着周期误差变大,各级次衍射光强的变化程度不同。在该误差范围内,±1级处的衍射光强下降程度相对较小,而其它级次的衍射光强度随着级次和误差的增大,衍射光强的下降程度越来越大。由此可知,该周期误差会降低矩形光栅各级次的衍射效率,而且随着误差增大,级次越高的衍射效率下降得越快。
当加工误差使周期变小时,设其中两个周期因加工出现误差0.5μm或1.0μm的误差。则利用(8)式进行计算,得到的衍射光分布结果如图 4所示。
Figure 4. Comparison of diffraction light intensity distribution between ideal grating and error grating with the decrease of cylce by formula (8)
图 4中曲线的变化显示了存在周期误差时, ±1, ±2, ±3和±4级次的衍射光强均会发生变化,随着周期误差的增加,各衍射主极大将远离零级方向朝两边移动。另外衍射光强会有相应降低,具体降低的值列于表 2中。
Table 2. Variations of all order diffraction light intensity under different error rates with the decrease of cylce by formula (8)
cycle error rate ±1 ±2 ±3 ±4 -5.00% -2.12% -8.51% -19.41% -37.68% -10.00% -7.30% -26.93% -53.07% -63.10% 表 2中给出了光栅周期存在-5%和-10%的误差时,各级次衍射光强的变化率。相较于理想光栅,随着误差增大,各级次衍射光强会有不同程度的减小,级次越高的衍射光强下降得越快。对所得数据进行归一化处理,如图 5所示。
Figure 5. Relative diffraction light intensity at ±1 order, ±2 order, ±3 order and ±4 order of each group with the decrease of cycle by formula (8)
从图 5中可知,对比理想光栅可得,当加工误差使得光栅部分周期变小时,同样会造成(除零级外)的各主极大衍射光强度变小;但随着光栅周期误差变大时,各级次衍射光强度的变化程度不同。在该误差范围内,±1级处的衍射光强下降程度相对较小,而其它级次的衍射光强度随着级次和误差的增大,衍射光强的下降程度越来越大。由此可知,该周期误差会降低矩形光栅各级次的衍射效率,而且随着误差增大,级次越高的衍射效率下降得越快。
-
当N=5、入射光强λ=632.8nm、理想周期(光栅常数)d=0.0100mm、理想狭缝宽a=0.0020mm时,如果有3个狭缝是理想的,而有两个狭缝因加工出现0.5μm或1.0μm的误差。针对这一情况,则利用(11)式进行计算,得到的衍射光分布结果如图 6所示。
Figure 6. Comparison of diffraction light intensity distribution between ideal grating and error grating with the increase of slit width by formula (11)
图 6中曲线的变化显示了存在缝宽误差时, ±1, ±2, ±3和±4级次的衍射光强均会发生变化,随着缝宽误差的增加,各衍射主极大光强会有相应降低,具体降低的值列于表 3中。
Table 3. Variations of all order diffraction light intensity under different error rates with the increase of slit width by formula (11)
slits width error rate ±1 ±2 ±3 ±4 25.00% -2.97% -12.24% -29.55% -61.13% 50.00% -6.45% -24.76% -52.33% -65.90% 表 3中给出了光栅缝宽存在25%和50%的误差时,各级次衍射光强的变化率。相较于理想光栅,随着误差增大,各级次衍射光强会有不同程度的减小,级次越高的衍射光强下降得越快。对所得数据进行归一化处理,如图 7所示。
Figure 7. Relative diffraction light intensity at ±1 order, ±2 order, ±3 order and ±4 order of each group with the increase of slit width by formula (11)
从图 7可知,在加工制作矩形光栅过程中产生的缝宽误差,对比理想光栅可得,当加工误差使得光栅部分狭缝宽度变大时,会造成光栅各主极大(除零级外)衍射光强度减小,但随着光栅缝宽误差变大时,各级次衍射光强度的变化程度不同。在该误差范围内,±1级处的衍射光强下降程度相对较小,而其它级次的衍射光强度随着级次和误差的增大,衍射光强的下降程度越来越大。由此可知,该缝宽误差会降低矩形光栅各级次的衍射效率,而且随着误差增大,级次越高的衍射效率下降得越快。
当加工误差使缝宽变小时,设其中两个缝宽因加工出现0.5μm或1μm的误差。则利用(11)式进行计算,得到的衍射光分布结果如图 8所示。
Figure 8. Comparison of diffraction light intensity distribution between ideal grating and error grating with the decrease of slit width by formula (11)
图 8中曲线的变化显示了存在缝宽误差时, ±1, ±2, ±3和±4级次的衍射光强均会发生变化,随着缝宽误差的增加,各衍射主极大光强会有相应增加,具体增加的值列于表 4中。
Table 4. Variations of all order diffraction light intensity under different error rates with the decrease of slit width by formula (11)
slit width error rate ±1 ±2 ±3 ±4 -25.00% 2.40% 10.99% 32.92% 112.06% -50.00% 4.15% 19.71% 63.56% 254.46% 表 4中给出了光栅缝宽存在-25%和-50%的误差时,各级次衍射光强的变化率。相较于理想光栅,随着误差增大,各级次衍射光强会有不同程度的减小,级次越高的衍射光强增大得越快。对所得数据进行归一化处理,如图 9所示。
Figure 9. Relative diffraction light intensity at ±1 order, ±2 order, ±3 order and ±4 order of each group with the decrease of slit width by formula (11)
从图 9可知,对比理想光栅可得,当加工制作光栅出现部分狭缝宽度变小的误差时,会造成光栅各主极大(除零级外)的衍射光强度变大,但随着光栅缝宽误差变大时,各级次衍射光强度的变化程度不同。在该误差范围内,±1级处的衍射光强增大程度相对较小,而其它级次的衍射光强度随着级次和误差的增大,衍射光强的增大程度越来越大。由此可知,该缝宽误差会提高矩形光栅各级次的衍射效率,而且随着误差增大,级次越高的衍射效率增大得越快。
矩形光栅局部结构误差对衍射效率的影响
Influence of partial structure error of rectangular grating on diffraction efficiency
-
摘要: 为了提高高光谱遥感技术的分析精度, 利用标量衍射理论, 对矩形光栅局部结构误差的衍射效率进行了分析, 并计算了光栅局部周期和缝宽误差对衍射效率的影响。结果表明, 当加工误差使得光栅局部缝宽变大时, 会造成光栅各主极大(除零级)衍射光强变小, 这一误差对±1级处的衍射光强影响相对较小, 随着误差的增加, 越高的衍射级次其光强下降得越快; 而当加工误差使得光栅局部缝宽变小时, 会造成光栅各主极大(除零级)衍射光强变大, 同样, 当误差变大时, 相对±1级的衍射光受到的影响而言, 级次越高对应的主极大衍射光强增大得越快。该研究对于加工矩形光栅减小局部周期和缝宽误差控制提供了参考。Abstract: In order to improve the analysis accuracy of hyperspectral remote sensing technology, the diffraction efficiency of partial structural errors of rectangular gratings was analyzed by using scalar diffraction theory. The influence of grating local period and slit width error on diffraction efficiency was calculated. The results show that, when the processing error makes the local slit width of grating wider, the main maxima intensity (excluding zero order) of the grating decreases. This error has a relatively small effect on the intensity of the diffracted light at ±1 order. With the increase of the error, the higher the diffraction order, the faster the intensity of the light decreases. With the increase of the error, relative to the effect on ±1 order diffraction light, the higher the order, the faster the principal maximum diffraction intensity increases. This study provides a reference for the control of local period and slit width error in processing rectangular gratings.
-
Key words:
- gratings /
- diffraction efficiency /
- scalar diffraction theory /
- partial structural error
-
Table 1. Variations of all order diffraction light intensity under different error rates with the increase of cylce by formula (8)
cycle error rate ±1 ±2 ±3 ±4 5.00% -0.66% -2.41% -4.19% -0.90% 10.00% -3.69% -13.86% -27.63% -38.29% Table 2. Variations of all order diffraction light intensity under different error rates with the decrease of cylce by formula (8)
cycle error rate ±1 ±2 ±3 ±4 -5.00% -2.12% -8.51% -19.41% -37.68% -10.00% -7.30% -26.93% -53.07% -63.10% Table 3. Variations of all order diffraction light intensity under different error rates with the increase of slit width by formula (11)
slits width error rate ±1 ±2 ±3 ±4 25.00% -2.97% -12.24% -29.55% -61.13% 50.00% -6.45% -24.76% -52.33% -65.90% Table 4. Variations of all order diffraction light intensity under different error rates with the decrease of slit width by formula (11)
slit width error rate ±1 ±2 ±3 ±4 -25.00% 2.40% 10.99% 32.92% 112.06% -50.00% 4.15% 19.71% 63.56% 254.46% -
[1] JIANG Y X. The research on design meshods and fabricating technology of varied line-space holographic plane grating[D]. Changchun: University of Chinese Academy of Sciences(Changchun Institute of Optics, Fine Mechanics and Physics), 2015: 84-93(in Chinese). [2] LOU J. Design and fabrication of variable-line-space gratings and its application to position sensor[D].Hefei: University of Science and Technology of China, 2006: 76-80(in Chinese). [3] LI X T, BAYANHESHIG, QI X D, et al. Influence and revising method of machine-ruling grating line's curve error, location error on plane grating' performance[J]. Chinese Journal of Lasers, 2013, 40(3): 0308009(in Chinese). doi: 10.3788/CJL201340.0308009 [4] QIAN G L, LI Ch M, CHEN X R, et al. Error analysis of holographic mosaic gratings[J]. Laser Technology, 2013, 37(6): 747-751(in Chinese). [5] YANG R F, ZHU X L, CHEN J N, et al. Error simulation of Ronchi gratings[J]. Laser Technology, 2012, 36(1): 37-41(in Chinese). [6] WANG Y. Research on error compensation technology of grating tiled[D]. Chongqing: Chongqing University, 2017: 13-20(in Chinese). [7] FAN Sh W, ZHOU Q H. Analysis of diffraction characteristics of blazed gratings[J]. Laser Technology, 2010, 34(1): 41-44(in Chinese). [8] POMMET D A, GRANN E B, MOHARAM M G. Effects of process errors on the diffraction characteristics of binary dielectric gratings[J]. Applied Optics, 1995, 34(14): 2430-2435. doi: 10.1364/AO.34.002430 [9] CAO Zh L, LU Zh W, LI F Y, et al. Analysis of fabrication error of subwavelength dielectric gratings[J]. Acta Photonica Sinica, 2004, 33(1): 76-80(in Chinese). [10] GE J P, SHEN W M, LIU Q, et al. Effect of fabrication errors on the diffraction efficiency of sawtooth blazed grating[J]. Infrared and Laser Engineering, 2013, 42(6): 1557-1561(in Chinese). [11] BAYANHESHIG, QI X D, TANG Y G. The general formula to the diffraction efficiency of rectangular grating and its phenomenon of missing orders[J]. Journal of Optoelectronics·Laser, 2003, 14(10): 1021-1024(in Chinese). [12] WANG W M, RUAN D Sh, JING X F. Analysis of accuracy of scalar diffraction theory and effective medium theory for sinusoidal grating[J]. Acta Photonica Sinica, 2016, 45(7): 0705001(in Chinese). [13] ZHENG G G, ZHAN Y, CAO K, et al. Fabrication of subwavelength metal grating and analysis with vector diffraction theory[J]. Chinese Journal of Luminescence, 2013, 34(7): 935-939(in Ch-inese). doi: 10.3788/fgxb20133407.0935 [14] YAN Sh H. Research on design theory and parallel manufacturing technology of binary optical elements[D]. Changsha: School of Mechatronical Engineering & Automation National University of Defense Technology, 2004: 16-17(in Chinese). [15] GUO F Y, LI L H. Comparison on the scalar diffraction integral formulae[J]. Acta Optica Sinica, 2013, 33(2): 0226001(in Chinese). doi: 10.3788/AOS201333.0226001 [16] LIU Q, WU J H. Analysis and comparison of the scalar diffraction theory and coupled-wave theory about grating[J]. Laser Journal, 2004, 25(2): 31-34(in Chinese). [17] JIANG T F, WEN J, JIANG Ch Y. Analysis and comparrison of the vector and scalar diffraction theories about phase gratings[J]. Journal of Guangxi Normal University(Natural Science Edition), 2001, 19(3): 15-18(in Chinese). [18] REN N, LI Zh Ch, YANG H. The study of vectorial diffraction theories for plane waves diffracted at rectangular hole[J]. Laser Journal, 2015, 36(9): 26-29(in Chinese). [19] DENG X J, LI G X. The vector theory and its scalar approximation of the diffraction at a plane aperture in the far-field[J]. Journal of Hefei University of Technology(Natural Science Edition), 2000, 23(6): 999-1002(in Chinese). [20] LI B Sh, WU Zh. The general formula of the light strength distribution for a varied-line space grating[J]. Sensor World, 2004, 10(5): 19-22(in Chinese).