Dynamic Polarization M athematica Platform Realized with Poynting Vectors

Dynamic Polarization M athematica Platform Realized with Poynting Vectors

Hee-Joong Yun ^∗

Korean Institute of Science and Technology Information, Daejeon 305-806, Korea

Yong-Dae Choi

Department of Microbiology & Nanomaterials, Mokwon University, Daejeon 302-729, Korea (Received 17 January 2013 : revised 3 August 2013 : accepted 10 October 2013)

The fundamental concept of polarization propagation modes of electromagnetic waves is a newly understood cardinal keyword in the age of speed transport technology and the Standard Model in modern physics. Accordingly, the propagation mechanism of polarized electromagnetism and its helicity have received attention in modern technology and science. We have provided a dy- namic polarization platform presenting various propagation modes interactively with M athematica system. In the platform, we confirm that the propagation mechanism satisfies the Maxwell’s two vector equations graphically by realizing the Poynting vector in time-domain vector functions in a Graphics3D scheme.

PACS numbers: 07.05.Tp, 41.20.Jb, 03.50.De

Keywords: Polarization propagation platform, Mathematica simulation, Vector fields, Helicity, Poynting vector, Maxwell’s vector equations

ƒ

ºß à ÅÊ S Ës ð ' [z º  Œ”  ô” X ¢   Å° Ë Ñà à Å{  E 8 ý MAT HEMAT ICA þ u § » ˜  Û

*

× <r )^ ï B ^∗

ô

 Dz D G õ † < Æl Õ ü t& ñ ˜ Ѓ  ½ ¨" é ¶, @ /„   305-806

L

| ÷ 7 B6 0

3

l q" é ¶ @ /† < Ɠ § p Ò q tÓ ü t
” ¸™ èF † < Æõ , @ /„   302-729

(2013¸   1 Z 4 17{ 9  ~ à Î6 £ §, 2013¸   8 Z 4 3{ 9  à º& ñ ‘ : r ~ à Î6 £ §, 2013¸   10 Z 4 10{ 9  > F  S X ‰& ñ )

F 

g  _  ¼ # F  g : £ ¤$ í “ É r ‰ & ³@ /_  œ í“ ¦5 Å q & ñ ˜ Ð: Ÿ x’   r Û ¼% 7 › ½ ¨» ¡ ¤ _  Ù þ ˜d ” כ ¹^ ‰s  9 ³ ðï  r — ¸+ þ A (Standard Model) \ " f ™ èw n  \  ¦ ½ ©" î   H l ‘ : r כ ¹™ ès  . s ] j ¼ # F  g _  7 ˜' & h  ”  ' Ÿ B j& m 7 £ § õ  Õ ª_ 
‚  • ¸ (he- licity)  H ‰ & ³@ /Ó ü t o † < Æ`  ¦ s K  “ ¦ t d ”  & ñ ˜ Ð or Û ¼% 7 › ½ ¨» ¡ ¤ \  ×  æ כ ¹ô  Ç v 0 >× ¼ ÷ &% 3  . Ä ºo   H ¼ # F  g   _

 7 ˜' & h  ”  ' Ÿ õ & ñ `  ¦ % i 1 l x& h Ü ¼– Ð › ' a ¹ 1 Ï “ ¦ ¼ # F  g _ 
‚  • ¸• ¸ ~ 1 >  ó ø ÍZ > ½ + É Ã º e ”   H ¼ # F  g   ”  ' Ÿ  e  ¦Ï ? @

;

Ÿ

§`  ¦ M athematica r Û ¼% 7 ›\ " f ] j Œ • % i  . e  ¦Ï ? @; Ÿ §“ É r Ÿ í“  h A 7 ˜' _  ”  ' Ÿ õ & ñ `  ¦ r ç ß – • ¸B j“   7 ˜' 

#

QA s – Ð ½ ¨‰ & ³† < ÊÜ ¼– Ð+ ‹ ¼ # F  g  _  ”  ' Ÿ  õ & ñ s  Ð  oÛ ¼R / ÷ 7 ˜' ~ ½ Ó& ñ d ” `  ¦ ë ß –7 á ¤   H „    e ” `  ¦ Õ ªA i ” Ü ¼

–

Ð S X ‰ “  ½ + É Ã º e ” >  % i  .

PACS numbers: 07.05.Tp, 41.20.Jb, 03.50.De

Keywords: ¼ # F  g   ”  ' Ÿ e  ¦Ï ? @; Ÿ §, B Û ¼B jw   r Ó ý t Y Us ‚  , 7 ˜'  © œ,
‚  • ¸, Ÿ í“  h A 7 ˜' , Ð  oÛ ¼R / ÷ ~ ½ Ó& ñ d ” 

∗

E-mail: [email protected] 1118

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License

(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any

medium, provided the original work is properly cited.

I. " e  ] Ø

F 

g   $ 3 % ò s 
 ~ ½ ÓK $ 3 õ  ° ú  “ É r q 1 p x ~ ½ Ó$ í Ó ü t| 9 `  ¦ : Ÿ x õ

½ + É M : 4 Ÿ ¤ Ï ã J] X ÷ &€  " f ~ ½ ӆ ¾ Ó\       É r 5 Å q • ¸– Ð : Ÿ x õ 

÷

&€   ”  1 l x€  s  { 9 & ñ ô  Ç ¼ # F  g (polarization)`  ¦ % 3 `  ¦ à º e ” 



. ¼ # F  g s  µ 1 Ï|  ) a Ê ê t è ß – 190¸   1 l x î ß – ¼ # F  g`  ¦ % 3 l  0 A ô

 Ç  € ª œô  Ç ¼ # F  g   > hµ 1 Ï÷ &“ ¦ ¼ # F  g: £ ¤$ í `  ¦ & h 6   x   H ~ ½ Ó Z O

[ þ t s  F  g# 3 0 A >  ƒ  ½ ¨÷ &“ ¦ z  ´6   x  o÷ &% 3   [1]. ¼ # F  g: £ ¤

$ í

“ É r ¼ # F  g ç ß –[ O > , ¼ # F  g‰ & ³p  â `  ¦ q 2 Ÿ © ô  Ç F  g † < Æl l \  { 9  n ”

 & h 6   x ÷ &% 3 Ü ¼ 9  n š ¸ü < TV  H ‚  + þ A¼ # F  g`  ¦, GPS ü < s  1

l

x: Ÿ x’  “ É r " é ¶+ þ A¼ # F  g „    \  ¦  6   x ô  Ç . ‰ & ³F _  “ ¦¾ ¡ §| 9  _

   É r F  g : Ÿ x’  • ¸ „    _  ¼ # F  g: £ ¤$ í `  ¦ s 6   x   H l Õ ü t

–

Ð | 9 €  •÷ &“ ¦ e ”   [2–5]. [ j@ / F  g @ /% i  Å Ò à º @ /% i “   _

… K ‰Ø ÔÞ Ô  µ 1 ÏÒ q t`  ¦ 0 Aô  Ç Ÿ íž Ðb ” d ç “ É r “ ¦5 Å q F  g„  • ¸ ™ è



\  1 l x{ 9 ô  Ç ¼ # F  g _  Å Ò à º   É r ¿ º Y Us $ \  ¦  6   x ô  Ç



. _ …  K ‰Ø ÔÞ Ô ¼ # F  g% ò  © œ, 3D% ò  © œõ  ¼ # F  g î ß – â 1 p x“ É r s  p

 z  ´6   x  o÷ &% 3 Ü ¼ 9 [ jŸ í_  — ¸m ' a A`  ¦ 0 AK  [ jŸ í_  ¼ #  F 

g ì ø Í6 £ x`  ¦ › ' a ¹ 1 Ïô  Ç .  Œ •¸   2012¸   7 Z 4 CERN\ " f j Ë ²Û ¼ (Higgs) ˜ Д > r _  µ 1 Ï| `  ¦ µ 1 ϳ ð >  H † d Ü ¼– Ð+ ‹ Ä ºÅ Ò_  l ‘ : r

½

¨› ¸\  ¦ l Õ ü t   H ³ ðï  r — ¸+ þ A (Standard Model)s  ¢ - a$ í é ß –

>

\  s Ø Ô! 3   [6–8]. ³ ðï  r — ¸+ þ A\ " f W ^± ˜ Д > r“ É r ¢ , a’ < H


‚

 • ¸ (left-handed helicity)_  ×  æ$ í p  ü < š ¸ É r’ < H
‚  • ¸ (right-handed helicity) _  ì ø Í×  æ$ í p  \ ë ß –  Œ •6   x ô  Ç . Õ ª



 X < s  ×  æ$ í p  _  ì ø Í{ 9  [ þ t s  ¢ ¸ µ 1 Ï|  ) a  €   | 9 | ¾ Ә Ð

”

> r s  L :ä ¼ 9t Ù ¼– Ð t F K  t  à ºw n ÷ &% 3 ~   ³ ðï  r — ¸+ þ A`  ¦   r

 + ‹ ë ß – ô  Ç . { 9  _ 
‚  • ¸ (helicity)  H Ä ºÅ Ò_  l ‘ : r Ó

ü t| 9 `  ¦ ó ø ÍZ >    H l ‘ : r כ ¹™ è H † d Ü ¼– Ð+ ‹ õ † < Æ_  — ¸Ž  H ì  r  

 { 9  Ó ü t o † < Æ\ " f & ñ _ ô  Ç
‚  • ¸– Ð & ñ o ÷ &“ ¦ e ”  . s  ]

j ¼ # F  g : £ ¤$ í “ É r F  g † < Æ, „   l † < Æ, …  ;ë  H † < Æ, Ò q tÓ ü t _  Ä »„  † < Æ,



o S X ‰ ì  r  \ " f ÷  r ë ß –  m   õ † < Æl Õ ü t „   ì  r  \ " f ˜ м # 

&

h Ü ¼– Ð & h 6   xH † d Ü ¼– Ð Õ ª_  Ó ü t o † < Æ& h  > h¥ Æ s  D h\  v >  “  d ” 

÷

&“ ¦ e ”  . Õ ª 1 l x î ß – y Œ • ì  r  \ " f & ñ _   ) a 6   x # Q[ þ t s  í ß –ë ß – 

%

i Ü ¼ 9 „   B j& m 7 £ §`  ¦ l Õ ü tÙ þ ¡~   r y Œ • o ³ ð‰ & ³• ¸ x  © œ& h  s

“ ¦ + þ Ad ” & h Ü ¼– Ð l Õ ü t ÷ &# Q ¼ # F  g _  Ó ü t o † < Æ& h  > h¥ Æ s  ç ß –õ 

÷ &% 3   [9–13]. y n C“ É r „   l  – Ð" f „  l  © œ 7 ˜' ü <  l 



© œ 7 ˜'  ¨ î €    \  ¦ s À Ҁ  " f Ÿ í“  h A 7 ˜' ~ ½ ӆ ¾ ÓÜ ¼– Ð ”   '

Ÿ ÷ &  H 7 ˜' & h   1 l x s Ù ¼– Ð s _  7 ˜' & h  : £ ¤$ í s  Ù þ ˜d ” כ ¹

^

‰s  . Õ ª  X < ¼ # o $ í `  ¦ 0 A # Œ „  l  © œ 7 ˜' – Ðë ß – ³ ðr 

€  " f ×  æ^ o ? ) a › ¸ o† < Êà º\  ¦ Animate † < Êà º– Ð s 1 l x r v   H

~

½ ÓZ O Ü ¼– Ð „    _  „   õ & ñ `  ¦ { 9  _ 
‚  î  r1 l x õ  Ä » 

>  l Õ ü t ô  Ç  â Ä º @ /Â Òì  r s % 3  .
‚  • ¸_  & ñ _ • ¸ “ § F

\      1 l x _  ”  ' Ÿ ~ ½ ӆ ¾ Ó`  ¦   ˜ Ѐ Œ ¤`  ¦ M :_ 
‚  ~ ½ Ó

†

¾ ÓÜ ¼– Ð & ñ _ ô  Ç  â Ä º• ¸ e ” “ ¦,  " é ¶ A á ¤ \ " f ˜ Ð  & ñ _ ô  Ç  â Ä

º• ¸ e ” # Q Ó ü t o † < ƕ ¸[ þ t`  ¦ ™ D ¥ ê ø ÍÛ ¼X O >  “ ¦ e ”   [1,9,14, 15].  8¹ ¡ ¤ s  ¼ # F  g  \  ¦ s À ҍ  H ¿ º 7 ˜'  ~ E, ~ B  Ð  oÛ ¼R / ÷ _ 

7 ˜' ~ ½ Ó& ñ d ” `  ¦ ë ß –7 á ¤ €  " f ”  ' Ÿ ÷ &  H õ & ñ `  ¦ Õ ªA i ” Ü ¼– Ð S X

‰ “  ô  Ç  â Ä º  H  _  \ O % 3  . t F K“ É r ¼ # F  g _  „   õ & ñ s  Ð 

oÛ ¼R / ÷ _  7 ˜' ~ ½ Ó& ñ d ” `  ¦ ë ß –7 á ¤  9 ”  ' Ÿ ÷ &  H „    _  „   

B j& m 7 £ §`  ¦ & ñ S X ‰ >  ³ ðr K  Šғ ¦ : £ ¤ y 
‚  • ¸• ¸ " î S X

‰ >  ó ø ÍZ > ½ + É Ã º e ”   H r  o  Œ •\ O s  € 9 כ ¹  “ ¦ Ò q ty Œ • ô

 Ç . Ä ºo   H Mathematica r Û ¼% 7 ›\ " f ¼ # F  g  _  7 ˜' & h 

”

 ' Ÿ õ & ñ `  ¦ % i 1 l x& h Ü ¼– Ð ³ ð‰ & ³½ + É Ã º e ”   H ¼ # F  g   ”  ' Ÿ  e  ¦ Ï ?

@; Ÿ § hDPP (dynamic polarization platform helicityn)`  ¦ ]

j Œ • % i  . s  e  ¦Ï ? @; Ÿ § \ " f  H ¼ # F  g  _  7 ˜' & h  ”  ' Ÿ õ 

&

ñ `  ¦ Ÿ í“  h A 7 ˜' – Ð ½ ¨‰ & ³† < ÊÜ ¼– Ð+ ‹ ¼ # F  g _  Ó ü t o & h  : £ ¤$ í

`

 ¦ " î S X ‰ >  s K  “ ¦  Ö ¸6   x ½ + É Ã º e ” >  % i  . Ä ºo   H l

  ñ> í ß –õ  Õ ªA i ”  ½ ¨‰ & ³s  „ à Ì Z 4ô  Ç Mathematica r Û ¼% 7 ›

\

" f ¼ # F  g  _  „   — ¸× ¼_  7 ˜' & h  ”  ' Ÿ õ & ñ `  ¦ % i 1 l x& h Ü ¼

–

Ð ³ ðr K  Šҍ  H e  ¦Ï ? @; Ÿ §`  ¦ Manipulate † < Êà º\  ¦  6   x # Œ Graphic3D \ " f ½ ¨‰ & ³ % i  . s  á Ԗ ÐÕ ªÏ þ ›“ É r nb <  ʓ É r cdf 

{ 9 – Ð ] j Œ • H † d Ü ¼– Ð Mathematica r Û ¼% 7 ›\ " f % i 1 l x& h Ü ¼

–

Ð Ã º' Ÿ  ) a   [16–18]. ¢ ¸ô  Ç Mathematica r Û ¼% 7 ›s   © œ‚ à Ì

÷

&t  · ú §“ É r pc r Û ¼% 7 ›\ " f• ¸ r Ó ý t Y Us ‚  `  ¦ z  ´' Ÿ r ~  ´ à º e ”

  H cdf  { 9 • ¸ † < Êa  ] j/ B N ) a  .

II. Mathematica _{r Û ¼%} 7 ›\ " f ¼ # F  g — ¸× ¼ r Ó ý t Y Us ‚  

1) ¼ # F  g  _  7 ˜' & h  : £ ¤$ í

1 p

x ~ ½ Ó$ í B | 9 \ " f Ð  oÛ ¼R / ÷ ~ ½ Ó& ñ d ” _  7 ˜' d ” `  ¦ ë ß –7 á ¤   H 

1 l x ~ ½ Ó& ñ d ” “ É r [14]

∇ ² U = 1 v ²

∂ ² U

∂t ² (1)

s

 . # Œl " f v = (µ) ^−1/2 “   1 p x ~ ½ Ó$ í B | 9 \ " f  1 l x _  5

Å

q • ¸s  9 U(~r, t)  H ( J $ ™[ >  † < Êà º . s  ~ ½ Ó& ñ d ” _  K   H ½ ¨

€

   <  ʓ É r ¨ î €    — ¸¿ º 0 p x  . # Œl " f  H  " é ¶ s  Y O  o

 b  # Q4 R e ” “ ¦ 1 p x ~ ½ Ó$ í ç  H| 9  B | 9 \ " f „     ”  ' Ÿ 

 )

a  “ ¦ & ñ # Œ ¨ î €   – Ð 2 [/ å L ô  Ç . ¿ º 4 Ÿ ¤ ™ è © œ 7 ˜' :

E(~ ˆ r, t) = ~ E 0 e ^{i(~ k.~ r−ωt+φ)} ü < ˆ B(~ r, t) = ~ B 0 e ^{i(~ k.~ r−ωt+φ)}   H d ”  (1) ü < Ð  oÛ ¼R / ÷ _  ¿ º 7 ˜' ~ ½ Ó& ñ d ” , d ”  (2)ü < d ”  (3)`  ¦ ë ß –7 á ¤ K 



 ô  Ç .

~ k · ˆ E = 0, ~ k · ˆ B = 0, ˆ B = √

µ ₀  ₀ ~ k × ˆ E (2) S = ~ ~ E × ~ B (3)

#

Œl " f ˆ E, ˆ B  H 4 Ÿ ¤ ™ è 7 ˜' † < Êà º\  ¦ ~ E, ~ B  H z  ´Ã º 7 ˜' † < Êà º

\

 ¦ ~ k  H  à º 7 ˜' \  ¦ ´ ú ˜ô  Ç . „   l    H s  W 1 7 ˜'  E, ~ ~ B, ~ S, ~ k  d ”  (2)ü < d ”  (3)`  ¦ ë ß –7 á ¤ €  " f ”  ' Ÿ ÷ &  H 7 ˜ '

& h   1 l x s  . Ä ºo   H s  [ þ t 7 ˜' & h   1 l x _  Ó ü t o ‰ & ³ © œ`  ¦

M athematica _  Graphic3D\ " f r  o % i  . ¼ # F  g“ É r :

£ ¤Z > y  ¿ º 7 ˜'  ~ E, ~ B _  y Œ •y Œ •_  $ í ì  r[ þ t _  0 A © œ › ' a > – Ð Ò q t

$ í

÷ &  H 7 ˜' & h   1 l x s  . „  l  © œ 7 ˜'  ~ E ü <  l  © œ 7 ˜'  B ~ _  7 ˜'  Y  L“   Ÿ í“  h A 7 ˜'  ~ S  f ” “ §ý a³ ð >  (orthogo- nal system) \ " f ”  ' Ÿ ÷ &  H „   õ & ñ `  ¦ 7 ˜'  > í ß – # Œ r  y

Œ

• o % i  . „  l  © œ 7 ˜'  ~ E  H ~ E = E x e ˆ x + E y e ˆ y – Ð ³ ðr 

÷

&“ ¦ ~ B • ¸ > á ¤ ° ú  s  ° ú  “ É r xy ¨ î €   ? /\ " f ¿ º 7 ˜'  s À ҍ  H

¨ î

€  \  à ºf ” “   ~ ½ ӆ ¾ ÓÜ ¼– Ð ¨ î €    ”  ' Ÿ  ) a  . s M : ~ E _ 

$ í

ì  r q  b = E ^y /E x   “ ¦ 7 ˜'  ~ E _  ¿ º $ í ì  r _  0 A © œ 

 φ{ 9  M : 4 Ÿ ¤ ™ è„  l  © œ 7 ˜'  ˆ E\  ¦  6 £ § õ  ° ú  s  x-$ í ì  r õ  y-$ í ì  r Ü ¼– Ð
¾ º# Q æ ¼€  

E(~ ˆ r, t) = E 0 [ ˆ e x e ^{i(~ k.~ r−ωt)} + b ˆ e y e ^{i(~ k.~ r−ωt±φ)}

= E 0 e ^{i(~ k.~ r)} [ ˆ e x e ^−iωt + b ˆ e y e ^{−i(ωt∓φ)} (4) s

 ÷ &“ ¦  l  © œ 7 ˜'   H d ”  (2)– Ð > í ß – €    ) a  . # Œl " f i = √

−1 s  . ¨ î €   _   â Ä º, # QÖ ¼ { 9 & ñ ô  Ç 0 Au  ~r\ " f

› '

a ¹ 1 Ï €   e ^{i(~ k.~ r)} _  z  ´Ã º ° ú כ“ É r  © œÃ º– Ð 2 [/ å L ½ + É Ã º e ”  .   

² D

G 4 Ÿ ¤ ™ è„  l  © œ 7 ˜'  ˆ E ü < 4 Ÿ ¤ ™ è l  © œ 7 ˜'  ˆ B _  z  ´Ã º 7 ˜ '

 ~ E ü < ~ B  H y Œ •y Œ •  6 £ § õ  ° ú  s  ³ ðr ½ + É Ã º e ”  .

E(~ ~ r, t) = E 0 [ ˆ e x Cosωt + b ˆ e y Cos(ωt ∓ φ)]

B(~ ~ r, t) = √

µ ₀  ₀ (~ k × ~ E)

= B ₀ [ ˆ e _y Cosωt − b ˆ e _x Cos(ωt ∓ φ)] (5) s

 d ”  (5)  H d ” (2)ü < d ” (3)`  ¦ ë ß –7 á ¤   H z  ´Ã º 7 ˜' † < Ê Ã

º– Ð" f f ” “ §ý a³ ð > \ " f z-~ ½ ӆ ¾ Ó`  ¦ o v   H Ÿ í“  h A 7 ˜ '

 ~ S\  ¦ ³ ðr K  ï  r  . Õ ª  X < ¼ # o $ í `  ¦ 0 AK  zx ¨ î €   

 y-~ ½ ӆ ¾ ÓÜ ¼– Ð ”  ' Ÿ ÷ &  H ¨ î €   \  ¦ ³ ðr   9€   ( ˆ e x → ˆ

e y → ˆ e z ) í  H " fC \ P `  ¦ ( ˆ e z → ˆ e x → ˆ e y ) – Ð í  H¨ 8 Š  o 



À D K(cyclic permutation) €   zx¨ î €   ? /\ " f ¿ º 7 ˜'  © œ E, ~ ~ B  f ” “ § €  " f Õ ª ”  1 l x€  \  à ºf ” “   ~ ½ ӆ ¾ Ó 7 £ ¤ y- ~ ½ Ó

†

¾ Ó ( ˆ e _y ~ ½ ӆ ¾ Ó)Ü ¼– Ð ”  ' Ÿ K 
  H  1 l x`  ¦ Õ ªw n = à º e ”  .

Õ

ªo  # Œ ~ E(x, t, z) = E 0 (bCos(ωt ∓ φ) ˆ e x + Cosωt ˆ e z ), B(x, t, z) = B ~ 0 (Cosωt ˆ e x ) − bCos(ωt ∓ φ) ˆ e z )   ) a  . Ä º o

  H Mathematica _  Graphic3D\ " f t \  ¦ y» ¡ ¤ Ü ¼– Ð ¸ ú š  e ”

_ _  r y Œ • t & h \ " f zx¨ î €  \  ~ E ü < ~ B 7 ˜' \  ¦  o¶ ú ˜³ ð– Ð Õ

ª 9º ¡ § Ü ¼– Ð+ ‹ ¿ º 7 ˜'  © œ_  r ç ß – • ¸B j“   Õ ªA i ”  # QY Us 

\

 ¦ Õ ª 9
y Œ ¤ .   õ & h Ü ¼– Ð Graphic3D ï` ç `  ¦ 0 Aô  Ç r ç ß –

•

¸B j“   7 ˜'  # QY Us  † < Êà º\  ¦  6 £ § õ  ° ú  s  j þ t à º e ”  .

E(t) = E ~ 0 (bCos(ωt ∓ φ) ˆ e x , t ˆ e y , Cosωt ˆ e z ) B(t) = B ~ 0 (Cosωt ˆ e x , t ˆ e y , −bCos(ωt ∓ φ) ˆ e z ) (6) d ”

(6)“ É r Mathematica r Û ¼% 7 ›\ " f 3 " é ¶ Õ ªA i ” `  ¦ Õ ª 9 Å

ҍ  H r ç ß – • ¸B j“   † < Êà º– Ð r ç ß –s  ”  ' Ÿ H † d \     ¿ º € 9 

Fig. 1. (Color online) hDPP block diagram plotting the Poynting vector ~ S in the orthogonal system.

×

¼ 7 ˜'  ƒ  ½ + Ë # Œ Ÿ í“  h A 7 ˜' ~ ½ ӆ ¾ ÓÜ ¼– Ð ”  ' Ÿ    H ¼ # F  g 

\  ¦ ³ ðr K  ï  r  . þ j   H \  ¼ # o ô  Ç E m B js ‚   Ú  ¦`  ¦ s  6

 

x # Œ % i 1 l x& h “   ¼ # F  g   ”  ' Ÿ  E m B js ‚   1 p x s  R / Û  © œ\  ]

jr ÷ &“ ¦ e ”   [9–13]. @ / Òì  r ¼ # F  g  _  x  © œ& h “   1 l x& h  õ

& ñ `  ¦ E m B js ‚   “ ¦ e ” Ü ¼ 9 { 9  ҍ  H  l  © œ“ É r Ò q t| Ä Ì 

“

¦ „  l  © œ 7 ˜' ë ß – ³ ðr   
 [11,13] ¼ # F  g _ 
‚   • ¸\  ¦ ì

ø Í@ /– Ð & ñ _  [9] “ ¦ Ÿ í| à Ì& h _  s 1 l x ë ß – ³ ðr ÷ &“ ¦ e ” # Q" f

”

 & ñ ô  Ç „    _  Ó ü t o † < Æ& h  „   õ & ñ `  ¦ l Õ ü t t  3 l w “ ¦ e ”

 . @ / Òì  r E m B js ‚  [ þ t s  ¿ º > h_  › ¸ o† < Êà º\  ¦ ½ + Ë$ í

# Œ á ÔY Ue ” `  ¦ p o  Õ ªo “ ¦ s \  ¦ Animate † < Êà º– Ð s 1 l x r

v   H l Z O `  ¦ 2 [ “ ¦ e ”  . s   H à º† < Æ& h  † < Êà º_  E m B j s

‚  { 9  ÷  r, Ð  oÛ ¼R / ÷ ~ ½ Ó& ñ d ” `  ¦ ë ß –7 á ¤   H ¿ º 7 ˜'  © œ_  Ó ü t o 

&

h

 „   õ & ñ `  ¦ l Õ ü t   H  כ õ   H   H‘ : r& h Ü ¼– Ð  Ø Ô .   1

l

x s  t 
  H  2 [_  s 1 l x`  ¦
 è ­ q ÷  r „     B | 9 `  ¦ :

Ÿ

x # Œ # Qb  G>  „   ÷ &  H t  Ó ü t o † < Æ& h  õ & ñ `  ¦ ˜ Ð# Œ Å Òt 



 H 3 l w “ ¦ e ”  . s  hdpp á Ԗ ÐÕ ªÏ þ ›“ É r ¿ º 7 ˜'  ~ E, ~ B _  7 ˜ '

& h   1 l x`  ¦ & ñ S X ‰ >  ˜ Ð# Œ º ¡ § Ü ¼– Ð+ ‹ z  ´| 9 & h “   Ÿ í“  h A 7 ˜' _  â ì2 £ §`  ¦ ³ ðr K  ï  r  . à º† < Æ& h  † < Êà º_  E m B js ‚   õ

  H l ‘ : r& h Ü ¼– Ð  Ø Ô . ¼ # F  g  _  `  ¦   É r K $ 3 õ  & h 6   x

“ É

r Ð  oÛ ¼R / ÷ 7 ˜' ~ ½ Ó& ñ d ” `  ¦ ë ß –7 á ¤ €  " f ”  ' Ÿ ÷ &  H ¿ º 7 ˜'  © œ

Fig. 2. (Color online) Dynamic polarization platform (DPP). The simulation will run while you click the ¶ after click the ⊕ of the t control of the platform. The picture shown is the snapshot of the left-handed circular polarized (LHCP) propagation mode. While you click the another mode of the platform then simulator will show the another modes promptly indicating the combination of the present mode. The snapshot is the LHCP EM wave viewed at ViewPoint (1.647,1.137,1.0775). You may decide precisely the helicity of the polarization mode on the platform.

E, ~ ~ B  s À ҍ  H Ÿ í“  h A 7 ˜' _   1 l x`  ¦ & ñ S X ‰ >  l Õ ü t 



 H  כ Ü ¼– РÒ'  r  Œ •  ) a   [19–22]. Ä ºo   H s  [ þ t ¼ # F  g  _  Ó

ü

t o † < Æ& h  „   B j& m 7 £ §`  ¦ Mathematica _  Graphics3D

\

" f r  o % i  .

2) ¼ # F  g   Mathematica r Ó ý t Y Us ‚  

Mathematica _  Graphics3D\  ¦ s 6   x # Œ d ” (6)Ü ¼– Ð

³

ðr   ) a 7 ˜' † < Êà º\  ¦ r ç ß – t\  @ /ô  Ç ~ E ü < ~ B _  7 ˜'  Õ ª 9 Å

Òl \  ¦ ì ø Í4 Ÿ ¤ # Œ ¼ # F  g  _  ”  ' Ÿ õ & ñ `  ¦  z  ´& h Ü ¼– Ð ˜ Ð

#

ŒÅ ҍ  H r  o   H á Ԗ ÐÕ ªÏ þ ›`  ¦ ] j Œ • % i   (Fig. 1 ‚ à Ð

›

¸) 7 ˜' \  ¦ ³ ðr  l  0 A # Œ Mathematica _  Arrow† < Êà º

\

 ¦  6   x % i Ü ¼ 9 x 9 | 9  ) a ì  rì  r \ " f• ¸  o¶ ú ˜³ ð ³ ðr ÷ &

•

¸2 Ÿ ¤ Line Ü ¼– Ð  y Œ •+ þ A`  ¦ Õ ª 9 & ñ “ §ô  Ç 7 ˜'  € 9 × ¼\  ¦ ³ ð‰ & ³

% i  . Mathematica \ " f  H à ºd ” `  ¦ Z …z   H d ” Ü ¼– Ð  ï

` ç

K  ° ú ˜ à º e ” 6 £ § Ü ¼– Ð d ” (6)_  ~ E ü < ~ B _  7 ˜'  # QY Us \  ¦ Arrow – Ð Õ ªo l  0 Aô  Ç  ï` ç `  ¦ In[2]:= _  gr311=ü < ° ú   s

 % i  . Hue[0.6]( ê ø ÍÒ  o) 6 £ § \  „  l  © œ $ í ì  r ° ú כ[ þ t

`

 ¦, Hue[0.016](Ô  „“ É rÒ  o)  6 £ § \   l  © œ $ í ì  r ° ú כ[ þ t`  ¦ d ”  Õ

ª@ /– Ð  s i ç ô  Ç  כ s  . s X O >  €    ê ø ÍÒ  o“ É r ~ E\  ¦, Ô

 „“ É rÒ  o“ É r ~ B\  ¦ ³ ðr  >  H † d Ü ¼– Ð ¿ º 7 ˜'  © œ _  î  r1 l x`  ¦ › ' a

¹

1 Ï   H X < ¼ # o   . Ñ þ ˜'  © œ ¿ º $ í ì  r _  0 A © œ  φ Å Ò

#

Qt “ ¦ r y Œ • t Å Ò# Qt €   [ j $ í ì  r ý a³ ð (E x , t, E _z ) í  H " f

Š

© œs    & ñ ÷ &“ ¦ & h  p1 t (0, t, 0) \ " f & h  p2 t (E x , t, E y )  s 

\

 ¦  o¶ ú ˜³ ð– Ð Õ ªw n = à º e ”  . Ä ºo   H 7 á §  8 " î S X ‰ >  ¿ º 7 ˜ '

\  ¦ Õ ªo l  0 A # Œ Lineõ  PolygonÜ ¼– Ð ¿ º 7 ˜'  ~ E ü <

B\ ~  ¦ B  í  H ç ß – > í ß – # Œ Õ ª 9º ¡ § Ü ¼– Ð+ ‹ ¼ # F  g  _  ”  ' Ÿ õ 

&

ñ `  ¦ o \ O  >  ³ ð‰ & ³ % i  . s  Õ ªA i ”  · ú ˜“ ¦o 7 £ §“ É r  Ò2 Ÿ ¤ _

 hdpp.nb\ " f gr321— ¸Ñ ý t`  ¦ ‚ à Г ¦ # Œ . ~ E ü < ƒ  ½ + ˝ ) a B ~ • ¸ 1 l x r \  Õ ª 9Å Ò>  H † d Ü ¼– Ð ¢ - a„  ô  Ç „   l  _  ¼ # F  g B

j& m 7 £ §`  ¦ l Õ ü t ½ + É Ã º e ”  . ~ E ü < ƒ  ½ + ˝ ) a B ~ ü < † < Êa  ³ ð

Fig. 3. (Color online) Dynamic polarization modes at ViewPoint(1.647,1.137,1.0775): (a) Unpolarized , (b) Linearly polarized, (c)Left-handed circularly polarized(LHCP), (d) Left-handed elliptically polarized(LHEP), (e) Right-handed circularly polarized (RHCP), (f) Right-handed elliptically polarized(RHEP) EM waves. The helicity of the polarization mode is shown on the platform panel.

r

K  º ¡ § Ü ¼– Ð" f Ð  oÛ ¼R / ÷ ~ ½ Ó& ñ d ” _  7 ˜' ~ ½ Ó& ñ d ” s  ë ß –7 á ¤ ÷ &   H  כ `  ¦ 3D Õ ªA i ” Ü ¼– Ð S X ‰ “   ½ + É Ã º e ”   [15].

In[1]:= Mainpulate[

coord = {{Thickness[0.003], Arrowheads[0.01], Arrow[{{{0,0,0},{0,4.5Pi,0}},{{0,0,0},{0,0,1.3}}, {{0,0,0},{1.2,0,0}}}]}};

In[2]:=

gr311= {(Thickness[0.005], Arrowheads[0.01], {Hue[0.6], Arrow[{(0,t,0},

{b Cos[omega t- phi[[1]]],t,Cos[omega t]}}]}, {Hue[0.016],Arrow[ {(0,t,0},

{Cos[omega t],t,-b Cos[omega t-phi[[1]]]}}]}, ControllerLinking -> All} } ;

d ”

 (6)_  & ñ _ \        p '  bü < 7 ˜'  © œ $ í ì  r _

 0 A © œ  φ\    É r ¼ # F  g — ¸× ¼\  ¦ & ñ _ ½ + É Ã º e ”  . b = 1 s “ ¦ φ = 0s €   ‚  + þ A¼ # F  g (linearly polarized, LP)  

  ) a  . b = 1s “ ¦ φ = −π/2s €   ¢ , a’ < H " é ¶¼ # F  g (left- handed circularly polarized, LCP)  s “ ¦ φ = π/2s 

€

  š ¸ É r’ < H " é ¶¼ # F  g (right-handed circularly polarized, RHCP)   . ¢ ¸ b > 0 s “ ¦ φ = ±π/2s €    " é ¶¼ # F  g s 

÷

&  H X < " é ¶+ þ A¼ # F  g õ   ð ø Ít – Ð φ = −π/2s €   ¢ , a’ < H  

"

é

¶¼ # F  g (left-handed elliptically polarized, LHEP)  s “ ¦ φ = π/2 s €   š ¸ É r’ < H  " é ¶¼ # F  g (right-handed elliptically polarized, LHEP)    ) a  .

s

 e  ¦Ï ? @; Ÿ §“ É r Mathematica 8.0 Window ¨ 8 Š â \ " f nb   { 9

õ  cdf  { 9 – Ð ] j Œ •÷ &% 3 Ü ¼ 9 — ¸Ž  H e  ¦Ï ? @; Ÿ § _  Mathemat- ica ¨ 8 Š â \ " f z  ´' Ÿ  ) a  . Mathematica 8.0s  [ O u ÷ &t  · ú §

“ É

r r Û ¼% 7 ›\ " f  H CDF Player(Wolfram  – РÒ'  Á º« і Ð



î  r) [18] \ " f z  ´' Ÿ ½ + É Ã º e ” l  M :ë  H \  á Ԗ ÐÕ ªÏ þ ›“ É r — ¸Ž  H PC ¨ 8 Š â \ " f z  ´' Ÿ  ½ + É Ã º e ”  . hdpp á Ԗ ÐÕ ªÏ þ ›“ É r  s à Ô [23] \ " f  î  r ~ à Î  à º' Ÿ ½ + É Ã º e ”  .

3) ¼ # F  g — ¸× ¼ e  ¦Ï ? @; Ÿ § hDPP _  z  ´' Ÿ õ  î  r6   x

hDPP\  ¦ z  ´' Ÿ r v l  0 AK " f  H Mathematica r Û ¼% 7 › ¨ 8 Š

 â

\ " f hddp.nb <  ʓ É r hddp.cdf  { 9 `  ¦ z  ´' Ÿ  €    ) a  . z  ´ '

Ÿ s  ÷ &€   Fig. 2ü < ° ú  s  œ íl  o€  Ü ¼– Ð LHCP z  ´' Ÿ  @ /l 



o€  s 
 è ß – . e  ¦Ï ? @; Ÿ §“ É r 5 > h_  † à Ô\  ¦ J V , – Ð ½ ¨$ í ÷ &

%

3 Ü ¼ 9 Helicity, Polarization Modes, ViewPointJ  V ,

_  Û ¼0 Au – Ð " é ¶   H ¼ # F  g — ¸× ¼\  ¦ ‚  × þ ˜ ½ + É Ã º e ”  . e  ¦Ï ? @

;

Ÿ

§ \ " f tJ V , _  Ä º8 £ ¤ ⊕ é ß –Æ Ò\  ¦ ¾ ºØ Ԁ   [ jÂ Ò B j¾ »
 

  H X < # Œl \ " f ¶`  ¦ 9 þ ta Ë : €   ¼ # F  g   ”  ' Ÿ  ) a  . s 

†

à Ô\  ¦“ É r  1 l x _  ”  ' Ÿ `  ¦  Ø Ô>  <  ʓ É r Ö ¼o > , r ç ß –ç ß –  

`

 ¦  ú ª>  <  ʓ É r U  ´> , ”  ' Ÿ `  ¦ + ' <  ʓ É r · ú ¡ é ß –> – Ð s 1 l x r ~  ´ Ã

º e ” Ü ¼ 9 ”  ' Ÿ `  ¦ ×  æ t Ù þ ¡   r  > 5 Å q >  ½ + É Ã º e ”  .

omega† à Ô\  ¦ – Ð ”  ' Ÿ  _  Å Ò à º\  ¦    or ~  ´ à º e ”  .

Figure 2  H hDPP à º' Ÿ õ & ñ _  ô  Ç í  H ç ß –`  ¦ Û ¼è ­ s g › ô  Ç  כ s

 . — ¸Ž  H — ¸× ¼_  ”  ' Ÿ õ & ñ `  ¦ í  H ç ß –& h Ü ¼– Ð  Ü ã J à º e ” Ü ¼ 9 # QÖ ¼ í  H ç ß –_  Û ¼è ­ s g ›• ¸ á ԏ 2 ;à Ô ½ + É Ã º e ” Ü ¼ 9 $  © œ½ + É Ã

º e ”  . Ø  ¦§ 4  ) a “   W ¾ ¡ §| 9 “ É r $ V , \  & h 6   x   H X <\ • ¸

’

< HÒ  os  \ O `  ¦ & ñ • ¸– Ð & ñ “ § “ ¦ ‚  " î  . s   7 Hë  H \   6   x

 )

a Õ ªa Ë >“ É r — ¸¿ º hDPP– РÒ'  Û ¼è ­ s # Œ EPS Õ ªa Ë > { 9 – Ð

$

 © œK " f  6   x ô  Ç  כ [ þ t s  . Fig. 3  H hDPP \ " f à º' Ÿ ÷ &



 H 6 > h_  ¼ # F  g — ¸× ¼\  ¦ Û ¼è ­ s g › ô  Ç Ò  re  ¦ s  . — ¸Ž  H Û ¼è ­ s g ›“ É r ViewPoint(1.647,1.137,1,0775) \ " f Û ¼è ­ s ô  Ç  כ s  9 & " f

\

 ¦  Œ •1 l x r &  ¼ # F  g — ¸× ¼\  ¦ e ” _ _  ~ ½ ӆ ¾ Ó\ " f › ' a ¹ 1 Ͻ + É Ã º e ” 



. Fig. 3(a)  H ¼ # F  g ÷ &t  · ú §“ É r „    _  ”  ' Ÿ — ¸_ þ v s  .

s

  H W 1 > h_  è ß –à º\  ¦ µ 1 ÏÒ q tr &  0 A © œ ° ú כ\  ½ + É{ © œ # Œ á Ԗ Ð Õ

ªA b ç ô  Ç  1 l x _  Û ¼è ­ s g ›s  . Fig. 3(b)  H b = 1, φ = 0 _ 

‚

 + þ A¼ # F  g (LP)`  ¦, (c)  H b = 1, φ = −π/2 _  ¢ , a’ < H " é ¶¼ #  F 

g (LHCP)`  ¦, (d)  H b = 0.74, φ = −π/2 _  ¢ , a’ < H  " é ¶

¼

# F  g (LHEP)`  ¦, (e)  H b = 1, φ = π/2 _  š ¸ É r’ < H " é ¶¼ # F  g (RHCP)`  ¦, (f)  H b = 0.74, φ = π/2 _  š ¸ É r’ < H  " é ¶¼ # F  g (RHEP)`  ¦ ³ ðr ô  Ç  כ s  .  1 l x“ É r ¿ º > h_  7 ˜'  © œ`  ¦ ‚  

•

¸   H [ j > h_  s × ¼ 7 ˜'  7 ˜'  © œ_  ”  ' Ÿ `  ¦ “  • ¸ô  Ç



. s  s × ¼ 7 ˜'  ”  ' Ÿ    H ~ ½ ӆ ¾ ÓÜ ¼– Ð ¢ , a’ < H <  ʓ É r š ¸ É r

’

< H`  ¦ y Œ ™  % 3 t ’ < H | à Ìs  o v   H ~ ½ ӆ ¾ Ós  ¼ # F  g  _  ”   '

Ÿ ~ ½ ӆ ¾ Óõ  { 9 u ½ + É M :_  ’ < H s  ¢ , a’ < H s €   ¼ # F  g _ 
‚  • ¸

“¢ , a’ < H” s  ÷ &“ ¦ š ¸ É r ’ < H s €  
‚  • ¸ “š ¸ É r’ < H” s   ) a  .

Figure 3 _  (c), (d)  H ¢ , a’ < H Ü ¼– Ð, (e), (f)  H š ¸ É r’ < H Ü ¼– Ð

¼

# F  g  _ 
‚  • ¸   & ñ  ) a  . Fig. 3(a)_ 
‚   • ¸  H   

&

ñ ½ + É Ã º \ O Ü ¼ 9 Fig. 3(b)_ 
‚  • ¸  H “0” s  . e  ¦Ï ? @; Ÿ §

\

" f ¼ # F  g — ¸× ¼ ”  ' Ÿ ÷ &  H ×  æ s  
 & ñ t ÷ &# Q e ” `  ¦ M : e ”

_ _  ¼ # F  g — ¸× ¼ü < Helicityü < ViewPoint\  ¦ ‚  × þ ˜ # Œ

—

¸× ¼\  ¦  Ë ¨# Q" f › ' a ¹ 1 Ͻ + É Ã º e ” Ü ¼ 9 „  ”  , Ê ê@, & ñ t  ½ + É Ã

º e ” Ü ¼ 9  Ø Ôl \  ¦ › ¸& ñ ½ + É Ã º e ”  . e  ¦Ï ? @; Ÿ § \ " f † à Ô X O

 J V , “ É r e ” _ _  › ¸½ + ËÜ ¼– Ð ‚  × þ ˜½ + É Ã º e ” Ü ¼
 ‚  + þ A— ¸× ¼

 ‚  × þ ˜÷ &% 3 `  ¦ M : (φ = 0)  H Helicity† à ÔX O “ É r ì ø Í6 £ x  t

 · ú §  H  .

Figure 4  H [ j > h_  ¼ # F  g   (LP, RHCP, RHEP)

\

 ¦ ¿ º > h_  › ' a& h , 7 £ ¤ ViewPoint(1.647,1.137,1,0775) ü <

ViewPoint(0,1,0) \ " f ‘ : r Û ¼è ­ s g ›s  . Fig. 4(a),(d)  H

‚

 + þ A¼ # F  g`  ¦, (b),(e)  H š ¸ É r’ < H " é ¶+ þ A¼ # F  g`  ¦, (c),(f)  H

š

¸ É r’ < H  " é ¶¼ # F  g`  ¦ ViewPoint(1.647,1.137,1,0775) ü <

ViewPoint(0,1,0) \ " f y Œ •y Œ • Û ¼è ­ s g › ô  Ç  כ s  . ‚  + þ A¼ # F  g

“ É

r ¿ º 7 ˜'  ~ E, ~ B  ‚  + þ AÜ ¼– Ð ”  1 l x  9 š ¸ É r’ < H " é ¶+ þ A¼ # F  g

“ É

r ¿ º 7 ˜'  " f– Ð f ” y Œ •`  ¦ s À Ҁ  " f ì ø Í r >  ~ ½ ӆ ¾ ÓÜ ¼– Ð

‚

  r† < Ê`  ¦ › ' a ¹ 1 Ï ½ + É Ã º e ”  . š ¸ É r’ < H  " é ¶¼ # F  g • ¸ ¿ º 7 ˜' 



© œs  " f– Ð f ” y Œ •`  ¦ s À Ҁ  " f ì ø Í r >  ~ ½ ӆ ¾ ÓÜ ¼– Ð ‚   r   H

Fig. 4. (Color online) Polarized modes viewed at Viewpoint(1.647,1.137,1,0775) and ViewPoint(0,1,0)on different phase differences φ s : (a) LP (b) RHCP, (c) RHEP at Viewpoint(1.647,1.137,1,0775), (d) LP, (e) RHCP (b = 1, φ = π/2) (f) RHEP (b = 1.217, φ = π/2) at ViewPoint(0,1,0).

X

< ¿ º 7 ˜'   " é ¶+ þ AÜ ¼– Ð ‚   r† < Ê`  ¦ ^  ¦ à º e ”  . š ¸ É r’ < H

"

é

¶+ þ A¼ # F  g “    â Ä ºe ” \ • ¸ r > ~ ½ ӆ ¾ Ós   _ ” `  ¦ Ä »_  # Œ !

ViewPoinr(0,-1,0) _  ~ ½ ӆ ¾ Ó\ " f ˜ Ѐ   r > ~ ½ ӆ ¾ Ós  . s  M : hDPP _  J V , \  ¼ # F  g — ¸× ¼ü < Helicityü < ViewPoint

³

ðr  H † d Ü ¼– Ð ¼ # F  g — ¸× ¼\  ¦ & ñ S X ‰ >  S X ‰ “   ½ + É Ã º e ”  . — ¸

Ž

 H ¼ # F  g — ¸× ¼  H ¿ º 7 ˜'  ~ E, ~ B  † ½ Ó © œ à ºf ” `  ¦ s À Ҁ  " f

”

 ' Ÿ H † d`  ¦ S X ‰ “  ½ + É Ã º e ”  . s   H Ð  oÛ ¼R / ÷ _  ¿ º 7 ˜'  ~ ½ Ó& ñ d ”

, d ” (2)ü < d ” (3)`  ¦ ë ß –7 á ¤ “ ¦ e ” 6 £ §`  ¦ Õ ªA i ” Ü ¼– Ð S X ‰ “   ô

 Ç  כ s  . s  hDDP J V , “ É r M athematica  [ O u ÷ &t 

· ú

§“ É r PC \ " f• ¸ CDF z  ´' Ÿ s  0 p x ô  Ç — ¸Ž  H OS _  PC ¨ 8 Š â

\

" f J V , _  † à Ô\  ¦`  ¦ 9 þ ta Ë :† < ÊÜ ¼– Ð+ ‹ þ j™ èô  Ç 48> h s  © œ _

 ¼ # F  g — ¸× ¼_  „   õ & ñ `  ¦ 
_  e  ¦Ï ? @; Ÿ § \ " f r Ó ý t Y U s

‚   ½ + É Ã º e ”  . Mathematica   © œ‚ Ã Ì  ) a r Û ¼% 7 ›\ " f  H 0

A © œ ° ú כ φ, $ í ì  r q  b, à º' Ÿ r ç ß – t, „   _  Å Ò à º ω\  ¦ e ”  _

– Ð    or &  à º' Ÿ r v €  " f „    _  ”  ' Ÿ õ & ñ `  ¦  z  ´

&

h Ü ¼– Ð › ' a ¹ 1 Ï  9 & ñ “ §ô  Ç ¼ # F  g  _  ”  ' Ÿ  2 [\  ¦ l 2 Ÿ ¤ “ ¦

$

 © œ½ + É Ã º e ”  .

III. + s Ç Â ] Ø

„

   _  ¼ # F  g: £ ¤$ í “ É r ‰ & ³@ /_  & ñ ˜ Ð: Ÿ x’   B j& m 7 £ § õ  ‰ & ³

@

/õ † < Æ`  ¦ s K  l  0 Aô  Ç l ‘ : r כ ¹™ è– Ð Å Ò3 l q ÷ &“ ¦ e ” # Q s  _

 r  o ³ ð‰ & ³ Œ •\ O • ¸ „     l Õ ü t > hµ 1 Ï â Ô q tr @ /\ " f ×  æ כ

¹ô  Ç õ ] j– Ð Â Òy Œ •÷ &“ ¦ e ”  . Ä ºo   H Mathematica r Û ¼

% 7

›\ " f ¼ # F  g  _  7 ˜' & h  ”  ' Ÿ õ & ñ `  ¦ % i 1 l x& h Ü ¼– Ð › ' a ¹ 1 Ͻ + É Ã

º e ”   H ¼ # F  g   ”  ' Ÿ  e  ¦Ï ? @; Ÿ § hDPP\  ¦ ] j Œ • % i  . e  ¦Ï ? @

;

Ÿ

§“ É r ¼ # F  g  _  7 ˜' & h  ”  ' Ÿ õ & ñ õ  Ÿ í“  h A 7 ˜' ü < † < Êa 

¼

# F  g  _ 
‚  • ¸• ¸ ì  r" î >  ½ ¨ì  r ½ + É Ã º e ” >  r y Œ • o† < Ê Ü

¼– Ð+ ‹ ¼ # F  g _  Ó ü t o & h  : £ ¤$ í `  ¦ & ñ S X ‰ >  s K  “ ¦  Ö ¸6   x

½

+ É Ã º e ” >  % i  . ] j Œ •  ) a hDPP J V , “ É r cdf z  ´' Ÿ s   0

p

x ô  Ç — ¸Ž  H PC¨ 8 Š â \ " f  € ª œô  Ç — ¸× ¼_  ¼ # F  g`  ¦  – Ð r Ó ý t Y

Us ‚   ½ + É Ã º e ” Ü ¼ 9, Mathematica   © œ‚ Ã Ì  ) a r Û ¼% 7 ›\ 

"

f  H 0 A © œ ° ú כ, 7 ˜'  € 9 × ¼_  $ í ì  r q , à º' Ÿ r ç ß –, „   _  Å Ò 

à º\  ¦    or &  €  " f „    _  ”  ' Ÿ õ & ñ `  ¦ z  ´y Œ ™ e ” 

>

 › ' a ¹ 1 Ͻ + É Ã º e ”  . ¢ ¸ ViewPoint\  ¦ › ¸& ñ # Œ e ” _ _  ~ ½ Ó

†

¾ Ó\ " f „    _  ”  ' Ÿ õ & ñ `  ¦ › ' a ¹ 1 φ < ÊÜ ¼– Ð „    _  „   

õ

& ñ s  Ð  oÛ ¼R / ÷ ~ ½ Ó& ñ d ” `  ¦ ë ß –7 á ¤   H ”  ' Ÿ   e ” `  ¦ S X ‰ “  ½ + É Ã

º e ”  . s  e  ¦Ï ? @; Ÿ §“ É r s / B N >  “ §Ã º† < Æ_ þ v õ & ñ \ " f <  ʓ É r „  



  > µ 1 σ  ½ ¨ ‰ & ³ © œ\ " f Ä »6   x ô  Ç • ¸½ ¨– Ð  Ö ¸6   x| ¨ c à º e ” `  ¦

 כ

Ü ¼– Ð l @ /ô  Ç .

P

c p 8 ý ò k >

‘

: r ƒ  ½ ¨  H ô  Dz D G õ † < Æl Õ ü t& ñ ˜ Ѓ  ½ ¨" é ¶ s  “ §¹ ¢ ¤ õ † < Æl Õ ü t

”

 < É ª l F K Ü ¼– Ð Ã º' Ÿ    H 2012 ReSEAT á Ԗ ÐÕ ªÏ þ › (@ /† < Æ /

B

N1 l x ƒ  ½ ¨) t " é ¶ \  _  # Œ à º' Ÿ ÷ &% 3 Ü ¼Ù ¼– Ð s \  y Œ ™  × ¼ w n

m  .

APPENDIX A: ” ¼   ×

hdpp.nb

Dynamic polarization platform (ver. 1.03) July 5,2013, c H.J.Yun In[1]:=

Manipulate[

coord = {{Thickness[0.003], Arrowheads[0.01], Arrow[{{{0,0,0},{0,4.5Pi,0}},{{0,0,0},{0,0,1.3}}, {{0,0,0},{1.2,0,0}}}]}};

In[2]:=

gr311= {(Thickness[0.005], Arrowheads[0.01], {Hue[0.6], Arrow[{(0,t,0},

{b Cos[omega t- phi[[1]]],t,Cos[omega t]}}]}, {Hue[0.016],Arrow[ {(0,t,0},

{Cos[omega t],t,-b Cos[omega t-phi[[1]]]}}]}, ControllerLinking -> All} } ;

In[3]:=

gr321 = Join[ Line /@ Flatten[

Table[ {{{ 0, t, 0}, {b Cos [omega t - phi[[1]]], t, Cos [omega t]}}, {{ 0, t, 0}, {Cos [omega t], t, -b Cos [omega t - phi[[1]]]}}},

{t, 0, t + 0.3, 0.1}], 1, ControllerLinking -> All] , Polygon /@ Flatten[

Table[ {{ { b Cos [omega t - phi[[1]]], t, Cos [omega t]} (* E *)

, {b Cos [ omega t - phi[[1]]] - Sign[b Cos [omega t - phi[[1]]]] del2, t + del1 , Cos[omega t] - Sign[Cos [omegat]] del2 },

{b Cos [omega t - phi[[1]]] - Sign[b Cos [omega t - phi[[1]]]] del2, t - del1 , Cos [omega t] - Sign[Cos [omega t]] del2 }} ,

{{ Cos [omega t], t, -b Cos [ omega t - phi[[1]]]} (* B *) , {Cos [omega t] - Sign[ Cos [omega t ]] del2, t + del1 ,

-b Cos [ omega t +phi[[1]]]-Sign[-b Cos [omega t - phi[[1]]]] del2} , {Cos[omega t] - Sign[Cos [omega t]] del2 , t - del1 ,

-b Cos [ omega t - phi[[1]]] - Sign[-b Cos [omega t - phi[[1]]]] del2}}} , {t, 0, t + 0.3, 0.1} ], 1, ControllerLinking -> All ]

/. {del1 -> 0.012, del2 -> 0.025} ];

In[4]:=

gr4 = MapThread[Text, {MapThread[StyleForm,

{{"x", "y", "z", "E", "B", "" }, opts1, opts1, opts1, opts3, opts2, opts4}}], {{1 + 0.60433, 0.5, 0.21}, {0, .85 Pi + 2.25, 0.22}, {0, 0,0.41 + 1.25},

{0, 3 Pi/20 + 0.25 , Cos[ 3 Pi/20] + 0.52, {Cos[ Pi/30] + 0.995, 3.61 Pi/10 + 0.44, -0.1 } ,

{0, 3.6 Pi + 1.8 , 2.43221}}} ]/.

{opts1 -> Sequence[ FontFamily -> "Times", FontSize -> 22, FontColor -> GrayLevel[0], FontSlant -> "Italic"], opts2 -> Sequence[FontFamily -> "Times", FontSize -> 22,

FontColor -> Hue[0.001], FontWeight -> "Bold"], opts3 -> Sequence[FontFamily -> "Times", FontSize -> 22,

FontColor -> Hue[0.6], FontWeight -> "Bold"],

opts4 -> Sequence[FontFamily -> "Italic", FontFace -> Italic, FontSize -> 13, FontColor -> Hue[0.1], FontSlant -> "Italic"] };

In[5]:=

polarAni = Graphics3D[Join[coord, gr311, gr321, gr4],

BoxRatios -> {0.3, 1.0, 0.41}, Boxed -> False, Axes -> False, ControllerLinking -> All, PlotRange -> {{-1.2, 1.2}, {0, 5.35 Pi}, {-1.2, 1.2}} , Lighting -> Automatic,

view = {{0, 200, 0},{0, -200, 0},{1.647, 1.011, 1.176},{1.147, -1.1782, 1.386},{ 3.647, 2.137,1.076}};

ViewPoint -> view[[i]], AspectRatio -> 1, ImageSize -> 560, DisplayFunction ->

$DisplayFunction ];

In[6]:=

Show[polarAni] ],

Delimiter, { t, 0, 4.43 Pi, 0.1}, {omega, 1, 5, 1}

Delimiter , {{b, 1, Style["Polarization Modes", 14, Italic, Bold, Hue[0.6]]}, { 0 ->

Style["Linearly ", 14, Italic], 1 -> Style["Circularly", 14, Italic], 0.63 ->

Style["Elliptically", 14, Italic] }},

Delimiter, {{phi, { Pi/2, Pi/2, Pi/2, Pi/2, Pi/2}, Style["Helicity", 14, Bold, Italic, Hue[0.6]]}, {{0, 0, 0, 0, 0} -> Style["0 ", 14, Hue[0.66], Bold, Italic] , { Pi/2, Pi/2, Pi/2, Pi/2, Pi/2} -> Style[" Right-handed", 14, Bold, Hue[0.70], Italic], {-Pi/2, -Pi/2, -Pi/2, -Pi/2, -Pi/2} -> Style["Left-handed", 14, Bold, Hue[0.76], Italic] , {Pi/3.2, Pi/4.1, Pi/1.5, Pi/1.1, Pi/0.8} -> Style["Undetermined", Hue[0.56], 14, Italic]}}

,Delimiter, {{i, 3, Style["ViewPoint", 14, Italic, Bold, Hue[0.591]]}, {1 -> Style[" {0,1,0}

", 14, Italic], 2 -> Style["{0,-1,0 } ", 14, Italic], 3 -> Style["{x1,y1,z1}", 14, Italic], 4 ->

Style["{x2,y2,z2}", 14, Italic]}}]

REFERENCES

[1] F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (MacGraw-Hill, Inc., 1976), Chap.

24.

[2] M. Kukutsu and Y. Kado, NTT Technology Review 7, 1 (2009).

[3] D. Manandhar, R. Shibasaki and D. Mandhar, in The 2004 International Symposium on GNNS/GPS (December, 2004).

[4] J. P. Gordon and H. Gogelnik, PNAS 97, 4541 (2000).

[5] M. Tonouchi, Nat. Photon. 1, 97 (2007).

[6] Y. Liu, G. Bian, T. Miller and T.-C.Chiang, Phys.

Rev. Lett. 107, 166803 (2011).

[7] Quantum Diaries, http://www.quantumdiaries.org /2011/06/19/helicity-chirality-mass-and-the-higgs/

(accessed July 27, 2013).

[8] G. Aad, B. Abbott, J. Adallab, S. Abdelkhalek and A. Abdesselam et al., PRB 108, 111803 (2012).

[9] Understanding circular dichroism, http://www.

photophysics.com/tutorials/circular-dichroism-cd- spectroscopy/1-understanding-circular-dichroism (accessed July 27, 2013).

[10] Circularly and linear polarized light and optical activity, http://ja01.chem.buffalo.edu/jochena/ re- search/opticalactivity.html (accessed July 27, 2013).

[11] Right Hand Circularpolarization animation- Youtube, http://www.youtube.com/watch?v=jY9 hnDzA6Ps (accessed July 27, 2013).

[12] Wolfram Demonstration Project: Circular and Elliptic Polarization of Light Waves, http://demonstrations.wolfram.com /CircularAn- dEllipticPolarizationOfLightWaves/ (accessed July 27, 2013).

[13] Wolfram Demonstration Project: Polarization of Optical Wave through Polarizer and WavePlates, http://demonstrations.wolfram.com/Polarization OfAnOpticalWaveThroughPolarizersAndWave- Plates/(accessed July 27, 2013).

[14] G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Holt, Reinhart and Winston, 1975), Chap. 2.

[15] M. Born and E. Wolf, Principle of Opics, 7th ed.

(Cambridge University Press, Cmbridge, 1999), p.

30; ibid p. 795.

[16] Mathematica System Used to Process and Typeset Formulas in U.S. Patents,

http://www.wolfram.com/news/patents.html (accessed July 27, 2013).

[17] Wolfram|Alpha:Computational Knowledge Engine, http://www.wolframalpha.com/ (accessed July 27, 2013).

[18] Wolfram CDF Player for Interactive Computable Document Format, http://www.wolfram.com/cdf- player/ (accessed July 27, 2013).

[19] R. Ehrlich, J. Tuszynski, L. Roelofs and R. Stoner, Electricity and Magnetism Simlations The Consor- thium for Upper-level Physics Software (John Willey

& Sons, Inc, 1995), p. 154.

[20] P. T. Tamm, A Physicist’s Guide to Mathematica (Academic Press, San Diago 1997), p. 291.

[21] H. J. Yun, Sae Mulli 50, 134 (2005).

[22] H. J. Yun, Visualizing Electromagnetic Vec- tor Fields in the Matter, 1st Korea Math- ematica User Conference (Seoul, Nov. 2007), http://home.mokwon.ac.kr/ ˜ heejy/program.htm. † the polarization simulation in ver. 5.2 presented (un- published).

[23] hdpp programs, http://home.mokwon.ac.kr/ ^∼ heejy

/program.htm (accessed July 27, 2013).