Image Quality Improvement of Volume Hologram Printer with Digital Filter ର੷ܗϙϫήϧϞϕϨϱν͹υΡζνϩ

Image Quality Improvement of Volume Hologram Printer with Digital Filter ର੷ܗϙϫήϧϞϕϨϱν͹υΡζνϩ

ϓΡϩνͶΓΖժ࣯րવͶؖͤΖݜڂ

January, 2021

Computer Science Major

Graduate School of Science and Technology Doctor Course

Nihon University

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O(x, y) =Oo(x, y)2tT[iφo(x, y)] UkXRV R(x, y) =Rr(x, y)2tT[iφr(x, y)] UkXkV

e

Hologram plane Reference beam

Object

T

y

x

O

z Object beam

( x

, y

, z

)

6B;m`2 kXR, +H+mHiBQM Q7 ?QHQ;`KX

r?2`2 O<sub>o</sub>(x, y), R<sub>r</sub>(x, y),̵<sub>o</sub>(x, y) M/ ̵<sub>r</sub>(x, y) `2 i?2 KTHBim/2 M/ T?b2 /Bb@

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U(x, y) =O(x, y) +R(x, y) UkXjV i?2 BMi2MbBiv /Bbi`B#miBQM 7i2` i?2 BMi2`72`2M+2 Bb b 7QHHQrbƕ

I(X, Y) = |O(X, Y) +R(X, Y)|²

= |O|²+|R|²+ 2{OR^∗}

= |O|²+R²_r+ 2O_oR_r+Qb[φ_r(x, y)−φ_o(x, y)] UkX9V AM i?2 #Qp2 7Q`KmH- i?2 }`bi i2`K M/ i?2 b2+QM/ i2`K `2 i?2 BMi2MbBiB2b Q7 i?2 `272`2M+2 HB;?i M/ i?2 Q#D2+i HB;?i r?2M i?2v `2 b2T`i2Hv QM i?2 THM2 ?X i?2B` bmK `2T`2b2Mib i?2 p2`;2 BMi2MbBiv Q7 i?2 BMi2`72`2M+2 7`BM;2bX h?2 i?B`/

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t(x, y) = ̢(|O|²+R²_r

+2OoRr+Qb[φr(x, y)−φo(x, y)]) UkX8V

d

kXj qp2@7`QMi /BbTHv

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i?`Qm;? i?2 ?QHQ;`K THM2 >,

Uo(X, Y) = R(x, y)t(x, y)

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1+? TQBMi QM i?2 ?QHQ;`K `2+Q`/b HH i?2 BM7Q`KiBQM Q7 i?2 rp2 2KBii2/

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kX9 *QKTmi2`@;2M2`i2/ ?QHQ;`T?v

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Laser Mirror PC

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Green Laser Shutter

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L2

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CGH on LCOS (Elemental

hologram) Holographic plate PBS2

PC PBS: Polarizing beam splitter

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/BK2MbBQMH Q#D2+i Bb HQ+i2/ QM i?2 ?QHQ;`K THM2- M/ i?2 pQHmK2 ?QHQ;`K Bb }MHHv QmiTmiX AM //BiBQM- #27Q`2 +QKTmiBM; i?2 i`;2i /i b i?2 2H2K2Mi

?QHQ;`K- i?2 pB`imH rBM/Qr Bb mb2/ iQ b2;K2Mi i?2 i`;2i /iX b b?QrM BM i?2 };m`2- i?2 pB`imH rBM/Qr bTHBib i?2 ?QHQ;`K THM2 i 2[mH BMi2`pHbX AM //BiBQM- M 2H2K2Mi ?QHQ;`K ?b M QT2M pB`imH rBM/QrX PMHv Q#D2+ib TbbBM;

i?`Qm;? i?2 pB`imH rBM/Qr +M #2 mb2/ iQ +H+mHi2 i?2 2H2K2Mi ?QHQ;`KX AM Qi?2` rQ`/b- i?2 2H2K2Mi ?QHQ;`K `2+Q`/b i?2 Q#D2+i /i b2;K2Mi2/ #v i?2 pB`imH rBM/QrX AM Q`/2` iQ b2;K2Mi i?2 i`;2i /i BM T`+iB+H TTHB+iBQM- i?2 `2+Q`/BM; `2 Bb //2/ iQ i?2 i`;2i TQBMi BM7Q`KiBQM QM2 #v QM2X h?2

`2+Q`/BM; `2 Bb i?2 `2 iQ #2 +H+mHi2/ 7Q` i?2 i`;2i TQBMi QM i?2 2H2K2Mi

?QHQ;`KX h?2 };m`2 jXj b?Qrb i?2 KQ/2H 7Q` +H+mHiBM; i?2 `2+Q`/BM; `2X A7 i?2 HB;?i 7`QK i?2 i`;2i TQBMi Tbb2b i?`Qm;? #Qi? i?2 QT2M pB`imH rBM/Qr M/

i?2 2H2K2Mi ?QHQ;`K- i?2 TQBMi Bb `2+Q`/2/ QM i?2 2H2K2Mi ?QHQ;`KX AM Q`/2`

iQ +H+mHi2 i?2 `2+Q`/BM; `2 ;2QK2i`B+HHv- r2 QMHv +QMbB/2` i?2 HB;?i TbbBM;

i?`Qm;? #Qi? 2M/b Q7 i?2 QT2M pB`imH rBM/QrX

jXj *H+mHiBQM K2i?Q/

h?2 T`BK`v THM2 ?QHQ;`K Bb +H+mHi2/ b i?2 H2MbH2bb 6Qm`B2` ?QHQ;`K b b?Qr BM 6B;X jX9Xh?2 pQHmK2 ?QHQ;`K +M #2 QTiB+HHv `2+Q`/2/ 7`QK i?2 *:>

T`BMi2/ #v i?2 ?QHQ;`T?B+ 7`BM;2 T`BMi2` FMQrM b i?2 QTiB+H i`Mb72`X h?2 BK@

;2 `2+QMbi`m+i2/ 7`QK i?2 *:> Bb `2+Q`/2/ BM T?QiQ b2MbBiBp2 Ki2`BH b i?2 pQHmK2 `2~2+iBQM ?QHQ;`KX Hi?Qm;? i?2 *:> +M #2 `2+Q`/2/ miQKiB+HHv- i?2 QTiB+H i`Mb72` b?QmH/ #2 /QM2 KMmHHvX h?2 #bB+ B/2 Q7 i?2 pQHmK2 ?QHQ@

;`K T`BMi2` Bb `2TH+BM; i?2 T`BMi2/ *:> iQ i?2 aGJX AM i?Bb };m`2- Bi b?Qrb KQ/2H iQ +H+mHi2 i?2 2H2K2MiH ?QHQ;`K b i?2 H2MbH2bb 6Qm`B2` ?QHQ;`KX `272`2M+2 TQBMi bQm`+2 Bb HQ+i2/ mM/2` i?2 QT2M pB`imH rBM/Qr- M/ i?2 Q#@

D2+i TQBMi bQm`+2 Bb mb2/ 2t+2Ti 7Q` i?2 /`F xQM2X h?2 2H2K2MiH ?QHQ;`K Bb +H+mHi2/ #v mbBM; i?2 `2H T`i Q7 i?2 6`2bM2H ?QHQ;`K +H+mHiBQM b 7QHHQrb,

I(x, y) = N

i=1

ai

r_i +Qb{kri+φR(x, y) +φi}, UjXRV r?2`2 N Bb i?2 MmK#2` Q7 i?2 Q#D2+i TQBMib- a_i Bb i?2 `2H@pHm2/ KTHBim/2 M/ φi Bb BMBiBH T?b2 Q7 i?2 i@i? Q#D2+i TQBMiX h?2 rp2 MmK#2` k Bb /2}M2/ b k = 2π/λ- r?2`2 λ Bb i?2 7`22@bT+2 rp2H2M;i? Q7 i?2 HB;?iX ri Bb i?2 /BbiM+2

#2ir22M i?2 i@i? Q#D2+i TQBMi M/ i?2 TQBMi Ux- yV QM i?2 2H2K2MiH ?QHQ;`K- M/ φR Bb i?2 T?b2 /Bbi`B#miBQM Q7 i?2 `272`2M+2 HB;?iX h?2 BMi2MbBiv Tii2`M Q7 i?2 HB;?i I Bb +QKTmi2/ QMHv BM i?2 `2+Q`/ `2 Q7 i?2 Q#D2+i TQBMiX AM //BiBQM- i?Bb +H+mHiBQM Bb M2+2bb`v iQ biBb7v i?2 7QHHQrBM; 2[miBQM,

2d(bBMθobj−bBMθref)≤λ, UjXkV

R9

r?2`2dBb i?2 TBt2H TBi+? Q7 i?2 2H2K2MiH ?QHQ;`K-θobj Bb i?2 M;H2 Q7 i?2 Q#D2+i

#2K M/ θref Bb i?2 M;H2 Q7 i?2 `272`2M+2 #2K iQ i?2 ?QHQ;`KX KQM; i?2K- θ<sub>obj</sub> M/θ<sub>ref</sub> `2 p`B#H2 /2T2M/BM; QM i?2 bBx2 Q7 i?2 pB`imH rBM/QrX h?2`27Q`2- i?2 bBx2 Q7 i?2 pB`imH rBM/Qr Bb /2i2`KBM2/ iQ biBb7v 1[X UjXkVX

AM iQiHHv- i?2 j. Q#D2+i KQ/2H Bb bbmK2/ b +QHH2+iBQM Q7 b2H7@BHHmKBMi2/

TQBMibX h?2 bmK Q7 +QKTH2t KTHBim/2b 7`QK Q#D2+i TQBMib Bb BMi2`72`2/ rBi? i?2 +QKTH2t KTHBim/2 7`QK i?2 TQBMi `272`2M+2 bQm`+2 TH+2/ BM i?2 Q`B;BM- Q` bB/2 Q7 i?2 Q#D2+iX 6B;m`2 jX8 b?Qrb pB`imH rBM/Qr M/ `272`2M+2 TQBMi bQm`+2 7Q`

*:> +H+mHiBQMX h?2 pB`imH rBM/Qr ?b bK2 bBx2 M/ TQbBiBQM Q7 i?2 b2+QM/`v pQHmK2 ?QHQ;`K iQ pQB/ mMM2+2bb`v +H+mHiBQMX h?2 +H+mHiBQM Bb /QM2 QMHv 7Q` HB;?i rp2b r?B+? Tbb i?2 pB`imH rBM/QrX h?2 `2+QMbi`m+i2/ BK;2 rBHH TT2` i i?2 bK2 TQbBiBQM Q7 i?2 Q`B;BMH Q#D2+i r?2M i?2 *:> Bb BHHmKBMi2/

rBi? i?2 +QMDm;i2 rp2 Q7 i?2 `272`2M+2 rp2X h?2 +QMDm;i2 BK;2 rBHH HbQ TT2` i i?2 TQBMi bvKK2i`B+ TQbBiBQM iQ i?2 Q`B;BM b b?QrM BM 6B;bX jX8X

R8

6B;m`2 jXk, JQ/2H iQ +`2i2 Q#D2+i /iX

6B;m`2 jXj, a+?2KiB+ Q7 i?2 2H2K2MiH ?QHQ;`KX

Re

Reference beam

for transfer Object and Reconstructed image

Conjugate image

Holographic plate

CGH Lens

Illumination beam

d Reference

point source for CGH

z y

6B;m`2 jX9, *QM};m`iBQM Q7 +QKTmi2`@;2M2`i2/ ?QHQ;`K U*:>V +H+mHiBQM M/ i`Mb72`X

y

x H W

M O

Reconstructed Image (Virtual Window)

Conjugate Image b

b

a a

(- x , y )

( x , y )

Reference point source for CGH

6B;m`2 jX8, *QQ`/BMi2b 7Q` pB`imH rBM/Qr BM i?2 }MH ?QHQ;`K THM2X h?2

`272`2M+2 TQBMi bQm`+2 7Q` *:> Bb TH+2/ i i?2 Q`B;BMX

Rd

*?Ti2` 9

oQHmK2 ?QHQ;`K T`BMi2` bvbi2K

h?2 }2H/ Q7 biiB+ ?QHQ;`T?B+ i?`22@/BK2MbBQMH /BbTHv `2[mB`2K2Mib Bb p2`v rB/2- bm+? b pB`imH /BbTHv Q7 T`2+BQmb +mHim`H `2HB+b- biiB+ i?`22@/BK2MbBQMH /p2`iBbBM;- ?QHQ;`T?B+ i?`22@/BK2MbBQMH iQTQ;`T?B+ KTb M/ ?QHQ;`T?B+

T`BMiBM; (9dĜ8k)X .m2 iQ i?2 `TB/ /2p2HQTK2Mi Q7 +QKTmi2` i2+?MQHQ;v M/

QTiQ2H2+i`QMB+ i2+?MQHQ;v- i?2 #2bi bQHmiBQM 7Q` ?QHQ;`T?B+ /BbTHv Bb iQ mb2 +QK@

Tmi2` M/ QTiQ2H2+i`QMB+ i2+?MQHQ;v- +QK#BM2/ rBi? i?2 +?`+i2`BbiB+b Q7 i?2

?QHQ;`K bi`m+im`2- iQ /2+QKTQb2 i?2 ?QHQ;`K BMiQ BM/BpB/mH ?QHQ;`K mMBib- M/ i?2M mb2 bTHB+BM; i2+?MQHQ;v _2+Q`/2/ QM T?QiQb2MbBiBp2 Ki2`BH- i?Bb Bb i?2 bQ@+HH2/ ?QHQ;`K T`BMiBM;X >QHQ;`K T`BMiBM; i2+?MQHQ;v ?b /2p2HQT2/ KMv i2+?MQHQ;B2b- M/ i?Bb `iB+H2 KBMHv BMi`Q/m+2b i?2 7mHH@+QHQ` pQHmK2 ?QHQ;`K T`BMi2`X

9XR PTiB+H ?QHQ;`T?B+ T`BMiBM; bvbi2K

h?2 bi`m+im`2 /B;`K Q7 KQMQ+?`QKiB+ bQm`+2 pQHmK2 ?QHQ;`T?B+ T`BMiBM;

bvbi2K Bb b?QrM BM 6B;X jXRX 7i2` TbbBM; i?`Qm;? i?2 >qS- i?2 HB;?i 2KBii2/

#v i?2 Hb2` Bb /BpB/2/ #v i?2 TQH`BxBM; #2K bTHBii2` S"aRX PM2 rv Bb i?2 Q#D2+i HB;?i M/ i?2 Qi?2` Bb i?2 `272`2M+2 HB;?iX 7i2` +QHHBKiBM; M/ 2tTM/@

BM;- i?2 `272`2M+2 #2K Tbb2b i?`Qm;? B`Bb M/ `2+?2b i?2 ?QHQ;`T?B+ THi2X 7i2` +QHHBKiBM; M/ 2tTM/BM;- i?2 Q#D2+i HB;?i Bb `2~2+i2/ #v i?2 TQH`BxBM;

#2K bTHBii2` S"ak M/ i?2M p2`iB+HHv 2Mi2`b G*QaX h?2M i?2 #2K `2+?2b i?2

?QHQ;`T?B+ THi2 i?`Qm;? S"akX aBM+2 i?2 ?QHQ;`K Bb +QKTQb2/ Q7 KmHiBTH2 2H2K2MiH ?QHQ;`Kb- i?2 +QKTH2i2 ?QHQ;`K +M #2 T`BMi2/ #v +QKTmi2` +QMi`QH THi7Q`K s@uX

h?2 pQHmK2 ?QHQ;`K Bb QmiTmi r?2M i?2 `272`2M+2 #2K 7`QK QTTQbBi2 bB/2 BMi2`72`2b rBi? i?2 `2+QMbi`m+i2/ BK;2 Q7 i?2 2H2K2MiH ?QHQ;`K QM i?2 G*QaX h?2 BKTQ`iMi +QKTQM2Mib Q7 i?2 KBMBKH bvbi2K `2 b?QrM #2HQr, h?2 Gj- BiǶb 7Q+H H2M;i? Bb 8y KK M/ rBi? /BK2i2` Q7 98 KK- r?Qb2 +QMp2`;BM; M;H2 Bb T`QTQ`iBQMi2 iQ i?2 pB2rBM; M;2H Q7 QmiTmi ?QHQ;`Kc h?2 }Hi2` +M QMHv Tbb2b i?2 }`bi Q`/2` Q7 i?2 /Bz`+i2/ HB;?i 7`QK i?2 G*Qac i?2 G*QaǶb `2bQHmiBQM Bb R-Nky

#v R-y3y TBt2Hb M/ Bi ?b 3 #Bib H2p2Hb- r?B+? +M /BbTHvb i?2 2H2K2MiH ?QHQ;`Kc i?2 s@u bi;2Ƕb `2bQHmiBQM Bb 9 μK M/ BiǶb KQpBM; `2 Bb kyy KK #v kyy KK- r?B+? +M i`MbHi2b i?2 ?QHQ;`T?B+ THi2c a?mii2` +M +QMi`QH i?2 2tTQbm`2 iBK2X h?2 #2?pBQ`b Q7 i?2 pQHmK2 ?QHQ;`K T`BMi2` `2- +?M;BM; i?2 /BbTHv2/

R3

2H2K2MiH ?QHQ;`K- brBi+?BM; i?2 b?mii2` iQ 2tTQb2 i?2 ?QHQ;`K- i`MbHiBM;

i?2 ?QHQ;`T?B+ THi2 M/ rBiBM; i?2 b2iiHBM; iBK2 iQ 2HBKBMi2 pB#`iBQM b2iiH2X h?2 T2`bQMH +QKTmi2` Bb +QMi`QHHBM; b2`B2b Q7 i?Qb2 #2?pBQ`bX h?2`2#v- i?2 pQHmK2 ?QHQ;`K +M #2 QmiTmi miQKiB+HHvX

AM i?2 KBMBKH bvbi2K- i?2 `2+QMbi`m+i2/ BK;2 Q7 i?2 2H2K2MiH ?QHQ;`K

?b /BbiQ`iBQM /m2 iQ i?2 /Bz2`2M+2 #2ir22M i?2 +?`+i2`BbiB+ Q7 i?2 BHHmKBMiBQM

#2K M/ i?2 +H+mHiBQM T`K2i2` Q7 i?2 `272`2M+2 #2KX 6B;m`2 \\ b?Qrb i?2 TT2`M+2 Q7 `2+QMbi`m+iBQM 7`QK i?2 2H2K2MiH ?QHQ;`KX A7 i?2 G2Mb UGjV Bb HQ+i2/ QM i?2 bK2 TQbBiBQM b i?2 2H2K2MiH ?QHQ;`K- i?2 T`HH2H BHHmKBMiBQM +QMp2`;2b QM i?2 `272`2M+2 TQBMi M/ KF2b i?2 Q#D2+i TQBMi `2+QMbi`m+i2/ i Bib Q`B;BMH TQbBiBQMX >Qr2p2`- i?2 /BbiM+2 r #2ir22M Gj M/ i?2 2H2K2MiH

?QHQ;`K +imHHv Bb MQi #H2 iQ #2 x2`Q /m2 iQ i?2 i?B+FM2bb Q7 i?2 H2Mb M/ S"aX h?2`27Q`2- i?2 `2+QMbi`m+i2/ BK;2 Bb Q#b2`p2/ i i?2 /Bz2`2Mi TQbBiBQM 7`QK i?2 Q`B;BMH TQbBiBQMX h?2 Q#b2`p2/ /2Ti? z M/ i?2 Hi2`H K;MB}+iBQM m `2 b?QrM #v i?2 7QHHQrBM; 2[miBQM,

z = 1 1 z+ r

f²

, U9XRV

m= z +f z+f+ rz

f

, U9XkV

r?2`2 z Bb i?2 Q`B;BMH /2Ti? Q7 i?2 Q#D2+i TQBMiX AM Qi?2` rQ`/b- i?2 KQ`2 i?2

`2+QMbi`m+i2/ BK;2 ?b i?2 /2Ti? M/ i?2 KQ`2 Bi #2+QK2b /BbiQ`i2/X hQ +M+2H i?2 /BbiQ`iBQM BM i?2 `2+QMbi`m+iBQM- i?2 j. Q#D2+i b?QmH/ #2 T`2@/BbiQ`i2/ #v i?2

#bBb Q7 1[X U9XRV M/ U9XkV #27Q`2 i?2 +H+mHiBQM Q7 i?2 2H2K2MiH ?QHQ;`KX AM Qm` 2tT2`BK2Mi- r 4 Ryy KKX

6B;m`2 9XR, .BbiQ`iBQM Q7 i?2 `2+QMbi`m+i2/ BK;2X

9Xk >QHQ;`T?B+ rp27`QMi T`BMiBM; bvbi2K

b r2 FMQr- i?2 `2~2+iBQM ?QHQ;`K +M #2 `2T`Q/m+2/ mM/2` r?Bi2 HB;?i- M/

i?2 ?QHQ;`T?B+ rp27`QMi T`BMiBM; bvbi2K +M +?B2p2 i?Bb ;QHX h?2 T`BM+BTH2 Bb RN

6B;m`2 9Xk, a+?2KiB+ Q7 7mHH@+QHQ` pQHmK2 ?QHQ;`K T`BMi2` bvbi2KX iQ HQ/ i?2 +QKTmi2` ?QHQ;`K QM i?2 bTiBH HB;?i KQ/mHiQ` G*Qa Ub i?2 aGJ V- /Bz`+i i?2 BK;2 7i2` Hb2` BHHmKBMiBQM- M/ i?2M Tbb i?2 #M/@Tbb }Hi2`

iQ ;2i i?2 rp27`QMi Q7 i?2 BK;2X h?2M iF2 i?Bb rp27`QMi b i?2 Q#D2+i HB;?iX h?`Qm;? BMi2`72`BM; i?2 `272`2M+2 HB;?i rBi? Bi- r2 +M ;2i i?2 `2~2+iBQM ?QHQ;`K M/ `2+Q`/ BiX h?2 };m`2 #2HQr b?Qrb i?2 HB;?i Ti? bi`m+im`2 /B;`KX Ai +M

#2 b22M i?i i?2 ?QHQ;`T?B+ rp27`QMi T`BMi2` BMTmib i?2 2H2K2MiH ?QHQ;`Kb BMiQ GB[mB/ *`vbiH QM aBHB+QM UG*QaV- M/ i?2 H2Mb2b GR M/ Gk 7Q`K 96 bvbi2K- r?B+? `2/m+2b i?2 BK;2 /BbTHv2/ QM i?2 G*Qa QMiQ i?2 `2+Q`/BM; THM2X KQM; i?2K- i?2`2 Bb bTiBH bT2+i`mK THM2 BM i?2 97 bvbi2K- M/ #M/@

Tbb }Hi2` Bb TH+2/ QM i?Bb THM2 iQ Tbb QMHv i?2 bT2+i`mK Q7 i?2 `2T`Q/m+2/

BK;2X AM i?Bb rv- i?2 HB;?i /Bbi`B#miBQM QM i?2 `2+Q`/BM; THM2 Bb MQ HQM;2`

2H2K2MiH ?QHQ;`Kb- #mi Q#D2+i HB;?i rp2bX i i?2 bK2 iBK2- i?2 `272`2M+2 HB;?i Bb BMi`Q/m+2/ #2?BM/ i?2 `2+Q`/BM; Ki2`BH iQ BMi2`72`2 rBi? i?2 Q#D2+i HB;?i rp2- M/ i?2 `2~2+iBQM ?QHQ;`K +M #2 `2+Q`/2/X Aib #B;;2bi /pMi;2 Bb i?i Bi +M +?B2p2 r?Bi2 HB;?i `2T`Q/m+iBQMX

9Xj 6mHH@+QHQ` pQHmK2 ?QHQ;`K T`BMi2` bvbi2K

6mHH@+QHQ` pQHmK2 ?QHQ;`K ?b #22M BMi`Q/m+2/ #v MQi?2` `2b2`+? ;`QmT (8j)- Bi Bb bvMi?2bBx2/ #v i?`22 KQMQ+?`QKiB+ HB;?i bQm`+2b- bQ i?2 bvbi2K M22/b iQ bTHBi M/ +QK#BM2 i?2 `2/- ;`22M M/ #Hm2 Hb2`b i i?2 bK2 iBK2X 6B;m`2 9Xk b?Qrb Qm` T`QTQb2/ bvbi2KX

9X9 oQHmK2 ?QHQ;`K T`BMi2` bvbi2K

h?2 pQHmK2 ?QHQ;`K +M #2 QTiB+HHv `2+Q`/2/ 7`QK i?2 *:> T`BMi2/ #v i?2

?QHQ;`T?B+ 7`BM;2 T`BMi2` b b?QrM BM 6B;X jX9- FMQrM b i?2 QTiB+H i`Mb72`X h?2 ky

6B;m`2 9Xj, JQ/B}2/ 97 QTiB+b bvbi2K #2ir22M i?2 S"ak M/ i?2 ?QHQ;`T?B+

THi2X

BK;2 `2+QMbi`m+i2/ 7`QK i?2 *:> Bb `2+Q`/2/ BM T?QiQ b2MbBiBp2 Ki2`BH b i?2 pQHmK2 `2~2+iBQM ?QHQ;`KX Hi?Qm;? i?2 *:> +M #2 `2+Q`/2/ miQKiB+HHv- i?2 QTiB+H i`Mb72` b?QmH/ #2 /QM2 KMmHHvX h?2 #bB+ B/2 Q7 i?2 pQHmK2

?QHQ;`K T`BMi2` Bb `2TH+BM; i?2 T`BMi2/ *:> iQ i?2 aGJX T`i Q7 i?2 *:> Bb /BbTHv2/ QM i?2 G*Qa b i?2 aGJ M/ i?2 `2+QMbi`m+i2/ BK;2 7`QK i?2 G*Qa Bb i`Mb72``2/ iQ i?2 mM2tTQb2/ ?QHQ;`T?B+ THi2X h?2 *:> Bb +H+mHi2/ b i?2 H2MbH2bb 6Qm`B2` ?QHQ;`KX h?2 +H+mHiBQM K2i?Q/ Bb bK2 b i?2 6`2bM2H

?QHQ;`K rBi? TQBMi `272`2M+2 HB;?i HQ+i2/ #2bB/2 Q7 i?2 j. Q#D2+iX Hi?Qm;?

M KTHBim/2 KQ/mHiBQM ivT2 G*Qa +M #2 mb2/ 7Q` i?2 aGJ- T?b2@QMHv G*Qa Bb HbQ mb2/ iQ T`QpB/2 +QKTH2t KTHBim/2 KQ/mHiBQMX aBKBH` iQ i?2

?QHQ;`T?B+ 7`BM;2 T`BMi2`- i?2 bBM;H2 i`Mb72``2/ ?QHQ;`K Bb b bKHH b b[m`2 KBHHBK2i2`bX h?2`27Q`2- i?2 s@u bi;2 i`MbHi2b i?2 THi2 M/ T`BMiBM; MmK#2`b Q7 i?2 2H2K2MiH i`Mb72``2/ ?QHQ;`K iQ KF2 i?2 `2bQM#H2 bBx2 pQHmK2 ?QHQ;`KX h?2 bK2 +QMi`QH bvbi2K /2bB;M2/ 7Q` i?2 ?QHQ;`T?B+ 7`BM;2 T`BMi2` +M #2 mb2/

7Q` i?2 pQHmK2 ?QHQ;`K T`BMi2`X

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0 (x <−x1)

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