Volume 61, Number 10, 2011¸ 10 Z 4, pp. 972∼975

Volume 61, Number 10, 2011¸   10 Z 4, pp. 972∼975

New Physics: Sae Mulli (The Korean Physical Society), DOI: 10.3938/NPSM.61.972

Æ

V Ø6 “ ˜ m; c 8 ýA 0 ù m É ’ Ò ×c Ü R – ¤­ Ž° Ë Ñ8 ý •  ×V ê s • ¤ 

† ç

¡¬ £

ô

 Çz Œ ™@ /† < Ɠ § F  g„   Ó ü t o † < Æõ , @ /„   306-791

(2011¸   7 Z 4 14{ 9  ~ à Î6 £ §, 2011¸   8 Z 4 5{ 9  à º& ñ ‘ : r ~ à Î6 £ §, 2011¸   10 Z 4 5{ 9  > F  S X ‰& ñ )

î 

r @ /ó ø Í\  _ K " f | 9 5 Å q ) a Ä ºÛ ¼ F  g _  » ¡ ¤  © œ à º \  @ /K " f  7 H _ ô  Ç . €  $ , S ~ ½ ӆ ¾ Ó ý a³ ð_  4  † ½ Ó  t

 Ÿ í† < ʝ ) a 4 Ÿ ¤ ™ èà º & h  ½ ¨€   _  î  r @ /ó ø Í\  _ ô  Ç  r] X `  ¦ > í ß –ô  Ç . Õ ª  6 £ §,  r] X  & h ì  rd ” Ü ¼– РÒ'  » ¡ ¤  © œ Ã

º _  ½ ¨^ ‰& h  ³ ð‰ & ³`  ¦ Ä »• ¸ “ ¦, M

“   \  p u   H à º _  % ò † ¾ Ó`  ¦ ¨ î ô  Ç . ‘ : rë  H \  Å Ò# Q”   4  ˜ Ð

&

ñ  ) a  1 l x † < Êà º  H î  r @ /ó ø Í\  _ K " f | 9 5 Å q ) a Ä ºÛ ¼ F  g _  M

“   \  ¦   & ñ   H X <  6   x| ¨ c à º e ”  .

Ù þ

˜d ” # Q: 4 Ÿ ¤ ™ èà º & h  ½ ¨€   , Ä ºÛ ¼ F  g, à º , M

“   

On-axis Aberration of a Gaussian Beam Focused by Using a Zone Plate

Soo Chang ^∗

Department of Physics, Hannam University, Taejon 306-791 (Received 14 July 2011 : revised 5 August 2011 : accepted 5 October 2011)

We discuss the on-axis aberration of a Gaussian beam focused by using a zone plate. First, we examine the diffraction of a complex-source-point spherical wave through the zone plate, and terms of up to fourth order in the aperture variables are considered. Then, we find an explicit formula for the on-axis aberration by solving the diffraction integral. We also evaluate the effect of the aberration on the M

factor. The wave function with fourth-order corrections presented here may be applied to evaluate the degradation in the M

factor of a Gaussian beam focused by using a zone plate.

PACS numbers: 42.30.Va

Keywords: Complex-source-point spherical wave, Gaussian beam, Aberration, M

factor

I. " e  ] Ø



 H» ¡ ¤ Ä ºÛ ¼ F  g \  @ /ô  Ç — ¸Ž  H “ ¦  ˜ Ð& ñ † ½ Ó`  ¦  8 €   4 Ÿ ¤

™

èà º & h  ½ ¨€     ) a    H  z  ´s  · ú ˜ 94 R e ”   [1,2]. Ä º o

  H S ~ ½ ӆ ¾ Ó ý a³ ð_  4  † ½ Ó t  “ ¦ 9ô  Ç 4 Ÿ ¤ ™ èà º & h  ½ ¨€   

\  ¦  6   x K " f E $ ™Ý ¼ ¢ ¸  H  Ö  ¦ \  _ K  | 9 5 Å q ) a Ä ºÛ ¼ F  g _

 à º \  ¦ Ä »• ¸ “ ¦, M ² “   \  p u   H % ò † ¾ Ó`  ¦ ¨ î  

%

i   [3–5]. ô  Ǽ # ,  r] X F  g † < Æ ™ è “   î  r @ /ó ø Í(zone plate)“ É r E $

™Ý ¼
  Ö  ¦ õ   ð ø Ít – Ð y n C`  ¦ — ¸`  ¦ à º e ” `  ¦ ÷  r ë ß –  m 

∗

E-mail: [email protected]



, E $ ™Ý ¼ ] j Œ •s  Ô  ¦ 0 p x ô  Ç   © œ % ò % i _  „   l  `  ¦ | 9 5 Å q

  H X <  6   x| ¨ c à º e ”   [6,7]. Õ ª Q
 Ä ºÛ ¼ F  g \  @ /ô  Ç î  r

@

/ó ø Í_  à º   H  f ”  ì  r$ 3  ) a & h s  \ O % 3  .

‘

: r  7 Hë  H“ É r î  r @ /ó ø Í\  _ K " f | 9 5 Å q ) a Ä ºÛ ¼ F  g _  » ¡ ¤  © œ Ã

º \  @ /K " f  7 H _ ô  Ç . €  $ , á ÔY U3 A q-v Ø Ôy   ñá Ô  r] X 

&

h

ì  r [8]`  ¦  6   x K " f î  r @ /ó ø Í`  ¦ t 
  H 4  ˜ Ð& ñ  ) a 4 Ÿ ¤ ™ è Ã

º & h  ½ ¨€   _  F  g‚   F  g † < Æ& h    H  K \  ¦ ½ ¨ô  Ç . Õ ª  6 £ §,



r] X F  g \  Ÿ í† < ʝ ) a » ¡ ¤  © œ à º _  ½ ¨^ ‰& h  ³ ð‰ & ³`  ¦ Ä »• ¸ “ ¦, M <sup>2</sup> “   \  p u   H à º _  % ò † ¾ Ó`  ¦ ¨ î ô  Ç . { 9   F  g _ 

œ

í& h  ì ø Í â w ₀  7 £ x   
  r] X  à º ms  9 þ t à º2 Ÿ ¤ Ä º Û

¼ F  g \  @ /ô  Ç î  r @ /ó ø Í_  » ¡ ¤  © œ à º  / å L  y  7 £ x † < Ê`  ¦ ˜ Ð

-972-

î 

r @ /ó ø Í\  _ K  | 9 5 Å q ) a Ä ºÛ ¼ F  g _  » ¡ ¤  © œ à º  –  © œÃ º · Soo Chang -973-

Fig. 1. A Gaussian beam is incident upon a zone plate separating two media of refractive indices n and n ⁰ . The zone plate consists of a set of concentric annuli, alter- nately transparent and opaque. The Gaussian beam diffracted through the zone plate converges to (or ap- pears to diverge from) several foci corresponding to a certain finite number of diffracted orders. The Cartesian coordinate system is referenced to the center of the zone plate normal to the z axis. The incident (or diffracted) beam of Rayleigh range b (or b ⁰ _m ) is equivalent paraxi- ally to the spherical wave originating from (0, 0, z + ib) (or (0, 0, z ⁰ _m + ib ⁰ _m )).

#

Ε  r  .

II. Æ V Ø6 “ ˜ m ù p § —  ÞÊ Ý” X ¢ – ¤­ Ž° Ë Ñ; c 6 ” X ¢ 4 

| ºX N Ë

Figure 1“ É r   © œ λ“   Ä ºÛ ¼ F  g s  î  r @ /ó ø Í(zone plate)`  ¦ :

Ÿ

x õ ô  Ç Ê ê,  r] X  à º m\  K { © œ   H œ í& h \  | 9 5 Å q ÷ &  H — ¸ _

þ

v`  ¦ ˜ Ð# Œï  r  . ý a³ ð> _  " é ¶& h “ É r î  r @ /ó ø Í_  ×  æd ” \  e ” “ ¦, z» ¡ ¤“ É r î  r @ /ó ø Íõ  à ºf ” s  . ¢ ¸ô  Ç î  r @ /ó ø Í „  õ  Ê ê B | 9 _  Ï

ã J] X Ò  ¦`  ¦ y Œ •y Œ • nõ  n ⁰ s  “ ¦ & ñ ô  Ç .

ë

ß –{ 9  { 9   Ä ºÛ ¼ F  g _  œ í& h s  (0,0,z)\  e ” “ ¦, œ í& h  ì ø Í

 â

s  w ⁰ s  9, Y U{ 9 o  % ò % i `  ¦ b = nπw ₀ ² /λ  “ ¦ ¿ º€  , { 9 



 F  g“ É r 4 Ÿ ¤ ™ èà º & h  (0,0,z+ib)\ " f
š ¸  H ½ ¨€   ü < ° ú    [1,2].   H» ¡ ¤ % ò % i \ " f S ~ ½ ӆ ¾ Ó ý a³ ð_  4  † ½ Ó t  “ ¦ 9 

€

  { 9   F  g _   1 l x † < Êà º  H

ψ(x, y, z) ' C exp

 ink

 x ² + y ²

2(z − z − ib) − (x ² + y ² ) ² 8(z − z − ib) ³



(1)

  ) a  . # Œl " f C  H x ü < y\  Á º › ' a ô  Ç “   s  9, i  H ) ‡Ã º l

  ñs “ ¦, k(= 2π/λ)  H  1 l x  © œÃ ºs  .

ô

 Ǽ # , î  r @ /ó ø Í_  È Òõ > à º  H { 9 ì ø Í& h Ü ¼– Ð

τ (x, y) =

∞

X

m=−∞

b m exp [−ikmφ(x, y)] (2)

Fig. 2. Transmission coefficient τ of a Fresnel zone plate plotted as a function of position in the radial direction, where we let α = -1.

ü

< ° ú  s  ³ ð‰ & ³ ) a  . # Œl " f m = 0 { 9  M : b 0 = 1/2 s “ ¦, m 6= 0 s €   b _m = sin(mπ/2)/mπ s  . ¢ ¸ô  Ç î  r @ /ó ø Í_  1  œ í& h  o \  ¦ f  “ ¦ ¿ º€  , 0 A © œ† < Êà º  H

φ(x, y) = n ⁰

2f x ² + y ²
+ n ⁰

8f ³ (1 + α) x ² + y ^2
2 (3)

s

“ ¦, ፠ H " é ¶ Æ Ò  © œÃ ºs  . Fig. 2  H α = −1“   á ÔY U3 A q î  r

@

/ó ø Í_  t 2 £ § ~ ½ ӆ ¾ Ó 0 Au \    É r È Òõ > à º_     o\  ¦ ˜ Ð# Œ ï

 r  .

s

] j î  r @ /ó ø Í`  ¦ : Ÿ x õ ô  Ç y n Cs  › ' a ¹ 1 Ï& h \  • ¸² ú ˜ô  Ç “ ¦ 



. î  r @ /ó ø Í 0 A_  & h  (x, y, 0)\ " f › ' a ¹ 1 Ï t & h  (x ⁰ , y ⁰ , z ⁰ )   t

  o \  @ /ô  Ç 4    H  d ” “ É r

r ⁰ ' z ⁰ + (x ⁰ − x) ² + (y ⁰ − y) ²

2z ⁰ − [(x ⁰ − x) ² + (y ⁰ − y) ² ] ² 8z ⁰³

(4) s

Ù ¼– Ð,  1 l x † < Êà º  H á ÔY U3 A q-v Ø Ôy   ñá Ô  r] X  & h ì  r Ü ¼– Ð

 è ­ q à º e ”   [8].

ψ ⁰ (x ⁰ , y ⁰ , z ⁰ ) = 1 iλz ⁰

Z Z

dxdyψ(x, y, 0)τ (x, y) exp(ikn ⁰ r ⁰ )

=

∞

X

m=−∞

b m U _m ⁰ (x ⁰ , y ⁰ , z ⁰ ) (5)

0

Ad ” \ " f U m ⁰ (x ⁰ , y ⁰ , z ⁰ )“ É r m   r] X  ) a  1 l x $ í ì  r`  ¦
 



· p . ‚ à Г ¦ë  H‰  ³ [3]_  ~ ½ ÓZ O Ü ¼– Ð % 3 “ É r d ” (5)\  @ /ô  Ç F  g‚   F 

g † < Æ& h    H  K   H

-974- ô  Dz D GÓ ü t o † < Æ rt  “D hÓ ü t o ”, Volume 61, Number 10, 2011¸   10 Z 4

U _m ⁰ (x ⁰ , y ⁰ , z _m ⁰ )

' C _m ⁰ exp



 

  ikn ⁰







x ⁰² + y ⁰² 2 (z ⁰ − z ⁰ _m − ib ⁰ _m ) −



x ⁰² + y ⁰²  2

8 (z ⁰ − z ⁰ _m − ib ⁰ _m ) ³





 − ikW ^(m)



 

 

(6)

s

 . # Œl " f à º † < Êà º  H

W ^(m) = 1

8 S ^(m) (z ⁰ _m + ib ⁰ _m ) ⁴ (z ⁰ − z ⁰ _m − ib ⁰ _m ) ⁴



x ⁰² + y ⁰²  ² (7) s

“ ¦, B > h  à º  H n ⁰

z ⁰ _m + ib ⁰ _m = n

z + ib + n ⁰

(f /m) (8) Õ

ªo “ ¦

S ^(m) = n ⁰

(z ⁰ _m + ib ⁰ _m ) ³ − n

(z + ib) ³ + n ⁰ m

f ³ (1 + α) (9) ü

< ° ú  s  & ñ _ ÷ &% 3  . z ⁰ m “ É r m  œ í& h _  z» ¡ ¤ ý a³ ðs “ ¦, m  œ í& h  ì ø Í â w ⁰ _m _  6   x # Q– Ð ³ ð‰ & ³ ) a Y U{ 9 o  % ò % i “ É r b ⁰ _m = n ⁰ πw _m ^{0 2} /λ s  . d ” (6)\ " f à º † < Êà º W ^(m) \  ¦ ] jü @r v 

€

 , 4 Ÿ ¤ ™ èà º & h  (0, 0, z ⁰ _m + ib ⁰ _m ) \  à º§ 4    H ½ ¨€   _    H

»

¡

¤   H  d ” õ  ° ú   .

III. m  > H± n Ç° Ë Ñ8 ý M ² ß Ã Å 

m  œ í& h \  à º§ 4    H 4  ˜ Ð& ñ  ) a  1 l x † < Êà º s  © œ& h 

“

  Ä ºÛ ¼ † < Êà º– РÒ'  \ O  
 # Á # Qz Œ ¤  H t \  ¦ ¶ ú ˜( R˜ Ðl  0 A K

" f M x ² “   \  ¦ ì  r$ 3  % i   [9]. d ” (6)_   1 l x \  @ /ô  Ç x x 9

 y ~ ½ ӆ ¾ Ó / B N ç ß –Å Ò à º f x ⁰ , f _y ⁰
_  ì  r Ÿ í  H V _m ⁰ f _x ⁰ , f _y ⁰ , z ⁰ _m

= Z Z

dx ⁰ dy ⁰ U _m ⁰ (x ⁰ , y ⁰ , z ⁰ _m ) exp −i2π f _x ⁰ x ⁰ + f _y ⁰ y ⁰


(10)

  ) a  . s ] j x ~ ½ ӆ ¾ Ó 0 Au  x 9  / B N ç ß –Å Ò à º\  @ /ô  Ç 2  — ¸ F '

pà Ô\  ¦ y Œ •y Œ •

hx ⁰² i = R R dx ⁰ dy ⁰ x ⁰² |U _m ⁰ | ² R R dx ⁰ dy ⁰ |U _m ⁰ | ² , hf _x ^{0 2} i = R R df _x ⁰ df _y ⁰ f _x ^{0 2} |V _m ⁰ | ²

R R df _x ⁰ df _y ⁰ |V _m ⁰ | ² (11)



“ ¦ ¿ º€  , M _x ² “     H M _x ² = 4π

q

hx ⁰² ihf _x ^{0 2} i (12)

Fig. 3. Variations of M _x ² for the diffracted beams con- verging to the foci of orders m = 1 (solid line), 3 (dashed line), and 5 (dotted line). The beam waist radius w 0 is taken as a variable, while other parameters are fixed at z = 0, λ = 632.8 nm, n = n ⁰ =1, α = -1, and f = 900 mm.

ü

< ° ú  s  & ñ _   ) a  . M y ² “     H d ” (11)õ  (12)\ " f x ⁰ õ  f _x ⁰ `  ¦ y <sup>0</sup> õ  f y <sup>0</sup> Ü ¼– Ð  Ë ¨€    ) a  . 4  ˜ Ð& ñ † ½ Ó`  ¦ Á ºr ½ + É  â Ä

º, d ” (6)\  @ /ô  Ç M _x ² ( ¢ ¸  H M _y ² ) “     H 1 s   ) a    H & h `  ¦ Å

Ò3 l q K   ô  Ç . 7 £ ¤, s  © œ& h “   Ä ºÛ ¼ F  g _  M x ² ( ¢ ¸  H M _y ² )

“

    H 1 s  . ô  Ǽ # , î  r @ /ó ø ͓ É r z» ¡ ¤ \  › ' a K " f @ /g As Ù ¼

–

Ð z» ¡ ¤`  ¦    { 9  ô  Ç Ä ºÛ ¼ F  g _   â Ä º, M x ² = M _y ² s  .

Figure 3“ É r { 9  ô  Ç Ä ºÛ ¼ F  g _  œ í& h  ì ø Í â w ₀ s     o½ + É M

:, m = 1(z  ´‚  ), 3( W‚  ), 5(& h ‚  )“   œ í& h \  à º§ 4    H  r ] X

F  g _  M x ² / B G‚  `  ¦ ˜ Ð# Œï  r  .   É r   à º[ þ t“ É r z = 0, λ = 632.8 nm, n = n ⁰ = 1, α = -1, and f = 900 mm  “ ¦ 

&

ñ % i  . d ” (7)_  à º † < Êà º  H S ~ ½ ӆ ¾ Ó ý a³ ð (x ⁰ , y ⁰ ) _  W 1 ]

jY  L \  q Y Vô  Ç .   " f { 9   F  g _  œ í& h  ì ø Í â w ₀  0\  ] X

  H ½ + Éà º2 Ÿ ¤ à º  / å L5 Å q y  y Œ ™™ è l  M :ë  H \  M x ² “     H 1 s   ) a  . ¢ ¸ô  Ç, α = -1s “ ¦ b  z, (f/m)s €  , à º  >  Ã

º  H à º m_  [ j] jY  L \  q Y Vô  Ç .   " f m = 5“   / B G

‚

 s  m = 3“   / B G‚  ˜ Ð  Õ ªo “ ¦ m = 3“   / B G‚  s  m =1“   /

B G‚  ˜ Ð   8 / å L  y  7 £ x    H  כ `  ¦ ^  ¦ à º e ”  .

Figure 4  H î  r @ /ó ø Í_  0 A © œ† < Êà º\  Ÿ í† < ʝ ) a " é ¶ Æ Ò  © œÃ º α _  † < Êà º– Ð > í ß –ô  Ç 1  œ í& h \  à º§ 4    H  r] X F  g _  M x ²

/

B G‚  `  ¦ ˜ Ð# Œï  r  . î  r @ /ó ø Í_  1  œ í& h  o  f  H y Œ •y Œ • 300

î 

r @ /ó ø Í\  _ K  | 9 5 Å q ) a Ä ºÛ ¼ F  g _  » ¡ ¤  © œ à º  –  © œÃ º · Soo Chang -975-

Fig. 4. Variations of M _x ² for the diffracted beams con- verging to the focus of order m = 1. The solid, dashed, and dotted lines are of f = 300 mm, 600 mm, and 900 mm, respectively. The conic constant α is taken as a variable, while other parameters are fixed at z = 0, λ = 632.8 nm, n = n ⁰ = 1, and w 0 = 10 mm.

mm(z  ´‚  ), 600 mm( W‚  ), 900 mm(& h ‚  ) “ ¦ ¿ º% 3 Ü ¼ 9,



 É r   à º[ þ t“ É r z = 0, λ = 632.8 nm, n = n ⁰ = 1, w ₀ = 10 mm  “ ¦ & ñ % i  . d ” (8)õ  (9)\  ¦ ˜ Ѐ  , b zü < (f/m)

˜

Ð  B Ä º 9 þ t M :, z ⁰ m ' f /m õ  b ⁰ m b ' (n/n ⁰ )(f /m) ² s “ ¦, S ^(m) = (n ⁰ m/f ³ )(m ² + α + 1) s   ) a  .   " f

α ' −m ² − 1 (13)

\

" f à º  > à º  H ¢ - a„  y    ”   . Fig. 4  H 1  œ í& h 

\

 à º§ 4    H  r] X F  g _  M x ²  î  r @ /ó ø Í_  œ í& h  o \  Á º › ' a

>  α ' −2\ " f 1\  à º§ 4    H — ¸_ þ v`  ¦ ˜ Ð# ŒÅ ғ ¦ e ”  .

{ 9

ì ø Í& h Ü ¼– Ð d ” (6)\  Å Ò# Q”   4  ˜ Ð& ñ  ) a Ä ºÛ ¼ F  g _    1

l

x † < Êà º  H î  r @ /ó ø Í\  _ K " f | 9 5 Å q ) a Ä ºÛ ¼ Y Us $  F  g _  M _x ² “   \  ¦   & ñ   H X <  6   x| ¨ c à º e ”  .

IV. + s Ç Â ] Ø

î

 r @ /ó ø Í\  _ K " f | 9 5 Å q ) a Ä ºÛ ¼ F  g _  » ¡ ¤  © œ à º \  @ / K

" f  7 H _  % i  . €  $ , á ÔY U3 A q-v Ø Ôy   ñá Ô  r] X  & h ì  r`  ¦



6   x K " f î  r @ /ó ø Í`  ¦ t 
  H 4  ˜ Ð& ñ  ) a 4 Ÿ ¤ ™ èà º & h  ½ ¨€    _

 F  g‚   F  g † < Æ& h    H  K \  ¦ % 3 % 3  . Õ ª  6 £ §,  r] X F  g \  Ÿ í

†

< ʝ ) a » ¡ ¤  © œ à º _  ½ ¨^ ‰& h  ³ ð‰ & ³`  ¦ Ä »• ¸ “ ¦, M ² “   \  p

u   H à º _  % ò † ¾ Ó`  ¦ ¨ î  % i  . { 9   F  g _  œ í& h  ì ø Í â w ₀  7 £ x   
  r] X  à º ms  9 þ t à º2 Ÿ ¤ Ä ºÛ ¼ F  g \  @ / ô

 Ç î  r @ /ó ø Í_  » ¡ ¤  © œ à º  / å L  y  7 £ x ô  Ç . ‘ : rë  H \  Å Ò# Q

”

  4  ˜ Ð& ñ  ) a  1 l x † < Êà º  H î  r @ /ó ø Í\  _ K " f | 9 5 Å q ) a Ä º Û

¼ F  g _  M ² “   \  ¦   & ñ   H X <  6   x| ¨ c à º e ”  .

P

c p 8 ý ò k >

‘

: r ƒ  ½ ¨  H 2011¸   ô  Çz Œ ™@ /† < Ɠ § † < ÆÕ ü tƒ  ½ ¨q _  t " é ¶ Ü ¼– Ð Ã

º' Ÿ ÷ &% 3 _ þ v m  .

Y

c p w Š à U Ø ”  ô

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[2] M. Couture and P. A. Belanger, Phys. Rev. A 24, 355 (1981).

[3] S. Chang, Optik, doi:10.1016/j.ijleo.2011.01.003.

[4] S. Chang, Optik, doi:10.1016/j.ijleo.2011.06.013.

[5] S. Chang, Optik in press (2011).

[6] F. A. Jenkins and H. E. White, Fundamentals of Op- tics (McGraw-Hill, London, 1981), p. 44, 384.

[7] M. Young, J. Opt. Soc. Am. 62, 972 (1972).

[8] M. Born and E. Wolf, Principles of Optics (Perga- mon, Oxford, 1980), p. 378.

[9] A. E. Siegman, Proc. SPIE 1224, 2 (1990).