Er:YAG Y Us $ _ Q-switching\ @ /ô Ç q Ö ¦ ~ ½ Ó& ñ d ` ¦ Ã ºu K $ 3 % i . : £ ¤ y Er

(1)

»* å ¸

é

ß ² D G @ / < Æ §, ' é ß õ < ÆÂ Ò, ;î ß 330-714 (2006¸ 11 Z 4 13{ 9 ~ Ã Î6 £ §)

Er:YAG Y Us $ _ Q-switching\ @ /ô Ç q Ö ¦ ~ ½ Ó& ñ d ` ¦ Ã ºu K $ 3 % i . : £ ¤ y Er

³⁺

_ [ þ t H © I (

⁴

I

11/2

) ü < { © I (

⁴

I

13/2

) _ Û ¼ ß ¼ ! QF Kï r 0 A (Stark sublevel) s \ " f F g µ 1 Ï ;s Ö ¦ Q

1

õ Q

2

_ o\ É r Q-switching ) a Er:YAG Y Us $ ` O Û ¼_ ì ø Íu ; ¤, ì ø Í4 ¤Ò ¦ x 9 Ø ¦§ 4 \ -t _ o\ ¦

¸ % i . Õ ªo ¦ / B N l < Hz ´\ É r Ø ¦§ 4 \ -t _ o\ ¦ s : r& h Ü ¼ Ð ¸ % i .

PACS numbers: 42.55.P, 42.60G, 07.05.T

Keywords: Er:YAG Y Us $ , Q-switching, Ã ºu K $ 3

I. " e Â ] Ø

þ

j H H& h ü @ % ò % i _ © \ " f µ 1 Ï H ¦^ Y Us $

\

@ /ô Ç ½ ¨ Ö ¸ µ 1 Ïy ' ÷ & ¦ e . t F K t Dy:CaF

& ñ ` ¦ s 6 x ô Ç 2 ∼ 3 µm% ò % i _ Y Us $ > hµ 1 Ï÷ &% 3 ¦ [1], Cr, Tm, Ho:YAG & ñ ` ¦ s 6 x ô Ç 2.1 µm_ Y Us $

Ð ¦÷ &% 3 [2]. Õ ªo ¦ Cr, Tm:YSGG (1.862 µm), Cr, Tm, Ho:YSGG (1.924, 2.010, 2.086 µm), Cr, Ho, Er:YAG (2.696 µm), Er:YAG (1.646, 1.640 µm) 1 p x _ Y U s

$ z ´ : r \ " f ½ ¨ & ³ ) a e [3]. ô Ç¼ # ì ø Í ¸^ Y Us

$

Ð * 3 i ç # 3 µ\ " f µ 1 Ï H F g$ 3 Ä » Y Us $ \ @ /ô Ç

½ ¨ Ã º' ÷ &% 3 [4].

s

Qô Ç H& h ü @ Y Us $ [ þ t × æ \ " f 2.94 µm_ © Ü ¼

Ð µ 1 Ï H Er:YAG Y Us $ [5] _ « Ñ6 x Ü ¼ Ð ´ ú § É r ' a d

`

¦ ~ Ã Î ¦ e [6]. s כ É r Tm:YAG (2.01 µm), Ho:YAG (2.12 µm), Er:YAG (2.94 µm) Y Us $ _ © É r y y Ó ü t

\

" f 200 µm, 400 µm, Õ ªo ¦ 5 µm î ß ¼ 1 Ú_ U ·s \ ¦ È Òõ K

" f [2,3], Ã ºì r s ´ ú § É r ¸f [ j í\ Er:YAG Y Us $ c

`

¦ Å Ò ½ + É â Ä º É r Y Us $ [ þ t Ð È Òõ U ·s · û Ü ¼Ù ¼

Ð Ã ºì r` ¦ í < Êô Ç ¸f [ j í\ ¦ u « Ñ H X < © & h ] X

¦ ¸f _ & ñ S X ô Ç ] X > hü < ] j 0 p x ô Ç Y Us $ s l M : ë

H s .

Er:YAG Y Us $ H { © I ü < [ þ t H © I _ ¨ î ç H Ã º" î s

y y 0.12 msü < 2 ms Ð 5 Å q 1 l x ¢ ¸ H Q-switching

∗

E-mail: [email protected]

1 l

x s Ô ¦ 0 p x ô Ç כ Ü ¼ Ð # 4 R M ® o . Õ ª Q þ j H Er ³⁺

s

: r _ Å Ò{ 9 0 l x ¸\ ¦ 50 % t Z } > # ì ø Í ¸^ Y Us $

Ð * 3 i ç ô Ç 5 Å q 1 l x Er:YAG Y Us $ \ @ /ô Ç ½ ¨ õ

µ 1 Ï³ ð÷ &% 3 [7]. ¢ ¸ô Ç l F g < Æ ´ òõ [8], 6 £ § ¾ Ó F g < Æ ´ ò õ

[9], FTIR (Frustrated Total Internal Reflection) ~ ½ ÓZ O [10], r Ö ¦ [11] 1 p x` ¦ s 6 x ô Ç 0 p x1 l x+ þ A Q-switching ÷ r

m , Ó ü t õ \ » 1 Ï : r [12], InAs í o f ¨ Ã º^ [13]\ ¦ s 6 x ô

Ç Ã º1 l x+ þ A Q-switching\ @ /ô Ç ½ ¨ õ [ þ t s Ð ¦÷ &% 3

.

: r ½ ¨z ´\ " f H LiNbO 3 íÖ 0 q ! s q_ l F g < Æ ´ òõ

\

¦ s 6 x ô Ç Er:YAG Y Us $ _ Q- switching\ @ /ô Ç

½

¨\ ¦ Ã º' ¦ e . : r 7 Hë H \ " f H Er:YAG Y Us $ _ Q-switching \ @ /ô Ç q Ö ¦ ~ ½ Ó& ñ d ` ¦ Ã ºu K $ 3 # s : r

&

h Ü ¼ Ð ì r$ 3 % i . : £ ¤ y Er ³⁺ _ [ þ t H © I ( ⁴ I _11/2 ) ü < {

© I ( ⁴ I 13/2 ) _ Û ¼ ß ¼ ! QF Kï r 0 A (Stark sublevel) s

\ " f F g µ 1 Ï ;s Ö ¦ Q 1 õ Q 2 _ o\ É r Q- switching ) a Er:YAG Y Us $ _ ` O Û ¼_ ì ø Íu ; ¤, ì ø Í4 ¤Ò ¦ x 9 Ø

¦§ 4 \ -t _ o\ ¦ ¸ % i . ¢ ¸ô Ç / B N l < Hz ´Ò ¦

\

É r ` O Û ¼_ + þ AI ü < ` O Û ¼; ¤ õ ` O Û ¼\ -t _ oü <

Q ₂ /Q ₁ q Ö ¦ _ o\ " f ` O Û ¼_ ì ø Í4 ¤Ò ¦ _ o\ ¦ ¸

% i .

II. Q-switched Er:YAG 7 _T $ [ X ê sV

Fig. 1 É r Er:YAG Y Us $ _ Q-switching` ¦ 0 Aô Ç © u _

½

¨$ í ¸s . / B N l _ ^ U ´s H 40 cm s 9, Er:YAG

-573-

(2)

& ñ É r Er ³⁺ s : r _ 0 l x ¸ 50 %, t 2 £ § s 5 mm, U ´s

100 mm s . & ñ _ ª É r 2.94 µm \ @ /ô Ç Á ºì ø Í ï h A` ¦ % i ¦, \ P E $ Ý ¼\ _ ô Ç ´ òõ \ ¦ y èr v l 0 A # 5 m _ / B GÒ ¦ ì ø Ít 2 £ § Ü ¼ Ð ¸3 l q > / B N % i . ì ø Í Ö

¦ õ Ø ¦§ 4 Ö ¦ É r ¸¿ º ¨ î Ü ¼ Ð ÷ &# Q e ¦, 2.94 µm\

"

f_ ì ø Í Ö ¦ É r y y 99 % s © õ 95 %\ ¦ כ ` ¦ 6

x % i .

Er:YAG & ñ É r Xe $ 3 F g1 p x Ü ¼ Ð * 3 i ç ¦, Õ ª כ É r ? /

â

5 mm, ü @ â 6 mm, ~ ½ Ó U ´s 4.4 u s 9 Xes 450 Torr G 0 >4 R e . Xe $ 3 F g1 p x` ¦ 1 l x r v l 0 Aô Ç

"

é

¶ / B N/ å L © u Ð H Qr PCI _ 02PS165\ ¦ 6 x

%

i ¦, ` O Û ¼{ © 150 J_ \ -t \ ¦ / B N/ å L ½ + É Ã º e ¦, 1 ∼ 20 Hz _ ì ø Í4 ¤Ò ¦ Ð 1 l x ô Ç .

íÖ 0 q ! s q É r p ² D G LINOS _ LM7IR` ¦ 6 x % i ¦, 2

\

" f È Òõ Ö ¦ s 98 % LiNbO 3 & ñ Ü ¼ Ð ½ ¨$ í ÷ &# Q e .

s

M : Q-switching\ 9 כ ¹ô Ç λ/4 · ú É r 2.94 µm \ " f 6 kV s .

III. Er ³⁺ T Æ X Ø8 ý ; c .U Ç U Ø üÑ ÷ Q 1 Ê Ý Q 2 ë ÅT

ôV ê s

YAG & ñ î ß _ Er ³⁺ s : r _ ï r 0 A × æ Y Us $ µ 1 Ï \ l

# H ï r 0 A\ ¦ Fig. 2 \ ? /% 3 . Y Us $ 1 l x É r

Ö

¸$ í ï r 0 A ⁴ I 11/2 - ⁴ I 13/2 ? /_ Stark $ í ì r X 2 - Y 7 ü <

X _5,6 - Y ₇ s _ ;s \ _ K " f 2936.4 nm ü < 2831.0 nm @ /_ © ` ¦ ~ ½ ÓØ ¦ 9, Õ ª × æ 2936.4 nm © @ /_ Ø ¦

§

4 s 40C & ñ ¸ ß ¼ . { © I ( ⁴ I _13/2 ) _ ¨ î ç H Ã º" î É r 2 ms Ð [ þ t H © I ( ⁴ I _11/2 ) _ ¨ î ç H Ã º" î 0.12 ms Ð B Ä

º U ´# Q" f 5 Å q 1 l x õ Q-switching 1 l x s # Q 9î r כ Ü ¼

Ð · ú 94 R M ® o . Õ ª Q V. Lupei 1 p x [14] _ ½ ¨ õ \ _

Er:YAG Y Us $ ï r 0 A_ [ þ t H © I \ " f © 0 Aï r 0

A Ð( ⁴ I _11/2 - ⁴ F _7/2 ) _ ¨ 8 (up-conversion) õ { © I

⁴ I 13/2 \ " f © 0 Aï r 0 A Ð( ⁴ I 13/2 - ⁴ I 9/2 ) _ ¨ 8 & ³ © s

Fig. 2. Diagram for the energy levels of Er ion and the photon flux transition rates Q 1 , Q 2 between the Stark sublevels of the upper level ( ⁴ I 11/2 ) and the lower level ( ⁴ I 13/2 ).

H X <, s H × æ í 7 H _ % ò ¾ Ó\ _ ô Ç כ s . s ü

< ° ú É r © 0 Aï r 0 A Ð_ ¨ 8 & ³ © õ Er ³⁺ s : r _ 0 l x ¸

50 % s © Er:YAG & ñ ` ¦ 6 x # 5 Å q [7] 1 l x x 9 Q-switching [8-13] 1 l x \ @ /ô Ç ½ ¨ õ Ð ¦÷ &% 3 .

Fig. 2 \ ³ ðr ô Ç Q 1 õ Q ² _ Ó ü t o | ¾ Ó É r Er:YAG Y Us $

\

¦ Q-switching r ~ ´ M : H ;s & ³ © Ü ¼ Ð · ú 94 R e

[15]. ¢ ¸ô Ç ⁴ I _9/2 ü < ⁴ F _9/2 ï r 0 A[ þ t ÐÂ Ò' _ # l ;s

H ´ ú § É r í 7 H ;s [ þ t M :ë H \ Ò q tl ¦ ⁴ I 11/2 ü < ⁴ I 13/2 ï r 0 A [

þ

t _ µ 1 Ï# l \ _ ô Ç © ñ ¢ - a o 6 x (cross relaxation mechanism) \ _ K " f ⁴ S _3/2 Ð ;s ) a . s Qô Ç ;s

Q-Switching ´ òõ \ % ò ¾ Ó` ¦ p 2 ; .

IV. Er:YAG 7 _T $ [8 ý Q-Switching; c 6 X ¢ R

ù o Ú U ê sX N ËÅ k Ä

Er:YAG _ Q-switching` ¦ [ O " î l 0 Aô Ç q Ö ¦ ~ ½ Ó& ñ d

É

r A ü < ° ú É r d Ü ¼ Ð è q Ã º e [15].

dN 3

dt = Q 1 (t) − N 3

τ − γ ij

α in c hν φ N

N ₀ (1)

dN ₂

dt = Q ₂ (t) + β _j α i

N ₃ τ − γ ij

α _in c hν φ N

N 0

(2)

dφ dt = −α 0

c

χ φ + α in

c χ φ N

N 0

(3)

#

l \ " f N ³ ü < N ² H y y Er ³⁺ s : r s [ þ t H © I ( ⁴ I _11/2 ) ü < { © I ( ⁴ I _13/2 ) \ e H é ß 0 A Â Òx { © > hÃ ºs

¦, τ H ⁴ I _11/2 _ ¨ î ç H Ã º" î , Q ₁ ü < Q 2 H Fig. 2 \ ³ ðr ô Ç

כ

õ ° ú s ⁴ I _11/2 ü < ⁴ I _13/2 © I _ Û ¼ ß ¼ ! QF Kï r 0 A s

\ " f { 9 # Q H F g µ 1 Ï[ þ t _ ;s , α in H * 3 i ç s r

(3)

Fig. 3. Photon flux density and normalized population inversion density as a function of time when Q 1 = 0 and Q 2 = 0.

Fig. 4. Photon flux density as a function of time for various resonator loss (Q ₁ = 10 ⁵ /cm, Q ₂ = 10 ⁴ /cm.

÷

&l _ 6 £ § _ f ¨ Ã º > Ã º, c H B | 9 î ß \ " f_ y n C_ 5 Å q§ 4 , hν H Y Us $ F g \ -t , φ H / B N l î ß _ F g \ -t x 9

¸, α 0 H Ö ¦ _ < Hz ´` ¦ í < Êô Ç / B N l î ß _ ^ < H z

´, α _i ü < β j H y y ⁴ I _11/2 ü < ⁴ I _13/2 © I _ Û ¼ ß ¼ ! QF K ï

r 0 A\ K { © H ^ ¦ Þ Ôë ß > Ã º, N = N ³ − N 2 H ⁴ I _11/2 ü <

4 I _13/2 s _ é ß 0 A Â Òx { © _ x 9 ¸ ì ø Í s ¦, N ⁰ H * 3 i ç s

r ÷ &l _ N_ ° ú כs . Õ ªo ¦ γ ij = α i + β j , χ = [L(n − 1) + L ⁰ ]nL s 9 Lõ n É r Y Us $ & ñ _ U ´ s

ü < Ï ã J] X Ò ¦ s .

0 A_ ~ ½ Ó& ñ d (1) ∼ (3)[ þ t _ Ã ºu & h K \ ¦ ½ ¨ # Q- switching ) a Y Us $ ` O Û ¼ ì ø Í Ö ¦ s R Ö ¦` ¦ : x K Ø ¦

§

4 ÷ & H \ -t ü < ì ø Íu ; ¤ ∆τ (Full Width at Half Max- imum, FWHM) \ ¦ % 3 ` ¦ Ã º e Ü ¼ 9, 6 £ § õ ° ú s ³ ð & ³ ) a

[16].

E = hνcA

1 + (n − 1)(L/L ⁰ ) ln(1/R) Z

φ(t)dt (4)

Fig. 5. Pulse energy and pulse width (FWHM) variation as a function of resonator loss (Q 1 = 10 ⁵ /cm, Q 2 = 10 ⁴ /cm).

∆τ = 1 φ max

Z

φ(t)dt (5)

#

l \ " f A H Ö ¸$ í % ò % i _ é ß & h , Lõ L ⁰ É r Y Us $

&

ñ õ / B N l ^ U ´s , n H Y Us $ & ñ _ Ï ã J] X Ò ¦, R É r Ø

¦§ 4 Ö ¦ _ ì ø Í Ö ¦, φ max H F g \ -t x 9 ¸ φ(t)_ þ j Î .

° ú כ` ¦ · p .

V. R ù o Ú U ê sX N ËÅ k Ä8 ý ¤V A 0V Ä

4 Runge-Kutta ~ ½ ÓZ O ` ¦ s 6 x # q Ö ¦ ~ ½ Ó& ñ d (1) ∼ (3) \ " f N ³ , N 2 x 9 φ\ @ /ô Ç Ã ºu & h K \ ¦ ½ ¨ ¦, (4)ü <

(5) d ` ¦ s 6 x # Ø ¦§ 4 \ -t ü < ` O Û ¼; ¤` ¦ > í ß % i .

Fig. 3 É r Q 1 = 0, Q 2 = 0{ 9 M :, 7 £ ¤ Q-switching ´ òõ \ O

`

¦ â Ä º / B N l î ß _ F g \ -t x 9 ¸ φ(t)ü < [ þ t H © I

⁴ I _11/2 ï r 0 Aü < { © I ⁴ I _13/2 ï r 0 A s _ ½ © o

)

a x 9 ¸ ì ø Í N (t)/N ₀ \ ¦ > í ß ô Ç כ s . Fig. 3\ " f x 9

¸ ì ø Í ) a ª s þ jÎ . ° ú כ\ " f × ¦ # Q[ þ t l r ½ + É M :Â Ò' Y U s

$ 1 l x ¦ e 6 £ §` ¦ ^ ¦ Ã º e . s ü < ° ú É r õ H

` O

Û ¼ Y Us $ \ " f H { 9 ì ø Í& h & ³ © s . Fig. 3\

"

f Ðs H φ(t) \ ¦ (4) ü < (5) d ` ¦ s 6 x # \ -t ü < ` O Û

¼; ¤` ¦ ½ ¨ y y 0.86 J, 250 µs Ð z ¤ . s Qô Ç

õ H Q-switching ÷ &t · ú § É r Er:YAG Y Us $ _ { 9 ì ø Í& h

z ´+ « > õ [ þ t õ q 5 p w ô Ç כ s [5].

Fig. 4 H Q 1 = 10 ⁵ /cm, Q ₂ = 10 ⁴ /cm{ 9 M : / B N l < H z

´ α 0 ` ¦ or v " f F g \ -t x 9 ¸ φ(t)_ o\ ¦

¸ ô Ç כ s . Fig. 4\ " f φ(t)_ þ j@ /° ú כ É r α 0 = 0 â Ä

º © ß ¼ ¦, α 0 & | 9 Ã º2 ¤ y è 9 ` O Û ¼ + þ A$ í ÷ & H r

ç ß ¸ Ö ¼ 9t H כ ` ¦ ^ ¦ Ã º e . s כ É r / B N l < Hz ´ s

& t Y Us $ µ 1 Ï ë H) 3 ° ú כs Z } t Ù ¼ Ð Y Us $

µ

1 Ï l 0 Aô Ç x 9 ¸ ì ø Í s s À Ò# Qt H r ç ß s U ´# Qt

(4)

Fig. 6. Photon flux density as a function of time : (a) Q 1 = 10 ⁵ /cm, Q 2 = 10 ¹ /cm, (b)Q 1 = 10 ⁵ /cm = 10 ³ /cm (resonator loss α 0 = 0.016 /cm).

¦ " f Y Us $ ` O Û ¼ + þ A$ í ÷ & H r ç ß ¸ 0 p y # Qt l M : ë

H s .

Fig. 5 H / B N l < Hz ´ α ₀ \ ¦ or ~ ´ M : ` O Û ¼ \ -t ü

< ` O Û ¼; ¤ _ o\ ¦ · p כ s . s M : Q 1 = 10 ⁵ /cm, Q 2 = 10 ⁴ /cm Ð ¿ º% 3 . α 0 = 0.0{ 9 M : ` O Û ¼ \ -t H 50 mJ, ` O Û ¼; ¤ É r 70 ns כ s α 0 = 0.1{ 9 M : y y 30 mJ õ

90 ns Ð o % i . / B N l < Hz ´s & | 9 Ã º2 ¤ ` O Û ¼ \

-t H × ¦ # Q× ¼ H ì ø Í ` O Û ¼; ¤ É r V , # Qt H â ¾ Ó` ¦ ^ ¦ Ã º e

. s Qô Ç & ³ © \ @ /ô Ç כ ¸ Fig. 4\ @ /ô Ç K $ 3 õ ° ú s

½ + É Ã º e .

Fig. 6 H Q 1 = 10 ⁵ /cm Ð ¦ Q ² \ ¦ 10 ¹ /cm (a) ü <

10 ³ /cm (b) Ð or ~ ´ M : r ç ß \ É r F g \ -t x 9

¸ φ(t)\ ¦ > í ß ô Ç כ Ü ¼ Ð, / B N l < Hz ´ α = 0.016/cms

. s M : Q 2 /Q ₁ _ q Ö ¦ s 10 ⁻⁴ (Fig. 6(a)) \ " f 10 ⁻² (Fig. 6(b)) Ð & t Y Us $ ` O Û ¼ Ò q t$ í ÷ & H ì ø Í4 ¤Ò ¦ s

± ú

f ` ¦ · ú Ã º e . s Qô Ç & ³ © É r Q 2 /Q ₁ _ q Ö ¦ s

&

t © @ /& h Ü ¼ Ð Q 2 s 7 £ x H כ s ¦, Fig. 2\ " f

4 I 11/2 ï r 0 Aü < ⁴ I 13/2 ï r 0 A s _ x 9 ¸ ì ø Í s t Ù ¼

Fig. 7. Pulse width (FWHM) and pulse energy as a function of the ratio of Q 1 to Q 2 : (a) Q 1 10 ⁵ /cm, (b) Q 1 = 10 ⁶ /cm (resonator loss α 0 = 0.016 /cm.

Ð Y Us $ ` O Û ¼ Ò q t$ í ÷ & H r ç ß s U ´# Qt ¦ " f ` O Û

¼ ì ø Í4 ¤Ò ¦ s ± ú t H כ Ü ¼ Ð Ò q ty ½ + É Ã º e .

Fig. 7(a) H / B N l < Hz ´ α 0 = 0.016/cm s ¦, Q ¹ = 10 ⁵ /cm{ 9 M : Q ² /Q 1 _ q Ö ¦` ¦ or v " f ` O Û ¼_ \

-t ü < ì ø Íu ; ¤` ¦ > í ß ô Ç õ s . s M : Q 2 /Q ₁ 0.1

â

Ä º ì ø Íu ; ¤ É r 150 ns, \ -t H 0.9 J כ s Q 2 /Q ₁ 0.9 Ð 7 £ x y y 450 ns, 0.06 J Ð ô Ç . ô Ç¼ # Q ₁ = 10 ⁵ /cm Fig. 7(b)_ â Ä º\ H Q 2 /Q 1 0.1{ 9 M : ì ø Íu

;

¤ É r 50 ns, \ -t H 1.4 J כ s Q ² /Q 1 0.9{ 9 M : y y

150 ns, 0.6 J Ð > í ß ÷ &% 3 . s Qô Ç õ H Q 2 /Q ₁ s

&

t ⁴ I _11/2 ï r 0 Aü < ⁴ I _13/2 ï r 0 A s _ x 9 ¸ ì ø Í s

t l M :ë H \ % 3 ` ¦ Ã º e H Y Us $ ` O Û ¼_ \ -t

t ¦ ` O Û ¼; ¤ É r Z þ t # Q l M :ë H s .

Fig. 7(a) ü < (b)\ ¦ q § Q 1 ° ú כs & t ° ú É r Q 2 /Q 1 \ " f Y Us $ ` O Û ¼_ ; ¤ É r Â ú ª ¦ \ -t H & t H

כ

` ¦ · ú Ã º e . s כ É r Fig. 2\ ¦ Ð ° ú É r Q 2 /Q ₁ ¦

¦ " f x 9 ¸ ì ø Í s & t l M :ë H s .

(5)

Fig. 8. Pulse energy dependence on the ratio of Q 2 to Q 1

for various values of Q 1 (resonator loss α = 0.016 /cm).

/

B N l < Hz ´ α ₀ = 0.016/cm Ð ¿ º ¦ # Q t Q 1 ° ú כ\

@

/ # Q 2 /Q ₁ _ q Ö ¦` ¦ or v " f Q-switching ) a ` O Û

¼_ \ -t ü < ` O Û ¼; ¤` ¦ > í ß ô Ç כ ` ¦ y y Fig. 8õ Fig.

9 \ ? /% 3 . s õ [ þ t \ " f ° ú É r Q 2 /Q ₁ ¦ 8

¸ Q 1 s & | 9 Ã º2 ¤ ` O Û ¼ \ -t H 7 £ x ¦ ` O Û ¼; ¤ É r a % v

9 Q 1 _ ° ú כ` ¦ ß ¼> Õ ªo ¦ Q ² _ ° ú כ` ¦ > H כ s

| Ã Ðf H כ ` ¦ · ú Ã º e .

VI. + s Ç Â ] Ø õ m Í º8 ý

: r 7 Hë H \ " f H Er:YAG Y Us $ _ Q-switching\ @ /ô Ç q

\ " f F g µ 1 Ï ;s Ö ¦ Q 1 õ Q ² _ o\ É r Q- switching ) a Er:YAG Y Us $ ` O Û ¼_ ì ø Íu ; ¤, ì ø Í4 ¤Ò ¦ x 9 Ø

¦§ 4 \ -t _ o\ ¦ ¸ % i . Õ ªo ¦ / B N l < H z

´\ É r Ø ¦§ 4 \ -t _ o\ ¦ ¸ % i . Õ ª õ Q ₂ /Q ₁ _ q Ö ¦ s 7 £ x ½ + ÉÃ º2 ¤ Ø ¦§ 4 \ -t H y è ¦,

` O

Û ¼; ¤ É r 7 £ x 9, ` O Û ¼ ì ø Í4 ¤Ò ¦ É r ± ú t H כ Ü ¼ Ð

z ¤ . ¢ ¸ô Ç / B N l < Hz ´Ò ¦ s 7 £ x ` O Û ¼ \ -t H y

è ¦, ` O Û ¼; ¤ É r Z þ t # Q H כ Ü ¼ Ð z ¤ . s ü < ° ú s

Q ¹ õ Q ² H Q-switching ) a Y Us $ ` O Û ¼_ \ -t ü < ` O Û

¼; ¤ \ × æ כ ¹ô Ç % ò ¾ Ó` ¦ p u ¦, ` O Û ¼_ \ -t \ ¦ ß ¼> ; ¤

`

¦ Â ú ª> 9 Q 1 _ ° ú כ` ¦ ß ¼> , Q ² ° ú כ` ¦ > H כ s

| Ã Ðf . & ³F LiNbO ³ íÖ 0 q ! s q_ l F g < Æ ´ òõ

\

¦ s 6 x ô Ç Er:YAG Y Us $ _ Q- switching\ @ /ô Ç z ´+ « >

`

¦ ' ¦ e . : r 7 Hë H \ " f % 3 É r õ \ ¦ z ´+ « > õ ü

< q §½ + É \ V& ñ s .

Fig. 9. Pulse width dependence on the ratio of Q ₂ to Q ₁ for various values of Q ₁ (resonator loss α ₀ = 0.016 /cm).

P

c p 8 ý ò k >

: r ½ ¨ H é ß ² D G @ / < Æ § _ < ÆY Us $ ½ ¨G ' p' _ t " é ¶ \ _

# s À Ò# Q כ e .

Y

c p w Ã U Ø ô

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Quantum Electron. 24, 920 (1988).

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24, 1133 (1999).

[5] C. E. Hamilton, R. J. Beach, S. B. Sutton, et al.

Opt. Lett. 19, 1627 (1994).

[6] M. Mrochen, V. Semshichen, R. H. W. Funk and T.

Seiler, J. of Refractive Surgery 16, 51 (2000).

[7] D. Chen, C. L. Fincher, T. S. Rose, F. L. Vernon and R. A. Fields, Opt. Lett. 24, 385 (1999).

[8] A. Zajac, M. Skorczawski, J. Swiderski and P. Nyga, Opics Express 12, 5125 (2004).

[9] M.G. Jani, N.P. Barnes and K.E. Murray, Appl.

Opt. 26, 3357 (1997).

[10] H. J. Eichler, B. Liu, M. Kayser and S. I. Khomenko, Optics Materials 5, 259 (1996).

[11] M. Lukac, IEEE J. Quantum Electron. 27, 2094

(1991).

(6)

Sang Ho Lee and Gyu Ug Kim ^∗

School of Natural Science, Kumoh National Institute of Technology, Gyeongbuk 730-701

Yeong-Sik Kim

Division of Advanced Science, Dankook University, Cheonan 330-714 (Received 13 November 2006)

We have analyzed numerically the rate equations for a Q-switched Er:YAG laser. We have investigated, especially, the effects of the Q

1

and Q

2

on the output energy, the pulse width, and the repetition rate of the Q-switched Er:YAG laser, where Q

1

and Q

2

are the photon flux transition rates between the Stark sublevels of the upper level (

⁴

I

_11/2

) and the lower level (

⁴

I

_13/2

) of the Er

³⁺

ion. We have also examined the dependence of the output energy on the resonator loss.

PACS numbers: 42.55.P 07.05.Tp

Keywords: Er:YAG laser, Q-switching, Numerical calculation

∗

E-mail: [email protected]

Er:YAG Y Us $ _ Q-switching\  @ /ô Ç q Ö ¦ ~ ½ Ó& ñ d  ` ¦ Ã ºu K $ 3 % i . : £ ¤ y Er

 »* å    ¸

é

ß ² D G @ / < Æ §, '  é ß õ  < ÆÂ Ò,  ;î ß  330-714 (2006¸  11 Z 4 13{ 9  ~ Ã Î6 £ §)

Er:YAG Y Us $ _  Q-switching\  @ /ô  Ç q Ö  ¦ ~ ½ Ó& ñ d  `  ¦ Ã ºu  K $ 3 % i  . : £ ¤ y  Er

_  [ þ t  H  © I  (

I

) ü <  {   © I  (

I

) _  Û ¼  ß ¼ ! QF Kï  r 0 A (Stark sublevel)  s \ " f F g   µ 1 Ï  ;s Ö  ¦ Q

õ  Q

_    o\    É r Q-switching ) a Er:YAG Y Us $  ` O Û ¼_  ì ø Íu ;  ¤, ì ø Í4  ¤Ò  ¦ x 9 Ø  ¦§ 4 \  -t _    o\  ¦



¸  % i  . Õ ªo  ¦ / B N  l   < Hz ´\    É r Ø  ¦§ 4 \  -t _    o\  ¦ s  : r& h Ü ¼ Ð  ¸  % i  .

PACS numbers: 42.55.P, 42.60G, 07.05.T

Keywords: Er:YAG Y Us $ , Q-switching, Ã ºu  K $ 3

I. " e Â ] Ø

þ

j   H   H& h ü @  % ò % i _   © \ " f µ 1 Ï    H  ¦^  Y Us $ 

\

 @ /ô  Ç   ½ ¨  Ö ¸ µ 1 Ïy    '  ÷ & ¦ e   . t F K  t  Dy:CaF

 

& ñ `  ¦ s 6 x ô  Ç 2 ∼ 3 µm% ò % i _  Y Us $  > hµ 1 Ï÷ &% 3  ¦ [1], Cr, Tm, Ho:YAG   & ñ `  ¦ s 6 x ô  Ç 2.1 µm_  Y Us $ 

  Ð ¦÷ &% 3   [2]. Õ ªo  ¦ Cr, Tm:YSGG (1.862 µm), Cr, Tm, Ho:YSGG (1.924, 2.010, 2.086 µm), Cr, Ho, Er:YAG (2.696 µm), Er:YAG (1.646, 1.640 µm) 1 p x _  Y U s

$  z ´ : r \ " f ½ ¨ & ³ ) a   e    [3]. ô  Ç¼ # ì ø Í ¸^  Y Us 

$

 Ð * 3 i ç #  3 µ\ " f µ 1 Ï    H F g$ 3 Ä » Y Us $ \  @ /ô  Ç



 ½ ¨ Ã º'  ÷ &% 3   [4].

s

 Qô  Ç   H& h ü @  Y Us $ [ þ t ×  æ \ " f 2.94 µm_   © Ü ¼



Ð µ 1 Ï    H Er:YAG Y Us $  [5] _ « Ñ6 x Ü ¼ Ð ´ ú § É r  ' a d  

`

 ¦ ~ Ã Î ¦ e    [6]. s  כ  É r Tm:YAG (2.01 µm), Ho:YAG (2.12 µm), Er:YAG (2.94 µm) Y Us $ _   ©  É r y  y   Ó ü t

\

" f 200 µm, 400 µm, Õ ªo  ¦ 5 µm î ß ¼ 1 Ú_  U ·s \  ¦ È Òõ  K

" f [2,3], Ã ºì  r s  ´ ú § É r  ¸f  [ j í\  Er:YAG Y Us $  c  

`

 ¦ Å Ò ½ + É  â Ä º  É r Y Us $ [ þ t  Ð  È Òõ  U ·s  · û Ü ¼Ù ¼



Ð Ã ºì  r`  ¦  í < Êô  Ç  ¸f  [ j í\  ¦ u « Ñ   H X <  ©  & h ] X  



¦  ¸f  _  & ñ S X  ô  Ç ] X > hü < ] j  0 p x ô  Ç Y Us $ s l  M : ë

 H s  .

Er:YAG Y Us $   H  {   © I ü < [ þ t  H  © I _  ¨ î ç  H Ã º" î s

 y  y   0.12 msü < 2 ms Ð   5 Å q 1 l x    ¢ ¸  H Q-switching

E-mail: [email protected]

1 l

x   s  Ô  ¦ 0 p x ô  Ç  כ Ü ¼ Ð #  4 R M ® o . Õ ª Q  þ j   H Er 3+

s

 : r _  Å Ò{ 9  0 l x  ¸\  ¦ 50 %  t  Z  } >  #  ì ø Í ¸^  Y Us $ 



Ð * 3 i ç ô  Ç   5 Å q 1 l x    Er:YAG Y Us $ \  @ /ô  Ç   ½ ¨   õ 

 µ 1 Ï³ ð÷ &% 3   [7]. ¢ ¸ô  Ç   l  F g  < Æ ´ òõ  [8], 6 £ §  ¾ Ó F g  < Æ ´ ò õ

 [9], FTIR (Frustrated Total Internal Reflection) ~ ½ ÓZ O  [10], r   Ö  ¦ [11] 1 p x`  ¦ s 6 x ô  Ç 0 p x1 l x+ þ A Q-switching ÷  r



m  , Ó ü t õ  \ » 1 Ï : r [12], InAs  í o f  ¨ Ã º^  [13]\  ¦ s 6 x ô

 Ç Ã º1 l x+ þ A Q-switching\  @ /ô  Ç   ½ ¨   õ [ þ t s   Ð ¦÷ &% 3 

.



: r   ½ ¨z ´\ " f  H LiNbO 3  íÖ 0 q ! s q_    l  F g  < Æ ´ òõ 

\

 ¦ s 6 x ô  Ç Er:YAG Y Us $ _  Q- switching\  @ /ô  Ç  

½

¨\  ¦ Ã º'    ¦ e   .  : r  7 Hë  H \ " f  H Er:YAG Y Us $ _  Q-switching \  @ /ô  Ç q Ö  ¦ ~ ½ Ó& ñ d  `  ¦ Ã ºu  K $ 3 #  s  : r

&

h Ü ¼ Ð ì  r$ 3 % i  . : £ ¤ y  Er 3+ _  [ þ t  H  © I ( 4 I 11/2 ) ü <   {



 © I ( 4 I 13/2 ) _  Û ¼  ß ¼ ! QF Kï  r 0 A (Stark sublevel)   s

\ " f F g   µ 1 Ï  ;s Ö  ¦ Q 1 õ  Q 2 _    o\    É r Q- switching ) a Er:YAG Y Us $ _  ` O Û ¼_  ì ø Íu ;  ¤, ì ø Í4  ¤Ò  ¦ x 9 Ø

 ¦§ 4 \  -t _    o\  ¦  ¸  % i  . ¢ ¸ô  Ç / B N  l   < Hz ´Ò  ¦

\

   É r ` O Û ¼_  + þ AI ü < ` O Û ¼;  ¤ õ  ` O Û ¼\  -t _    oü <

Q 2 /Q 1 q Ö  ¦ _    o\    " f ` O Û ¼_  ì ø Í4  ¤Ò  ¦ _    o\  ¦  ¸



 % i  .

II. Q-switched Er:YAG 7 _T $ [ X ê sV 

Fig. 1 É r Er:YAG Y Us $ _  Q-switching`  ¦ 0 Aô  Ç  © u _ 

½

Er:YAG Y Us $ _ Q-switching\ @ /ô Ç q Ö ¦ ~ ½ Ó& ñ d ` ¦ Ã ºu K $ 3 % i . : £ ¤ y Er

»* å ¸

ß ² D G @ / < Æ §, ' é ß õ < ÆÂ Ò, ;î ß 330-714 (2006¸ 11 Z 4 13{ 9 ~ Ã Î6 £ §)

Er:YAG Y Us $ _ Q-switching\ @ /ô Ç q Ö ¦ ~ ½ Ó& ñ d ` ¦ Ã ºu K $ 3 % i . : £ ¤ y Er

_ [ þ t H © I (

) ü < { © I (

) _ Û ¼ ß ¼ ! QF Kï r 0 A (Stark sublevel) s \ " f F g µ 1 Ï ;s Ö ¦ Q

õ Q

_ o\ É r Q-switching ) a Er:YAG Y Us $ ` O Û ¼_ ì ø Íu ; ¤, ì ø Í4 ¤Ò ¦ x 9 Ø ¦§ 4 \ -t _ o\ ¦

¸ % i . Õ ªo ¦ / B N l < Hz ´\ É r Ø ¦§ 4 \ -t _ o\ ¦ s : r& h Ü ¼ Ð ¸ % i .

Keywords: Er:YAG Y Us $ , Q-switching, Ã ºu K $ 3

j H H& h ü @ % ò % i _ © \ " f µ 1 Ï H ¦^ Y Us $

@ /ô Ç ½ ¨ Ö ¸ µ 1 Ïy ' ÷ & ¦ e . t F K t Dy:CaF

& ñ ` ¦ s 6 x ô Ç 2 ∼ 3 µm% ò % i _ Y Us $ > hµ 1 Ï÷ &% 3 ¦ [1], Cr, Tm, Ho:YAG & ñ ` ¦ s 6 x ô Ç 2.1 µm_ Y Us $

Ð ¦÷ &% 3 [2]. Õ ªo ¦ Cr, Tm:YSGG (1.862 µm), Cr, Tm, Ho:YSGG (1.924, 2.010, 2.086 µm), Cr, Ho, Er:YAG (2.696 µm), Er:YAG (1.646, 1.640 µm) 1 p x _ Y U s

$ z ´ : r \ " f ½ ¨ & ³ ) a e [3]. ô Ç¼ # ì ø Í ¸^ Y Us

Ð * 3 i ç # 3 µ\ " f µ 1 Ï H F g$ 3 Ä » Y Us $ \ @ /ô Ç

½ ¨ Ã º' ÷ &% 3 [4].

Qô Ç H& h ü @ Y Us $ [ þ t × æ \ " f 2.94 µm_ © Ü ¼

Ð µ 1 Ï H Er:YAG Y Us $ [5] _ « Ñ6 x Ü ¼ Ð ´ ú § É r ' a d

¦ ~ Ã Î ¦ e [6]. s כ É r Tm:YAG (2.01 µm), Ho:YAG (2.12 µm), Er:YAG (2.94 µm) Y Us $ _ © É r y y Ó ü t

" f 200 µm, 400 µm, Õ ªo ¦ 5 µm î ß ¼ 1 Ú_ U ·s \ ¦ È Òõ K

" f [2,3], Ã ºì r s ´ ú § É r ¸f [ j í\ Er:YAG Y Us $ c

¦ Å Ò ½ + É â Ä º É r Y Us $ [ þ t Ð È Òõ U ·s · û Ü ¼Ù ¼

Ð Ã ºì r` ¦ í < Êô Ç ¸f [ j í\ ¦ u « Ñ H X < © & h ] X

¦ ¸f _ & ñ S X ô Ç ] X > hü < ] j 0 p x ô Ç Y Us $ s l M : ë

H s .

Er:YAG Y Us $ H { © I ü < [ þ t H © I _ ¨ î ç H Ã º" î s

y y 0.12 msü < 2 ms Ð 5 Å q 1 l x ¢ ¸ H Q-switching

x s Ô ¦ 0 p x ô Ç כ Ü ¼ Ð # 4 R M ® o . Õ ª Q þ j H Er ³⁺

: r _ Å Ò{ 9 0 l x ¸\ ¦ 50 % t Z } > # ì ø Í ¸^ Y Us $

Ð * 3 i ç ô Ç 5 Å q 1 l x Er:YAG Y Us $ \ @ /ô Ç ½ ¨ õ

µ 1 Ï³ ð÷ &% 3 [7]. ¢ ¸ô Ç l F g < Æ ´ òõ [8], 6 £ § ¾ Ó F g < Æ ´ ò õ

[9], FTIR (Frustrated Total Internal Reflection) ~ ½ ÓZ O [10], r Ö ¦ [11] 1 p x` ¦ s 6 x ô Ç 0 p x1 l x+ þ A Q-switching ÷ r

m , Ó ü t õ \ » 1 Ï : r [12], InAs í o f ¨ Ã º^ [13]\ ¦ s 6 x ô

Ç Ã º1 l x+ þ A Q-switching\ @ /ô Ç ½ ¨ õ [ þ t s Ð ¦÷ &% 3

.

: r ½ ¨z ´\ " f H LiNbO 3 íÖ 0 q ! s q_ l F g < Æ ´ òõ

¦ s 6 x ô Ç Er:YAG Y Us $ _ Q- switching\ @ /ô Ç

¨\ ¦ Ã º' ¦ e . : r 7 Hë H \ " f H Er:YAG Y Us $ _ Q-switching \ @ /ô Ç q Ö ¦ ~ ½ Ó& ñ d ` ¦ Ã ºu K $ 3 # s : r

h Ü ¼ Ð ì r$ 3 % i . : £ ¤ y Er ³⁺ _ [ þ t H © I ( ⁴ I _11/2 ) ü < {

© I ( ⁴ I 13/2 ) _ Û ¼ ß ¼ ! QF Kï r 0 A (Stark sublevel) s

\ " f F g µ 1 Ï ;s Ö ¦ Q 1 õ Q 2 _ o\ É r Q- switching ) a Er:YAG Y Us $ _ ` O Û ¼_ ì ø Íu ; ¤, ì ø Í4 ¤Ò ¦ x 9 Ø

¦§ 4 \ -t _ o\ ¦ ¸ % i . ¢ ¸ô Ç / B N l < Hz ´Ò ¦

É r ` O Û ¼_ + þ AI ü < ` O Û ¼; ¤ õ ` O Û ¼\ -t _ oü <

Q ₂ /Q ₁ q Ö ¦ _ o\ " f ` O Û ¼_ ì ø Í4 ¤Ò ¦ _ o\ ¦ ¸

% i .

II. Q-switched Er:YAG 7 _T $ [ X ê sV

Fig. 1 É r Er:YAG Y Us $ _ Q-switching` ¦ 0 Aô Ç © u _

¨$ í ¸s . / B N l _ ^ U ´s H 40 cm s 9, Er:YAG

& ñ É r Er ³⁺ s : r _ 0 l x ¸ 50 %, t 2 £ § s 5 mm, U ´s

¦ õ Ø ¦§ 4 Ö ¦ É r ¸¿ º ¨ î Ü ¼ Ð ÷ &# Q e ¦, 2.94 µm\

f_ ì ø Í Ö ¦ É r y y 99 % s © õ 95 %\ ¦ כ ` ¦ 6

x % i .

Er:YAG & ñ É r Xe $ 3 F g1 p x Ü ¼ Ð * 3 i ç ¦, Õ ª כ É r ? /

â

5 mm, ü @ â 6 mm, ~ ½ Ó U ´s 4.4 u s 9 Xes 450 Torr G 0 >4 R e . Xe $ 3 F g1 p x` ¦ 1 l x r v l 0 Aô Ç

¶ / B N/ å L © u Ð H Qr PCI _ 02PS165\ ¦ 6 x

i ¦, ` O Û ¼{ © 150 J_ \ -t \ ¦ / B N/ å L ½ + É Ã º e ¦, 1 ∼ 20 Hz _ ì ø Í4 ¤Ò ¦ Ð 1 l x ô Ç .

íÖ 0 q ! s q É r p ² D G LINOS _ LM7IR` ¦ 6 x % i ¦, 2

" f È Òõ Ö ¦ s 98 % LiNbO 3 & ñ Ü ¼ Ð ½ ¨$ í ÷ &# Q e .

M : Q-switching\ 9 כ ¹ô Ç λ/4 · ú É r 2.94 µm \ " f 6 kV s .

III. Er ³⁺ T Æ X Ø8 ý ; c .U Ç U Ø üÑ ÷ Q 1 Ê Ý Q 2 ë ÅT

ôV ê s

YAG & ñ î ß _ Er ³⁺ s : r _ ï r 0 A × æ Y Us $ µ 1 Ï \ l

# H ï r 0 A\ ¦ Fig. 2 \ ? /% 3 . Y Us $ 1 l x É r

Ö

¸$ í ï r 0 A ⁴ I 11/2 - ⁴ I 13/2 ? /_ Stark $ í ì r X 2 - Y 7 ü <

X _5,6 - Y ₇ s _ ;s \ _ K " f 2936.4 nm ü < 2831.0 nm @ /_ © ` ¦ ~ ½ ÓØ ¦ 9, Õ ª × æ 2936.4 nm © @ /_ Ø ¦

4 s 40C & ñ ¸ ß ¼ . { © I ( ⁴ I _13/2 ) _ ¨ î ç H Ã º" î É r 2 ms Ð [ þ t H © I ( ⁴ I _11/2 ) _ ¨ î ç H Ã º" î 0.12 ms Ð B Ä

º U ´# Q" f 5 Å q 1 l x õ Q-switching 1 l x s # Q 9î r כ Ü ¼

Ð · ú 94 R M ® o . Õ ª Q V. Lupei 1 p x [14] _ ½ ¨ õ \ _

Er:YAG Y Us $ ï r 0 A_ [ þ t H © I \ " f © 0 Aï r 0

A Ð( ⁴ I _11/2 - ⁴ F _7/2 ) _ ¨ 8 (up-conversion) õ { © I

⁴ I 13/2 \ " f © 0 Aï r 0 A Ð( ⁴ I 13/2 - ⁴ I 9/2 ) _ ¨ 8 & ³ © s

Fig. 2. Diagram for the energy levels of Er ion and the photon flux transition rates Q 1 , Q 2 between the Stark sublevels of the upper level ( ⁴ I 11/2 ) and the lower level ( ⁴ I 13/2 ).

H X <, s H × æ í 7 H _ % ò ¾ Ó\ _ ô Ç כ s . s ü

< ° ú É r © 0 Aï r 0 A Ð_ ¨ 8 & ³ © õ Er ³⁺ s : r _ 0 l x ¸

50 % s © Er:YAG & ñ ` ¦ 6 x # 5 Å q [7] 1 l x x 9 Q-switching [8-13] 1 l x \ @ /ô Ç ½ ¨ õ Ð ¦÷ &% 3 .

Fig. 2 \ ³ ðr ô Ç Q 1 õ Q ² _ Ó ü t o | ¾ Ó É r Er:YAG Y Us $