Analysis of saturated absorption spectra of the Cs-D2 line

J. Quant.Spectrosc.Radiat.TransferVol. 36, No. 5, pp. 459-469, 1986

0022-4073/86 $3.00+ 0.00 Copyright © 1986PergamonJournals Ltd

Printed in Great Britain. All rights reserved

ANALYSIS OF SATURATED ABSORPTION SPECTRA OF THE C s - D 2 LINE ISAO HIRANO National Research Laboratory of Metrology, Sakura, Niihari, Ibaraki 305, Japan

(Received 12 March 1986) Abstract--The saturated absorption spectra of the C s - D 2 line are analyzed using four-level and six-level rate equations, including not only radiative transitions but also collisional relaxations. The parameters allowed for are the intensity of the saturation beam and the proportionality constant common to all relaxations. The spectra obtained from the two models agree quite well with our experimental results.

INTRODUCTION Measurements of the saturated absorption spectra ~ of the C s - D 2 line and stabilization of the diode lasers using these spectra 2-4 have been reported previously. Profiles of the saturated absorption spectra obtained theoretically from polarization spectroscopy 5'6 were also published 7.s. In this paper, saturated absorption spectra are obtained in good agreement with experimental results by using a method that is completely different from that used before. Exact solutions of the four-level rate equations are obtained to calculate the saturated absorption profiles of the Cs-D2 line. The spectral intensity and its width are obtained by using saturation parameters and also by considering only absorption and induced emission of the probe beam. These two cases yield almost the same results. F r o m the solution o f the four-level rate equations, it is found that the populations approach equilibrium. Assuming a stationary state, the six-level populations are also found to be applicable to an analysis of the spectra. Both models yield almost the same spectral profiles. OPTICAL PUMPING

OF T H E F O U R - L E V E L

SYSTEM

The Cs-D2 line constitutes a six-level system with two levels in the ground 62St/2 state and four levels in the excited 62P3/2 state. However, the incident laser beam interacts at a particular frequency with one level of the ground state and with one of the four levels in the excited state. The three levels in the excited state that do not interact with the laser light may be treated as a single level. With these assumptions, optical pumping on the Cs-D2 line will now be analyzed with a four-level system. Generally, relaxations due to atomic or wall collisions exert strong effects on the pumping efficiency. Figure 1 shows the four-level system. The level populations are designated as N~, N2, N3, and N4 and the time changes of these populations are then

dN1/dt = - (A + L ) NI + BN2 + UN3 + HN4, dN2/dt

ANI-(B+D+E)N2+FN4,

dN3/dt

LNm + DN2 - UN3 + GN4,

dN4/dt

E N 2 - ( F +G + H ) N 4.

i

~ /

(1)

i

3

The solutions o f these differential equations are

Nl = aMoR . e x p ( - 0 t ) + bMvS.exp(-vt) + cM~T.exp(--#t) + c&, N2=aA(2-O)R'exp(-Ot)+bA(2-v)Sexp(-vt)+cA(2-1~)

T'exp(-#t)-cA2E f

N3 = - a N o R - e x p ( - 0t) - bN,,S-exp(- vt) - cNuT.exp(-t~t) + No - c& + cA2E + e, N4 = A E R . e x p ( - O t ) + b A E S . e x p ( - v t ) + c A E T . e x p ( - # t ) - c. 459

(2)

460

ISAO HIRANO

4

El

2

N

Fig. 1. The four-level system. The solid lines show resonance transitions. The wavy lines show relaxations. The nos I and 3 denote ground-state hyperfine structures and the nos 2 and 4 belong to the excited state. The incident laser beam interacts with levels I and 2, the transition probability of which is A. The symbols B, D, G, and H indicate transition probabilities between the excited and ground states. The symbols U and L are relaxation rates between the ground hyperfine levels. The symbols E and F show those within exited levels.

~I

IANo_N t

(ANo-N~+N2)

I

f

(N,- N2)f

Fig. 2. The levels 1 and 2 show populations. The lowest curve shows the absorption coefficient.

The values of the coefficients in these equations are sumamarized in Appendix 1. Hereafter, it is assumed that Nj, N2 etc. indicate populations at t = 0% the validity of which is shown in Appendix 2. In the stationary state, N) - N2 = (2a - E F - 2 A ) U N o / y .

(3)

SATURATED ABSORPTION SPECTRA In measurements of saturated absorption spectra, the strong saturation and weak probe beams are sent in opposite directions through an absorption cell. Two beams of equal frequency interact with atoms having approximately zero axial velocity. The probe beam passes through the Lamb dip produced by the saturation beam. The saturation beam produces stationary populations in levels 1 and 2, as is shown in Fig. 2. For the ground state 1, the population is A N o . The Bennett hole is produced at the center of the Doppler distribution. The depth of the hole is A N o - N l . The excited state has population N2. For this condition, the probe beam absorption is proportional to N~ while it induces stimulated emission proportional to N2. Therefore, assuming f to be proportional to the oscillator strength, the line strength of the saturated absorption spectra becomes ( A N o - N j + N 2 ) f . The saturated absorptionspectra profile is then approximately expressed as follows: I(v) = ANof

e x p - [(v - Vo)/Avn] 2 - - f ( A N o

- - N~ + N 2 ) ( F / 2 ) 2 / [ ( v - - v0)2 + (F/2)2],

(4)

where v0 denotes the central frequency of the resonance or the cross-over resonance line. The Doppler-width is AvD. In this approximate equation, the spectral width is F in the Doppler limit (F <
- N2"exp - [(v' - v " ) / Z A v o ] 2 } ,

(5)

where the frequencies of the resonance lines relating to the cross-over resonance are v' and v", N o

Saturated absorption spectra

461

and N o are populations in the absence of the laser beam. F o r the cross-over resonances, the line width is 7so = 7 (1 + CR) 1/2. In profile-fitting calculations, the spectral widths of the incident laser light and broadening by the intersection angle between the saturation and probe beams 9 are folded in F and 7s,..

A P P L I C A T I O N OF T H E F O U R - L E V E L S Y S T E M TO T H E Cs-D 2 S P E C T R A L L I N E Figure 3 shows the energy levels of the Cs-D2 line. When the laser frequency is swept, the light beam induces twelve transitions, including cross-over resonances. By applying the four-level approximation to each of the twelve cases, the entire spectrum is obtained for overlap of the twelve cases. When the laser beam induces a transition between the hyperfine structures Fg = 3 and F e = 2, the value of B in Fig. 1 becomes 20/128 with D = 0, G = 722/(27. 128), which are averaged by the weighting factors considering the Zeeman levels, H is 141/(8. 128). The symbols E, F, U, and L express relaxation probabilities and are assumed to have the same proportionality constant (V I) while obeying the equipartition law E = 1/'1.27/32, F = I/'1.5/32, L = 11"1.9/16, and U = II1.7/16. It is also assumed that one perturbation causes one relaxation for both excited and ground levels because the excited state and ground states reach stationary population distributions. In this case, the cross-over resonances are produced between excited levels Fe = 2 and Fe = 3 and between F~ = 2 and Fe = 4. The separation of the energies E¢ of F~ = 2 and Ee of F¢, = 3 is 152.3 M H z and, for E,(Fe = 2) and E,,(Fe = 4), the separation is 355.1 MHz. All other transitions are similarly clarified. Table 1 shows a s u m m a r y for the twelve cases. In Table 1, P denotes the population distributions for Fg = 3 or Fg = 4 before onset o f the laser beam but assuming equipartition. No is the number of atoms. X ( I ) is the spectral strength constituting the Doppler background.

F 5

./ 4

~9:

6 ZP3/2

Sum 7

2 \

Cs-D2

2O 2t

15

7 21 44 x

t

128

4 Sum t6

6 2~/2 3

Fig. 3. Energy levels of the Cs-D2 line and transition probabilities of the resonance lines.

ISAO HIRANO

462

Table 1. Parameters for the C s - D 2 line B(I)

D(I)

G(I)

H(I)

E(I)/V!

F(I)/VI

I

Fg

Fe

Fe'

Ee-Ee'

"128

"128

*128

"128

• 128

*32

*32

L(I)/VI

U(I)/VI

P(I)

1 2 3 4 5 6 7 8 9 10 11 12

3

2

3 4 2 4 2 3 4 5 3 5 3 4

152.3 355.1 152.3 202.8 355.1 202.8 202.8 455.8 202.8 253.0 455.8 253.0

20

20

0

722/27

141/8

27

5

9/16

7/16

7No/16

21

21

7

673/20

235/14

25

7

15

15

21

533/18

247/12

23

9

7

7

21

235/14

673/20

25

7

7/16

9/16

9No/16

21

21

15

247/12

533/18

23

9

44

44

0

382/21

238/16

21

I1

3 4 4

3 4 5

X(I)

The l t h parameters (I = 1. . . . . 12) are assumed to be A (I), B (I) . . . . . 2 (I), a (I), etc.

and

A (I) = XI. B (I),

XI denotes the photon number in the saturation beam interacting with the atoms. The line strength of the resonance line is given by I ( I ) = a (I) P(I){1 - 1/[1 + S(I)] 1/2} or

A (I)[P(I) - Nl(I ) + N2(I)],

(6)

where S (I) is the saturation parameter. For cross-over resonances, the strength is obtained by replacing I ( I ) by C(I), S ( I ) by CR(I), and Nz(I)'by N z ( I ) ' E E ( I ) , where

EE (I) = exp - [(Ee,- E~)/2Avo] 2.

(7)

It should be noted that cross-over resonances are contributed by two resonance lines. The cross-over spectral strengths CC o. (i and j denote Fe and Fe,) are given by

CC32 = C(1) + C ( 3 ) , . . . , etc.

(8)

The cross-over spectral widths 7,j are given by the average 732 = [7,,(1) + 7~c(3)]/2. . . . . etc.

(9)

RESULTS Figure 4 shows a calculated line strength for the Fg = 3 component. In this case, the absorption-cell temperature is T = 293 K, XI = 0.1, and VI = 0.01. The spectral intensities in the upper half show the Doppler background. The lower half shows the resonance and cross-over resonance lines. The numbers show the relative intensities of the lines. The widths of the spectra lie between 5 and 7 MHz. The theoretical profile is shown in Fig. 5. The group of lines show relative positions of lines with respect to the spectral profile. Comparison between Fig.'5 and the measured result in Fig. 6 indicates good agreement. In Fig. 5, the spectral width including the laser spectral width and others is taken to be about 130 MHz. Figure 7 shows the line-strength distribution for the Fg = 4 component. The cell temperature is 293 K, XI = 0.1, and VI = 0.01. The upper half denotes linear absorption strengths. In the lower half, the line-strength distribution seems to be largely modified by cross-over resonances. In this case, the numbers show relative intensities. The widths of the spectra are between 5 and 7 MHz. Figure 8 shows the calculated profile. In this case, the spectral width, including the laser width, is about 120 MHz. The profile is in close agreement with measured results in Fig. 9. The spectra are good agreement with measured results and are obtained for values of 3(1 and VI in the enclosed part of Fig. 10 and in the neighbouring regions. P O P U L A T I O N S FOR SIX L E V E L S As shown in Fig. 11, the Cs-D2 line is composed of six levels. In the figure, A', B', C', D', E', and F' express transition probabilities. Symbols G' and H ' denote relaxation probabilities within

Saturated absorption spectra

Fg:3 T:293K

463

XI=0.1 VI = 0.01

20 t5

i

5i00

5t O0

5183.731

5082.326

50'00

49'00

48'00

4980.920 L/ MHz 5006.t52 4904.750

4828.581

I

608 )'o = 5.4 t MHz

| 278 )'s = 6.56 MHz

1468

2690 T3.4=6.34 MHz

1698 Ts--6.19 MHz T2.3= 5.89 MHz 2017 )'32= 5.77 MHz

Fig. 4. Line-strength distribution for the Fg = 3 component.

I

I

6000

5000

4000

MHz 6000 I

5001

4000 I

Fig. 5. Calculated profile for the

Fg=

3 component.

ISAO HIRANO

464

106mA

~-~

< Fig. 6. Experimental results for the Fg = 3 component. The cell temperature is around 293 K. The injection current is 106 mA. The laser power is typically 10 mW (measured by the manufacturing company). As the result of reflection and scattering, the power in the absorption cell, in this experiment, is estimated to be reduced to a few hundredth of a/~W for the saturation beam and to a few tenths of a/~W for the probe beam.

the ground-state hyperfine structures. I ' . . . T' are relaxation rates within the excited sublevels. We define the population of levels as N1... N6, and then obtain the equations dN1/dt =

-

( A ' + J' + P" + T') NI + S ' N 2 + O'N3 + I'N4 + AX" Ns,

dN2/dt = T'N, - (B' + D" + L ' + R ' + S') N2 + Q'N3 + K'N4 + B X "N5 + DX" N6, dN3/dt = P'N~ + R'Nz - (C' + E' + N ' + O' + Q') N 3 + M'N4 + C X ' N 5 + E X ' N 6 , dN4/dt = J'Nl + L'Nz + N'N3 -- (F' + I" + K' + i ' )

(lo)

N4 + FX" N6,

dN~/dt = A 'N~ + B'N2 + C'N3 - (H" + A X + B X + C X ) N5 + G'N6, dN6/dt = D'Nz + E'N3 + F'N4 + H'N5 - (G' + O X + E X + F X ) N6, X[=0.1 Vl= 0.01

Fg=4 T:293K

44

21

I

7

I

-42'00 -421t.71

-4t'00

-40'00 -3983.80

-4tt0.3t

-4008.90

-39b0

-38b0

)-Y MHz

-588L39

-37b0 -3755.88

I

523 7° = 5.93 MHz

1656 ~3=5.56 MHz 2587 2865 ~3=6.26 MHz

~'s = 6 . 6 5 MHz

t 760 T. = 5.42 MHz o

5582 74.5= 5.97

Fig. 7. Line-strength distribution for F~ = 4.

MHz

Saturated absorption spectra

465

I

I

-3000

-4000

-5000 )./

MHz

- 4000

-3000

- 5000

I

I

I

Fig. 8. The calculated profile for the Fg = 4 component.

where A X , B X , . . . . F X are the probabilities when the laser beam induces the transitions A',B',

.... F'.

For equipartition, the relaxation rate is proportional to VI, which is common to the ground and excited states. The relaxation probabilities are G ' = 9. V I / 1 6 , H ' = 7. V I / 1 6 , I ' = 11. V I / 3 2 . . . . .

etc.

The level populations become N, = ~'{11WU0-- [(11 W - - A X ) h

+ 11Wi]~/},

N2

fl'{9 W N o -- [(9 W -- B X ) h + (9 W - D X ) i]/j},

N3

7'{7 W N o - [(7 W - C X ) h + (7 W - E X ) i]/j},

N4 = 6'{5

WUo -

[5

Wh +

(5

W - FX)

i]/j},

1

/ (11)

/

J

N 5 = h/j,

N6=i#,

the coefficients are listed in Appendix 3. In this case, as in Table 1, the number I (I = 1 . . . 12) is assumed, Then, for transitions caused by the saturation beam, the following assumptions are made: AX(I) AX(ll)

QS.R.T. 36/~-E

= 0, I = 1 . . . 10, AX(12) = X I . 4 4 / 1 2 8 ,

BX(I)

0,1=1...8,1=11...12,

BX(9)

BX(10) =)(1.21/128 . . . . .

"]

F

etc.j

(12)

ISAO HIRANO

466 F=4

~~/~~.....~

~i7mA)J V

u

<

Fig. 9. A measured example for the Fg 4 component. The cell temperature is around 293 K. The injection current is 107 mA. =

0.015

VI

001

0005

0001

0

O' .I

0.2

~Xl 04

013

Fig. 10. This figure shows the parameters

X1

and

F/5

VL

T ST I1'1

QR I~, [J,

Cs,D2 852. lnm

\

T

[J II,

KL I~,

Nt

253 0 MHz

I,I,

N2 202.8 MHz

OP

\5

T

N3

t 52.3 MHz I~,

N4

ABC N~

9t9 GHz

6 2Pt/2 E

N6

Fig. 11. The energy diagram of the

Cs-D 2

line.

Saturated absorption spectra

467

G2:5.421 4782

G3=6.195 G4=6. 561

3722

G34= 6. 542 G2=5.408

6 levels

3093 2655

G23= 5. 779 G24= ,5.895 2256

I

t191

I

4785

G3 =6. t94

3695 3065 2636

G4 =6.562 G23 =5. 772 2257

4 levels

1163

G24:5 889 634 =6.543 4,5

2,4 5

G2=5.421

2,5

I

2

N o - N t .t-N 2

G3=6.t95 G4=6. 561 G23=5.779

2688

G24=5.895 G34=6.342 1277 G2=5.408

G3=6.t94 G4 =6. 562 G23=5.772 G24=5.889 G34=6.343

17t 51467

1

I t 278

205t

26901

16981468 I

6 levels

624

I 2017

4 levels 608

4,5

2,4 5 (2o ( t - t / J t + S

)

I 2,3

2

XI=0.t Vl:O.01 Fg=5 T=295 K

Fig. 12. Comparison of the distribution of spectral line strengths. where X I is proportional to the laser-beam intensity. In this way, the line strengths are obtained for saturated absorption spectra. Comparison between line strengths obtained for the six-level and four-level approximations are shown in Fig. 12 for Fg = 3 and in Fig. 13 for Fg = 4. The upper halves of both figures show the spectral intensity distribution without the saturation parameter, while the lower halves denote those calculated from the saturation parameter. It is assumed that X I = 0.1, 111 = 0.01, and T = 293 K. The G-values shown at the left are the spectral widths, the units of which are MHz. The line strengths obtained for the four-level approximations have almost the same values as those for six-level populations. The same behaviour holds for the spectral widths. D I S C U S S I O N AND C O N C L U D I N G

REMARKS

The saturated absorption spectra of the C s - D 2 line can be calculated by using a four-level approximation and also from a six-level model. In both cases, the calculated profiles agree well with our experimental results. In Figs 12 and 13, line strengths obtained by using the saturation parameter are found to be different from those obtained for absorption and induced emission. The spectral profile is determined by the line-strength ratio. Different values of the strengths do not change in the profiles if their relative strengths are almost the same.

468

ISAO HIRANO G3 =5,936

7074

G4 =6. 648 G5=5. 439 G43 =6,267

4161

3468

5052

6 levels

32781

635 = 5. 573

957

G45= 5. 977

I 6958

63=5,935

5046

64=6. 645 65=5 420

4156

336t

643= 6.265

4

levels 956

635 = 5. 564

4,5

645 = 5. 966

3,5

4

4,3

I

3

No-Nt-t-N2

63=5. 936

3895

G4=6. 648 G5=5 439 643 = 6.267

2390

2869

6 levels

1820

G35= 5. 575

523

I

G45: 5. 977 3832 G3=5,935 G4=6. 645 G5=5.420

2387 1760

2865 4 levels

t656 r

643 = 6.2 65

523

635 = 5.5 64 G45= 5. 966

,5

'3,5

ao ( t - t l . 5 - ~

4 )

4,3

I

3

X[ =0. t VI = 0.0t Fg: 4 T : 2 9 3 K

Fig. 13. Comparison of line strength distribution for the Fg = 4 component.

I n Fig. 5, the spectral w i d t h including the laser b a n d - w i d t h is a s s u m e d to be 130 M H z a n d in Fig. 8 a b o u t 120 M H z . Values for the w i d t h s o f a b o u t 6 M H z are c a l c u l a t e d f r o m the s a t u r a t i o n p a r a m e t e r a n d d o n o t influence profile fitting in this case. F o r s h o r t - t i m e e v o l u t i o n o f p o p u l a t i o n s , the exact s o l u t i o n o f the four-level rate e q u a t i o n can be readily used. In this p a p e r , the s h o r t - t i m e e v o l u t i o n is n o t discussed because o u r m e a s u r e d results d o n o t v a r y with time. T h e t i m e - v a r i a b l e p o p u l a t i o n s can be t r e a t e d b y the four-level model. This a p p r o x i m a t i o n is also a p p l i c a b l e to the m a n y level system, for e x a m p l e to the Z e e m a n split levels o f the Cs-D2 line. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

I. Hirano, Opt. Commun. 57, 331 (1986). I. Hirano, Bull. N.R.L.M. 34, 203 (1985). T. Yabuzaki, A. Ibaragi, H. Hori, M. Kitano and T. Ogawa, Jap. J. appl. Phys. 20, L451 (1981). H. Hori, Y. Kitayama, M. Kitano, T. Yabuzaki and T. Ogawa, IEEE. J. Quantum Electron. QE-19, 169 (1983). C. Wieman and T. W. H~insh, Phys. Rev. Lett 36, 1170 (1976). S. Nakayama, G. E. Series and W. Gaulik, Opt. Commun. 34, 382 (1980). S. Nakayama, Jap. J. appl. Phys. 23, 836 (1984). S. Nakayama, Jap. J. appl. Phys. 24, 1 (1985). I. Hirano, Bull. N.R.L.M. 34, 207 (1985).

Saturated absorption spectra

APPENDIX

469

1

Parameters o f Solutions for Four-Level System The parameters of equation (2) are assumed to be a = 1/ A E ( v - O)(lt - 0), b = I / A E O -- v ), c = 1/AE, d = - 2 a + EF, 2 = F + G + H , a = B + D + E, N o = ( 2 - O ) ( a - O ) - E F + A ( 2 - O ) + AE, N v = ( 2 - - v ) ( a - v ) - E F + A(2-v)+ AE , N~ = ( 2 - # ) ( t r - - / ~ ) - - E F + A ( A - # ) + A E , M u = (2--/t)(a-#)-EF, Mo = (2-O)(a--O)-EF, M~ = (2 - v)(a - v) -- EF. Assuming ~ = 2 + a + A + L + U , fl=2a-EF+(A+L+U)(A+a)-AB+AU, 7=(A+L+U)(Aa-EF)AB2 + AU2 + A E U - AEH, 6 = - A E U N o , E = 6/7, O, v, a n d / z satisfy the following equations:

03--~xO2-~-fl 0 --7 = 0 , V3--OtV2-~-flV --7 = 0 , ]23 -- ~(#2 "~- fl~ --~/ : 0 . Other parameters are given by R = { N o ( Y 2 W 3 - - W z Y 3 ) + Yo(U2W3--U3Wz)+ Y 2 ( U 3 W o - U o W 3 ) + Y3(UoWz--U2Wo)}/J, S = { N o ( Y 3 W , - Y, W3)+ Y o ( U 3 W I - UIW3) + Y t ( U o W 3 - U3Wo) + Y 3 ( U I W o - UoWj)}/J, T = { N o ( Y I W2 -- Y 2 W I ) + Y o ( U ~ W : - U:W~)+ Yl(U:Wo - UoW2)+ Y : ( U o W I - UiWo)}/J, J = Y , ( U 3 W : - U: W3) + Y2(U~W3 - U3W~) + Y 3 ( U 2 W I - UIW2), where Uo=No+cA2E +E, U, = a ( M o - N o ) , U : = b ( M u - N u ) , U3=c(M~,--N~) , Yo = - c A 2 , , Y ~ = a A ( 2 - 0 ) , Y 2 = b A ( A - v ) , Y 3 = c A ( A - I z ) , W0=--~, W I = a A E , W2=bAE, W3=cAE.

APPENDIX

2

The Reason for Reaching a Stationary State In the solutions of the equations, there appear the factors exp ( - O t ) , e x p ( - v t ) , e x p ( - # t ) , where the summation of the coefficients for each factor from N~ to N 4 is always zero. The sum of the constant terms is N 0. In the exponential factors, 0, v, # are generally complex numbers. Their real parts are positive numbers because the populations diverge if they are negative. If 0, v, # were pure imaginary numbers, ~fl = 7, or y = 0. Because 7 > 0 in general and assuming q~ = A + L + U, ~fl -- ~ = (2 + ~r + q~)[~b(2 + a) - AB] + A ( E H + B2) + (2 + a)(2a - EF) + AU(cr + (9 - E) > O, and it follows that 0, v, # are not pure imaginary numbers, i.e. the population of each level converges to the stationary state. The factors 0, v, a n d / z express complex velocities to reach an equilibrium. At t = 0, i.e. before the onset of the incident laser light, N I + N 3 = No, N 2 + N 4 = 0, and N 1(t = ~ ) = cd~ = (2a - EF) UNo/y > O, N2(t = ~ ) = --cA2E = 2AUNo/y > O.

APPENDIX

3

Coefficients for Six-Level Solutions Assuming that the populations reach an equilibrium state as t --, ~ , dNi/dt = O, for i = 1 . . . 6. Then, putting W = VI/32 and using the relation N, + N z + N 3 + N 4 + N s + N 6 = N o , while assuming a ' = 1/(A" + 32W), f l ' = 1/(B' + D" + 32W), 7 ' = 1/(C" + E' + 32W), 6"= I/(F" + 32W), Y = 11A "~" + 9B'fl" + 7C'y', Z = 9D'[3" + 7E'y' + 5F'5", and a = W ( 1 4 + Y ) + A X ( 1 - A ' c t ' ) + B X ( I - B ' f l ' ) + C X ( I - C'7'), b = W ( - 1 8 + Y ) - D X . B ' f l ' - E X . C ' v ' , c = WNoY, d = W ( - 1 4 + Z ) - B X . D'fl' -- C X . E'v ', e = W (18 + Z ) + D X ( 1 - D'fl') + E X (I - E'7 ") + F X ( I - F'5'), f = WNoZ, h =ce-bf, i =af-ed, j=ae-bd.