- Email: [email protected]
Density operator for unpolarized radiation
Volume 34A, number 1
P H Y S I C S L E T TE RS
A ± = 670 ± 60MHz;
All = 1400 + 60MHz; a=ll2+3MHz;
g
2.007+0.001;
S
The i n t e r s t i t i a l i m p u r i t y spin r e s o n a n c e had i s o t r o p i c g - f a c t o r and was fitted by the s a m e form of spin H a m i l t o n i a n when anisotropic hyperfine i n t e r a c t i o n s with eight f i r s t shell 19F- ions i were included, (axial components a , ,a±) in addition to a n i s o t r o p i c i n t e r a c t i o n s with 207pb2+ ions j located in the second coordination shell. The p a r a m e t e r s found for the i m p u r i t y s p e c t r u m were: Auj = 630 + 20 MHz;
A± = 250 + 30MHz;
a,,=
a.=
98+
3MHz;
g = 2.0012 + 0.0003;
62+
3MHz;
S = ½.
As was pointed out e l s e w h e r e [3] the i r r a d i a =
DENSITY
OPERATOR
FOR
25 January 1971
tion induced defect exhibits the s a m e anisotropy in the ESR hyperfine splitting as would be expected for a simple F - c e n t r e , c o n s i s t i n g of a single e l e c t r o n trapped in a f l u o r i n e vacancy. An a l t e r native i n t e r p r e t a t i o n as a s p h e r i c a l l y s y m m e t r i c t r a p p e d - h o l e c e n t r e will be d e s c r i b e d elsewhere. It is hoped that ENDOR m e a s u r e m e n t s will decide between the two models. This work is supported by the Science R e s e a r c h Council.
References [1] K.K. Chan and L.Shields, J.Phys. C2 (1969) 1978. [2] A. V. Patankar, J. Sci. Instrum. 44 (1967) 354. [3] B. F. Rose and E. E. Schneider, XVth Colloque AMPERE, Grenoble, (September, 1968), Magnetic Resonance and Radiofrequency Spectroscopy, ed. P.Averbuch (North Holland, Amsterdam) p. 341.
UNPOLARIZED
RADIATION
H. PRAKASH and N. CHANDRA Department o3 Physics, University of Allahabad, Allahabad, India Received 23 October 1970
The form of density operator for unpolarized radiation is obtained by a simple method.
In a r e c e n t note [1], the a u t h o r s calculated c r o s s - s e c t i o n s for second h a r m o n i c g e n e r a t i o n in s c a t t e r ing of radiation of a g e n e r a l n a t u r e by free e l e c t r o n s and pointed out that the c u r r e n t l y used p r o c e d u r e [2] for studying n o n l i n e a r i n t e r a c t i o n s of unpolarized radiation is not c o r r e c t . The p u r p o s e of the p r e s e n t note is to r e p o r t s the form of the d e n s i t y o p e r a t o r for u n p o l a r i z e d r a d i a t i o n obtained by a s i m p l e method and to point out that the "unpolarized r a d i a t i o n " , n o n l i n e a r i n t e r a c t i o n s of which have been studied by s e v e r a l authors [2], is not actually unpolarized. An u n p o l a r i z e d beam of r a d i a t i o n is defined as one which is (i) s y m m e t r i c about the d i r e c t i o n of propagation, and (ii) r e m a i n s so even on i n t r o ducing a r b i t r a r y phase changes in its two orthogonally p o l a r i z e d components. To avoid c o m p l i c a t i o n s , we c o n s i d e r a single mode of r a d i a t i o n , say, (k, ~), and e x p r e s s the d e n sity o p e r a t o r in the f o r m
=
Ivac>
(1)
w h e r e ~ and 66 a r e the c r e a t i o n and annihilation o p e r a t o r s for this mode. If e x , e y , e z a r e the t h r e e r e a l u n i t v e c t o r s along the coordinate axes, and if e z is along k , we can e x p r e s s the annihilation p a r t of the o p e r a t o r A (x, t) as
SThe authors have obtained the same results also by a different mathematically rigorous but complicated method; the calculations wiLLbe reported elsewhere. 28
Volume34A, number 1
A (-)(x,t)
PHYSICS LETTERS
= (2~o/V)l/2[ex~x+ey~y]expi(k.x-o)t), = (2,o~/r)1/2[~(~
25 January 1971
(0~=
tkl)
x + ~y* .ay) + e±(Cy ax - exay)] exp i(k. x -cot),
(2)
where fix and fly a r e a n n i h i l a t i o n o p e r a t o r s for the modes (k, ex) and (k, ey) r e s p e c t i v e l y , e = ex e x + e y e y is a complex unit v e c t o r (i.e., e . e* = 1), e± is a unit v e c t o r orthogonal to e (i.e., e. e* = 0). F r o m eq. (2), it is evident that
at
= ~ x• a^x + e y*^a y ,
S t J. = E y a x - C x h y
(3)
a r e a n n i h i l a t i o n o p e r a t o r s for the modes (k, 6) and (k, 6±). Substitution of the expansion of fig in eq. (1) gives the e x p r e s s i o n of ~ e in the space of the s t a t e s Jr, s) - (r:s'.) -1/2 ~ r ~ TS[vac>. Since, for u n p o l a r i z e d r a d i a t i o n , • can take r a n d o m l y any value which sYalisfies the r e l a t i o n t . c* = 1, the d e n s i t y o p e r a t o r for u n p o l a r i z e d r a d i a t i o n can be obtained by a v e r a g i n g P e over e . This is done by i n t e g r a t i n g f3 r over E~c and E~ with the weight f u n c t i o n ~ P ( e ) = A 5(~. 6" - 1~ N o r m a l i z a t i o n condition / d 2 ~xd2Ey P(~) = 1, gives ~l = 2~ -2. The density operator for unpolarized" radiation thus has the form
p~
= / d 2E xd 2£y27r-25(~.~* - 1)/~ = ~n (n+l)-lpn;~
Ir,n -~'}
(4)
Eq. (4) shows that unpolarized radiation can exist in infinite different states having different values of {Pnn~, and Pe cannot be specified by any single parameter, e.g., the total number of photons The commonly used procedure for studying nonlinear interactions of unpolarized radiation [2] of averaging the results for plane polarized coherent radiation over the direction of polariz~ttion is evidently incorrect. It is interesting to see that the thus obtained "unpolarized radiation" (which is completely specified by a single parameter, the intensity) is not actually unpolarized as its nature changes * on introducing arbitrary phase changes in the x- and y-components.
(=~nnpnn).
The a u t h o r s a r e thankful to P r o f e s s o r Vachaspati for his i n t e r e s t in this work. One of the a u t h o r s (N. C.) g r a t e f u l l y acknowledges the f i n a n c i a l support of Indian National Science Academy, New Delhi. ~t The weight function satisfies the requirements (i) and (ii) in the definition of unpolarized radiation. This is because the authors [2] obtained this "unpolarized radiation" by averaging a plane polarized coherent radiation over the direction of polarization, and this does not involve separate averagings over the phases of Ex and
Ey.
Refe~'ences [1] H. Prakash and N. Chandra, Phys. Letters 31A (1970) 331. [2] 7.. Fried, Nuovo Cimento 22 (1961) 1303; Vaehaspati, Phys. Rev. 128 (1962) 664; 130 (1963) 2598(E); Vachaspati and S. L. Punhani, Prec. Nat. Inst. Sci., India, A29 (1963) 129; N. C.Rastogi and Vachaspati, Indian J. Pure Appt. Phys. 2 (1964) 426; L. M. Bali and J. Dutt, Nuovo Cimento 35 (1965) 805; J°Dutt, L.M. Bali and Vachaspati, Indian J. Pure App[. Phys. 4 (1966) 137; 5 (1967) 267; M. K. Srivastava and Vachaspati, Indian J. Pure Appl. Phys. 4 (1966) 141 ; S.S. Jha, Phys. Rev. 140 (1966) A2020; R. C.Garg and Vachaspati, Indian J. Pure Appl. Phys. 6 (1966) 329; H. Prakash and Vachaspati, Huovo Cimento 53B (1968) 24.
29
P H Y S I C S L E T TE RS
A ± = 670 ± 60MHz;
All = 1400 + 60MHz; a=ll2+3MHz;
g
2.007+0.001;
S
The i n t e r s t i t i a l i m p u r i t y spin r e s o n a n c e had i s o t r o p i c g - f a c t o r and was fitted by the s a m e form of spin H a m i l t o n i a n when anisotropic hyperfine i n t e r a c t i o n s with eight f i r s t shell 19F- ions i were included, (axial components a , ,a±) in addition to a n i s o t r o p i c i n t e r a c t i o n s with 207pb2+ ions j located in the second coordination shell. The p a r a m e t e r s found for the i m p u r i t y s p e c t r u m were: Auj = 630 + 20 MHz;
A± = 250 + 30MHz;
a,,=
a.=
98+
3MHz;
g = 2.0012 + 0.0003;
62+
3MHz;
S = ½.
As was pointed out e l s e w h e r e [3] the i r r a d i a =
DENSITY
OPERATOR
FOR
25 January 1971
tion induced defect exhibits the s a m e anisotropy in the ESR hyperfine splitting as would be expected for a simple F - c e n t r e , c o n s i s t i n g of a single e l e c t r o n trapped in a f l u o r i n e vacancy. An a l t e r native i n t e r p r e t a t i o n as a s p h e r i c a l l y s y m m e t r i c t r a p p e d - h o l e c e n t r e will be d e s c r i b e d elsewhere. It is hoped that ENDOR m e a s u r e m e n t s will decide between the two models. This work is supported by the Science R e s e a r c h Council.
References [1] K.K. Chan and L.Shields, J.Phys. C2 (1969) 1978. [2] A. V. Patankar, J. Sci. Instrum. 44 (1967) 354. [3] B. F. Rose and E. E. Schneider, XVth Colloque AMPERE, Grenoble, (September, 1968), Magnetic Resonance and Radiofrequency Spectroscopy, ed. P.Averbuch (North Holland, Amsterdam) p. 341.
UNPOLARIZED
RADIATION
H. PRAKASH and N. CHANDRA Department o3 Physics, University of Allahabad, Allahabad, India Received 23 October 1970
The form of density operator for unpolarized radiation is obtained by a simple method.
In a r e c e n t note [1], the a u t h o r s calculated c r o s s - s e c t i o n s for second h a r m o n i c g e n e r a t i o n in s c a t t e r ing of radiation of a g e n e r a l n a t u r e by free e l e c t r o n s and pointed out that the c u r r e n t l y used p r o c e d u r e [2] for studying n o n l i n e a r i n t e r a c t i o n s of unpolarized radiation is not c o r r e c t . The p u r p o s e of the p r e s e n t note is to r e p o r t s the form of the d e n s i t y o p e r a t o r for u n p o l a r i z e d r a d i a t i o n obtained by a s i m p l e method and to point out that the "unpolarized r a d i a t i o n " , n o n l i n e a r i n t e r a c t i o n s of which have been studied by s e v e r a l authors [2], is not actually unpolarized. An u n p o l a r i z e d beam of r a d i a t i o n is defined as one which is (i) s y m m e t r i c about the d i r e c t i o n of propagation, and (ii) r e m a i n s so even on i n t r o ducing a r b i t r a r y phase changes in its two orthogonally p o l a r i z e d components. To avoid c o m p l i c a t i o n s , we c o n s i d e r a single mode of r a d i a t i o n , say, (k, ~), and e x p r e s s the d e n sity o p e r a t o r in the f o r m
=
Ivac>
(1)
w h e r e ~ and 66 a r e the c r e a t i o n and annihilation o p e r a t o r s for this mode. If e x , e y , e z a r e the t h r e e r e a l u n i t v e c t o r s along the coordinate axes, and if e z is along k , we can e x p r e s s the annihilation p a r t of the o p e r a t o r A (x, t) as
SThe authors have obtained the same results also by a different mathematically rigorous but complicated method; the calculations wiLLbe reported elsewhere. 28
Volume34A, number 1
A (-)(x,t)
PHYSICS LETTERS
= (2~o/V)l/2[ex~x+ey~y]expi(k.x-o)t), = (2,o~/r)1/2[~(~
25 January 1971
(0~=
tkl)
x + ~y* .ay) + e±(Cy ax - exay)] exp i(k. x -cot),
(2)
where fix and fly a r e a n n i h i l a t i o n o p e r a t o r s for the modes (k, ex) and (k, ey) r e s p e c t i v e l y , e = ex e x + e y e y is a complex unit v e c t o r (i.e., e . e* = 1), e± is a unit v e c t o r orthogonal to e (i.e., e. e* = 0). F r o m eq. (2), it is evident that
at
= ~ x• a^x + e y*^a y ,
S t J. = E y a x - C x h y
(3)
a r e a n n i h i l a t i o n o p e r a t o r s for the modes (k, 6) and (k, 6±). Substitution of the expansion of fig in eq. (1) gives the e x p r e s s i o n of ~ e in the space of the s t a t e s Jr, s) - (r:s'.) -1/2 ~ r ~ TS[vac>. Since, for u n p o l a r i z e d r a d i a t i o n , • can take r a n d o m l y any value which sYalisfies the r e l a t i o n t . c* = 1, the d e n s i t y o p e r a t o r for u n p o l a r i z e d r a d i a t i o n can be obtained by a v e r a g i n g P e over e . This is done by i n t e g r a t i n g f3 r over E~c and E~ with the weight f u n c t i o n ~ P ( e ) = A 5(~. 6" - 1~ N o r m a l i z a t i o n condition / d 2 ~xd2Ey P(~) = 1, gives ~l = 2~ -2. The density operator for unpolarized" radiation thus has the form
p~
= / d 2E xd 2£y27r-25(~.~* - 1)/~ = ~n (n+l)-lpn;~
Ir,n -~'}
(4)
Eq. (4) shows that unpolarized radiation can exist in infinite different states having different values of {Pnn~, and Pe cannot be specified by any single parameter, e.g., the total number of photons The commonly used procedure for studying nonlinear interactions of unpolarized radiation [2] of averaging the results for plane polarized coherent radiation over the direction of polariz~ttion is evidently incorrect. It is interesting to see that the thus obtained "unpolarized radiation" (which is completely specified by a single parameter, the intensity) is not actually unpolarized as its nature changes * on introducing arbitrary phase changes in the x- and y-components.
(=~nnpnn).
The a u t h o r s a r e thankful to P r o f e s s o r Vachaspati for his i n t e r e s t in this work. One of the a u t h o r s (N. C.) g r a t e f u l l y acknowledges the f i n a n c i a l support of Indian National Science Academy, New Delhi. ~t The weight function satisfies the requirements (i) and (ii) in the definition of unpolarized radiation. This is because the authors [2] obtained this "unpolarized radiation" by averaging a plane polarized coherent radiation over the direction of polarization, and this does not involve separate averagings over the phases of Ex and
Ey.
Refe~'ences [1] H. Prakash and N. Chandra, Phys. Letters 31A (1970) 331. [2] 7.. Fried, Nuovo Cimento 22 (1961) 1303; Vaehaspati, Phys. Rev. 128 (1962) 664; 130 (1963) 2598(E); Vachaspati and S. L. Punhani, Prec. Nat. Inst. Sci., India, A29 (1963) 129; N. C.Rastogi and Vachaspati, Indian J. Pure Appt. Phys. 2 (1964) 426; L. M. Bali and J. Dutt, Nuovo Cimento 35 (1965) 805; J°Dutt, L.M. Bali and Vachaspati, Indian J. Pure App[. Phys. 4 (1966) 137; 5 (1967) 267; M. K. Srivastava and Vachaspati, Indian J. Pure Appl. Phys. 4 (1966) 141 ; S.S. Jha, Phys. Rev. 140 (1966) A2020; R. C.Garg and Vachaspati, Indian J. Pure Appl. Phys. 6 (1966) 329; H. Prakash and Vachaspati, Huovo Cimento 53B (1968) 24.
29