Simultaneous oscillation of competing laser transitions

S. E. Schacham; M. E. Marhic; M. Epstein

doi:10.1364/AO.17.000459

Applied Optics
Vol. 17,
Issue 3,
pp. 459-462
(1978)
•https://doi.org/10.1364/AO.17.000459

Simultaneous oscillation of competing laser transitions

S. E. Schacham, M. E. Marhic, and M. Epstein

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S. E. Schacham, M. E. Marhic, and M. Epstein, "Simultaneous oscillation of competing laser transitions," Appl. Opt. 17, 459-462 (1978)

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Abstract

The conditions necessary to obtain simultaneous oscillation of two laser transitions sharing a common energy level are examined. The general form of the equations is the same for all three possible cases, and it is shown that the operating point should lie within the area limited by the axes of coordinates and two straight lines in the transmittance plane. Parameters determining those lines can either be calculated or measured experimentally; their slopes depend only upon the transition parameters and are thus the same for all lasers operating on given transitions.

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Case	i	j	k	l	a	b	c	d	e	f	g	h
I	1	3	1	2	a₀	a_o + 1	d₀ + 1	d_o	0	$- \frac{g_{2}}{g_{1} t_{2}}$	$- \frac{g_{3}}{g_{1} t_{3}}$	$- \frac{g_{2}}{g_{1} t_{23}}$
II	1	3	2	3	a₀	−1	d_o + 1	1	$- \frac{g_{2}}{g_{1} t_{2}}$	$\frac{1}{t_{2}}$	$\frac{g_{2}}{g_{1} t_{23}} - \frac{g_{3}}{g_{1} t_{3}}$	$- \frac{1}{t_{23}}$
III	1	2	2	3	a₀ + 1	−1	d_o	1	$- \frac{g_{2}}{g_{1} t_{2}}$	0	$\frac{g_{2}}{g_{1} t_{23}} - \frac{g_{3}}{g_{1} t_{3}}$	$- \frac{g_{3}}{g_{2} t_{3}}$

Case	i	j	A_ij	B_ij	C_ij
I	1	3	d_o(R₁ + R₂) − (1 + a_o)R₃	$\frac{g_{2}}{g_{1} t_{23}} (1 + a_{o}) + \frac{g_{2}}{g_{1} t_{2}} d_{o}$	$\frac{g_{3}}{g_{1} t_{3}} (1 + a_{o})$
I	1	2	a_o(R₁+R₃) − (1 + d_o)R₂	$\frac{g_{3}}{g_{1} t_{3}} a_{o}$	$\frac{g_{2}}{g_{1} t_{23}} a_{o} + \frac{g_{2}}{g_{1} t_{2}} (1 + d_{o})$
II	1	3	(a_o + d_o)R₁ − R₂ − R₃	$\frac{1}{t_{20}}$	$\frac{g_{2}}{g_{1} t_{20}} + \frac{g_{3}}{g_{1} t_{30}}$
II	2	3	(1 + d_o)R₂ − a_o(R₁ + R₃)	$(\frac{g_{2}}{g_{1} t_{23}} - \frac{g_{3}}{g_{1} t_{3}}) a_{o} + \frac{g_{2}}{g_{1} t_{2}} (1 + d_{o})$	$\frac{a_{o}}{t_{23}} + \frac{1}{t_{2}} (1 + d_{o})$
III	1	2	(a_o + d_o)R₁ − R₂ − R₃	$- \frac{g_{3}}{g_{2} t_{3}}$	$\frac{g_{2}}{g_{1} t_{20}} + \frac{g_{3}}{g_{1} t_{30}}$
III	2	3	d_o(R₁ + R₂) − (1 + a_o)R₃	$- [(\frac{g_{3}}{g_{1} t_{3}} - \frac{g_{2}}{g_{1} t_{23}}) (1 + a_{o}) - \frac{g_{2} d_{o}}{g_{1} t_{2}}]$ *	$\frac{g_{3}}{g_{2} t_{3}} (1 + a_{o})$

Case	i	j	k	l	a	b	c	d	e	f	g	h
I	1	3	1	2	a₀	a_o + 1	d₀ + 1	d_o	0	$- \frac{g_{2}}{g_{1} t_{2}}$	$- \frac{g_{3}}{g_{1} t_{3}}$	$- \frac{g_{2}}{g_{1} t_{23}}$
II	1	3	2	3	a₀	−1	d_o + 1	1	$- \frac{g_{2}}{g_{1} t_{2}}$	$\frac{1}{t_{2}}$	$\frac{g_{2}}{g_{1} t_{23}} - \frac{g_{3}}{g_{1} t_{3}}$	$- \frac{1}{t_{23}}$
III	1	2	2	3	a₀ + 1	−1	d_o	1	$- \frac{g_{2}}{g_{1} t_{2}}$	0	$\frac{g_{2}}{g_{1} t_{23}} - \frac{g_{3}}{g_{1} t_{3}}$	$- \frac{g_{3}}{g_{2} t_{3}}$

Case	i	j	A_ij	B_ij	C_ij
I	1	3	d_o(R₁ + R₂) − (1 + a_o)R₃	$\frac{g_{2}}{g_{1} t_{23}} (1 + a_{o}) + \frac{g_{2}}{g_{1} t_{2}} d_{o}$	$\frac{g_{3}}{g_{1} t_{3}} (1 + a_{o})$
I	1	2	a_o(R₁+R₃) − (1 + d_o)R₂	$\frac{g_{3}}{g_{1} t_{3}} a_{o}$	$\frac{g_{2}}{g_{1} t_{23}} a_{o} + \frac{g_{2}}{g_{1} t_{2}} (1 + d_{o})$
II	1	3	(a_o + d_o)R₁ − R₂ − R₃	$\frac{1}{t_{20}}$	$\frac{g_{2}}{g_{1} t_{20}} + \frac{g_{3}}{g_{1} t_{30}}$
II	2	3	(1 + d_o)R₂ − a_o(R₁ + R₃)	$(\frac{g_{2}}{g_{1} t_{23}} - \frac{g_{3}}{g_{1} t_{3}}) a_{o} + \frac{g_{2}}{g_{1} t_{2}} (1 + d_{o})$	$\frac{a_{o}}{t_{23}} + \frac{1}{t_{2}} (1 + d_{o})$
III	1	2	(a_o + d_o)R₁ − R₂ − R₃	$- \frac{g_{3}}{g_{2} t_{3}}$	$\frac{g_{2}}{g_{1} t_{20}} + \frac{g_{3}}{g_{1} t_{30}}$
III	2	3	d_o(R₁ + R₂) − (1 + a_o)R₃	$- [(\frac{g_{3}}{g_{1} t_{3}} - \frac{g_{2}}{g_{1} t_{23}}) (1 + a_{o}) - \frac{g_{2} d_{o}}{g_{1} t_{2}}]$ *	$\frac{g_{3}}{g_{2} t_{3}} (1 + a_{o})$

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Equations (47)

Applied Optics