Theory of Condensing Lenses for Raman Tubes of Small Volume*

J. Rud Nielsen

doi:10.1364/JOSA.37.000494

Journal of the Optical Society of America
Vol. 37,
Issue 6,
pp. 494-499
(1947)
•https://doi.org/10.1364/JOSA.37.000494

Theory of Condensing Lenses for Raman Tubes of Small Volume

J. Rud Nielsen

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J. Rud Nielsen, "Theory of Condensing Lenses for Raman Tubes of Small Volume*," J. Opt. Soc. Am. 37, 494-499 (1947)

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Abstract

For a spatially extended light source of uniform cross section, such as a Raman tube, the most favorable focal length of the condensing lens is F = hf/(b − s), where 2h is the extension of the source in the direction of the spectrograph slit, f the focal length of the collimator lens, 2b the height of the prism, and 2s the length of the slit. With such a condensing lens, placed a distance of (h+s)f/(b−s) from the slit, the greatest possible length, l_max = h²f/bs, of the source is fully effective in filling the spectrograph. This was proved by the writer a number of years ago under the assumption that h≫s. Often, however, the small amount of the scattering material available necessitates the use of a Raman tube of small cross section. The case h≦s is therefore investigated here. It is shown that the theorem stated holds also in this case. However, in order that the full amount of light may enter the spectrograph, some of the scattered light must pass out through the wall of the Raman tube and pass through the condensing lens at a distance from the center which is greater than the radius of the Raman tube. This has certain disadvantages. It is therefore often advantageous to make h=s, in which case the maximum effective length of the Raman tube can be attained more satisfactorily by using a condensing lens of focal length sf/(b+s) immediately in front of the slit and placing the front end of the Raman tube right against the lens.

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Region	l
I	(1) $0,$
II	(2) $(\frac{z + (1 + h / s) z_{1}}{z - F + z_{1}} - \frac{z - (1 + h / s) z_{0}}{z - F - z_{0}}) F,$
III	(3) $\frac{2 h z_{1} F}{s (z - F + z_{1})},$
IV	(4) $(\frac{z - (1 + h / s) z_{0}}{z - F - z_{0}} - \frac{z + (1 - h / s) z_{1}}{z - F + z_{1}}) F,$
where
	(5) $\begin{array}{l} z_{0} = s f / (b - s), & and & z_{1} = s f / (b + s), \end{array}$

Point	F	z	I
P	hf/(b+h)	0	0
Q	hf/(b−s)	(h+s)f/(b−s)	h²f/bs
R	hf/(b−2s)	(h+2s)f/(b−2s)	2h²sf/3bs
S	hf/(2b+h)	0	2h²sf/b(4s²−h²)
T	hf/(2b−s)	0	2h²sf/3bs
U	hf/2b	hf/2b	h²f/3bs
V	$\frac{[s^{2} {(b + h) - b s (s^{2} - h^{2})}^{\frac{1}{2}}] f}{2 b s + b^{2} h + s h^{2}}$	0	$\frac{2 {[s (b + h) - b {(s^{2} - h^{2})}^{\frac{1}{2}}]}^{2} f}{b {[(b + s) {(s + h)}^{\frac{1}{2}} - (b - s) {(s - h)}^{\frac{1}{2}}]}^{2}}$

Point	F	z	l
P	sf/(b+s)	0	0
Q	sf/(b−s)	2sf/(b−s)	sf/b
R	sf/(b−2s)	3sf/(b−2s)	2sf/3b
S	sf/(2b + s)	0	2sf/3b
T	sf/(2b−s)	2sf/(2b−s)	2sf/3b
U	sf/2b	sf/2b	sf/2b
V	sf/(b+s)	0	sf/b

Region	l
I	(1) $0,$
II	(2) $(\frac{z + (1 + h / s) z_{1}}{z - F + z_{1}} - \frac{z - (1 + h / s) z_{0}}{z - F - z_{0}}) F,$
III	(3) $\frac{2 h z_{1} F}{s (z - F + z_{1})},$
IV	(4) $(\frac{z - (1 + h / s) z_{0}}{z - F - z_{0}} - \frac{z + (1 - h / s) z_{1}}{z - F + z_{1}}) F,$
where
	(5) $\begin{array}{l} z_{0} = s f / (b - s), & and & z_{1} = s f / (b + s), \end{array}$

Point	F	z	I
P	hf/(b+h)	0	0
Q	hf/(b−s)	(h+s)f/(b−s)	h²f/bs
R	hf/(b−2s)	(h+2s)f/(b−2s)	2h²sf/3bs
S	hf/(2b+h)	0	2h²sf/b(4s²−h²)
T	hf/(2b−s)	0	2h²sf/3bs
U	hf/2b	hf/2b	h²f/3bs
V	$\frac{[s^{2} {(b + h) - b s (s^{2} - h^{2})}^{\frac{1}{2}}] f}{2 b s + b^{2} h + s h^{2}}$	0	$\frac{2 {[s (b + h) - b {(s^{2} - h^{2})}^{\frac{1}{2}}]}^{2} f}{b {[(b + s) {(s + h)}^{\frac{1}{2}} - (b - s) {(s - h)}^{\frac{1}{2}}]}^{2}}$

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Journal of the Optical Society of America