- Email: [email protected]
Studies on the spectral width, monochromatic degree and spectral resolving power of prism monochromator
Optik 123 (2012) 2213–2217
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Studies on the spectral width, monochromatic degree and spectral resolving power of prism monochromator Qiankai Wang ∗ Department of Physics, Anhui Normal University, Wuhu, Anhui, 241000, China
a r t i c l e
i n f o
Article history: Received 5 May 2011 Accepted 15 October 2011
Keywords: Prism monochromator Coherent imaging and incoherent imaging Geometric line width Spectral width Spectral resolving power Monochromatic degree
a b s t r a c t The angular width and geometric line width in the spectral plane for the output incoherent imaging spectral line are discussed respectively on the basis of structure of prism monochromator, the relationship between the spectral width and line width of output incoherent imaging spectral line in the spectral plane is presented. The expression of monochromatic degree of output light in the exit slit plane is presented. The spectral resolving power of incoherent imaging spectral line and coherent imaging spectral line of the diffractive waves are discussed respectively, and the expressions of the spectral resolving power are presented respectively. © 2012 Elsevier GmbH. All rights reserved.
1. Introduction Reflecting prism monochromator is widely used in teaching and scientific research. Its advantage is that it possesses two reflecting concave mirrors to collimate light and focus light respectively so as to avoid chromatic aberration. However, some concepts such as the spectral width, monochromatic degree and spectral resolving power, which reveal the basic characteristics of prism monochromator, have not been clearly or correctly expounded in many textbooks. In the present study, the spectral width and line width of the output incoherent imaging spectral line in spectral plane will be discussed on the basis of structure of prism monochromator, and the relationship between the spectral width and geometric line width will be presented. The expression of monochromatic degree of output light in the exit slit plane will be presented. The spectral resolving power of incoherent imaging spectral line and coherent imaging spectral line of diffractive waves will be discussed respectively, the expressions of the spectral resolving power will be also presented respectively. 2. Spectral width The optical configuration of reflecting prism monochromator is shown in Fig. 1. It consists of incident collimation system, Wadsworth dispersion system and output focusing system. As is
∗ Corresponding author. E-mail address: [email protected] 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.10.040
shown in the Fig. 2, the entrance slit S1 with the width a1 is located in the focal plane of the concave mirror M1 with the focal length f1 . Light rays coming from the slit light source are collimated by M1 so as to form parallel light rays propagating in different directions. The maximum angle interval of the parallel light rays is
ı =
a1 f1
(1)
ı is small, for example, in some prism monochromators, f1 ≈ 34.4 cm, if taking a1 = 65 m, then ı ≈ 1.9 × 10−4 rad. As is shown in the Fig. 1, Wadsworth dispersion system is a device with constant deviation angle, the output parallel light coming out of the prism with the minimum deviation angle is always parallel to the incident parallel light toward the plane mirror M. The refractive prism is shown in the Fig. 3. In Fig. 3, A is the refracting angle of the prism, the dashed line denotes the light path that the parallel light with wavelength go through the prism and come out of the prism with the minimum deviation angle, i1 represent the incident angle, i2 represent the refractive angle. The solid line denotes the light path that the parallel light of wavelength + ı go through the prism with incident angle i1 + ıi1 , in which ıi1 represent the angle interval deviating from i1 . Now we calculate the angle interval ıi2 deviating from i2 for the output parallel light. According to refraction law [1–4], we obtain sin i1 = n() sin i1 ,
(2)
2214
Q. Wang / Optik 123 (2012) 2213–2217
According to Eqs. (3) and (7) and by using the relationships [2] i1 = i2 ,
A , 2
i1 = i2 =
we obtain the relation between ıi2 and ıi1 as ıi2 = −ıi1 + D ı
(8)
in which D =
2 sin(A/2) 1 − n2
dn d sin (A/2)
is the angular dispersion of the prism. As is seen from Eq. (8), the sign and value of ıi2 is related to the propagation direction of incident parallel light toward the prism. Now, we give a discussion on Eq. (8). We consider the conditions of ı > 0, and of the normal dispersive material, i.e. dn/d < 0, therefore D < 0. If ıi1 > 0, then the incident parallel lights with wavelength + ı reach to the prism with the incident angle i1 + ıi1 which is greater than i1 . In this condition, we obtain ıi2 < 0, and ıi2 = ıi1 − D ı. If ıi1 < 0, then the incident parallel lights with wavelength + ı reach to the prism with the incident angle i1 − |ıi1 | which is smaller than i1 . In this condition, if ıi1 < D ı, then ıi2 < 0; if ıi1 > D ı, then
Fig. 1. Optical configuration of prism monochromator.
Fig. 2. Collimated parallel light rays with maximum angle interval ı = a1 /f1 .
where n() denotes the refraction index of light with wavelength in the prism. By differentiating the two sides of Eq. (2), we obtain cos i1 ıi1 = sin i1 ın()+n() cos i1 ıi1 = sin i1
dn ı+n() cos i1 ıi1 (3) d
In Eq. (3), if incident angle or refraction angle increase, then ıi1 > 0 or ıi1 > 0. If incident angle or refraction angle decrease, then ıi1 < 0 or ıi1 < 0. It is easily proved that i0 = i2 − ıi1 ,
i0 = i2 + ıi2
i0
(4) i2 ,
ıi2
In Eq. (4), if is greater than then > 0, if then ıi2 < 0. According to the refraction law n( + ı) sin i0 = sin i0 ,
n() sin i2 = sin i2 ,
i0
is smaller than i2 , (5)
using the approximate relation n( + ı) = n() +
dn ı, d
(6)
taking small angle approximation value of sine and cosine function, and neglecting the quadratic small quantity (dn/d) · ı · ıi1 , we obtain n() cos i2 ıi1 = sin i2
dn ı − cos i2 ıi2 d
(9)
2
(7)
ıi2 > 0, and we obtain ıi2 = ıi1 − D ı. Now, we discuss the angular width and geometric line width of the output incoherent imaging spectral line, and the relationships between the line width and spectral width for the output spectral line in the spectral plane. When the parallel light rays with angular interval ı = a1 /f1 which is collimated by the concave mirror M1 reach to the prism. The direction of incident parallel light rays toward the prism is related to the structure of prism monochromator. If the incident parallel light toward the plane mirror M come from the parallel light rays collimated by M1 for the light rays giving off from the center of entrance slit light source, and if the output parallel light coming out of the prism with minimum deviation angle is parallel to the incident parallel light toward the plane mirror M, it is easily proved that the parallel light rays with angular interval ı = a1 /f1 always illuminate the prism symmetrically with the angular ı/2 around the dashed line shown in Fig. 3. In the conditions as stated above, when the parallel lights with angular interval ı = a1 /f1 (ı > 0) and with the wavelengths range of ∼ + ı go through the refractive prism, according to Eq. (8), we calculate the angular interval of the output parallel lights of single wavelength as ı
ı ı + D · ı − + D · ı = ı = − 2 2
(10)
The output parallel light rays is focused by M2 to the focal plane of M2 . Therefore, in the focal plane, i.e. spectral plane, the line width of the output incoherent imaging spectral line of single wavelength light is expressed as ıl = ı · f2 = ı · f2 =
a1 · f2 f1
(11)
As is known from Eq. (11), the line width of the output incoherent imaging spectral line of single wavelength light in the spectral plane is just the image width of the entrance slit. In this condition, the spectral width of light source ıS = 0, and the spectral width of the output spectral line is also zero, i.e. ıSL = 0. Now, we calculate the maximum angular interval ı of the output parallel light rays for the condition that the parallel light rays with angular interval ı = a1 /f1 and consecutive wavelengths ∼ + ı go through the prism. According to Eq. (8) and the successive analyses, we obtain the relationships Fig. 3. Light paths that parallel light rays with different incident angles toward the prism pass through the prism.
ı − D · ı ı = 2
ı < D · ı 2
,
(12a)
Q. Wang / Optik 123 (2012) 2213–2217
ı =
ı ı − D · ı + + D · ı 2 2
ı > D · ı 2
(12b)
The relationship between the line width ıl and angular width ı in the spectral plane is expressed as ıl = ı · f2
(13)
For the condition that the parallel light rays with angular interval ı = a1 /f1 and consecutive wavelengths − ı/2∼ + ı/2 go through the prism, the maximum angular interval ı is calculated as
ı
ı = −
2
+ D ·
ı 2
−
ı ı − D · 2 2
= ı − D · ı
ıl = ı · f2 = (ı − D · ı) · f2
(a1 /f1 ) − (ıl/f2 ) 2 sin(A/2)(dn/d)
(15)
A 2
(16)
3. Monochromatic degree We may obtain the monochromatic light from the polychromatic lights with the prism monochromator. In prism monochromator, the plane of exit slit S2 is located in the focal plane of concave mirror M2 , i.e. the spectral plane. If the consecutive wavelengths range of the output monochromatic light is ∼ + in the plane of slit S2 , then the wavelength interval represents the monochromatic degree of the monochromatic light beam. If the linear dispersion is dl/d in the spectral plane, is expressed as =
d · S2 , dl
(17)
where S2 is the width of exit slit S2 and is expressed as S2 = rect
x − x 0
2a
× 2 |x − x0 |
(|x − x0 | ≤ a),
(18)
where rect(· · ·) is the rectangular function, x0 represent the emergent position of output light focused by M2 for the parallel light coming out of the prism with minimum deviation angle, x represents the position of a edge of the adjustable exit slit S2 , 2a is a small quantity. For prism monochromator, dl = D · f2 = d
2 sin(A/2)
1 − n2
dn · f2 d sin (A/2) 2
(19)
Thus the monochromatic degree of output monochromatic light beam is expressed as
=
1 − n2 sin2 (A/2) 1 1 · S2 · · 2 sin(A/2) (dn/d) f2
(20)
If the light source, as an example of that with line-shape spectral line, has a finite spectral width of S , the expression of Eq. (20) is only valid on the condition of S2 ≤
dl · S d
dl · ı, d
(22)
where dl/d = D f2 is the line dispersion in the spectral plane. The condition that two spectral lines of single wavelength with wavelength interval ı is just resolved is expressed as (23)
From relationship l = ıl , the resolvable wavelength interval ı is derived as a1 1 a1 · = · ı = f1 D f1
1 − n2 sin2 (A/2) 1 · 2 sin(A/2) (dn/d)
(24)
The spectral resolving power of incoherent imaging spectral line is expressed as
1 − n2 sin2
l =
l ≥ ıl
The spectral width ıSL is further calculated as ıSL = ı =
of the entrance slit S1 which is shown in Eq. (11). The line interval between the two monochromatic incoherent imaging spectral lines with single wavelength and + ı respectively is expressed as
(14)
The relation between the line width ıl and spectral width ıSL = ı in the spectral plane is expressed as
2215
(21)
4. Spectral resolving power Now, we discuss spectral resolving power of incoherent imaging spectral line and coherent imaging spectral line of diffractive waves respectively. In the spectral plane, the line width of output incoherent imaging spectral line of single wavelength is equal to the image width
f1 · = a1 ı
2 sin(A/2) 1 − n2
2
sin (A/2)
·
dn d
(25)
As is known from Eq. (25), we may increase the spectral resolving power by decreasing the width of entrance slit. However, when the width of entrance slit become small enough, the resolving power will be related to the diffraction effect. According to Huygens’ principle [1–4], when a light wave propagates in a homogeneous medium, if it meets another medium and its wave surface is partly disturbed by the medium, any point of the wave surface will emit diffractive secondary waves. So when the width of entrance slit become small enough, the wave surface of a light wave will be partly disturbed by the edges matter of slit, and the diffractive secondary waves will be emitted. The diffractive waves spread out laterally. According to Huygens’–Fresnel principle [1–4], the diffractive secondary waves emitted from a point of the wave surface have the same epoch; the value of amplitude of diffractive secondary wave is inversely proportional to its propagating distance, and is related to the diffractive tilt angle. The slit plane is located in the focal plane of M1 , so the diffractive secondary waves emitted from any point of the wave surface in the slit plane will be collimated as plane wave. We assume that the incident light is parallel light and is perpendicular to the slit plane, and neglect the obliquity factor for the smaller acceptance angle of light rays from the slit to the whole surface of M1 . The amplitude at the diffractive point x is assumed as U0 (x ). Thus, the amplitude distribution of plane wave along the direction which is vertical to the propagating direction is approximately expressed as U0 (x )/¯r , where r¯ is the mean distance through which the diffractive waves reach to M1 . We assume that the beam width of the plane wave is l. Now we calculate the optical path of the plane wave during its propagation to the exit slit plane. The reflection of plane wave by the plane mirror M does not change the property of plane wave. Next we investigate the optical path of the plane wave passing through the prism. As is shown in Fig. 4, the dashed line denotes the light path that the incident plane wave of wavelength go through the prism and come out of the prism with the minimum deviation angle, i1 represent the incident angle, i2 represent the refractive angle. The solid line denotes the light path that the incident plane wave of wavelength go through the prism with the incident angle i1 + ıi1 . The refractive angle of output light is i2 + ıi2 . KK is the wave surface of equal phase which is vertical to the propagation direction of the incident plane wave. When the incident plane wave has gone through the prism, the difference of optical path between the two points X X in the plane which is vertical to the propagation direction of the
2216
Q. Wang / Optik 123 (2012) 2213–2217
where l ≈ l, k = (2/). The resultant light intensity of the wavelength is approximately expressed as
output parallel light is expressed approximately as h − h0 cos 1 cos(A/2)
= S(K → X ) − S(K → X) =
A A A + n 2h0 · tg − 2h0 · tg · ıi1 · tg 2 2 2 +
(1 − 2tg(A/2)ıi1 ) (h − h0 ) cos(A/2)
cos 2 − n 2h · tg(A/2)
−2h · tg(A/2) · ıi1 · tg(A/2) ,
(26)
where A is refraction angle of the prism, h0 = AF0 , h = AF. F0 , F are the intersection of the dashed line with the height respectively. n denote the refraction index of light with wavelength in the prism, the refraction index of light in air is approximately taken as 1. According to the relationships 1 =
− i1 + ıi1 , 2
2 =
− (i2 + ıi2 ), 2
(27)
i1 = i2 ,
(28)
sin i1 = n sin i1 = n sin
A , 2
sin i2 = n sin i2 = n sin
A 2
(29)
and using the relationship ıi1 = −ıi2 for single wavelength in Eq. (8), we calculate Eq. (26) as = 0.
(30)
In Fig. 4, if ıi1 < 0, i.e. the incident plane waveof wavelength go through the prism with the incident angle i1 − ıi1 , it may also be proved that the difference of optical path between any two points in the plane which is vertical to the propagation direction of the output parallel light equals zero. So, when the plane wave passes through the prism, the output light is also the plane wave in the condition of first-order approximation of ıi1 . ıi1 is easily calculated as:
1/2
ıi1 = ıi1 1 − n2 sin2 (A/2) /n cos(A/2). The output plane wave is focused on the focal plane by M2 through equal optical path 0 . We calculate the focused light intensity at the focal point x in the exit slit plane. The main loss of light wave during propagation is caused by reflection. We assume that the approximate aggregate propagation coefficient of light intensity is R. The resultant complex amplitude at the focal point is approximately expressed as [1–4]
U(x) =
dU ≈
l
2
I(x) = U(x) = R ·
√ R · U0 (x ) d · eiϕ · eik·0 · e−iωt r¯
0 √ l R · U0 (x ) iϕ ik·0 −iωt ≈ ·e ·e ·e , r¯
(31)
l 2 r¯
· I(x ),
(32)
where I(x ) = U0 (x )2 is the intensity of a incident wave at the diffractive point x in the entrance slit plane. If the slit is illuminated by incoherent light source, then the total approximate intensity at point x is the accumulation of each diffractive intensity produced by the diffractive waves of each light wave at the point x , and is expressed as It (x) =
N
Ii (x) =
i=1
N l 2 i
Ri
i=1
r¯ i
Ii (x ) = R
l 2 r¯
It (x ),
(33)
where It (x) is the total intensity of wavelength at the point x in Ii (x ) is the total the focal plane, i.e. the exit slit plane, It (x ) = intensity of the incident light at the point x in the entrance slit plane for the wavelength . So, the intensity at a point in the exit slit plane corresponds to the intensity at a point in the entrance slit plane. As is shown in Eq. (30), for prism monochromator, the intensities of output spectral line of in the spectral plane do not show the variation like the Fig. 12.5 in Ref. [5]. So, the spectral resolving power of diffractive spectral line may also be expressed by that of incoherent imaging spectral line, i.e. Eq. (25). Now we give a comment on the calculation of spectral resolving power for prism monochromator or spectrometer. In prism monochromator or spectrometer, if the illuminating light on the entrance slit is incoherent light, when the slit width becomes enough small, the coherent diffractive light will be produced, the emitting diffractive secondary waves at any point of the wavefront in the slit plane are collimated as the coherent plane wave with a light beam width l by the collimator, the maximum angle interval of the plane waves along different directions is expressed by Eq. (1) as ı = a1 /f1 . The plane waves go through the prism and are finally focused on the focal plane by a concave mirror or a lens, the distribution of the resultant intensity is related to the path differences between the points in the plane which is vertical to the propagation direction of the output parallel light coming out of the prism, and is related to the light beam width l. But the present cases differ from the Fraunhofer diffraction with a limiting aperture in Fig. 12.5 in Ref. [5]. So in Ref. [5], for the prism spectrometer, the resultant intensity distribution cannot be expressed simply with the formula (12.4). In order to give the distribution of the resultant intensity, we must calculate the actual path differences between the points in the plane which is vertical to the propagation direction of the output parallel light coming out of the prism, therefore, the spectral resolving power also can not be expressed simply as the formula (12.13). But it is also complicated to calculate the distribution of the resultant intensity and spectral resolving power for the prism spectrometer. 5. Conclusions
Fig. 4. Light paths that the plane waves pass through the prism.
The angular width and the geometric line width in the spectral plane for the output incoherent imaging spectral line are related to the structure of prism monochromator. If the incident parallel light toward the plane mirror M come from the parallel light rays collimated by M1 for the light rays giving off from the center of entrance slit light source, and the output parallel light coming out of the prism with minimum deviation angle is parallel to the incident parallel light toward the plane mirror M, and if the parallel light rays with angular interval ı = a1 /f1 and consecutive wavelengths − ı/2∼ + ı/2 go through the prism, then the angular width and geometric line width is expressed respectively with Eq. (14)
Q. Wang / Optik 123 (2012) 2213–2217
and Eq. (15), and the relationship between the spectral width and line width is expressed with Eq. (16). The monochromatic degree of output light of prism monochromator is presented with Eq. (20) for the output light with consecutive wavelengths. The spectral resolving power of incoherent imaging spectral line is presented. If there is diffractive effect of the entrance slit with very small width, the spectral resolving power of coherent imaging spectral line of diffractive waves is also expressed with Eq. (25).
2217
References [1] E. Hecht, Optics, 4th ed., Addison-Wesley, MA, 2002. [2] M.H. Freeman, C.C. Hull, Optics, eleventh edition, Butterworth Heinemann, 2003. [3] M. Born, E. Wolf, Principles of Optics, 6th ed., Pergamon, Oxford, 1980. [4] F.A. Jenkins, H.E. White, Fundamentals of Optics, McGraw-Hill, New York, 1976. [5] W. Demtröder, Grundlagen und Techniken der Laserspektroskopie, SpringerVerlag, 1977.
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Studies on the spectral width, monochromatic degree and spectral resolving power of prism monochromator Qiankai Wang ∗ Department of Physics, Anhui Normal University, Wuhu, Anhui, 241000, China
a r t i c l e
i n f o
Article history: Received 5 May 2011 Accepted 15 October 2011
Keywords: Prism monochromator Coherent imaging and incoherent imaging Geometric line width Spectral width Spectral resolving power Monochromatic degree
a b s t r a c t The angular width and geometric line width in the spectral plane for the output incoherent imaging spectral line are discussed respectively on the basis of structure of prism monochromator, the relationship between the spectral width and line width of output incoherent imaging spectral line in the spectral plane is presented. The expression of monochromatic degree of output light in the exit slit plane is presented. The spectral resolving power of incoherent imaging spectral line and coherent imaging spectral line of the diffractive waves are discussed respectively, and the expressions of the spectral resolving power are presented respectively. © 2012 Elsevier GmbH. All rights reserved.
1. Introduction Reflecting prism monochromator is widely used in teaching and scientific research. Its advantage is that it possesses two reflecting concave mirrors to collimate light and focus light respectively so as to avoid chromatic aberration. However, some concepts such as the spectral width, monochromatic degree and spectral resolving power, which reveal the basic characteristics of prism monochromator, have not been clearly or correctly expounded in many textbooks. In the present study, the spectral width and line width of the output incoherent imaging spectral line in spectral plane will be discussed on the basis of structure of prism monochromator, and the relationship between the spectral width and geometric line width will be presented. The expression of monochromatic degree of output light in the exit slit plane will be presented. The spectral resolving power of incoherent imaging spectral line and coherent imaging spectral line of diffractive waves will be discussed respectively, the expressions of the spectral resolving power will be also presented respectively. 2. Spectral width The optical configuration of reflecting prism monochromator is shown in Fig. 1. It consists of incident collimation system, Wadsworth dispersion system and output focusing system. As is
∗ Corresponding author. E-mail address: [email protected] 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.10.040
shown in the Fig. 2, the entrance slit S1 with the width a1 is located in the focal plane of the concave mirror M1 with the focal length f1 . Light rays coming from the slit light source are collimated by M1 so as to form parallel light rays propagating in different directions. The maximum angle interval of the parallel light rays is
ı =
a1 f1
(1)
ı is small, for example, in some prism monochromators, f1 ≈ 34.4 cm, if taking a1 = 65 m, then ı ≈ 1.9 × 10−4 rad. As is shown in the Fig. 1, Wadsworth dispersion system is a device with constant deviation angle, the output parallel light coming out of the prism with the minimum deviation angle is always parallel to the incident parallel light toward the plane mirror M. The refractive prism is shown in the Fig. 3. In Fig. 3, A is the refracting angle of the prism, the dashed line denotes the light path that the parallel light with wavelength go through the prism and come out of the prism with the minimum deviation angle, i1 represent the incident angle, i2 represent the refractive angle. The solid line denotes the light path that the parallel light of wavelength + ı go through the prism with incident angle i1 + ıi1 , in which ıi1 represent the angle interval deviating from i1 . Now we calculate the angle interval ıi2 deviating from i2 for the output parallel light. According to refraction law [1–4], we obtain sin i1 = n() sin i1 ,
(2)
2214
Q. Wang / Optik 123 (2012) 2213–2217
According to Eqs. (3) and (7) and by using the relationships [2] i1 = i2 ,
A , 2
i1 = i2 =
we obtain the relation between ıi2 and ıi1 as ıi2 = −ıi1 + D ı
(8)
in which D =
2 sin(A/2) 1 − n2
dn d sin (A/2)
is the angular dispersion of the prism. As is seen from Eq. (8), the sign and value of ıi2 is related to the propagation direction of incident parallel light toward the prism. Now, we give a discussion on Eq. (8). We consider the conditions of ı > 0, and of the normal dispersive material, i.e. dn/d < 0, therefore D < 0. If ıi1 > 0, then the incident parallel lights with wavelength + ı reach to the prism with the incident angle i1 + ıi1 which is greater than i1 . In this condition, we obtain ıi2 < 0, and ıi2 = ıi1 − D ı. If ıi1 < 0, then the incident parallel lights with wavelength + ı reach to the prism with the incident angle i1 − |ıi1 | which is smaller than i1 . In this condition, if ıi1 < D ı, then ıi2 < 0; if ıi1 > D ı, then
Fig. 1. Optical configuration of prism monochromator.
Fig. 2. Collimated parallel light rays with maximum angle interval ı = a1 /f1 .
where n() denotes the refraction index of light with wavelength in the prism. By differentiating the two sides of Eq. (2), we obtain cos i1 ıi1 = sin i1 ın()+n() cos i1 ıi1 = sin i1
dn ı+n() cos i1 ıi1 (3) d
In Eq. (3), if incident angle or refraction angle increase, then ıi1 > 0 or ıi1 > 0. If incident angle or refraction angle decrease, then ıi1 < 0 or ıi1 < 0. It is easily proved that i0 = i2 − ıi1 ,
i0 = i2 + ıi2
i0
(4) i2 ,
ıi2
In Eq. (4), if is greater than then > 0, if then ıi2 < 0. According to the refraction law n( + ı) sin i0 = sin i0 ,
n() sin i2 = sin i2 ,
i0
is smaller than i2 , (5)
using the approximate relation n( + ı) = n() +
dn ı, d
(6)
taking small angle approximation value of sine and cosine function, and neglecting the quadratic small quantity (dn/d) · ı · ıi1 , we obtain n() cos i2 ıi1 = sin i2
dn ı − cos i2 ıi2 d
(9)
2
(7)
ıi2 > 0, and we obtain ıi2 = ıi1 − D ı. Now, we discuss the angular width and geometric line width of the output incoherent imaging spectral line, and the relationships between the line width and spectral width for the output spectral line in the spectral plane. When the parallel light rays with angular interval ı = a1 /f1 which is collimated by the concave mirror M1 reach to the prism. The direction of incident parallel light rays toward the prism is related to the structure of prism monochromator. If the incident parallel light toward the plane mirror M come from the parallel light rays collimated by M1 for the light rays giving off from the center of entrance slit light source, and if the output parallel light coming out of the prism with minimum deviation angle is parallel to the incident parallel light toward the plane mirror M, it is easily proved that the parallel light rays with angular interval ı = a1 /f1 always illuminate the prism symmetrically with the angular ı/2 around the dashed line shown in Fig. 3. In the conditions as stated above, when the parallel lights with angular interval ı = a1 /f1 (ı > 0) and with the wavelengths range of ∼ + ı go through the refractive prism, according to Eq. (8), we calculate the angular interval of the output parallel lights of single wavelength as ı
ı ı + D · ı − + D · ı = ı = − 2 2
(10)
The output parallel light rays is focused by M2 to the focal plane of M2 . Therefore, in the focal plane, i.e. spectral plane, the line width of the output incoherent imaging spectral line of single wavelength light is expressed as ıl = ı · f2 = ı · f2 =
a1 · f2 f1
(11)
As is known from Eq. (11), the line width of the output incoherent imaging spectral line of single wavelength light in the spectral plane is just the image width of the entrance slit. In this condition, the spectral width of light source ıS = 0, and the spectral width of the output spectral line is also zero, i.e. ıSL = 0. Now, we calculate the maximum angular interval ı of the output parallel light rays for the condition that the parallel light rays with angular interval ı = a1 /f1 and consecutive wavelengths ∼ + ı go through the prism. According to Eq. (8) and the successive analyses, we obtain the relationships Fig. 3. Light paths that parallel light rays with different incident angles toward the prism pass through the prism.
ı − D · ı ı = 2
ı < D · ı 2
,
(12a)
Q. Wang / Optik 123 (2012) 2213–2217
ı =
ı ı − D · ı + + D · ı 2 2
ı > D · ı 2
(12b)
The relationship between the line width ıl and angular width ı in the spectral plane is expressed as ıl = ı · f2
(13)
For the condition that the parallel light rays with angular interval ı = a1 /f1 and consecutive wavelengths − ı/2∼ + ı/2 go through the prism, the maximum angular interval ı is calculated as
ı
ı = −
2
+ D ·
ı 2
−
ı ı − D · 2 2
= ı − D · ı
ıl = ı · f2 = (ı − D · ı) · f2
(a1 /f1 ) − (ıl/f2 ) 2 sin(A/2)(dn/d)
(15)
A 2
(16)
3. Monochromatic degree We may obtain the monochromatic light from the polychromatic lights with the prism monochromator. In prism monochromator, the plane of exit slit S2 is located in the focal plane of concave mirror M2 , i.e. the spectral plane. If the consecutive wavelengths range of the output monochromatic light is ∼ + in the plane of slit S2 , then the wavelength interval represents the monochromatic degree of the monochromatic light beam. If the linear dispersion is dl/d in the spectral plane, is expressed as =
d · S2 , dl
(17)
where S2 is the width of exit slit S2 and is expressed as S2 = rect
x − x 0
2a
× 2 |x − x0 |
(|x − x0 | ≤ a),
(18)
where rect(· · ·) is the rectangular function, x0 represent the emergent position of output light focused by M2 for the parallel light coming out of the prism with minimum deviation angle, x represents the position of a edge of the adjustable exit slit S2 , 2a is a small quantity. For prism monochromator, dl = D · f2 = d
2 sin(A/2)
1 − n2
dn · f2 d sin (A/2) 2
(19)
Thus the monochromatic degree of output monochromatic light beam is expressed as
=
1 − n2 sin2 (A/2) 1 1 · S2 · · 2 sin(A/2) (dn/d) f2
(20)
If the light source, as an example of that with line-shape spectral line, has a finite spectral width of S , the expression of Eq. (20) is only valid on the condition of S2 ≤
dl · S d
dl · ı, d
(22)
where dl/d = D f2 is the line dispersion in the spectral plane. The condition that two spectral lines of single wavelength with wavelength interval ı is just resolved is expressed as (23)
From relationship l = ıl , the resolvable wavelength interval ı is derived as a1 1 a1 · = · ı = f1 D f1
1 − n2 sin2 (A/2) 1 · 2 sin(A/2) (dn/d)
(24)
The spectral resolving power of incoherent imaging spectral line is expressed as
1 − n2 sin2
l =
l ≥ ıl
The spectral width ıSL is further calculated as ıSL = ı =
of the entrance slit S1 which is shown in Eq. (11). The line interval between the two monochromatic incoherent imaging spectral lines with single wavelength and + ı respectively is expressed as
(14)
The relation between the line width ıl and spectral width ıSL = ı in the spectral plane is expressed as
2215
(21)
4. Spectral resolving power Now, we discuss spectral resolving power of incoherent imaging spectral line and coherent imaging spectral line of diffractive waves respectively. In the spectral plane, the line width of output incoherent imaging spectral line of single wavelength is equal to the image width
f1 · = a1 ı
2 sin(A/2) 1 − n2
2
sin (A/2)
·
dn d
(25)
As is known from Eq. (25), we may increase the spectral resolving power by decreasing the width of entrance slit. However, when the width of entrance slit become small enough, the resolving power will be related to the diffraction effect. According to Huygens’ principle [1–4], when a light wave propagates in a homogeneous medium, if it meets another medium and its wave surface is partly disturbed by the medium, any point of the wave surface will emit diffractive secondary waves. So when the width of entrance slit become small enough, the wave surface of a light wave will be partly disturbed by the edges matter of slit, and the diffractive secondary waves will be emitted. The diffractive waves spread out laterally. According to Huygens’–Fresnel principle [1–4], the diffractive secondary waves emitted from a point of the wave surface have the same epoch; the value of amplitude of diffractive secondary wave is inversely proportional to its propagating distance, and is related to the diffractive tilt angle. The slit plane is located in the focal plane of M1 , so the diffractive secondary waves emitted from any point of the wave surface in the slit plane will be collimated as plane wave. We assume that the incident light is parallel light and is perpendicular to the slit plane, and neglect the obliquity factor for the smaller acceptance angle of light rays from the slit to the whole surface of M1 . The amplitude at the diffractive point x is assumed as U0 (x ). Thus, the amplitude distribution of plane wave along the direction which is vertical to the propagating direction is approximately expressed as U0 (x )/¯r , where r¯ is the mean distance through which the diffractive waves reach to M1 . We assume that the beam width of the plane wave is l. Now we calculate the optical path of the plane wave during its propagation to the exit slit plane. The reflection of plane wave by the plane mirror M does not change the property of plane wave. Next we investigate the optical path of the plane wave passing through the prism. As is shown in Fig. 4, the dashed line denotes the light path that the incident plane wave of wavelength go through the prism and come out of the prism with the minimum deviation angle, i1 represent the incident angle, i2 represent the refractive angle. The solid line denotes the light path that the incident plane wave of wavelength go through the prism with the incident angle i1 + ıi1 . The refractive angle of output light is i2 + ıi2 . KK is the wave surface of equal phase which is vertical to the propagation direction of the incident plane wave. When the incident plane wave has gone through the prism, the difference of optical path between the two points X X in the plane which is vertical to the propagation direction of the
2216
Q. Wang / Optik 123 (2012) 2213–2217
where l ≈ l, k = (2/). The resultant light intensity of the wavelength is approximately expressed as
output parallel light is expressed approximately as h − h0 cos 1 cos(A/2)
= S(K → X ) − S(K → X) =
A A A + n 2h0 · tg − 2h0 · tg · ıi1 · tg 2 2 2 +
(1 − 2tg(A/2)ıi1 ) (h − h0 ) cos(A/2)
cos 2 − n 2h · tg(A/2)
−2h · tg(A/2) · ıi1 · tg(A/2) ,
(26)
where A is refraction angle of the prism, h0 = AF0 , h = AF. F0 , F are the intersection of the dashed line with the height respectively. n denote the refraction index of light with wavelength in the prism, the refraction index of light in air is approximately taken as 1. According to the relationships 1 =
− i1 + ıi1 , 2
2 =
− (i2 + ıi2 ), 2
(27)
i1 = i2 ,
(28)
sin i1 = n sin i1 = n sin
A , 2
sin i2 = n sin i2 = n sin
A 2
(29)
and using the relationship ıi1 = −ıi2 for single wavelength in Eq. (8), we calculate Eq. (26) as = 0.
(30)
In Fig. 4, if ıi1 < 0, i.e. the incident plane waveof wavelength go through the prism with the incident angle i1 − ıi1 , it may also be proved that the difference of optical path between any two points in the plane which is vertical to the propagation direction of the output parallel light equals zero. So, when the plane wave passes through the prism, the output light is also the plane wave in the condition of first-order approximation of ıi1 . ıi1 is easily calculated as:
1/2
ıi1 = ıi1 1 − n2 sin2 (A/2) /n cos(A/2). The output plane wave is focused on the focal plane by M2 through equal optical path 0 . We calculate the focused light intensity at the focal point x in the exit slit plane. The main loss of light wave during propagation is caused by reflection. We assume that the approximate aggregate propagation coefficient of light intensity is R. The resultant complex amplitude at the focal point is approximately expressed as [1–4]
U(x) =
dU ≈
l
2
I(x) = U(x) = R ·
√ R · U0 (x ) d · eiϕ · eik·0 · e−iωt r¯
0 √ l R · U0 (x ) iϕ ik·0 −iωt ≈ ·e ·e ·e , r¯
(31)
l 2 r¯
· I(x ),
(32)
where I(x ) = U0 (x )2 is the intensity of a incident wave at the diffractive point x in the entrance slit plane. If the slit is illuminated by incoherent light source, then the total approximate intensity at point x is the accumulation of each diffractive intensity produced by the diffractive waves of each light wave at the point x , and is expressed as It (x) =
N
Ii (x) =
i=1
N l 2 i
Ri
i=1
r¯ i
Ii (x ) = R
l 2 r¯
It (x ),
(33)
where It (x) is the total intensity of wavelength at the point x in Ii (x ) is the total the focal plane, i.e. the exit slit plane, It (x ) = intensity of the incident light at the point x in the entrance slit plane for the wavelength . So, the intensity at a point in the exit slit plane corresponds to the intensity at a point in the entrance slit plane. As is shown in Eq. (30), for prism monochromator, the intensities of output spectral line of in the spectral plane do not show the variation like the Fig. 12.5 in Ref. [5]. So, the spectral resolving power of diffractive spectral line may also be expressed by that of incoherent imaging spectral line, i.e. Eq. (25). Now we give a comment on the calculation of spectral resolving power for prism monochromator or spectrometer. In prism monochromator or spectrometer, if the illuminating light on the entrance slit is incoherent light, when the slit width becomes enough small, the coherent diffractive light will be produced, the emitting diffractive secondary waves at any point of the wavefront in the slit plane are collimated as the coherent plane wave with a light beam width l by the collimator, the maximum angle interval of the plane waves along different directions is expressed by Eq. (1) as ı = a1 /f1 . The plane waves go through the prism and are finally focused on the focal plane by a concave mirror or a lens, the distribution of the resultant intensity is related to the path differences between the points in the plane which is vertical to the propagation direction of the output parallel light coming out of the prism, and is related to the light beam width l. But the present cases differ from the Fraunhofer diffraction with a limiting aperture in Fig. 12.5 in Ref. [5]. So in Ref. [5], for the prism spectrometer, the resultant intensity distribution cannot be expressed simply with the formula (12.4). In order to give the distribution of the resultant intensity, we must calculate the actual path differences between the points in the plane which is vertical to the propagation direction of the output parallel light coming out of the prism, therefore, the spectral resolving power also can not be expressed simply as the formula (12.13). But it is also complicated to calculate the distribution of the resultant intensity and spectral resolving power for the prism spectrometer. 5. Conclusions
Fig. 4. Light paths that the plane waves pass through the prism.
The angular width and the geometric line width in the spectral plane for the output incoherent imaging spectral line are related to the structure of prism monochromator. If the incident parallel light toward the plane mirror M come from the parallel light rays collimated by M1 for the light rays giving off from the center of entrance slit light source, and the output parallel light coming out of the prism with minimum deviation angle is parallel to the incident parallel light toward the plane mirror M, and if the parallel light rays with angular interval ı = a1 /f1 and consecutive wavelengths − ı/2∼ + ı/2 go through the prism, then the angular width and geometric line width is expressed respectively with Eq. (14)
Q. Wang / Optik 123 (2012) 2213–2217
and Eq. (15), and the relationship between the spectral width and line width is expressed with Eq. (16). The monochromatic degree of output light of prism monochromator is presented with Eq. (20) for the output light with consecutive wavelengths. The spectral resolving power of incoherent imaging spectral line is presented. If there is diffractive effect of the entrance slit with very small width, the spectral resolving power of coherent imaging spectral line of diffractive waves is also expressed with Eq. (25).
2217
References [1] E. Hecht, Optics, 4th ed., Addison-Wesley, MA, 2002. [2] M.H. Freeman, C.C. Hull, Optics, eleventh edition, Butterworth Heinemann, 2003. [3] M. Born, E. Wolf, Principles of Optics, 6th ed., Pergamon, Oxford, 1980. [4] F.A. Jenkins, H.E. White, Fundamentals of Optics, McGraw-Hill, New York, 1976. [5] W. Demtröder, Grundlagen und Techniken der Laserspektroskopie, SpringerVerlag, 1977.