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Experimental observation of spectral switch of partially coherent light focused by a lens with chromatic aberration
ARTICLE IN PRESS
Optics & Laser Technology 39 (2007) 1226–1230 www.elsevier.com/locate/optlastec
Experimental observation of spectral switch of partially coherent light focused by a lens with chromatic aberration Biao Qu, Jixiong Pu, Ziyang Chen College of Information Science and Engineering, Institute of Optics and Photonics, Huaqiao University, Quanzhou, Fujian 362021, China Received 15 May 2006; received in revised form 17 July 2006; accepted 4 August 2006 Available online 15 September 2006
Abstract In this paper, the effects of chromatic aberration of a focusing lens on the normalized spectrum of the partially coherent light at the geometrical-image plane are investigated experimentally. The experimental results show that, compared to the source spectrum, the normalized spectrum of the partially coherent light at the geometrical-image plane shifts either towards the red side or towards the blue side, which is dependent on the position of the observation point. In particular, the normalized spectrum splits into two peaks at some points and the spectral switch occurs at the critical point. Furthermore, each of the two peaks of the normalized spectrum still split into two subpeaks respectively at some points and the spectral switch occurred in the short-wavelength range of the normalized spectrum is also observed at another critical point. r 2006 Elsevier Ltd. All rights reserved. Keywords: Spectral shift; Spectral switch; Chromatic aberration
1. Introduction It has been shown by Wolf [1] theoretically that the normalized spectrum of light is the same on propagation and is equal to the normalized spectrum of light at the quasi-homogeneous source provided that the spectral degree of coherence is a function of the variable kðr2 r1 Þ only, where ðr2 r1 Þ denotes the vectorial distance between two points on the source, k is the wave number. If the spectral degree of coherence depends on frequency only through the variable kðr2 r1 Þ, the source is said to satisfy the scaling law. Conversely, the normalized spectrum of the light emitted by the source, which violates the scaling law will be changed in the propagation of the light field. If the peak wavelength of the normalized spectrum of light field is increased compared to that of the source, the normalized spectrum is termed redshift. Conversely, if the peak wavelength of the normalized spectrum of light field is decreased compared to that of the source, the normalized Corresponding author. Tel.: +86 595 2269 1063; fax: +86 595 2268 6969. E-mail address: [email protected] (J. Pu).
0030-3992/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2006.08.001
spectrum is termed blueshift. It has shown theoretically [2] that the spectral shift is changed gradually as the degree of coherence changes gradually and the spectral shift changes rapidly from redshift (or blueshift) to blueshift (or redshift) as the degree of coherence takes the critical value. This phenomenon is defined as spectral switch [3]. There has been a great deal of investigations, both theoretically [4–8] and experimentally [9–13], on the spectral shift and spectral switch since the scaling law is presented. Kandpal et al. [14] have performed the experiment of measuring the spectral switch of a partially coherent light diffracted by a one-dimensional rectangular aperture. Another rectangular aperture is placed behind the first aperture to form the light field of which the spectral characteristics are investigated. A disadvantage emerged in the experiment is that the light field after passing through two apertures is quite weak and furthermore, the spectral switch is occurred at the dark fringe of the diffraction pattern [15]. The consequence is that the phenomenon of spectral switch is hard to observe. In this paper, we employ a geometrical-image system of a chromatic lens to achieve a high intensity of the light field. The experimental results of the normalized spectrum
ARTICLE IN PRESS B. Qu et al. / Optics & Laser Technology 39 (2007) 1226–1230
2. Experimental setups and qualitative analysis of the chromatic aberration The schematics of the experimental setups are shown in Fig. 1, in which S is a 250 W tungsten-halogen lamp acting as the primary source. The tungsten halogen lamp used in our experiments is Philips projection lamp of type 13163, and the lamp’s color temperature is around 3400 K. A ground glass D and a slit A are located in sequence in close proximity behind the primary source S. A polychromatic source of uniform brightness is obtained at slit A. The width of slit A is a ¼ 0.15 mm. A chromatic lens is placed at a distance of L0 ¼ 2 f behind slit A, here f ¼ 30 cm is the focal length of the imaging lens. According to the geometrical imaging theory, slit A forms a real image of the same size at a distance of L ¼ 2 f behind the lens where is also the position of the observation plane. Finally we employ a fiber spectrometer (HR2000, Ocean Optics) to measure the spectra of light at the geometrical-image plane and the results are recorded on a computer. Considering that the long side (specified as the y-coordinate) of slit A used in our experiment is much longer than the short side (specified as the x-coordinate) of it, the optical field in one dimension of slit A can be considered as uniform along y-coordinate, and the optical field can be considered as only the function of x. Therefore, it is only necessary to measure the normalized spectrum along the x-axis. Here we set the origin point o at the optical axis. It is assumed that the peak wavelength of the source spectrum is l0 and the corresponding focal length and A
Chromatic Lens Fiber x
a
Spectrometer
S D L'
L
Computer
Fig. 1. Schematics of the experimental setups for measuring the normalized spectrum of partially coherent light produced by an imaging system. The experimental parameters are: f ¼ 30 cm, a ¼ 0.15 mm, L ¼ L0 ¼ 60 cm.
160 120 -dn / d|0
measured at the geometrical-image plane of the lens show that the normalized spectrum of light is either blueshifted or redshifted compared to the source spectrum, which is dependent on the position of the observation point and particularly the normalized spectrum splits into two peaks at some points. At the critical point the normalized spectrum is changed from redshift to blueshift rapidly, and the spectral switch occurs. Furthermore, the two peaks of the normalized spectrum still split into two subpeaks respectively at some points and another spectral switch occurred in the short-wavelength range of the normalized spectrum is also observed.
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refractive index are f 0 and n0 , respectively. The material of the chromatic lens used in our experiment is crown glass whose refractive index is a function of wavelength. The longitudinal chromatic aberration of the lens can be expressed as df f 0 dn f ðlÞ ¼ f 0 þ ðl l0 Þ ¼ f 0 ðl l0 Þ, (1) dl 0 n0 1 dl 0 where dn=dlj0 is the dispersion of the lens material, which is variant to different wavelength. The relation of dn=dlj0 to wavelength is shown in Fig. 2 [16]. It can be seen that the negative of the dispersion of crown glass decreases as the wavelength increased. According to this point and making use of Eq. (1), we can find that, if the wavelength l is smaller than l0 , then f ðlÞof 0 , and if the wavelength l is bigger than l0 , then f ðlÞ4f 0 . 3. Experimental results The experimental measurements of the normalized spectrum of light at the geometrical-image plane are shown in Fig. 3(a–g) by solid curves. The dotted curves are the normalized spectra of the primary source, whose peak wavelength is 574.0 nm. The solid curve in Fig. 3(a) is the normalized spectrum of light for on-axis point, i.e. x ¼ 0. We can get that the peak wavelength of the normalized spectrum of light for on-axis point is 563.3 nm, which is smaller than that of the primary source, i.e., the normalized spectrum is blueshifted compared to the source spectrum. Fig. 3(b) presents the normalized spectrum of light as x ¼ 0.5 mm. The peak wavelength is shifted to 602.2 nm, which is bigger than that of the primary source, i.e., the normalized spectrum is redshifted compared to the source spectrum. The normalized spectrum of light as x ¼ 0.7 mm is shown in Fig. 3(c). It can be seen that the normalized spectrum splits into two peaks and the peak wavelength is 704.2 nm, i.e., the normalized spectrum is redshifted compared to the source spectrum. The gap of the wavelength between the two peaks gets to 219.1 nm, which is much larger than the half-width of the source spectrum (82.0 nm).
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Fig. 3. Normalized spectrum, as the observation distance x is gradually increased, shows the evolution from a gradual spectral shift to a rapid spectral switch. The dotted curves are the normalized spectra of the primary source. (a) x ¼ 0, (b) x ¼ 0.5 mm, (c) x ¼ 0.7 mm, (d) x ¼ 0.745 mm, (e) x ¼ 0.76 mm, (f) x ¼ 0.77 mm and (g) x ¼ 0.8 mm. The other parameters are the same as in Fig. 1.
Fig. 3(d) shows the normalized spectrum of light as x ¼ 0.745 mm. It can be seen that the height of the peak located in the short-wavelength range of the normalized
spectrum is increased further and is equal to that of the peak located in the long-wavelength range of the normalized spectrum. The peak wavelength shifts rapidly from
ARTICLE IN PRESS B. Qu et al. / Optics & Laser Technology 39 (2007) 1226–1230
Dl lmax l0 ¼ , l0 l0
(2)
where l0 is the peak wavelength of the source spectrum and is equal to 574.0 nm; lmax is the wavelength at which the normalized spectrum of the observation light is maximum. Fig. 4 shows the relative shift of the peak wavelength Dl=l0 as a function of x for the cases that L is equal to 2f (solid curve), 2f2 mm (dashed curve) and 2f+2 mm (dotted curve), here f ¼ 30 cm is the focal length of the chromatic lens. It can be seen that the normalized spectrum is redshifted as Dl=l0 40 and is blueshifted as Dl=l0 o0. When the relative shift of the peak wavelength changes rapidly, the spectral switch occurs. From Fig. 4 we can easily find that there are two rapid shifts of Dl=l0 as L is equal to 2f, i.e., the spectral switch occurs two times, and there is only one rapid shift of Dl=l0 as L is equal to both 2f+2 mm and 2f2 mm, i.e., the spectral switch occurs only one time.
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x (mm) Fig. 4. Relative shift of the peak wavelength as a function of the observation distance x. L ¼ 2f (solid curve); L ¼ 2f2 mm (dashed curve); L ¼ 2f+2 mm (dotted curve). The other parameters are the same as in Fig. 1.
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719.8 to 479.0 nm, and simultaneously the normalized spectrum is changed from redshift to blueshift at this point, i.e., the spectral switch occurs. Furthermore, the gap of the wavelength between the two peaks is increased to 240.8 nm. Looking into the normalized spectrum shown in Fig. 3(d), we can get that the two peaks are going to split into two subpeaks, respectively. When the observation distance x is increased to 0.76 mm, the two peaks are already split into two subpeaks, respectively, which are shown in Fig. 3(e). The peak wavelength of the normalized spectrum is 475.2 nm and the normalized spectrum of light is blueshifted compared to the source spectrum. The gap of the wavelength between the two peaks gets to 254.4 nm. Fig. 3(f) gives the normalized spectrum of light as x ¼ 0.77 mm. It can be seen that the normalized spectrum is blueshifted compared to the source spectrum and the peak wavelength located in the short-wavelength range of the normalized spectrum shifts rapidly from 473.0 to 439.2 nm, i.e., the spectral switch occurs again. Simultaneously, another peak wavelength located in the longwavelength range of the normalized spectrum shifts rapidly from 735.4 to 767.4 nm. The gap of the wavelength between the two subpeaks located in the short-wavelength and the long-wavelength range of the normalized spectrum respectively gets to 328.2 nm. The solid curve in Fig. 3(g) is the normalized spectrum of light as x ¼ 0.8 mm. It can be seen that the peak wavelength of the normalized spectrum is equal to 437.0 nm and the normalized spectrum is blueshifted compared to the source spectrum. Furthermore, there appears another subpeak in the middle of the normalized spectrum whose height is even higher than that of the subpeak located in the longwavelength range of the normalized spectrum. Increasing the observation distance x further, the intensity of the light field is quiet small and the optical light fades out gradually. To show the optical characteristics of the spectral shift in detail, we define the relative shift of the peak wavelength as
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Fig. 5. Relative shift of the peak wavelength as a function of the observation distance x. a ¼ 0.15 mm (solid curve); a ¼ 0.2 mm (dashed curve); a ¼ 0.25 mm (dotted curve). The other parameters are the same as in Fig. 1.
Fig. 5 plots the relative shift of the peak wavelength Dl=l0 as a function of x for three widths of slit A, i.e., a ¼ 0.15 mm (solid curve), a ¼ 0.2 mm (dashed curve) and a ¼ 0.25 mm (dotted curve). From the picture we can see that the spectral switch occurs two times for the three values of a. Comparing the three curves, we can find that the distance from the optical axis to the position at which the spectral switch occurs is increased with the increment of a. Fig. 6 gives the relations of the relative shift of the peak wavelength Dl=l0 to x for f ¼ 30 cm (solid curve), f ¼ 19 cm (dashed curve), f ¼ 15 cm (dotted curve), respectively. From the three curves we can find that the bigger the focal length of the imaging lens is, the more obvious the phenomena of the spectral switches are. 4. Conclusions In this paper, we present the experimental results of the measurement of the normalized spectrum of light at the
ARTICLE IN PRESS B. Qu et al. / Optics & Laser Technology 39 (2007) 1226–1230
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References
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[1] Wolf E. Invariance of the spectrum of light on propagation. Phys Rev Lett 1986;56:1370–2. [2] Pu J, Nemoto S. Spectral changes and 1 N spectral switches in the diffraction of partially coherent light by an aperture. J Opt Soc Am A 2002;19:339–44. [3] Pu J, Zhang H, Nemoto S. Spectral shifts and spectral switches of partially coherent light passing through an aperture. Opt Commun 1999;162:57–63. [4] Wolf E. Non-cosmological redshifts. Nature 1987;326:363–5. [5] Wolf E, James DFV. Correlation-induced spectral changes. Rep Prog Phys 1996;59:771–818. [6] Foley JT, Wang M. A Theoretical analysis of the coherence-induced spectral shift experiments of Kandpal, Vaishya, and Joshi. J Res Nat Inst Stand Technol 1994;99:267–80. [7] Pan L, Lu¨ B. The spectral switch of partially coherent light in Young’s experiment. IEEE J Quantum Electron 2001;37:1377–81. [8] Cai Y, Lin Q. Spectral shift of partially coherent twisted anisotropic Gaussian Schell-model beams in free space. Opt Commun 2002;204:17–23. [9] Morris GM, Faklis D. Effects of source correlation on the spectrum of light. Opt Commun 1987;62:5–11. [10] Faklis D, Morris GM. Spectral shifts produced by source correlation. Opt Lett 1988;13:4–6. [11] Gori F, Guattari G, Palma C. Observation of optical redshifts and blueshifts produced by source correlation. Opt Commun 1988;67: 1–4. [12] Kandpal HC, Wasan A, Vaishya JS. Spectroscopy of partially coherent fields at geometrical-image plane and Fourier transform plane of a lens. Opt Commun 1998;149:1–7. [13] Lin Z, Gu Z. Angular spectrum redistribution from a real image of a light as a secondary source. Opt Lett 2001;26:663–5. [14] Kandpal HC. Experimental observation of the phenomenon of spectral switch. J Opt A: Pure Appl Opt 2001;3:296–9. [15] Ponomarenko SA, Wolf E. Spectral anomalies in a Fraunhofer diffraction pattern. Opt Lett 2002;27:1211–3. [16] Liu S. Handbook of modern optics, Beijing Print Press, 1993.
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0.6
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Fig. 6. Relative shift of the peak wavelength as a function of the observation distance x. f ¼ 30 cm (solid curve); f ¼ 19 cm (dashed curve); f ¼ 15 cm (dotted curve). The other parameters are the same as in Fig. 1.
geometrical-image plane of a chromatic lens. The experimental results show that not only blueshift but also redshift occur at the geometrical-image plane of a chromatic lens. The normalized spectrum splits into two peaks at some off-axis points and the phenomenon of spectral switch is also observed. Particularly, the two peaks of the normalized spectrum still split into two subpeaks respectively at some off-axis points and the spectral switch occurred in the short-wavelength range of the normalized spectrum is also observed. Acknowledgements This work is supported by the National Natural Science Foundation of China (No. 60477041) and Fujian Natural Science Foundation of China (Grant No. A0510018).
Optics & Laser Technology 39 (2007) 1226–1230 www.elsevier.com/locate/optlastec
Experimental observation of spectral switch of partially coherent light focused by a lens with chromatic aberration Biao Qu, Jixiong Pu, Ziyang Chen College of Information Science and Engineering, Institute of Optics and Photonics, Huaqiao University, Quanzhou, Fujian 362021, China Received 15 May 2006; received in revised form 17 July 2006; accepted 4 August 2006 Available online 15 September 2006
Abstract In this paper, the effects of chromatic aberration of a focusing lens on the normalized spectrum of the partially coherent light at the geometrical-image plane are investigated experimentally. The experimental results show that, compared to the source spectrum, the normalized spectrum of the partially coherent light at the geometrical-image plane shifts either towards the red side or towards the blue side, which is dependent on the position of the observation point. In particular, the normalized spectrum splits into two peaks at some points and the spectral switch occurs at the critical point. Furthermore, each of the two peaks of the normalized spectrum still split into two subpeaks respectively at some points and the spectral switch occurred in the short-wavelength range of the normalized spectrum is also observed at another critical point. r 2006 Elsevier Ltd. All rights reserved. Keywords: Spectral shift; Spectral switch; Chromatic aberration
1. Introduction It has been shown by Wolf [1] theoretically that the normalized spectrum of light is the same on propagation and is equal to the normalized spectrum of light at the quasi-homogeneous source provided that the spectral degree of coherence is a function of the variable kðr2 r1 Þ only, where ðr2 r1 Þ denotes the vectorial distance between two points on the source, k is the wave number. If the spectral degree of coherence depends on frequency only through the variable kðr2 r1 Þ, the source is said to satisfy the scaling law. Conversely, the normalized spectrum of the light emitted by the source, which violates the scaling law will be changed in the propagation of the light field. If the peak wavelength of the normalized spectrum of light field is increased compared to that of the source, the normalized spectrum is termed redshift. Conversely, if the peak wavelength of the normalized spectrum of light field is decreased compared to that of the source, the normalized Corresponding author. Tel.: +86 595 2269 1063; fax: +86 595 2268 6969. E-mail address: [email protected] (J. Pu).
0030-3992/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2006.08.001
spectrum is termed blueshift. It has shown theoretically [2] that the spectral shift is changed gradually as the degree of coherence changes gradually and the spectral shift changes rapidly from redshift (or blueshift) to blueshift (or redshift) as the degree of coherence takes the critical value. This phenomenon is defined as spectral switch [3]. There has been a great deal of investigations, both theoretically [4–8] and experimentally [9–13], on the spectral shift and spectral switch since the scaling law is presented. Kandpal et al. [14] have performed the experiment of measuring the spectral switch of a partially coherent light diffracted by a one-dimensional rectangular aperture. Another rectangular aperture is placed behind the first aperture to form the light field of which the spectral characteristics are investigated. A disadvantage emerged in the experiment is that the light field after passing through two apertures is quite weak and furthermore, the spectral switch is occurred at the dark fringe of the diffraction pattern [15]. The consequence is that the phenomenon of spectral switch is hard to observe. In this paper, we employ a geometrical-image system of a chromatic lens to achieve a high intensity of the light field. The experimental results of the normalized spectrum
ARTICLE IN PRESS B. Qu et al. / Optics & Laser Technology 39 (2007) 1226–1230
2. Experimental setups and qualitative analysis of the chromatic aberration The schematics of the experimental setups are shown in Fig. 1, in which S is a 250 W tungsten-halogen lamp acting as the primary source. The tungsten halogen lamp used in our experiments is Philips projection lamp of type 13163, and the lamp’s color temperature is around 3400 K. A ground glass D and a slit A are located in sequence in close proximity behind the primary source S. A polychromatic source of uniform brightness is obtained at slit A. The width of slit A is a ¼ 0.15 mm. A chromatic lens is placed at a distance of L0 ¼ 2 f behind slit A, here f ¼ 30 cm is the focal length of the imaging lens. According to the geometrical imaging theory, slit A forms a real image of the same size at a distance of L ¼ 2 f behind the lens where is also the position of the observation plane. Finally we employ a fiber spectrometer (HR2000, Ocean Optics) to measure the spectra of light at the geometrical-image plane and the results are recorded on a computer. Considering that the long side (specified as the y-coordinate) of slit A used in our experiment is much longer than the short side (specified as the x-coordinate) of it, the optical field in one dimension of slit A can be considered as uniform along y-coordinate, and the optical field can be considered as only the function of x. Therefore, it is only necessary to measure the normalized spectrum along the x-axis. Here we set the origin point o at the optical axis. It is assumed that the peak wavelength of the source spectrum is l0 and the corresponding focal length and A
Chromatic Lens Fiber x
a
Spectrometer
S D L'
L
Computer
Fig. 1. Schematics of the experimental setups for measuring the normalized spectrum of partially coherent light produced by an imaging system. The experimental parameters are: f ¼ 30 cm, a ¼ 0.15 mm, L ¼ L0 ¼ 60 cm.
160 120 -dn / d|0
measured at the geometrical-image plane of the lens show that the normalized spectrum of light is either blueshifted or redshifted compared to the source spectrum, which is dependent on the position of the observation point and particularly the normalized spectrum splits into two peaks at some points. At the critical point the normalized spectrum is changed from redshift to blueshift rapidly, and the spectral switch occurs. Furthermore, the two peaks of the normalized spectrum still split into two subpeaks respectively at some points and another spectral switch occurred in the short-wavelength range of the normalized spectrum is also observed.
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Wavelength (nm) Fig. 2. Negative of the dispersion for crown glass as a function of wavelength.
refractive index are f 0 and n0 , respectively. The material of the chromatic lens used in our experiment is crown glass whose refractive index is a function of wavelength. The longitudinal chromatic aberration of the lens can be expressed as df f 0 dn f ðlÞ ¼ f 0 þ ðl l0 Þ ¼ f 0 ðl l0 Þ, (1) dl 0 n0 1 dl 0 where dn=dlj0 is the dispersion of the lens material, which is variant to different wavelength. The relation of dn=dlj0 to wavelength is shown in Fig. 2 [16]. It can be seen that the negative of the dispersion of crown glass decreases as the wavelength increased. According to this point and making use of Eq. (1), we can find that, if the wavelength l is smaller than l0 , then f ðlÞof 0 , and if the wavelength l is bigger than l0 , then f ðlÞ4f 0 . 3. Experimental results The experimental measurements of the normalized spectrum of light at the geometrical-image plane are shown in Fig. 3(a–g) by solid curves. The dotted curves are the normalized spectra of the primary source, whose peak wavelength is 574.0 nm. The solid curve in Fig. 3(a) is the normalized spectrum of light for on-axis point, i.e. x ¼ 0. We can get that the peak wavelength of the normalized spectrum of light for on-axis point is 563.3 nm, which is smaller than that of the primary source, i.e., the normalized spectrum is blueshifted compared to the source spectrum. Fig. 3(b) presents the normalized spectrum of light as x ¼ 0.5 mm. The peak wavelength is shifted to 602.2 nm, which is bigger than that of the primary source, i.e., the normalized spectrum is redshifted compared to the source spectrum. The normalized spectrum of light as x ¼ 0.7 mm is shown in Fig. 3(c). It can be seen that the normalized spectrum splits into two peaks and the peak wavelength is 704.2 nm, i.e., the normalized spectrum is redshifted compared to the source spectrum. The gap of the wavelength between the two peaks gets to 219.1 nm, which is much larger than the half-width of the source spectrum (82.0 nm).
ARTICLE IN PRESS B. Qu et al. / Optics & Laser Technology 39 (2007) 1226–1230
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Fig. 3. Normalized spectrum, as the observation distance x is gradually increased, shows the evolution from a gradual spectral shift to a rapid spectral switch. The dotted curves are the normalized spectra of the primary source. (a) x ¼ 0, (b) x ¼ 0.5 mm, (c) x ¼ 0.7 mm, (d) x ¼ 0.745 mm, (e) x ¼ 0.76 mm, (f) x ¼ 0.77 mm and (g) x ¼ 0.8 mm. The other parameters are the same as in Fig. 1.
Fig. 3(d) shows the normalized spectrum of light as x ¼ 0.745 mm. It can be seen that the height of the peak located in the short-wavelength range of the normalized
spectrum is increased further and is equal to that of the peak located in the long-wavelength range of the normalized spectrum. The peak wavelength shifts rapidly from
ARTICLE IN PRESS B. Qu et al. / Optics & Laser Technology 39 (2007) 1226–1230
Dl lmax l0 ¼ , l0 l0
(2)
where l0 is the peak wavelength of the source spectrum and is equal to 574.0 nm; lmax is the wavelength at which the normalized spectrum of the observation light is maximum. Fig. 4 shows the relative shift of the peak wavelength Dl=l0 as a function of x for the cases that L is equal to 2f (solid curve), 2f2 mm (dashed curve) and 2f+2 mm (dotted curve), here f ¼ 30 cm is the focal length of the chromatic lens. It can be seen that the normalized spectrum is redshifted as Dl=l0 40 and is blueshifted as Dl=l0 o0. When the relative shift of the peak wavelength changes rapidly, the spectral switch occurs. From Fig. 4 we can easily find that there are two rapid shifts of Dl=l0 as L is equal to 2f, i.e., the spectral switch occurs two times, and there is only one rapid shift of Dl=l0 as L is equal to both 2f+2 mm and 2f2 mm, i.e., the spectral switch occurs only one time.
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x (mm) Fig. 4. Relative shift of the peak wavelength as a function of the observation distance x. L ¼ 2f (solid curve); L ¼ 2f2 mm (dashed curve); L ¼ 2f+2 mm (dotted curve). The other parameters are the same as in Fig. 1.
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719.8 to 479.0 nm, and simultaneously the normalized spectrum is changed from redshift to blueshift at this point, i.e., the spectral switch occurs. Furthermore, the gap of the wavelength between the two peaks is increased to 240.8 nm. Looking into the normalized spectrum shown in Fig. 3(d), we can get that the two peaks are going to split into two subpeaks, respectively. When the observation distance x is increased to 0.76 mm, the two peaks are already split into two subpeaks, respectively, which are shown in Fig. 3(e). The peak wavelength of the normalized spectrum is 475.2 nm and the normalized spectrum of light is blueshifted compared to the source spectrum. The gap of the wavelength between the two peaks gets to 254.4 nm. Fig. 3(f) gives the normalized spectrum of light as x ¼ 0.77 mm. It can be seen that the normalized spectrum is blueshifted compared to the source spectrum and the peak wavelength located in the short-wavelength range of the normalized spectrum shifts rapidly from 473.0 to 439.2 nm, i.e., the spectral switch occurs again. Simultaneously, another peak wavelength located in the longwavelength range of the normalized spectrum shifts rapidly from 735.4 to 767.4 nm. The gap of the wavelength between the two subpeaks located in the short-wavelength and the long-wavelength range of the normalized spectrum respectively gets to 328.2 nm. The solid curve in Fig. 3(g) is the normalized spectrum of light as x ¼ 0.8 mm. It can be seen that the peak wavelength of the normalized spectrum is equal to 437.0 nm and the normalized spectrum is blueshifted compared to the source spectrum. Furthermore, there appears another subpeak in the middle of the normalized spectrum whose height is even higher than that of the subpeak located in the longwavelength range of the normalized spectrum. Increasing the observation distance x further, the intensity of the light field is quiet small and the optical light fades out gradually. To show the optical characteristics of the spectral shift in detail, we define the relative shift of the peak wavelength as
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Fig. 5. Relative shift of the peak wavelength as a function of the observation distance x. a ¼ 0.15 mm (solid curve); a ¼ 0.2 mm (dashed curve); a ¼ 0.25 mm (dotted curve). The other parameters are the same as in Fig. 1.
Fig. 5 plots the relative shift of the peak wavelength Dl=l0 as a function of x for three widths of slit A, i.e., a ¼ 0.15 mm (solid curve), a ¼ 0.2 mm (dashed curve) and a ¼ 0.25 mm (dotted curve). From the picture we can see that the spectral switch occurs two times for the three values of a. Comparing the three curves, we can find that the distance from the optical axis to the position at which the spectral switch occurs is increased with the increment of a. Fig. 6 gives the relations of the relative shift of the peak wavelength Dl=l0 to x for f ¼ 30 cm (solid curve), f ¼ 19 cm (dashed curve), f ¼ 15 cm (dotted curve), respectively. From the three curves we can find that the bigger the focal length of the imaging lens is, the more obvious the phenomena of the spectral switches are. 4. Conclusions In this paper, we present the experimental results of the measurement of the normalized spectrum of light at the
ARTICLE IN PRESS B. Qu et al. / Optics & Laser Technology 39 (2007) 1226–1230
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References
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Fig. 6. Relative shift of the peak wavelength as a function of the observation distance x. f ¼ 30 cm (solid curve); f ¼ 19 cm (dashed curve); f ¼ 15 cm (dotted curve). The other parameters are the same as in Fig. 1.
geometrical-image plane of a chromatic lens. The experimental results show that not only blueshift but also redshift occur at the geometrical-image plane of a chromatic lens. The normalized spectrum splits into two peaks at some off-axis points and the phenomenon of spectral switch is also observed. Particularly, the two peaks of the normalized spectrum still split into two subpeaks respectively at some off-axis points and the spectral switch occurred in the short-wavelength range of the normalized spectrum is also observed. Acknowledgements This work is supported by the National Natural Science Foundation of China (No. 60477041) and Fujian Natural Science Foundation of China (Grant No. A0510018).