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Tunable laser optics: Applications to optics and quantum optics
Progress in Quantum Electronics 37 (2013) 326–347 www.elsevier.com/locate/pquantelec
Review
Tunable laser optics: Applications to optics and quantum optics F.J. Duartea,b,n a Interferometric Optics, Rochester, New York, USA University of New Mexico, Albuquerque, New Mexico, USA
b
Available online 9 October 2013
Abstract Optics originally developed for tunable organic dye lasers have found applications in other areas of optics, laser optics, and quantum optics. Here, the salient aspects of the physics related to the cavity linewidth equation and the effects of intracavity beam expansion and intracavity dispersion on this equation are reviewed. Additionally, the generalized multiple-prism dispersion equation is applied to direct-vision prisms, also known as Amici prisms, to calculate dispersion configurations of practical interest. Then, the higher derivatives of the multiple-prism dispersion equation applicable to laser pulse compression are considered. From this perspective, a new compact and generalized equation for higher-order phase derivatives is introduced for the first time. Furthermore, it is shown how the N-slit interferometric equation, derived from quantum principles using Dirac's notation, gives rise to generalized versions of the diffraction grating equation and the law of refraction. The nexus between the N-slit interferometric equation and the cavity linewidth equation is also illustrated. Finally, various optical and quantum optical applications that have benefited from these developments are highlighted. & 2013 Elsevier Ltd. All rights reserved. Keywords: Amici prisms; Generalized multiple-prism dispersion; Prismatic pulse compression; N-slit interferometric equation; Organic lasers; Quantum entanglement
Contents 1. 2. 3.
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Laser wavelength tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Expanded beam illumination of diffraction gratings in Littrow configuration . . . . . . . . . . . . . 327
n
Correspondence address: Interferometric Optics, Rochester, New York, USA. E-mail address: [email protected]
0079-6727/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.pquantelec.2013.09.001
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4.
Feedback from gratings in grazing-incidence configuration . . . . . . . . . . . . . . . . . . . . . . . . . 329 4.1. Applications to tunable diode lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 5. Multiple-prism beam expansion and generalized multiple-prism dispersion equations . . . . . . . 330 5.1. Direct-vision, compound, and Amici prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 5.2. Applications to tunable lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 6. Generalized higher-order multiple-prism dispersion equations for laser pulse compression . . . . 338 6.1. Applications to laser pulse compression and coherent microscopy . . . . . . . . . . . . . . . 340 7. The quantum interferometric connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 7.1. Interference and quantum entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 8. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
1. Introduction Soon after the discovery of the organic dye laser in 1966 by Sorokin and Lankard [1] and Schäfer et al. [2], the tunability of this laser source inspired several developments in optics that have found widespread applications in the field of lasers and in optics in general. The development of these techniques and their link to major subsequent developments in physics and science is reviewed here. Although the text only refers to the interaction of these contributions with specific optics and laser developments, it should be kept in mind that the narrow-linewidth tunable lasers and the ultrashort pulse lasers that adopted the dispersive techniques described here have had major impacts in a plethora of important fields including astronomy [3,4], laser isotope separation [4–6], medicine [7,8], photochemistry [9], spectroscopy [10,11], and ultrashort pulse phenomena [12]. This paper is dedicated to the memory of F.P. Schäfer (1931–2011), one of the discoverers of the organic dye laser, who displayed a keen interest in the dispersive optical techniques applied to tunable organic dye lasers [13]. 2. Laser wavelength tuning The first broadband laser wavelength tuning was performed by Soffer and McFarland [14] in 1967. They used a grating in the Littrow configuration to demonstrate the tuning of a Rhodamine 6G laser. The tuning range was approximately 555 r λr 590 nm with a reported linewidth of Δλ 4 nm. For a cavity tuned with a grating in the Littrow configuration the emission wavelength is given by mλ ¼ 2d sin Θ
ð2:1Þ
where λ is the laser emission wavelength, m is the order of diffraction, and d is the groove density of the diffraction grating in lines/mm. This equation is a special case of the more general diffraction grating equation discussed later. Fig. 1 describes a telescopic tunable dye laser cavity incorporating a Littrow grating as the tuning element. 3. Expanded beam illumination of diffraction gratings in Littrow configuration Following the introduction of the dye laser [1,2] and the demonstration of its tunability by Soffer and McFarland [14], a powerful class of tunable laser sources in the visible was made available to the scientific community. This momentous discovery occurred in 1972 when Hänsch
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Fig. 1. Littrow cavity configuration incorporating two-dimensional intracavity telescopic beam expansion.
[15] realized that expansion of the intracavity beam illuminating the tuning diffraction grating yielded an enormous reduction in the laser emission linewidth. Hänsch utilized a two-dimensional astronomical telescope comprised of two convex lenses (see Fig. 1). What Hänsch discovered can be succinctly explained by the cavity linewidth equation for a single-pass laser 1 ∂Θ Δλ Δθ ð3:1Þ ∂λ where the beam divergence Δθ can be represented by the diffraction-limited relation Δθ
λ πw
ð3:2Þ
Here, λ is the laser emission wavelength, and w is the beam waist at the gain region. As explained later, in Section 5, the reduction in the emission linewidth is a consequence of the large increase in intracavity dispersion, this increase results from the illumination of the diffraction grating by the expanded beam. However, Hänsch interpreted the effect in an equivalent mathematical form corresponding to a reduction in the intracavity beam divergence [15] Δθ
λ πMw
ð3:3Þ
Because the magnification factor M can be rather large, the subsequent reduction in the laser linewidth can be rather dramatic. In fact, Hänsch was able to measure linewidths of Δν 2:5 GHz at λ 600 nm, or Δλ 0:003 nm. Further reductions were achieved through the introduction of an intracavity etalon [15]. In Eq. (3.1), the angular dispersion of the grating, deployed in Littrow configuration is given by [15] ∂Θ 2 tan Θ ð3:4Þ ¼ ∂λ λ where Θ is the angle of incidence, which equals the angle of diffraction for the Littrow-mounted grating. In addition to the technical accomplishment of high-power tunable narrow-linewidth emission, the significance of Hänsch's contribution was the realization that intracavity beam expansion, or
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illumination of the whole diffraction grating, led to a gigantic reduction in the laser emission linewidth. At this stage, it should be mentioned that both the expressions applied by Hänsch, the expression for the single-pass linewidth (Eq. (3.1)) and the diffraction-limited beam divergence (Eq. (3.2)), can be found in classical optics books. For instance, Eq. (3.1) can be referenced to Robertson in 1955 [16], and Eq. (3.2) can be traced to Jenkins and White in 1957 [17]. However, as will be shown later, these expressions can also be derived from the principles of quantum optics. 4. Feedback from gratings in grazing-incidence configuration The next important development in the optics of tunable lasers was the introduction of grazing-incidence grating configurations by Shoshan et al. [18] and Littman and Metcalf [19]. The physics of linewidth narrowing remains essentially the same as the narrowing considered by Hänsch except that instead of using an expanded beam to illuminate the entire diffractive surface of the grating, these authors utilized the divergent emission to illuminate a diffraction grating deployed at a grazing-incidence angle, as illustrated in Fig. 2 for an open cavity configuration and in Fig. 3 for a closed cavity configuration. Thus, the linewidth equation becomes 1 ∂Θ Δλ Δθ ð4:1Þ ∂λ GI where Δθ is the beam divergence given by Eq. (3.2), and the angular dispersion for the diffraction grating in the grazing-incidence configuration is given by ∂Θ ð sin Θ þ sin Θ′Þ ð4:2Þ ¼ ∂λ GI λ cos Θ or
∂Θ m ¼ ∂λ GI d cos Θ
Fig. 2. Grazing-incidence grating oscillator in an open cavity configuration.
ð4:3Þ
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Fig. 3. Grazing-incidence grating oscillator in a closed cavity configuration.
where λ is the laser emission wavelength, m is the order of diffraction, and d is the groove density of the diffraction grating in lines/mm. Certainly, these dispersion expressions are derived from the diffraction grating equation [19] mλ ¼ dð sin Θ7 sin Θ′Þ
ð4:4Þ
Again, some observations are in order. First, the dispersion equations applied here to grazingincidence grating cavities are often multiplied by a factor of 2 to indicate a double-pass configuration [18]. Additionally, as for beam divergence, the diffraction grating equation is adopted from classical literature. In this case, we can trace Eq. (4.4) all the way back to Michelson's 1927 book entitled Studies in Optics [20]. It will also be shown later that this expression can also be derived from quantum optics principles. 4.1. Applications to tunable diode lasers Grazing-incidence grating configurations, developed for the organic dye laser [18,19], have been extensively applied to other types of lasers, including gas lasers [21,22] and solid-state tunable lasers [23,24]. However, these configurations have had their greatest influence through their widespread application to tunable semiconductor lasers, in which straightforward open cavity designs have had a major impact on the development of the tunable diode lasers used in laser cooling experiments [25,26]. 5. Multiple-prism beam expansion and generalized multiple-prism dispersion equations Prismatic beam expansion and multiple-prism arrays were first disclosed by Newton in his timeless book Opticks [27], in 1704 (see Figs. 4 and 5). Although these ideas would be used sporadically following their introduction [28], it was the tunable organic dye laser that revived their use some 270 years later. Of particular interest here was the use of a single-prism beam expander by Hanna et al. [29] and a study on double-prism expanders by Kasuya et al. [30]. The success of intracavity multiple-prism beam expansion configurations [31,32], depicted here in tunable organic solid-state dye laser oscillators [33,34] in Figs. 6 and 7, inspired Duarte and Piper to introduce the generalized multiple-prism dispersion theory [35]. For generalized
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Fig. 4. Newton's single-prism diagram depicting expansion of a light beam from Newton's 1704 book Opticks [27].
Fig. 5. Newton's prism array from Newton's 1704 book Opticks [27].
multiple prism arrays of any geometry and material, see Fig. 8, the dispersion can be succinctly expressed as [35] ∇λ ϕ2;m ¼ H2;m ∇λ nm þ ðk 1;m k 2;m Þ 1 ðH1;m ∇λ nm 7∇λ ϕ2;ðm 1Þ Þ
ð5:1Þ
where ∇λ ¼
∂ ∂λ
ð5:2Þ
k1;m ¼
cos ψ 1;m cos ϕ1;m
ð5:3Þ
k2;m ¼
cos ϕ2;m cos ψ 2;m
ð5:4Þ
H1;m ¼
tan ϕ1;m nm
ð5:5Þ
H2;m ¼
tan ϕ2;m nm
ð5:6Þ
In these equations, for the mth prism, nm is the refractive index, ϕ1;m is the incidence angle, and
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Fig. 6. Hybrid multiple-prism pre-expanded near grazing-incidence (HMPGI) grating solid state dye laser oscillator, from Duarte [33].
Fig. 7. Multiple-prism Littrow (MPL) grating solid-state dye laser oscillator, from Duarte [34].
ψ 1;m is the corresponding refraction angle. Similarly, ϕ2;m is the emergence angle, and ψ 2;m is the corresponding refraction angle. Additionally, k1;m and k 2;m are the beam expansions at the incidence and the exit surfaces, respectively. Furthermore, H1;m and H2;m are geometrical identities related to ϕ1;m and ϕ2;m , respectively. Physically, the generalized single-pass multiple-prism dispersion equation tells us that the cumulative dispersion, at the mth prism (∇λ ϕ2;m ), depends on the geometry of this particular prism, the angle of incidence at this prism, the material of this prism, and the cumulative dispersion up to the previous prism, ∇λ ϕ2;ðm 1Þ [35,36]. The dispersion expression given in Eq. (5.1) applies to a single pass and is technically known as the single-pass generalized multiple-prism dispersion equation. To describe the physics of linewidth narrowing in a pulsed high-gain tunable laser oscillator, it is necessary to consider the double-pass multiple-prism dispersion. The double pass is constructed by unfolding the single pass through the multiple-prism grating, or multiple-prism mirror, assembly used to induce linewidth narrowing in the frequency-selective laser cavity as illustrated in Fig. 9. Thus, as
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Fig. 8. (a) Generalized multiple-prism array in a positive configuration, and (b) in a compensating configuration. Generalized prismatic configurations of this class were first introduced by Duarte and Piper [36].
explained by Duarte and Piper [35] the double-pass angular multiple-prism dispersion becomes ∇λ ϕ01;m ¼ H01;m ∇λ nm þ ðk 1;m0 k2;m0 Þ 1 ðH02;m ∇λ nm 7∇λ ϕ01;ðmþ1Þ Þ
ð5:7Þ
where k′1;m ¼
cos ψ′1;m cos ϕ′1;m
ð5:8Þ
k′2;m ¼
cos ϕ′2;m cos ψ′2;m
ð5:9Þ
H′1;m ¼
tan ′ϕ1;m nm
ð5:10Þ
H2;m0 ¼
tan ϕ2;m0 nm
ð5:11Þ
An important term in Eq. (5.7) is ∇λ ϕ′1;ðmþ1Þ , which links the angular dispersion of the doublepass, or single return-pass, to the last stage of the single-pass angular dispersion and includes the diffraction grating dispersion so that [37] ∇λ ϕ′1;ðmþ1Þ ¼ ð∇λ Θ7∇λ ϕ2;r Þ
ð5:12Þ
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Fig. 9. Unfolded multiple-prism grating configuration utilized to derive the double-pass, or single return-pass, multipleprism dispersion, adapted from Duarte and Piper [35].
If the diffraction grating is replaced by a mirror, ∇λ Θ ¼ 0, this equation leads directly to ∇λ ϕ′1;ðmþ1Þ ¼ ∇λ ϕ2;r
ð5:13Þ
In these equations, ∇λ ϕ2;r is the overall cumulative single-pass angular dispersion calculated from Eq. (5.1) at the last prism of a prism array composed of r prisms. More specifically, for an array of r prisms, at the last prism ∇λ ϕ2;r ¼ ∇λ ϕ2;m
ð5:14Þ
Subsequently, Duarte and Piper [37] showed that the multiple return-pass cavity linewidth equation becomes Δλ ΔθR ðMR∇λ Θ þ R∇λ ϕ′1;m Þ 1
ð5:15Þ
where ΔθR is the multi-pass beam divergence, which approaches the diffraction limit in a welldesigned cavity [38], Ris a finite number of intracavity return passes [37], and M is the overall intracavity beam expansion defined by [38,39] r
M 1 ¼ ∏ k1;m
ð5:16Þ
m¼1 r
M 2 ¼ ∏ k2;m
ð5:17Þ
m¼1
r
M ¼ M 1 M 2 ¼ ∏ k 1;m k2;m m¼1
ð5:18Þ
Here, we should indicate that in well-designed multiple-prism beam expanders with orthogonal beam exit angles, all the expansion is provided by M 1 , because M 2 1. Additionally, as highlighted recently [40], the relationship between single-pass and return-pass angular multiple-prism angular dispersion can
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be expressed explicitly as ∇λ ϕ2;m ¼ ð2MÞ 1 ð∇λ ϕ′1;m Þ
ð5:19Þ
Thus, the return-pass cavity linewidth equation, which is Eq. (5.15), can also be expressed as Δλ ΔθR ðMR∇λ Θ þ 2MR∇λ ϕ2;r Þ 1
ð5:20Þ
After removal of the MR factor out, Eq. (5.20) can also be written as ΔθR ð∇λ Θ þ 2∇λ ϕ2;r Þ 1 ð5:21Þ MR For R ¼ 1 and for a dispersionless telescopic beam expander, that is, for ∇λ ϕ2;r ¼ 0, Eq. (5.21) reduces to Δλ
Δθ ð∇λ ΘÞ 1 ð5:22Þ M and this result is consistent with the equations utilized by Hänsch. The primary difference is that the beam-expansion factor M is now systematically and consistently introduced through the dispersion theory and augments the intracavity dispersion rather than being introduced in an ad-hoc manner as a factor that reduces the beam divergence as for the original telescopic design [15]. Δλ
5.1. Direct-vision, compound, and Amici prisms Some applications require direct-vision prisms [17] in which the beam of light does not deviate after refraction by a single prism. These prisms are essentially multiple-prism configurations in which the prisms are deployed right next to each other. These multiple-prism configurations are known as direct-vision prisms [17], compound prisms, or Amici prisms. These prisms are important in corrections of atmospheric dispersion in adaptive optics systems for astronomical applications [41]. Here, the single-pass generalized multiple-prism dispersion equation, Eq. (5.1), is applied to illustrate the dispersive properties of these prismatic configurations. The discussion begins by introducing a fused silica prism (n1 ¼ 1:45838 at λ ¼ 590 nm) with the following angular dimensions α1 ¼ 451, ϕ1;1 ¼ 20:661, ψ 1;1 ¼ 14:001, ψ 2;1 ¼ 31:001, ϕ2;1 ¼ 48:681, which are
Fig. 10. Single fused silica prism with a negative dispersion of ð∂ϕ2;1 =∂λÞ 0:0384 at λ ¼ 590 nm. Drawing is approximately to scale.
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illustrated approximately to scale in Fig. 10. From Eq. (5.1), for m ¼ 1 and ðdn1 =dλÞ 0:0348 μm 1, the dispersion is ∂ϕ2;1 dn1 1:1036 0:0384 ð5:23Þ ∂λ dλ in units of μm 1. To prevent deviation in the beam path, a second prism is introduced in compensating configuration and is adjoined to the first prism to create a compound prism, or Amici prism, as illustrated in Fig. 11. The second prism is made of a higher refractive index material, in this case Schott SF10 (n2 ¼ 1:73020 at λ ¼ 590 nm) with an apex angle α2 ¼ 25:731. Here, it is noted that ϕ2;1 ¼ ϕ1;2 ¼ 48:681, ψ 1;2 ¼ 25:731, and ψ 2;2 ¼ ϕ2;2 ¼ 0. From Eq. (5.1), with m ¼ 2 and ðdn2 =dλÞ 0:1367 μm 1, the combined dispersion becomes ∂ϕ2;2 dn2 dn1 0:4818 0:8089 0:0377 ð5:24Þ ∂λ dλ dλ
Fig. 11. Amici prism configuration with two prisms in compensating configuration. The first prism is made of fused silica and the second prism is made of the higher refractive index material Schott SF10 (see text). This configuration yields negative dispersion of ð∂ϕ2;2 =∂λÞ 0:0377 at λ ¼ 590 nm. Drawing is approximately to scale.
Fig. 12. Amici prism, or direct-vision, configuration with three prisms in compensating configuration. The first and last prisms are identical and are made of fused silica whereas the center prism is made of the higher refractive index material Schott SF10 (see text). This configuration yields positive dispersion of ð∂ϕ2;3 =∂λÞ þ 0:0821 at λ ¼ 590 nm. Drawing is approximately to scale.
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in units of μm 1. Although the propagation of the beam is now straight, the exit beam is slightly displaced relative to its position at the incidence surface. To achieve collinear transmission, the compound prism in Fig. 11 is unfolded to create a three-prism direct-vision configuration [17], or a three-prism Amici configuration, as illustrated in Fig. 12. For this particular configuration: α1 ¼ 451, ϕ1;1 ¼ 20:661, ψ 1;1 ¼ 14:001, ψ 2;1 ¼ 31:001, α2 ¼ 51:461, ψ 1;2 ¼ ψ 2;2 ¼ 25:731, α3 ¼ 451, ψ 1;3 ¼ 31:001, ψ 2;3 ¼ 14:001, and ϕ2;3 ¼ 20:661. For Eq. (5.1) and m ¼ 3, the estimated combined dispersion is ∂ϕ2;3 dn1 dn2 1:7631 1:0503 þ 0:0821 ð5:25Þ ∂λ dλ dλ The change in the sign of the dispersion from Eq. (5.23), applicable to the single prism, to Eq. (5.25), applicable to the three-prism compensating configuration, illustrates that the usual dispersion wavelength order (from red to blue), is inverted (from blue to red) and is consistent with the qualitative description given by Jenkins and White [17]. Thus, the single-pass generalized multiple-prism dispersion equation has been shown to be naturally applicable to the dispersion of direct-vision and Amici prismatic configurations. Furthermore, it also been shown quantitatively that although these compound prisms do provide an exit beam collinear with the incident beam, these compound prisms are also quite dispersive; hence, these prisms are applied to the correction of atmospheric dispersion in astronomical optics. Next, a dispersionless multiple-prism array is described. Given the geometrical flexibility of multiple-prism beam expanders, a three-prism dispersionless configuration can be designed easily. From Eq. (5.7), for m ¼ 3 and for an orthogonal beam exit, ψ 2;1 ¼ ϕ2;1 ¼ 0, ψ 2;2 ¼ ϕ2;2 ¼ 0, ψ 2;3 ¼ ϕ2;3 ¼ 0; for a dispersionless configuration, the following condition must be satisfied [38,39] ðk 1;1 þ k 1;1 k 1;2 Þ tan ψ 1;1 ¼ ðk 1;1 k1;2 Þk 1;3 tan ψ 1;3
ð5:26Þ
For fused silica (n ¼ 1:45838 at λ ¼ 590 nm), the design of a dispersionless multiple-prism beam expander providing an overall beam magnification of M 42 is illustrated. In this illustration, the following expansions are selected: k1;1 ¼ k1;2 ¼ 5:2749 which implies that ϕ1;1 ¼ ϕ1;2 ¼ 821 and ψ 1;1 ¼ ψ 1;2 ¼ 42:76701. From Eq. (5.26) ðk 1;1 þ k 1;1 k 1;2 Þ tan ψ 1;1 tan ϕ1;3 ¼ k 1;1 k1;2 n
ð5:27Þ
Fig. 13. Quasi-achromatic fused silica multiple-prism beam expander yields zero dispersion and a beam expansion of M 42:7828 at λ ¼ 590 nm. Drawing is approximately to scale.
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and it can be directly deduced that ϕ1;3 58:06911, ψ 1;3 35:58701, and k 1;3 1:5376, so M ¼ k1;1 k 1;2 k 1;3 42:7828
ð5:28Þ
This class of quasi-achromatic beam expander, with zero dispersion at λ ¼ 590 nm, is directly applicable to the design of narrow-linewidth tunable lasers [38]. The three-prism beam expander is displayed approximately to scale in Fig. 13. Multiple-prism arrays, in their various forms, are applicable to optical instrumentation, such as spectrometers [38], and even astronomical instrumentation [42]. 5.2. Applications to tunable lasers Multiple-prism grating configurations, as developed for tunable organic lasers [31–34] have been successfully applied to tunable gas lasers [43], optical parametric oscillators [38], and tunable diode lasers [44,45]. In particular, research by Zorabedian [44] on tunable semiconductor lasers demonstrated the advantages of utilizing multiple-prism grating configurations that enable a closed cavity configuration, in which the laser output is coupled from the output coupler mirror rather than from the reflection losses of the diffraction grating. These advantages include reduced output noise and protection from unwanted external back reflections. Here, it should be noted that one of the first grating-tuned semiconductor lasers, introduced by Fleming and Mooradian [46], did use a closed cavity configuration and expanded the intracavity beam incident on a Littrow grating. 6. Generalized higher-order multiple-prism dispersion equations for laser pulse compression An additional area of optics created by developments in broadly tunable organic dye lasers was prismatic pulse compression [47] or multiple-prism pulse compression [48]. In essence, this area is a direct consequence of the pulse laser linewidth equation 1 ∂Θ Δλ Δθ ∂λ which indicates that greatly reduced intracavity dispersion leads to a very broad emission linewidth. Through a variant of Heisenberg's uncertainty principle, ΔνΔt 1
ð6:1Þ
this broad linewidth implies a very short pulse. This equation will be reconsidered in the next section. Control of the intracavity dispersion and the ability to introduce negative dispersion can enable a severe reduction of intracavity dispersion and cause very broadband emission or very short pulses, as explained by Eq. (6.1). Thus, learning to control the intracavity dispersion becomes very important. Returning to Eq. (5.1), the identity ∇n ϕ2;m ¼ ∇λ ϕ2;m ð∇λ nm Þ 1
ð6:2Þ
can be applied to restate the generalized multiple-prism dispersion as [49,50] ∇n ϕ2;m ¼ H2;m þ ðMÞ 1 ðH1;m 7∇n ϕ2;ðm 1Þ Þ
ð6:3Þ
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339
where the identity 1 1 k1;m k2;m ¼ ðM 1 Þ
ð6:4Þ
applies. Hence, the complete second derivative of the refraction angle, or first derivative of the dispersion ∇λ ϕ2;m , is given by [49] ∇2n ϕ2;m ¼ ∇n H2;m þ ð∇n M 1 ÞðH1;m 7∇n ϕ2;ðm 1Þ Þ þ ðM 1 Þð∇n H1;m 7∇2n ϕ2;ðm 1Þ Þ
ð6:5Þ
the third and higher derivatives are given by Duarte [40,50]. Based on the progression of the higher derivatives, a generalized expression can be found to be ∇rn ϕ2;m ¼ ∇rn 1 H2;m þ ðMÞ 1 ð∇n þ ζÞr 1
ð6:6Þ
where ζ s ¼ ∇sn H1;m 7∇sþ1 n ϕ2;ðm 1Þ
ð6:7Þ
ζ 0 ¼ 1 ¼ H1;m 7∇n ϕ2;ðm 1Þ
ð6:8Þ
Here, it is very important that when writing the expansion in r, in Eq. (6.6), the term ζ 0 ¼ 1 be included as defined in Eq. (6.8). Additionally, in Eq. (6.7) the maximum value of the exponent is s ¼ ðr 1Þ. As an initial example, we write the first derivative (r ¼ 1), or angular dispersion ∇1n ϕ2;m ¼ ∇0n H2;m þ ðMÞ 1 ð∇n þ ζÞ0 ∇1n ϕ2;m ¼ ∇0n H2;m þ ðMÞ 1 1 ∇n ϕ2;m ¼ H2;m þ ðMÞ 1 ðH1;m 7∇n ϕ2;ðm 1Þ Þ thus correctly reproducing Eq. (6.3). As a further example, we provide the sixth derivative, for which we set r ¼ 6 in Eq. (6.6) ∇6n ϕ2;m ¼ ∇5n H2;m þ ðMÞ 1 ð∇n þ ζÞ5 ∇6n ϕ2;m ¼ ∇5n H2;m þ ðMÞ 1 ð∇5n ζ 0 þ 5∇4n ζ 1 þ 10∇3n ζ 2 þ 10∇2n ζ 3 þ 5∇1n ζ 4 þ ∇0n ζ 5 Þ ∇6n ϕ2;m ¼ ∇5n H2;m þ ð∇5n M 1 ÞðH1;m 7∇n ϕ2;ðm 1Þ Þ þ 5ð∇4n M 1 Þð∇n H1;m 7∇2n ϕ2;ðm 1Þ Þ þ10ð∇3n M 1 Þð∇2n H1;m 7∇3n ϕ2;ðm 1Þ Þ þ 10ð∇2n M 1 Þð∇3n H1;m 7∇4n ϕ2;ðm 1Þ Þ
þ5ð∇n M 1 Þð∇4n H1;m 7∇5n ϕ2;ðm 1Þ Þ þ ðM 1 Þð∇5n H1;m 7∇6n ϕ2;ðm 1Þ Þ
ð6:9Þ
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Fig. 14. Double-prism pulse compressor.
For the seventh derivative, for which r ¼ 7 in Eq. (6.6), the equation becomes ∇7n ϕ2;m ¼ ∇6n H2;m þ ð∇6n M 1 ÞðH1;m 7∇n ϕ2;ðm 1Þ Þ þ 6ð∇5n M 1 Þð∇n H1;m 7∇2n ϕ2;ðm 1Þ Þ þ15ð∇4n M 1 Þð∇2n H1;m 7∇3n ϕ2;ðm 1Þ Þ þ 20ð∇3n M 1 Þð∇3n H1;m 7∇4n ϕ2;ðm 1Þ Þ þ15ð∇2n M 1 Þð∇4n H1;m 7∇5n ϕ2;ðm 1Þ Þ þ 6ð∇n M 1 Þð∇5n H1;m 7∇6n ϕ2;ðm 1Þ Þ
þðM 1 Þð∇6n H1;m 7∇7n ϕ2;ðm 1Þ Þ
ð6:10Þ
Thus, we have found a systematic, consistent, and even elegant approach to the expression of the higher-order phase derivatives, which are applicable to generalized prismatic pulse compression. 6.1. Applications to laser pulse compression and coherent microscopy The first and most direct application of the generalized multiple-prism dispersion theory [35,49] in the field of laser pulse compression, is the exact numerical evaluation of dispersion for multiple-prism pulse compressors [51]. In this area, the research of Osvay et al. [52,53] is particularly interesting because it demonstrates close agreement between calculated and measured dispersions for practical double-prism pulse compressors, illustrated in Fig. 14, used to generate femtosecond laser pulses with a duration of 18 fs [53]. In their work, these authors compared their measurements with the values calculated from Eqs. (6.3) and (6.5). In addition to direct applications to laser pulse compression, the generalized multiple-prism dispersion equations described here have been applied in the area of coherent microscopy. In this research, femtosecond pulsed lasers incorporating multiple-prism pulse compressors are used to generate three dimensional micrographs of GaAs/AlGaAs quantum wells [54] and V-groove quantum wires [55]. Further quantum optical applications of prismatic femtosecond lasers are reviewed by Diels and Rudolph [12]. 7. The quantum interferometric connection Dirac first applied quantum ideas to macroscopic optics [56] by connecting the concepts of states and probability amplitudes with light beams and two-beam macroscopic interference. In this regard, Dirac should be considered to be the father of quantum optics [38]. The second early example of the application of quantum concepts to macroscopic classical optics was by Feynman who used his quantum path integral formalism to describe the divergence of a Gaussian beam [57]. In Dirac's notation [56], the N-slit interferometric probability amplitude for single-photon propagation from a source (s) through the N-slit array (j) to the interferometric plane (x) is given
F.J. Duarte / Progress in Quantum Electronics 37 (2013) 326–347
341
by [58,59] N
〈xjs〉 ¼ ∑ 〈xjj〉〈jjs〉
ð7:1Þ
j¼1
which leads to the interferometric equation [58,59] N
N
j¼1
m¼1
j〈xjs〉j2 ¼ ∑ Ψ ðr j Þ ∑ Ψ ðr m ÞeiðΩm Ωj Þ or, in its alternative form, N
N
j〈xjs〉j ¼ ∑ Ψ ðr j Þ þ 2 ∑ Ψ ðr j Þ 2
j¼1
2
j¼1
N
∑ Ψ ðr m Þ cos ðΩm Ωj Þ
m ¼ jþ1
ð7:2Þ ! ð7:3Þ
where Ψ(rj ) are the wave function amplitudes of the ordinary wave optics as defined by Dirac [56], and the term in parenthesis describes the phase, which includes an exact description of the geometry of the N-slit interferometer [58,59]. Although originally derived for single-photon propagation, this equation has been found through a large number of experiments and measurements to accurately describe the interferometric propagation of ensembles of indistinguishable photons from narrow-linewidth lasers [59–62]. This point is illustrated, for N¼ 3, by the measured interferograms and the calculated interferometric distribution shown in Figs. 15–17. First, Fig. 15 shows a control interferogram measured at an intra interferometric distance of D〈xjj〉 ¼ 7:235 m. Then, Fig. 16 shows a measured interferogram at the same intra interferometric distance with the addition of the diffractive effect caused by a spider web silk fiber 15 cm from the interferometric plane. Next, Fig. 17 shows the interferometric distribution calculated from Eq. (7.3) and highlights the superimposed diffractive signal caused by the soft intersection of the silk fiber. The agreement between theory and experiment is remarkable [62]. Here, it should be indicated that the interferometric calculation is performed through a cascade approach, first introduced in [59]; this approach the silk fiber of the spider web to be introduced as two wide slits separated by the diameter of the fiber, which is approximately 30 μm [62].
Fig. 15. Measured control interferogram for N ¼ 3 at an intra-interferometric distance of D〈xjj〉 ¼ 7:235 m and a wavelength of λ ¼ 632:8 nm. The slits are 570 μm wide and are separated by 570 μm, from Duarte et al. [62].
342
F.J. Duarte / Progress in Quantum Electronics 37 (2013) 326–347
Fig. 16. Measured interferogram for N ¼ 3 at an intra-interferometric distance of D〈xjj〉 ¼ 7:235 m and a wavelength of λ ¼ 632:8 nm. In this case, however, a semi-transparent spider web silk fiber, approximately 30 μm in diameter, is inserted at an intra-interferometric distance of D〈xjj〉 ¼ 7:085 m (that is, 15 cm from the interference plane) to induce soft probing of the given interferometric character. The slits are 570 μm wide and are separated by 570 μm, from Duarte et al. [62].
Fig. 17. Calculated interferogram, using Eq. (7.2) for N ¼ 3 at an intra-interferometric distance of D〈xjj〉 ¼ 7:235 m and a wavelength of λ ¼ 632:8 nm. The presence of the spider web silk fiber, at a distance of D〈xjj〉 ¼ 7:085 m, is incorporated into the calculation through an interferometric cascade approach [59]. The slits are 570 μm wide and are separated by 570 μm, from Duarte et al. [62].
Thus, the interferogram produced by the interaction of the incident expanded laser beam with the N-slit array propagates for a distance of D〈xjj〉 ¼ 7:085 m prior to illumination of the two wide slits. The interaction with the second diffractive object produced the beautiful interferogram with the superimposed diffraction signal illustrated in Figs. 16 and 17. Here, we should mention that the application of quantum principles in the description of macroscopic optical phenomena is perfectly within the framework of van Kampen's quantum theorems [63].
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In addition to the ability to accurately predict, or reproduce, experimental results, Eq. (7.3) allows the derivation of the generalized diffraction grating equation [64,65] mλ ¼ dð7 sin Θ7 sin Θ′Þ
ð7:4Þ
and the generalized refraction equation for both positive and negative refraction [64,65] 7n1 sin Θ7n2 sin Φ ¼ 0
ð7:5Þ
This generalized refraction equation leads to an even more generalized form of the multipleprism dispersion equation, that is [65], ∇λ ϕ2;m ¼ 7H2;m ∇λ nm 7ðk 1;m k2;m Þ 1 ðH1;m ∇λ nm ð7Þ∇λ ϕ2;ðm 1Þ Þ
ð7:6Þ
where the free 7 sign refers to positive or negative refraction, and the alternative in parenthesis ð7Þ to either a positive ðþÞ or a compensating ð Þ configuration [64]. Furthermore, the N-slit interferometric equation has been used to derive the cavity linewidth equation [66] 1 ∂Θ Δλ Δθ ∂λ which has been shown to be a crucial concept in the development of tunable narrow-linewidth and tunable single-longitudinal-mode laser oscillators. To complete the story, it should be mentioned that the N-slit interferometric equation can also be used to derive the interferometric identity [38] Δλ
λ2 Δx
ð7:7Þ
which in conjunction with de Broglie's expression p ¼ ℏk, leads to Heisenberg's uncertainty principle [38,56] ΔpΔx h
ð7:8Þ
From this form of the uncertainty principle, it can be shown that ΔνΔt 1 follows. As described previously, this expression is central to our discussion of the interaction between intracavity dispersion, emission linewidth, and laser pulse duration. 7.1. Interference and quantum entanglement The generalized interferometric equation, Eq. (7.3), has been used to generate theoretical interferograms, called interferometric characters, for secure free-space optical communications [61,62,68]. In this application, interferometric characters are generated experimentally and are compared with their theoretical counterparts to determine the fidelity of the transmission. Any attempt to classically intercept the interferometric characters results in their catastrophic collapse [61,62,68] and, as discussed previously, even the interception of the interferometric character with subtle means, such as microscopic spider web silk fibers, can be detected. Ultimately, there is a nexus, at a fundamental notational level, between the approach based on the generalized interferometric equation and the more widely used quantum entanglement
344
F.J. Duarte / Progress in Quantum Electronics 37 (2013) 326–347
probability amplitude. Returning to the fundamental Dirac principle described by Eq. (7.1), N
〈xjs〉 ¼ ∑ 〈xjj〉〈jjs〉 j¼1
Duarte [69] has recently shown that, for N ¼ 2, this probability amplitude is directly applicable to an interferometric description of quantum entanglement. In this regard, replacing x with the notation for a detector d leads to 〈djs〉 ¼ 〈dj2〉〈2js〉 þ 〈dj1〉〈1js〉
ð7:9Þ
which can then be abstracted to js〉 ¼ j2〉〈2js〉 þ j1〉〈1js〉
ð7:10Þ
Experimentally, this equation can be applied to the description of two indistinguishable quanta that originate at a common source and travel in different directions while passing through different polarizers 1 and 2 [69]. Now, through the Dirac identity jϕ〉 ¼ jj〉〈jjϕ〉, Eq. (7.10) can be expressed as js〉 ¼ jB〉 þ jA〉
ð7:11Þ
where jB〉 ¼ j2〉〈2js〉 and jA〉 ¼ j1〉〈1js〉. Applying the Dirac identity for several particles [56] ja1 b2 c3 :::gn 〉 ¼ ja1 〉jb2 〉jc3 〉:::jgn 〉
ð7:12Þ
which can also be expressed as jX〉 ¼ ja〉1 jb〉2 jc〉3 :::jg〉n
ð7:13Þ
leads directly to jB〉 ¼ jx〉1 jy〉2
ð7:14Þ
jA〉 ¼ jy〉1 jx〉2
ð7:15Þ
and since, from Eq. (7.10), we know that jB〉 a jA〉. The substitution of Eqs. (7.14) and (7.15) into Eq. (7.11) leads to the probability amplitude js〉 ¼ jx〉1 jy〉2 þ jy〉1 jx〉2
ð7:16Þ
where x and y refer to different polarizations and the subscripts refer to quanta 1 and 2. The probability amplitudes of entangled states in this form were first discovered by Pryce and Ward [70] and Ward [71] in 1947–1949. Use of the usual normalization and replacement of js〉 by jψ〉 leads directly to a ubiquitous probability amplitude for entangled polarizations 1 jψ〉 ¼ pffiffiffiðjx〉1 jy〉2 7jy〉1 jx〉2 Þ 2
ð7:17Þ
where the 7 allows for different linear combinations. Use of the Dirac identity expressed in Eq. (7.13) leads to the more general Pryce-Ward probability amplitude 1 jψ〉 ¼ pffiffiffiðja〉1 jb〉2 jc〉3 :::jg〉n 7ja′〉1 jb′〉2 jc′〉3 :::jg′〉n Þ 2
ð7:18Þ
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8. Conclusions In this paper, we have highlighted various dispersive techniques crucial to the successful development of narrow-linewidth tunable organic dye lasers. The role of those dispersive techniques in tunable gas lasers, tunable solid state lasers, and tunable diode lasers has been indicated. Particular attention was paid to multiple-prism arrays and the generalized multiple-prism dispersion equation. That equation has been applied here, for the first time, to calculate the dispersion associated with direct-vision prisms, also known as Amici prisms, and compound prisms. This calculation was motivated by the use of these prisms in astronomical instrumentation. Following these dispersion calculations, a practical dispersionless multiple-prism beam expander was designed using the same equations, and the versatility of the generalized multiple-prism dispersion theory was thus demonstrated. Next, attention was focused on higher-order dispersions as needed and applied in prismatic pulse compressors for femtosecond lasers. In this regard, recent results obtained by Duarte [40,50] have been extended to obtain a succinct and elegant general equation for the generalized multiple-prism dispersion; this equation is valid up to the rth derivative ∇rn ϕ2;m ¼ ∇rn 1 H2;m þ ðMÞ 1 ð∇n þ ζÞr 1 and explicit versions of this equation are included for the higher derivatives r ¼ 6; 7. The discussion in this paper centers on the cavity linewidth equation 1 ∂Θ Δλ Δθ ∂λ which can be derived from the quantum interferometric equation [38,66] N
N
j〈xjs〉j ¼ ∑ Ψ ðr j Þ þ 2 ∑ Ψ ðr j Þ 2
j¼1
2
j¼1
N
!
∑ Ψ ðr m Þ cos ðΩm Ωj Þ
m ¼ jþ1
This discussion reinforces the interaction between classical optics and quantum optics; although highlighted by Dirac [56] and Feynman [57], this interaction is not always appreciated in today's literature, with the exception of notable contributions, such as those of Lamb [67]. It has been also shown that the generalized N-slit interferometric equation and the probability amplitude for quantum entanglement have a common origin.
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[61] F.J. Duarte, Interferometric imaging, in: F.J. Duarte (Ed.), Tunable Laser Applications, CRC, New York, 2009 (p. 341). [62] F.J. Duarte, T.S. Taylor, A.M. Black, I.E. Olivares, Journal of Modern Optics 60 (2013) 136. [63] N.G. van Kampen, Physica A 153 (1988) 97. [64] F.J. Duarte, American Journal of Physics 65 (1997) 637. [65] F.J. Duarte, Applied Physics B 82 (2006) 35. [66] F.J. Duarte, Applied Optics 31 (1992) 6979. [67] W.E. Lamb, Nuclear Physics B 6 (1989) 197. [68] F.J. Duarte, Optics Communications 205 (2002) 313. [69] F.J. Duarte, Quantum Optics for Engineers, CRC, New York, 2013. [70] M.L. Pryce, J.C. Ward, Nature 160 (1947) 435. [71] J.C. Ward, Some Properties of the Elementary Particles, Oxford University, Oxford, 1949.
Review
Tunable laser optics: Applications to optics and quantum optics F.J. Duartea,b,n a Interferometric Optics, Rochester, New York, USA University of New Mexico, Albuquerque, New Mexico, USA
b
Available online 9 October 2013
Abstract Optics originally developed for tunable organic dye lasers have found applications in other areas of optics, laser optics, and quantum optics. Here, the salient aspects of the physics related to the cavity linewidth equation and the effects of intracavity beam expansion and intracavity dispersion on this equation are reviewed. Additionally, the generalized multiple-prism dispersion equation is applied to direct-vision prisms, also known as Amici prisms, to calculate dispersion configurations of practical interest. Then, the higher derivatives of the multiple-prism dispersion equation applicable to laser pulse compression are considered. From this perspective, a new compact and generalized equation for higher-order phase derivatives is introduced for the first time. Furthermore, it is shown how the N-slit interferometric equation, derived from quantum principles using Dirac's notation, gives rise to generalized versions of the diffraction grating equation and the law of refraction. The nexus between the N-slit interferometric equation and the cavity linewidth equation is also illustrated. Finally, various optical and quantum optical applications that have benefited from these developments are highlighted. & 2013 Elsevier Ltd. All rights reserved. Keywords: Amici prisms; Generalized multiple-prism dispersion; Prismatic pulse compression; N-slit interferometric equation; Organic lasers; Quantum entanglement
Contents 1. 2. 3.
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Laser wavelength tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Expanded beam illumination of diffraction gratings in Littrow configuration . . . . . . . . . . . . . 327
n
Correspondence address: Interferometric Optics, Rochester, New York, USA. E-mail address: [email protected]
0079-6727/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.pquantelec.2013.09.001
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327
4.
Feedback from gratings in grazing-incidence configuration . . . . . . . . . . . . . . . . . . . . . . . . . 329 4.1. Applications to tunable diode lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 5. Multiple-prism beam expansion and generalized multiple-prism dispersion equations . . . . . . . 330 5.1. Direct-vision, compound, and Amici prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 5.2. Applications to tunable lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 6. Generalized higher-order multiple-prism dispersion equations for laser pulse compression . . . . 338 6.1. Applications to laser pulse compression and coherent microscopy . . . . . . . . . . . . . . . 340 7. The quantum interferometric connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 7.1. Interference and quantum entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 8. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
1. Introduction Soon after the discovery of the organic dye laser in 1966 by Sorokin and Lankard [1] and Schäfer et al. [2], the tunability of this laser source inspired several developments in optics that have found widespread applications in the field of lasers and in optics in general. The development of these techniques and their link to major subsequent developments in physics and science is reviewed here. Although the text only refers to the interaction of these contributions with specific optics and laser developments, it should be kept in mind that the narrow-linewidth tunable lasers and the ultrashort pulse lasers that adopted the dispersive techniques described here have had major impacts in a plethora of important fields including astronomy [3,4], laser isotope separation [4–6], medicine [7,8], photochemistry [9], spectroscopy [10,11], and ultrashort pulse phenomena [12]. This paper is dedicated to the memory of F.P. Schäfer (1931–2011), one of the discoverers of the organic dye laser, who displayed a keen interest in the dispersive optical techniques applied to tunable organic dye lasers [13]. 2. Laser wavelength tuning The first broadband laser wavelength tuning was performed by Soffer and McFarland [14] in 1967. They used a grating in the Littrow configuration to demonstrate the tuning of a Rhodamine 6G laser. The tuning range was approximately 555 r λr 590 nm with a reported linewidth of Δλ 4 nm. For a cavity tuned with a grating in the Littrow configuration the emission wavelength is given by mλ ¼ 2d sin Θ
ð2:1Þ
where λ is the laser emission wavelength, m is the order of diffraction, and d is the groove density of the diffraction grating in lines/mm. This equation is a special case of the more general diffraction grating equation discussed later. Fig. 1 describes a telescopic tunable dye laser cavity incorporating a Littrow grating as the tuning element. 3. Expanded beam illumination of diffraction gratings in Littrow configuration Following the introduction of the dye laser [1,2] and the demonstration of its tunability by Soffer and McFarland [14], a powerful class of tunable laser sources in the visible was made available to the scientific community. This momentous discovery occurred in 1972 when Hänsch
328
F.J. Duarte / Progress in Quantum Electronics 37 (2013) 326–347
Fig. 1. Littrow cavity configuration incorporating two-dimensional intracavity telescopic beam expansion.
[15] realized that expansion of the intracavity beam illuminating the tuning diffraction grating yielded an enormous reduction in the laser emission linewidth. Hänsch utilized a two-dimensional astronomical telescope comprised of two convex lenses (see Fig. 1). What Hänsch discovered can be succinctly explained by the cavity linewidth equation for a single-pass laser 1 ∂Θ Δλ Δθ ð3:1Þ ∂λ where the beam divergence Δθ can be represented by the diffraction-limited relation Δθ
λ πw
ð3:2Þ
Here, λ is the laser emission wavelength, and w is the beam waist at the gain region. As explained later, in Section 5, the reduction in the emission linewidth is a consequence of the large increase in intracavity dispersion, this increase results from the illumination of the diffraction grating by the expanded beam. However, Hänsch interpreted the effect in an equivalent mathematical form corresponding to a reduction in the intracavity beam divergence [15] Δθ
λ πMw
ð3:3Þ
Because the magnification factor M can be rather large, the subsequent reduction in the laser linewidth can be rather dramatic. In fact, Hänsch was able to measure linewidths of Δν 2:5 GHz at λ 600 nm, or Δλ 0:003 nm. Further reductions were achieved through the introduction of an intracavity etalon [15]. In Eq. (3.1), the angular dispersion of the grating, deployed in Littrow configuration is given by [15] ∂Θ 2 tan Θ ð3:4Þ ¼ ∂λ λ where Θ is the angle of incidence, which equals the angle of diffraction for the Littrow-mounted grating. In addition to the technical accomplishment of high-power tunable narrow-linewidth emission, the significance of Hänsch's contribution was the realization that intracavity beam expansion, or
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329
illumination of the whole diffraction grating, led to a gigantic reduction in the laser emission linewidth. At this stage, it should be mentioned that both the expressions applied by Hänsch, the expression for the single-pass linewidth (Eq. (3.1)) and the diffraction-limited beam divergence (Eq. (3.2)), can be found in classical optics books. For instance, Eq. (3.1) can be referenced to Robertson in 1955 [16], and Eq. (3.2) can be traced to Jenkins and White in 1957 [17]. However, as will be shown later, these expressions can also be derived from the principles of quantum optics. 4. Feedback from gratings in grazing-incidence configuration The next important development in the optics of tunable lasers was the introduction of grazing-incidence grating configurations by Shoshan et al. [18] and Littman and Metcalf [19]. The physics of linewidth narrowing remains essentially the same as the narrowing considered by Hänsch except that instead of using an expanded beam to illuminate the entire diffractive surface of the grating, these authors utilized the divergent emission to illuminate a diffraction grating deployed at a grazing-incidence angle, as illustrated in Fig. 2 for an open cavity configuration and in Fig. 3 for a closed cavity configuration. Thus, the linewidth equation becomes 1 ∂Θ Δλ Δθ ð4:1Þ ∂λ GI where Δθ is the beam divergence given by Eq. (3.2), and the angular dispersion for the diffraction grating in the grazing-incidence configuration is given by ∂Θ ð sin Θ þ sin Θ′Þ ð4:2Þ ¼ ∂λ GI λ cos Θ or
∂Θ m ¼ ∂λ GI d cos Θ
Fig. 2. Grazing-incidence grating oscillator in an open cavity configuration.
ð4:3Þ
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Fig. 3. Grazing-incidence grating oscillator in a closed cavity configuration.
where λ is the laser emission wavelength, m is the order of diffraction, and d is the groove density of the diffraction grating in lines/mm. Certainly, these dispersion expressions are derived from the diffraction grating equation [19] mλ ¼ dð sin Θ7 sin Θ′Þ
ð4:4Þ
Again, some observations are in order. First, the dispersion equations applied here to grazingincidence grating cavities are often multiplied by a factor of 2 to indicate a double-pass configuration [18]. Additionally, as for beam divergence, the diffraction grating equation is adopted from classical literature. In this case, we can trace Eq. (4.4) all the way back to Michelson's 1927 book entitled Studies in Optics [20]. It will also be shown later that this expression can also be derived from quantum optics principles. 4.1. Applications to tunable diode lasers Grazing-incidence grating configurations, developed for the organic dye laser [18,19], have been extensively applied to other types of lasers, including gas lasers [21,22] and solid-state tunable lasers [23,24]. However, these configurations have had their greatest influence through their widespread application to tunable semiconductor lasers, in which straightforward open cavity designs have had a major impact on the development of the tunable diode lasers used in laser cooling experiments [25,26]. 5. Multiple-prism beam expansion and generalized multiple-prism dispersion equations Prismatic beam expansion and multiple-prism arrays were first disclosed by Newton in his timeless book Opticks [27], in 1704 (see Figs. 4 and 5). Although these ideas would be used sporadically following their introduction [28], it was the tunable organic dye laser that revived their use some 270 years later. Of particular interest here was the use of a single-prism beam expander by Hanna et al. [29] and a study on double-prism expanders by Kasuya et al. [30]. The success of intracavity multiple-prism beam expansion configurations [31,32], depicted here in tunable organic solid-state dye laser oscillators [33,34] in Figs. 6 and 7, inspired Duarte and Piper to introduce the generalized multiple-prism dispersion theory [35]. For generalized
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Fig. 4. Newton's single-prism diagram depicting expansion of a light beam from Newton's 1704 book Opticks [27].
Fig. 5. Newton's prism array from Newton's 1704 book Opticks [27].
multiple prism arrays of any geometry and material, see Fig. 8, the dispersion can be succinctly expressed as [35] ∇λ ϕ2;m ¼ H2;m ∇λ nm þ ðk 1;m k 2;m Þ 1 ðH1;m ∇λ nm 7∇λ ϕ2;ðm 1Þ Þ
ð5:1Þ
where ∇λ ¼
∂ ∂λ
ð5:2Þ
k1;m ¼
cos ψ 1;m cos ϕ1;m
ð5:3Þ
k2;m ¼
cos ϕ2;m cos ψ 2;m
ð5:4Þ
H1;m ¼
tan ϕ1;m nm
ð5:5Þ
H2;m ¼
tan ϕ2;m nm
ð5:6Þ
In these equations, for the mth prism, nm is the refractive index, ϕ1;m is the incidence angle, and
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Fig. 6. Hybrid multiple-prism pre-expanded near grazing-incidence (HMPGI) grating solid state dye laser oscillator, from Duarte [33].
Fig. 7. Multiple-prism Littrow (MPL) grating solid-state dye laser oscillator, from Duarte [34].
ψ 1;m is the corresponding refraction angle. Similarly, ϕ2;m is the emergence angle, and ψ 2;m is the corresponding refraction angle. Additionally, k1;m and k 2;m are the beam expansions at the incidence and the exit surfaces, respectively. Furthermore, H1;m and H2;m are geometrical identities related to ϕ1;m and ϕ2;m , respectively. Physically, the generalized single-pass multiple-prism dispersion equation tells us that the cumulative dispersion, at the mth prism (∇λ ϕ2;m ), depends on the geometry of this particular prism, the angle of incidence at this prism, the material of this prism, and the cumulative dispersion up to the previous prism, ∇λ ϕ2;ðm 1Þ [35,36]. The dispersion expression given in Eq. (5.1) applies to a single pass and is technically known as the single-pass generalized multiple-prism dispersion equation. To describe the physics of linewidth narrowing in a pulsed high-gain tunable laser oscillator, it is necessary to consider the double-pass multiple-prism dispersion. The double pass is constructed by unfolding the single pass through the multiple-prism grating, or multiple-prism mirror, assembly used to induce linewidth narrowing in the frequency-selective laser cavity as illustrated in Fig. 9. Thus, as
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Fig. 8. (a) Generalized multiple-prism array in a positive configuration, and (b) in a compensating configuration. Generalized prismatic configurations of this class were first introduced by Duarte and Piper [36].
explained by Duarte and Piper [35] the double-pass angular multiple-prism dispersion becomes ∇λ ϕ01;m ¼ H01;m ∇λ nm þ ðk 1;m0 k2;m0 Þ 1 ðH02;m ∇λ nm 7∇λ ϕ01;ðmþ1Þ Þ
ð5:7Þ
where k′1;m ¼
cos ψ′1;m cos ϕ′1;m
ð5:8Þ
k′2;m ¼
cos ϕ′2;m cos ψ′2;m
ð5:9Þ
H′1;m ¼
tan ′ϕ1;m nm
ð5:10Þ
H2;m0 ¼
tan ϕ2;m0 nm
ð5:11Þ
An important term in Eq. (5.7) is ∇λ ϕ′1;ðmþ1Þ , which links the angular dispersion of the doublepass, or single return-pass, to the last stage of the single-pass angular dispersion and includes the diffraction grating dispersion so that [37] ∇λ ϕ′1;ðmþ1Þ ¼ ð∇λ Θ7∇λ ϕ2;r Þ
ð5:12Þ
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Fig. 9. Unfolded multiple-prism grating configuration utilized to derive the double-pass, or single return-pass, multipleprism dispersion, adapted from Duarte and Piper [35].
If the diffraction grating is replaced by a mirror, ∇λ Θ ¼ 0, this equation leads directly to ∇λ ϕ′1;ðmþ1Þ ¼ ∇λ ϕ2;r
ð5:13Þ
In these equations, ∇λ ϕ2;r is the overall cumulative single-pass angular dispersion calculated from Eq. (5.1) at the last prism of a prism array composed of r prisms. More specifically, for an array of r prisms, at the last prism ∇λ ϕ2;r ¼ ∇λ ϕ2;m
ð5:14Þ
Subsequently, Duarte and Piper [37] showed that the multiple return-pass cavity linewidth equation becomes Δλ ΔθR ðMR∇λ Θ þ R∇λ ϕ′1;m Þ 1
ð5:15Þ
where ΔθR is the multi-pass beam divergence, which approaches the diffraction limit in a welldesigned cavity [38], Ris a finite number of intracavity return passes [37], and M is the overall intracavity beam expansion defined by [38,39] r
M 1 ¼ ∏ k1;m
ð5:16Þ
m¼1 r
M 2 ¼ ∏ k2;m
ð5:17Þ
m¼1
r
M ¼ M 1 M 2 ¼ ∏ k 1;m k2;m m¼1
ð5:18Þ
Here, we should indicate that in well-designed multiple-prism beam expanders with orthogonal beam exit angles, all the expansion is provided by M 1 , because M 2 1. Additionally, as highlighted recently [40], the relationship between single-pass and return-pass angular multiple-prism angular dispersion can
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be expressed explicitly as ∇λ ϕ2;m ¼ ð2MÞ 1 ð∇λ ϕ′1;m Þ
ð5:19Þ
Thus, the return-pass cavity linewidth equation, which is Eq. (5.15), can also be expressed as Δλ ΔθR ðMR∇λ Θ þ 2MR∇λ ϕ2;r Þ 1
ð5:20Þ
After removal of the MR factor out, Eq. (5.20) can also be written as ΔθR ð∇λ Θ þ 2∇λ ϕ2;r Þ 1 ð5:21Þ MR For R ¼ 1 and for a dispersionless telescopic beam expander, that is, for ∇λ ϕ2;r ¼ 0, Eq. (5.21) reduces to Δλ
Δθ ð∇λ ΘÞ 1 ð5:22Þ M and this result is consistent with the equations utilized by Hänsch. The primary difference is that the beam-expansion factor M is now systematically and consistently introduced through the dispersion theory and augments the intracavity dispersion rather than being introduced in an ad-hoc manner as a factor that reduces the beam divergence as for the original telescopic design [15]. Δλ
5.1. Direct-vision, compound, and Amici prisms Some applications require direct-vision prisms [17] in which the beam of light does not deviate after refraction by a single prism. These prisms are essentially multiple-prism configurations in which the prisms are deployed right next to each other. These multiple-prism configurations are known as direct-vision prisms [17], compound prisms, or Amici prisms. These prisms are important in corrections of atmospheric dispersion in adaptive optics systems for astronomical applications [41]. Here, the single-pass generalized multiple-prism dispersion equation, Eq. (5.1), is applied to illustrate the dispersive properties of these prismatic configurations. The discussion begins by introducing a fused silica prism (n1 ¼ 1:45838 at λ ¼ 590 nm) with the following angular dimensions α1 ¼ 451, ϕ1;1 ¼ 20:661, ψ 1;1 ¼ 14:001, ψ 2;1 ¼ 31:001, ϕ2;1 ¼ 48:681, which are
Fig. 10. Single fused silica prism with a negative dispersion of ð∂ϕ2;1 =∂λÞ 0:0384 at λ ¼ 590 nm. Drawing is approximately to scale.
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illustrated approximately to scale in Fig. 10. From Eq. (5.1), for m ¼ 1 and ðdn1 =dλÞ 0:0348 μm 1, the dispersion is ∂ϕ2;1 dn1 1:1036 0:0384 ð5:23Þ ∂λ dλ in units of μm 1. To prevent deviation in the beam path, a second prism is introduced in compensating configuration and is adjoined to the first prism to create a compound prism, or Amici prism, as illustrated in Fig. 11. The second prism is made of a higher refractive index material, in this case Schott SF10 (n2 ¼ 1:73020 at λ ¼ 590 nm) with an apex angle α2 ¼ 25:731. Here, it is noted that ϕ2;1 ¼ ϕ1;2 ¼ 48:681, ψ 1;2 ¼ 25:731, and ψ 2;2 ¼ ϕ2;2 ¼ 0. From Eq. (5.1), with m ¼ 2 and ðdn2 =dλÞ 0:1367 μm 1, the combined dispersion becomes ∂ϕ2;2 dn2 dn1 0:4818 0:8089 0:0377 ð5:24Þ ∂λ dλ dλ
Fig. 11. Amici prism configuration with two prisms in compensating configuration. The first prism is made of fused silica and the second prism is made of the higher refractive index material Schott SF10 (see text). This configuration yields negative dispersion of ð∂ϕ2;2 =∂λÞ 0:0377 at λ ¼ 590 nm. Drawing is approximately to scale.
Fig. 12. Amici prism, or direct-vision, configuration with three prisms in compensating configuration. The first and last prisms are identical and are made of fused silica whereas the center prism is made of the higher refractive index material Schott SF10 (see text). This configuration yields positive dispersion of ð∂ϕ2;3 =∂λÞ þ 0:0821 at λ ¼ 590 nm. Drawing is approximately to scale.
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in units of μm 1. Although the propagation of the beam is now straight, the exit beam is slightly displaced relative to its position at the incidence surface. To achieve collinear transmission, the compound prism in Fig. 11 is unfolded to create a three-prism direct-vision configuration [17], or a three-prism Amici configuration, as illustrated in Fig. 12. For this particular configuration: α1 ¼ 451, ϕ1;1 ¼ 20:661, ψ 1;1 ¼ 14:001, ψ 2;1 ¼ 31:001, α2 ¼ 51:461, ψ 1;2 ¼ ψ 2;2 ¼ 25:731, α3 ¼ 451, ψ 1;3 ¼ 31:001, ψ 2;3 ¼ 14:001, and ϕ2;3 ¼ 20:661. For Eq. (5.1) and m ¼ 3, the estimated combined dispersion is ∂ϕ2;3 dn1 dn2 1:7631 1:0503 þ 0:0821 ð5:25Þ ∂λ dλ dλ The change in the sign of the dispersion from Eq. (5.23), applicable to the single prism, to Eq. (5.25), applicable to the three-prism compensating configuration, illustrates that the usual dispersion wavelength order (from red to blue), is inverted (from blue to red) and is consistent with the qualitative description given by Jenkins and White [17]. Thus, the single-pass generalized multiple-prism dispersion equation has been shown to be naturally applicable to the dispersion of direct-vision and Amici prismatic configurations. Furthermore, it also been shown quantitatively that although these compound prisms do provide an exit beam collinear with the incident beam, these compound prisms are also quite dispersive; hence, these prisms are applied to the correction of atmospheric dispersion in astronomical optics. Next, a dispersionless multiple-prism array is described. Given the geometrical flexibility of multiple-prism beam expanders, a three-prism dispersionless configuration can be designed easily. From Eq. (5.7), for m ¼ 3 and for an orthogonal beam exit, ψ 2;1 ¼ ϕ2;1 ¼ 0, ψ 2;2 ¼ ϕ2;2 ¼ 0, ψ 2;3 ¼ ϕ2;3 ¼ 0; for a dispersionless configuration, the following condition must be satisfied [38,39] ðk 1;1 þ k 1;1 k 1;2 Þ tan ψ 1;1 ¼ ðk 1;1 k1;2 Þk 1;3 tan ψ 1;3
ð5:26Þ
For fused silica (n ¼ 1:45838 at λ ¼ 590 nm), the design of a dispersionless multiple-prism beam expander providing an overall beam magnification of M 42 is illustrated. In this illustration, the following expansions are selected: k1;1 ¼ k1;2 ¼ 5:2749 which implies that ϕ1;1 ¼ ϕ1;2 ¼ 821 and ψ 1;1 ¼ ψ 1;2 ¼ 42:76701. From Eq. (5.26) ðk 1;1 þ k 1;1 k 1;2 Þ tan ψ 1;1 tan ϕ1;3 ¼ k 1;1 k1;2 n
ð5:27Þ
Fig. 13. Quasi-achromatic fused silica multiple-prism beam expander yields zero dispersion and a beam expansion of M 42:7828 at λ ¼ 590 nm. Drawing is approximately to scale.
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and it can be directly deduced that ϕ1;3 58:06911, ψ 1;3 35:58701, and k 1;3 1:5376, so M ¼ k1;1 k 1;2 k 1;3 42:7828
ð5:28Þ
This class of quasi-achromatic beam expander, with zero dispersion at λ ¼ 590 nm, is directly applicable to the design of narrow-linewidth tunable lasers [38]. The three-prism beam expander is displayed approximately to scale in Fig. 13. Multiple-prism arrays, in their various forms, are applicable to optical instrumentation, such as spectrometers [38], and even astronomical instrumentation [42]. 5.2. Applications to tunable lasers Multiple-prism grating configurations, as developed for tunable organic lasers [31–34] have been successfully applied to tunable gas lasers [43], optical parametric oscillators [38], and tunable diode lasers [44,45]. In particular, research by Zorabedian [44] on tunable semiconductor lasers demonstrated the advantages of utilizing multiple-prism grating configurations that enable a closed cavity configuration, in which the laser output is coupled from the output coupler mirror rather than from the reflection losses of the diffraction grating. These advantages include reduced output noise and protection from unwanted external back reflections. Here, it should be noted that one of the first grating-tuned semiconductor lasers, introduced by Fleming and Mooradian [46], did use a closed cavity configuration and expanded the intracavity beam incident on a Littrow grating. 6. Generalized higher-order multiple-prism dispersion equations for laser pulse compression An additional area of optics created by developments in broadly tunable organic dye lasers was prismatic pulse compression [47] or multiple-prism pulse compression [48]. In essence, this area is a direct consequence of the pulse laser linewidth equation 1 ∂Θ Δλ Δθ ∂λ which indicates that greatly reduced intracavity dispersion leads to a very broad emission linewidth. Through a variant of Heisenberg's uncertainty principle, ΔνΔt 1
ð6:1Þ
this broad linewidth implies a very short pulse. This equation will be reconsidered in the next section. Control of the intracavity dispersion and the ability to introduce negative dispersion can enable a severe reduction of intracavity dispersion and cause very broadband emission or very short pulses, as explained by Eq. (6.1). Thus, learning to control the intracavity dispersion becomes very important. Returning to Eq. (5.1), the identity ∇n ϕ2;m ¼ ∇λ ϕ2;m ð∇λ nm Þ 1
ð6:2Þ
can be applied to restate the generalized multiple-prism dispersion as [49,50] ∇n ϕ2;m ¼ H2;m þ ðMÞ 1 ðH1;m 7∇n ϕ2;ðm 1Þ Þ
ð6:3Þ
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where the identity 1 1 k1;m k2;m ¼ ðM 1 Þ
ð6:4Þ
applies. Hence, the complete second derivative of the refraction angle, or first derivative of the dispersion ∇λ ϕ2;m , is given by [49] ∇2n ϕ2;m ¼ ∇n H2;m þ ð∇n M 1 ÞðH1;m 7∇n ϕ2;ðm 1Þ Þ þ ðM 1 Þð∇n H1;m 7∇2n ϕ2;ðm 1Þ Þ
ð6:5Þ
the third and higher derivatives are given by Duarte [40,50]. Based on the progression of the higher derivatives, a generalized expression can be found to be ∇rn ϕ2;m ¼ ∇rn 1 H2;m þ ðMÞ 1 ð∇n þ ζÞr 1
ð6:6Þ
where ζ s ¼ ∇sn H1;m 7∇sþ1 n ϕ2;ðm 1Þ
ð6:7Þ
ζ 0 ¼ 1 ¼ H1;m 7∇n ϕ2;ðm 1Þ
ð6:8Þ
Here, it is very important that when writing the expansion in r, in Eq. (6.6), the term ζ 0 ¼ 1 be included as defined in Eq. (6.8). Additionally, in Eq. (6.7) the maximum value of the exponent is s ¼ ðr 1Þ. As an initial example, we write the first derivative (r ¼ 1), or angular dispersion ∇1n ϕ2;m ¼ ∇0n H2;m þ ðMÞ 1 ð∇n þ ζÞ0 ∇1n ϕ2;m ¼ ∇0n H2;m þ ðMÞ 1 1 ∇n ϕ2;m ¼ H2;m þ ðMÞ 1 ðH1;m 7∇n ϕ2;ðm 1Þ Þ thus correctly reproducing Eq. (6.3). As a further example, we provide the sixth derivative, for which we set r ¼ 6 in Eq. (6.6) ∇6n ϕ2;m ¼ ∇5n H2;m þ ðMÞ 1 ð∇n þ ζÞ5 ∇6n ϕ2;m ¼ ∇5n H2;m þ ðMÞ 1 ð∇5n ζ 0 þ 5∇4n ζ 1 þ 10∇3n ζ 2 þ 10∇2n ζ 3 þ 5∇1n ζ 4 þ ∇0n ζ 5 Þ ∇6n ϕ2;m ¼ ∇5n H2;m þ ð∇5n M 1 ÞðH1;m 7∇n ϕ2;ðm 1Þ Þ þ 5ð∇4n M 1 Þð∇n H1;m 7∇2n ϕ2;ðm 1Þ Þ þ10ð∇3n M 1 Þð∇2n H1;m 7∇3n ϕ2;ðm 1Þ Þ þ 10ð∇2n M 1 Þð∇3n H1;m 7∇4n ϕ2;ðm 1Þ Þ
þ5ð∇n M 1 Þð∇4n H1;m 7∇5n ϕ2;ðm 1Þ Þ þ ðM 1 Þð∇5n H1;m 7∇6n ϕ2;ðm 1Þ Þ
ð6:9Þ
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Fig. 14. Double-prism pulse compressor.
For the seventh derivative, for which r ¼ 7 in Eq. (6.6), the equation becomes ∇7n ϕ2;m ¼ ∇6n H2;m þ ð∇6n M 1 ÞðH1;m 7∇n ϕ2;ðm 1Þ Þ þ 6ð∇5n M 1 Þð∇n H1;m 7∇2n ϕ2;ðm 1Þ Þ þ15ð∇4n M 1 Þð∇2n H1;m 7∇3n ϕ2;ðm 1Þ Þ þ 20ð∇3n M 1 Þð∇3n H1;m 7∇4n ϕ2;ðm 1Þ Þ þ15ð∇2n M 1 Þð∇4n H1;m 7∇5n ϕ2;ðm 1Þ Þ þ 6ð∇n M 1 Þð∇5n H1;m 7∇6n ϕ2;ðm 1Þ Þ
þðM 1 Þð∇6n H1;m 7∇7n ϕ2;ðm 1Þ Þ
ð6:10Þ
Thus, we have found a systematic, consistent, and even elegant approach to the expression of the higher-order phase derivatives, which are applicable to generalized prismatic pulse compression. 6.1. Applications to laser pulse compression and coherent microscopy The first and most direct application of the generalized multiple-prism dispersion theory [35,49] in the field of laser pulse compression, is the exact numerical evaluation of dispersion for multiple-prism pulse compressors [51]. In this area, the research of Osvay et al. [52,53] is particularly interesting because it demonstrates close agreement between calculated and measured dispersions for practical double-prism pulse compressors, illustrated in Fig. 14, used to generate femtosecond laser pulses with a duration of 18 fs [53]. In their work, these authors compared their measurements with the values calculated from Eqs. (6.3) and (6.5). In addition to direct applications to laser pulse compression, the generalized multiple-prism dispersion equations described here have been applied in the area of coherent microscopy. In this research, femtosecond pulsed lasers incorporating multiple-prism pulse compressors are used to generate three dimensional micrographs of GaAs/AlGaAs quantum wells [54] and V-groove quantum wires [55]. Further quantum optical applications of prismatic femtosecond lasers are reviewed by Diels and Rudolph [12]. 7. The quantum interferometric connection Dirac first applied quantum ideas to macroscopic optics [56] by connecting the concepts of states and probability amplitudes with light beams and two-beam macroscopic interference. In this regard, Dirac should be considered to be the father of quantum optics [38]. The second early example of the application of quantum concepts to macroscopic classical optics was by Feynman who used his quantum path integral formalism to describe the divergence of a Gaussian beam [57]. In Dirac's notation [56], the N-slit interferometric probability amplitude for single-photon propagation from a source (s) through the N-slit array (j) to the interferometric plane (x) is given
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by [58,59] N
〈xjs〉 ¼ ∑ 〈xjj〉〈jjs〉
ð7:1Þ
j¼1
which leads to the interferometric equation [58,59] N
N
j¼1
m¼1
j〈xjs〉j2 ¼ ∑ Ψ ðr j Þ ∑ Ψ ðr m ÞeiðΩm Ωj Þ or, in its alternative form, N
N
j〈xjs〉j ¼ ∑ Ψ ðr j Þ þ 2 ∑ Ψ ðr j Þ 2
j¼1
2
j¼1
N
∑ Ψ ðr m Þ cos ðΩm Ωj Þ
m ¼ jþ1
ð7:2Þ ! ð7:3Þ
where Ψ(rj ) are the wave function amplitudes of the ordinary wave optics as defined by Dirac [56], and the term in parenthesis describes the phase, which includes an exact description of the geometry of the N-slit interferometer [58,59]. Although originally derived for single-photon propagation, this equation has been found through a large number of experiments and measurements to accurately describe the interferometric propagation of ensembles of indistinguishable photons from narrow-linewidth lasers [59–62]. This point is illustrated, for N¼ 3, by the measured interferograms and the calculated interferometric distribution shown in Figs. 15–17. First, Fig. 15 shows a control interferogram measured at an intra interferometric distance of D〈xjj〉 ¼ 7:235 m. Then, Fig. 16 shows a measured interferogram at the same intra interferometric distance with the addition of the diffractive effect caused by a spider web silk fiber 15 cm from the interferometric plane. Next, Fig. 17 shows the interferometric distribution calculated from Eq. (7.3) and highlights the superimposed diffractive signal caused by the soft intersection of the silk fiber. The agreement between theory and experiment is remarkable [62]. Here, it should be indicated that the interferometric calculation is performed through a cascade approach, first introduced in [59]; this approach the silk fiber of the spider web to be introduced as two wide slits separated by the diameter of the fiber, which is approximately 30 μm [62].
Fig. 15. Measured control interferogram for N ¼ 3 at an intra-interferometric distance of D〈xjj〉 ¼ 7:235 m and a wavelength of λ ¼ 632:8 nm. The slits are 570 μm wide and are separated by 570 μm, from Duarte et al. [62].
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Fig. 16. Measured interferogram for N ¼ 3 at an intra-interferometric distance of D〈xjj〉 ¼ 7:235 m and a wavelength of λ ¼ 632:8 nm. In this case, however, a semi-transparent spider web silk fiber, approximately 30 μm in diameter, is inserted at an intra-interferometric distance of D〈xjj〉 ¼ 7:085 m (that is, 15 cm from the interference plane) to induce soft probing of the given interferometric character. The slits are 570 μm wide and are separated by 570 μm, from Duarte et al. [62].
Fig. 17. Calculated interferogram, using Eq. (7.2) for N ¼ 3 at an intra-interferometric distance of D〈xjj〉 ¼ 7:235 m and a wavelength of λ ¼ 632:8 nm. The presence of the spider web silk fiber, at a distance of D〈xjj〉 ¼ 7:085 m, is incorporated into the calculation through an interferometric cascade approach [59]. The slits are 570 μm wide and are separated by 570 μm, from Duarte et al. [62].
Thus, the interferogram produced by the interaction of the incident expanded laser beam with the N-slit array propagates for a distance of D〈xjj〉 ¼ 7:085 m prior to illumination of the two wide slits. The interaction with the second diffractive object produced the beautiful interferogram with the superimposed diffraction signal illustrated in Figs. 16 and 17. Here, we should mention that the application of quantum principles in the description of macroscopic optical phenomena is perfectly within the framework of van Kampen's quantum theorems [63].
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In addition to the ability to accurately predict, or reproduce, experimental results, Eq. (7.3) allows the derivation of the generalized diffraction grating equation [64,65] mλ ¼ dð7 sin Θ7 sin Θ′Þ
ð7:4Þ
and the generalized refraction equation for both positive and negative refraction [64,65] 7n1 sin Θ7n2 sin Φ ¼ 0
ð7:5Þ
This generalized refraction equation leads to an even more generalized form of the multipleprism dispersion equation, that is [65], ∇λ ϕ2;m ¼ 7H2;m ∇λ nm 7ðk 1;m k2;m Þ 1 ðH1;m ∇λ nm ð7Þ∇λ ϕ2;ðm 1Þ Þ
ð7:6Þ
where the free 7 sign refers to positive or negative refraction, and the alternative in parenthesis ð7Þ to either a positive ðþÞ or a compensating ð Þ configuration [64]. Furthermore, the N-slit interferometric equation has been used to derive the cavity linewidth equation [66] 1 ∂Θ Δλ Δθ ∂λ which has been shown to be a crucial concept in the development of tunable narrow-linewidth and tunable single-longitudinal-mode laser oscillators. To complete the story, it should be mentioned that the N-slit interferometric equation can also be used to derive the interferometric identity [38] Δλ
λ2 Δx
ð7:7Þ
which in conjunction with de Broglie's expression p ¼ ℏk, leads to Heisenberg's uncertainty principle [38,56] ΔpΔx h
ð7:8Þ
From this form of the uncertainty principle, it can be shown that ΔνΔt 1 follows. As described previously, this expression is central to our discussion of the interaction between intracavity dispersion, emission linewidth, and laser pulse duration. 7.1. Interference and quantum entanglement The generalized interferometric equation, Eq. (7.3), has been used to generate theoretical interferograms, called interferometric characters, for secure free-space optical communications [61,62,68]. In this application, interferometric characters are generated experimentally and are compared with their theoretical counterparts to determine the fidelity of the transmission. Any attempt to classically intercept the interferometric characters results in their catastrophic collapse [61,62,68] and, as discussed previously, even the interception of the interferometric character with subtle means, such as microscopic spider web silk fibers, can be detected. Ultimately, there is a nexus, at a fundamental notational level, between the approach based on the generalized interferometric equation and the more widely used quantum entanglement
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probability amplitude. Returning to the fundamental Dirac principle described by Eq. (7.1), N
〈xjs〉 ¼ ∑ 〈xjj〉〈jjs〉 j¼1
Duarte [69] has recently shown that, for N ¼ 2, this probability amplitude is directly applicable to an interferometric description of quantum entanglement. In this regard, replacing x with the notation for a detector d leads to 〈djs〉 ¼ 〈dj2〉〈2js〉 þ 〈dj1〉〈1js〉
ð7:9Þ
which can then be abstracted to js〉 ¼ j2〉〈2js〉 þ j1〉〈1js〉
ð7:10Þ
Experimentally, this equation can be applied to the description of two indistinguishable quanta that originate at a common source and travel in different directions while passing through different polarizers 1 and 2 [69]. Now, through the Dirac identity jϕ〉 ¼ jj〉〈jjϕ〉, Eq. (7.10) can be expressed as js〉 ¼ jB〉 þ jA〉
ð7:11Þ
where jB〉 ¼ j2〉〈2js〉 and jA〉 ¼ j1〉〈1js〉. Applying the Dirac identity for several particles [56] ja1 b2 c3 :::gn 〉 ¼ ja1 〉jb2 〉jc3 〉:::jgn 〉
ð7:12Þ
which can also be expressed as jX〉 ¼ ja〉1 jb〉2 jc〉3 :::jg〉n
ð7:13Þ
leads directly to jB〉 ¼ jx〉1 jy〉2
ð7:14Þ
jA〉 ¼ jy〉1 jx〉2
ð7:15Þ
and since, from Eq. (7.10), we know that jB〉 a jA〉. The substitution of Eqs. (7.14) and (7.15) into Eq. (7.11) leads to the probability amplitude js〉 ¼ jx〉1 jy〉2 þ jy〉1 jx〉2
ð7:16Þ
where x and y refer to different polarizations and the subscripts refer to quanta 1 and 2. The probability amplitudes of entangled states in this form were first discovered by Pryce and Ward [70] and Ward [71] in 1947–1949. Use of the usual normalization and replacement of js〉 by jψ〉 leads directly to a ubiquitous probability amplitude for entangled polarizations 1 jψ〉 ¼ pffiffiffiðjx〉1 jy〉2 7jy〉1 jx〉2 Þ 2
ð7:17Þ
where the 7 allows for different linear combinations. Use of the Dirac identity expressed in Eq. (7.13) leads to the more general Pryce-Ward probability amplitude 1 jψ〉 ¼ pffiffiffiðja〉1 jb〉2 jc〉3 :::jg〉n 7ja′〉1 jb′〉2 jc′〉3 :::jg′〉n Þ 2
ð7:18Þ
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8. Conclusions In this paper, we have highlighted various dispersive techniques crucial to the successful development of narrow-linewidth tunable organic dye lasers. The role of those dispersive techniques in tunable gas lasers, tunable solid state lasers, and tunable diode lasers has been indicated. Particular attention was paid to multiple-prism arrays and the generalized multiple-prism dispersion equation. That equation has been applied here, for the first time, to calculate the dispersion associated with direct-vision prisms, also known as Amici prisms, and compound prisms. This calculation was motivated by the use of these prisms in astronomical instrumentation. Following these dispersion calculations, a practical dispersionless multiple-prism beam expander was designed using the same equations, and the versatility of the generalized multiple-prism dispersion theory was thus demonstrated. Next, attention was focused on higher-order dispersions as needed and applied in prismatic pulse compressors for femtosecond lasers. In this regard, recent results obtained by Duarte [40,50] have been extended to obtain a succinct and elegant general equation for the generalized multiple-prism dispersion; this equation is valid up to the rth derivative ∇rn ϕ2;m ¼ ∇rn 1 H2;m þ ðMÞ 1 ð∇n þ ζÞr 1 and explicit versions of this equation are included for the higher derivatives r ¼ 6; 7. The discussion in this paper centers on the cavity linewidth equation 1 ∂Θ Δλ Δθ ∂λ which can be derived from the quantum interferometric equation [38,66] N
N
j〈xjs〉j ¼ ∑ Ψ ðr j Þ þ 2 ∑ Ψ ðr j Þ 2
j¼1
2
j¼1
N
!
∑ Ψ ðr m Þ cos ðΩm Ωj Þ
m ¼ jþ1
This discussion reinforces the interaction between classical optics and quantum optics; although highlighted by Dirac [56] and Feynman [57], this interaction is not always appreciated in today's literature, with the exception of notable contributions, such as those of Lamb [67]. It has been also shown that the generalized N-slit interferometric equation and the probability amplitude for quantum entanglement have a common origin.
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