Spectral manipulation and complementary spectra with birefringence polarization control

Considering that the crystal is used as a half-wave plate at the central frequency, the induced phase difference between the fast and slow axes at the central frequency is equal to (N + 1/2) × 2π, where N is a positive integer. We choose λ₀ = 600 nm (ν₀ = 5* 10¹⁴ Hz), Γ = 0.05ν₀, and N = 8 as an example to show the spectrum shaping system in figure 1. This kind of crystal corresponds to a multiple-order wave plate, in which d, from equation (2), is equal to (8 + 1/2) λ₀/(n_e(λ₀) −n_o(λ₀)) ≈ 30 μm. Figure 2 shows the emergent spectra for different θ of Polarizer 2. By setting θ = 0, two major peaks exist, distributed symmetrically around the central frequency in figure 2(a). The spectral density of the central frequency component is 0 and it can be used as a notch filter at λ₀. With increasing θ, the central frequency spectral density component increases and the two peaks drop gradually, as in figures 2(b) and (c), until they reach the same level and a flat top occurs (see figure 2(d)), while θ = 30π/128. However, when θ = 32π/128, the spectral intensity tends to be a lower Gaussian-type distribution (see figure 2(e)). With a little increase in θ to 35π/128 (figure 2(f)) and to 40π/128 (figure 2(g)), a nearly triangle-type distribution is formed and then it changes into a nipple type. With continuous increase in θ, the central frequency component grows continuously, together with two frequency components on each side (figure 2(h)), until finally the central frequency dominates (figure 2(i)) at θ = 64π/128. The spectral intensity distribution evolution is a consecutive process while tuning θ from 0 to 64π/128. Although this kind of modulation is not comprehensive, the emergent spectrum can still be manipulated over a very wide range from figures 2(a)–(i), including a notch filter, a flat top, and triangle-type, nipple-type, and central-frequency-dominant distributions. While θ changes from 64π/128 to π, the induced spectral intensity distribution evolution is an inverse process. Note that only when ${\rm{\Delta }}\varphi (\nu )$ is a multiple of π can the polarization states be linearly polarized. To fully control the transmittance with a linear polarizer at the center frequency, as in figure 2, the condition ${\rm{\Delta }}\varphi ({\nu }_{0})=N\cdot \pi$ should be maintained; i.e., the crystal should be used as a half- or full-wave retarder at the center frequency. If the half-wave plate is changed to a full-wave plate, i.e., d = 8λ₀/(n_e(λ₀) −n_o(λ₀)) ≈ 28 μm, similar spectral density control can also be obtained, but for different values of θ.

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After examining the spectrum shaping for narrow polychromatic light, let us find some other possible applications for broader spectra. The authors suggest an effect called the optical frequency ruler [20], which utilizes the interference spectrum to give the constructive (color line) and destructive (dark line) frequency components for broadband light. Since the exact locations of the dark line are known, they can be used as the ticks of a ruler to refer to or estimate an unknown frequency (or wavelength). The interference spectrum usually contains many spikes in the frequency domain and in a previous work it is produced by a double-slit with moving fluid. By varying the fluid speed, the tick period in the ruler can be adjusted, producing a dynamical frequency ruler. Similar phenomena can be achieved using the Fabry–Perot etalon or the optical comb technique [21], which needs a resonator and where the line-width is usually fixed by the finesse and the Q-factor of the cavity [22].

Let us now turn our attention to the present scheme (figure 1) for a broader spectrum with Γ = 0.1ν₀ and the same ν₀ = 5* 10¹⁴ Hz. First set the angle θ of the Polarizer 2 to zero (along x-axis). The emergent spectrum (red line) is shown in figure 3(a), and again we find that many spikes (about four) exist. If more spikes are needed, we can either use a thicker crystal or increase the light bandwidth, as seen from equation (2). The positions of the peaks (marked with red triangles on the abscissa in the figure) and dark lines (marked with blue circles on the abscissa in the figure), from equation (5), satisfy

$$\begin{eqnarray}{\rm{\Delta }}\varphi & = & (2{m})\pi ({\rm{peaks}}),{\rm{\Delta }}\varphi =(2{m}+1)\pi \,({\rm{dark}}\,{\rm{lines}}),\\ {m} & = & 0,\,1,\,2,\,3\ldots ({\rm{for}}\,\theta =0).\end{eqnarray} \tag{ 6 }$$

It seems that this polarization scheme also gives many regular vanishing frequencies (dark lines) with exact known positions and can also be used as an optical frequency ruler, without resorting to the more complicated methods mentioned above, such as the interference spectrum or the Fabry–Perot etalon. Another interesting property of this scheme is illustrated by setting the angle θ to π/2; the emergent spectrum is shown in figure 3(b). The figure shows that the peaks and troughs (dark lines) in figure 3(a) turn into troughs (dark lines) and peaks in figure 3(b), respectively, as indicated by the two vertical arrows. The emergent spectrum undergoes a complete reverse transition and the spectra seem to complement each other. This phenomenon is proposed for the first time and we name the resulting spectra complementary spectra, as suggested in the title. According to equation (5), the positions of peaks and dark lines satisfy

$$\begin{eqnarray}{\rm{\Delta }}\varphi & = & (2m+1)\,\pi ({\rm{peaks}}),{\rm{\Delta }}\varphi =(2{m})\pi \,({\rm{dark}}\,{\rm{lines}}),\\ {m} & = & 0,\,1,\,2,\,3\ldots (\theta =\pi /2).\end{eqnarray} \tag{ 7 }$$

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Thus from equations (6) and (7) it is easy to see why the peaks in figure 3(a) are now troughs in figure 3(b), and vice versa. When light with a spectrum similar to that in figure 3 transmits through a dispersive element (like a prism or grating), the out-coming spectrum exhibits some regular dark lines, as shown in figure 4. Since the dark lines can be used as frequency ruler ticks to refer to an unknown frequency, complementary spectra facilitate identification of the unknown frequency, as explained below. Figure 4 schematically depicts a pair of complementary spectra (A) and (B), whose spectral colors (in the middle) and intensity distributions (at the top and bottom) are plotted separately for clarity. Note that the peaks and dark lines in (A) are in the positions of the dark lines and peaks in (B). As pointed out previously, the dark lines can be used as ticks, and we vertically extend the dark lines in intensity distribution into the spectrum, which are the ticks of the spectrum. Now if strong green light is emitted with unknown but specified wavelength (or frequency), as indicated in the figure, it will be located between the third and fourth ticks from the left in spectrum (A). To better locate the position of this light, the complementary spectrum (B) can be obtained by rotating the polarizer, and its third tick from the left is much closer to the green line, which facilitates identification of the location of the light frequency.

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Note that the polarization states caused by the calcite's birefringence in figure 1 are elliptically (or circularly) polarized, except for Δφ conditions being a multiple of π, where they are horizontally or vertically linearly polarized. Consequently the dark lines can only be obtained with a linear Polarizer 2 pointing along the x- (θ = 0) or y-axis (θ = π/2), as seen in equations (6) and (7). Other linear Polarizer 2 orientations cannot completely block the frequency components with elliptical (or circular) polarization. This statement is consistent with the result in figure 2, where only figure 2(a) (θ = 0) and figure 2(i) (θ = π/2) show the frequency components totally vanishing, as indicated by the blue circles.

At this point some comparisons can be made between our scheme and Zhu's work [17]. In order to obtain the specific TM transmission curve (see figure 6 in [17]), the crystal is prepared at precise thickness with the surface normal and the optic axis oriented and adjusted at specific angles. Our work is mainly intended for unpolarized polychromatic light, and thus two polarizers are used in addition to the birefringent crystal, with the optic axis set at an angle of 45° to the x-axis and the surface normal along z. Consequently the outgoing light is modified as equation (5) and has the same polarization as Polarizer 2 (see figure 1). While only the TM component of the outgoing light can be sustained with a resonant cavity for laser amplification, the TM transmission function has the form of the equation above figure (5) in Zhu's work, which is different from that of our modifier in equation (5). Thus, Zhu's work is more suitable for sophisticated spectral TM transmittance curve tuning for a short-pulse laser. Our scheme is favorable for shaping unpolarized light spectra and the emergent light has the same polarization as that of the second polarizer. It is more suitable for producing spectral dark lines (the blue circles in figure 3) for an optical frequency ruler or complementary spectra.