Octave-Spanning Phase Control for Single-Cycle Bi-Photons

The quantum correlation of octave-spanning time-energy entangled bi-photons can be as short as a single optical cycle. Many experiments designed to explore and exploit this correlation require a uniform spectral phase (transform-limited) with very low loss. So far, transform-limited single-cycle bi-photons were not demonstrated, primarily due to lack of precision control of the spectral phase. Here, we demonstrate precise correction of the spectral-phase of near-octave bi-photons to less than ($<\pi/20$) (residual phase) over nearly a full octave in frequency ($\approx1330-2600$ nm). Using a prism-pair with an effectively-negative separation, we obtain tuned, very low-loss compensation of both the 2nd and 4th dispersion orders. We verify the bi-photons spectral phase directly, using a non-classical bi-photon interference effect.

Since the introduction of the Brewster prism-pair as a mechanism for dispersion control [1], it became a major tool for tuned dispersion compensation in ultrafast optical systems. Specifically, a prism-pair is commonly used to compensate for second-order dispersion (group delay dispersion -GDD). The angular dispersion that is created by the separation between the prisms tips always introduces negative GDD, even when the prisms material produces positive GDD. Thus, any negative dispersion can be achieved by the appropriate separation between the prisms; and tuned dispersion can be achieved by controlling the amount of prism material in the beam. This technique is therefore most suitable for dispersion compensation in the visible and near IR spectrum, where most optical materials produce positive GDD. For longer wavelengths in the short-wavelength infra-red (SWIR) range and beyond, the GDD of materials tends to be negative, posing a hurdle for the prism-pair to produce an overall near zero or positive GDD.
For demanding intra-cavity applications, the prism pair is usually preferred over the grating pair [2], since it offers ultra-low internal loss, as well as precise, higher order dispersion control [3]. By tuning both degrees of freedom of the pair − the prisms separation and the insertion of the prisms into the beam, two orders of dispersion can be independently controlled, which is most useful for compensation of both 2nd and 3rd order dispersion in ultrafast cavities [4]. This higher-order compensation however, is not always possible, and requires a judicial choice of the prisms material for the specific application.
Our motivation for exploring new dispersion compensation techniques in the SWIR range stems from an experiment in quantum optics, where bi-photons interference is observed with ultra-broadband entangled photon pairs from spontaneous parametric down conversion (SPDC). In this experiment, an ultra-broad spectrum of bi-photons (∆ω ≥ 2π · 100T Hz) is produced around a center wavelength of λ 0 = 1760nm, spanning the entire 1400 − 2400nm range (nearly an octave) [5]. The exact signal-idler symmetry in this experiment indicates that the bi-photon interference is sensitive only to even orders of dispersion [6]. Thus, compensation is necessary for both 2nd and 4th order dispersion in order to observe a uniform interference across the entire spectrum [7,8].
While for visible and NIR light most optical materials produce positive GDD and may be compensated by the regular prism-pair separation, which introduces a negative GDD, at the SWIR range most optical materials produce negative GDD, indicating that the prims separation cannot compensate for most materials, limiting considerably the choice of materials that can match the experimental needs. Specifically, prism materials that enable compensation of both 2nd and 4th order dispersion are not available. For the standard materials with negative GDD in the SWIR range, the separation between the prisms must be also negative for the angular dispersion to compensate for the material dispersion. An effective negative separation is easily achievable by incorporating a telescope between the prisms that images the first prism to a location beyond the second prism, as was suggested earlier for grating-pairs [9,10]. Figure  1 shows prism-pair configurations for both the standard apparatus (a), and the prism-pair with negative separation (b).
For a clear understanding of the performance of the prism-pair for different dispersion orders in different wavelength ranges, we use the dispersion contour map in Fig.2. The axes of the map represent the two degrees of freedom of the prism-pair: the horizontal axis is the separation between the prisms tips R, and the vertical axis is the prism insertion H (see Fig.1(a)). The straight Fig. 1. Prism-pair configurations: (a) The standard prism-pair, where the main Brewster beam enters the 1st prism at minimum deviation (θ B , matched with the Brewster angle -red line). All other frequencies (blue line) will be deviated from the main beam by δθ = 0. R is the length of the segment AB, (b) A Brewster prismpair with negative separation. The telescope images the vertex of the 1st prism to a distance of 4f towards the second prism. Placing the second prism before the image plane results in an effective negative separation.
lines represent the sign-inversion borders of the even order terms in the Taylor expansion of the optical phase around the central frequency ω 0 , defined by: where δω = ω − ω 0 (ω 0 is the center of the spectrum). As a general rule, inserting the prisms into the beam (increasing H), produces GDD that corresponds to the material dispersion, whereas increasing the separation between the prisms (increasing R) induces negative dispersion. For visible and NIR wavelengths, the majority of optical materials produce positive GDD ( Fig.2(a), λ 0 = 600nm), and the sign inversion lines pass through the (+R, +H) first quarter of the map. For longer wavelengths, the majority of optical materials produce negative GDD ( Fig.2(b), λ 0 = 2000nm), and the sign inversion lines pass through the (−R, +H) second quarter of the map. Overall dispersion compensation can be obtained by choosing an appropriate sector in the dispersion contour map (Fig.2), in which the sign of the prism-pair dispersion order-terms are inverse to the contribution of the other dispersive elements in the system. Obviously, the introduction of a negative separation enables the prism-pair to produce a total positive GDD, even for negative prism material dispersion. Furthermore, a negative distance provides an important new knob for dispersion management − the ability to tune not only the magnitude but also the sign of any order of the optical dispersion function from far negative to far positive, allowing optimization of the optical phase by using the interplay between different orders. In our experiment this is the key for compensation of the 4th order dispersion, which is impossible with the standard prism-pair or grating-pair.
Let us now consider the dispersion of the prism-pair in some detail, focusing on the effects of negative separation. The prism-pair, illustrated in Fig.1(a), compensates for dispersion via two mechanisms: 1. by varying the geometrical path of different frequency components(angular dispersion) 2. by inserting a variable thickness of dispersive material into the optical beam path (material dispersion). The system is generally composed of two pairs of Brewster isosceles prisms in a symmetric configuration (or retro reflection through a single pair). The prisms apex angle is chosen such that the main optical axis is minimally deviated at a Brewster angle, eliminating reflection losses and refraction deformation of the beam.
In our analysis of the frequency dependent optical path in a prism-pair, we follow the original paper by Fork [1] using the concept of wave fronts. We divide the geometrical path through the prism-pair system into four free propagating sections and two refractions: 1. from a fixed wave front plane before the 1st prism to the 1st prism vertex (point A), 2. from the 1st prism vertex to an arbitrary wave front plane between the prisms, 3. from this arbitrary wave front plane to the 2nd prism vertex (point C), 4. from the 2nd prism vertex to a fixed wave front plane after the 2nd prism. Each segment represents a free propagation of a wave front without refraction. The two refractions are assigned to the vertices of the two prisms without contributing to the optical path. Normally, one would calculate the optical path of each frequency by tracing the propagation path of the relevant general beam through the two prisms with Snell's refraction [11]. However, this method usually results in complicated expressions for the optical path, leaving little room for intuition. For the purpose of tracking the optical phase, all geometrical rays normal to the wave fronts represent the same accumulated phase, indicating that one can choose any convenient ray in the four segments, and not necessarily a continuous path. Thus, we only choose rays that propagate through the vertices of the two prisms, eliminating the need to explicitly calculate the refraction [1].
We define the main-beam as the beam that is minimally deviated after the 1st prism (designed usually to match the center of the spectrum). Any other frequency will be deviated from the main-beam by an angle δθ that is frequency dependent. The optical path through the whole prism-pair setup is: (2) and the optical phase is given by: ϕ opt = (ω/c)l opt . Note that Eq.2 is exact without any approximations. Later, we will introduce several approximations, taking into account that δθ is small, and its explicit dependence on the refractive index and on the frequency.
The first term in Eq.2, defined as l R = R cos(δθ), is the optical path of the different frequency beams between two matching wave planes passing through the prims vertices, where the prisms are positioned at H = 0 (as originally expressed by Fork). Indeed, for the main-beam (δθ = 0) this part of the optical path is simply l R = R. Inserting the prisms into the main beam (H = 0) extends the optical path by: l H = l H,1 cos(δθ)+l H,2 sin(δθ), where we define l H,1 = H cos(π − 2θ B ) and l H,2 = H sin(2θ B ).
We may rewrite the optical path expression as: l opt (δθ) = (l R + l H,1 ) cos(δθ) + l H,2 sin(δθ) The first part: l a = l R + l H,1 , may be understood as the propagating distance between the prisms, and the second part: l b = l H,2 , is the propagation distance inside the prisms. Thus, the optical path is: Based on the fact that the deviation angles are small δθ ≪ 1, we use Snell's law to approximate δθ(ω) ≈ 2δn(ω), where δn(ω) = n(ω) − n(ω 0 ) is the variation of the refractive index with respect to the center optical frequency n(ω 0 ). Hence, cos(δθ) ≈ 1 − 2δn 2 and sin(δθ) ≈ 2δn. We can also define effective wave numbers:β a = (ω/c) cos(δθ) andβ b = (ω/c) sin(δθ), so the optical phase may be rewritten as: If we express δn as a power series in frequency around ω 0 (δn = n 1 δω + n 2 δω 2 + n 3 δω 3 + ...), we may approximate the prism-pair GDD as:  For our quantum interference experiment, we chose a prism pair of Sapphire material with a negative separation R < 0 in order to compensate for the overall dispersion in the setup, resulting from 6mm of periodicallypolled KTP crystal, ∼20mm of infrasil optics, and 4mm Silicon window. A contour map for the total dispersion is illustrated in Fig.3, plotting the zero level contour lines of the three dispersion orders (ϕ 2 , ϕ 4 and ϕ 6 ). An optimized compensation of the overall dispersion can be achieved by choosing a working point inside the highlighted triangle in the figure.
The uncorrected optical phase of the bi-photons (without the prism-pair) and the compensated phase are plotted in Fig.4. The small oscillations clearly indicate the tradeoff between the different orders in the phase expansion. In this wavelength range, the flattened phase profile is possible only when using a negative separation R < 0. As mentioned above, with a conventional positive separation it is possible to use only prisms highly dispersive materials, such as Silicon, which do not allow for multi-order compensation (both 2nd and 4th).
In conclusion, we showed that by enabling the Brewster prism-pair separation to be both positive and negative, tunable overall positive dispersion can be obtained, even when the material dispersion is negative. We derived a simple and intuitive expression for the prismpair optical phase, and illustrated a useful contour map for dispersion optimization, exploiting the negative separation possibility. Prism-pairs with negative separation widen the design freedom to obtain precise, low-loss dispersion control for broadband applications in the infrared wavelength range. This research was supported by the Israeli science foundation (grant 807/09).