Polarization coherence theorem

J. H. Eberly; X.-F. Qian; A. N. Vamivakas

doi:10.1364/OPTICA.4.001113

Centuries after Thomas Young’s famous double-slit interference experiment [1], quantification of coherence as a contextual resource (see [2]) is just now being examined (see [3,4]), and experimental evidence of polarization coherence in a previously unexplored context has been reported, exposing a new coherence triad [5].

The theorem of the title arises from the recognition that polarization is a two-party property, even in common usage. A polarized electorate means two things: (a) opinions within the population favor only a few political factions, and (b) this occurs within a specified background (the voting public, but not kindergarten school children). In much the same way, in optics a polarized field means that just one or two of the field’s intrinsic spin orientations are endowed with substantial field amplitude, so two independent degrees of freedom, spin and amplitude, are well correlated. Here we show how this leads to the new identity we refer to as the Polarization Coherence Theorem (PCT).

For simplicity we introduce the PCT in the most familiar context, Young’s double-slit scenario, as shown in Fig. 1. The combined amplitude of the light field arriving at the screen has a contribution from each of the two slits $a$ and $b$ :

F (r_{⊥}, z) = u_{a} (r_{⊥}, z) ϕ_{a} (q) + u_{b} (r_{⊥}, z) ϕ_{b} (q) .

Fig. 1. Double-slit interference setup. Classical light fields emerge from slits $a$ and $b$ and combine on screen $c$ .

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Here

u_{a} (r_{⊥}, z)

and

u_{b} (r_{⊥}, z)

are diffractive spatial mode functions unit-normalized and orthogonal in the intervening space from the slits to the screen, and

ϕ_{a}

and

ϕ_{b}

are the corresponding field strengths. They depend on degrees of freedom not identified and here loosely labeled

q

, such as temporal amplitude, spin (ordinary polarization), etc.

By adopting the conventional small-angle and distant-screen treatment of the Young signals, the propagation factors from slits $a$ and $b$ to screen $c$ will be the same in magnitude but differ in phase (which is absorbed into amplitudes $ϕ_{a}$ and $ϕ_{b}$ ). Then the intensity at the screen is obtained with the expected sinusoidal interference term,

I_{c} = I_{a} + I_{b} + 2 | ⟨ ϕ_{a}^{*} ϕ_{b} ⟩ | \cos [\arg (⟨ ϕ_{a}^{*} ϕ_{b} ⟩)],

with

I_{a} = ⟨ {| ϕ_{a} |}^{2} ⟩

and

I_{b} = ⟨ {| ϕ_{b} |}^{2} ⟩

, and the brackets for averaging are needed if the amplitudes are known only statistically, as is commonly the case.

Fringe visibility for field $F$ is given as

V_{F} = \frac{I_{c}^{\max} - I_{c}^{\min}}{I_{c}^{\max} + I_{c}^{\min}} = \frac{2 | ⟨ ϕ_{a}^{*} ϕ_{b} ⟩ |}{I_{a} + I_{b}},

where

I_{c}^{\max}

and

I_{c}^{\min}

are the maximum and minimum fringe intensities of

I_{c}

, as the cosine equals

+ 1

or

- 1

in Eq. (2).

When light is coming from only one of the two slits, then there can be no interference, and the origin of the light arriving at screen $c$ is known. Its intensity is distinguishable from the intensity observed when only the other slit is open. Consequently, the degree of distinguishability is also an indicator of a different coherence, and is defined as

D_{F} = \frac{| I_{a} - I_{b} |}{I_{a} + I_{b}} .

One can easily check (Schwarz inequality) that $⟨ {| ϕ_{a} |}^{2} ⟩ ⟨ {| ϕ_{b} |}^{2} ⟩ \geq | ⟨ ϕ_{a}^{*} ϕ_{b} ⟩ |^{2}$ , and conclude that $V_{F}$ and $D_{F}$ satisfy a strong constraint,

V_{F}^{2} + D_{F}^{2} \leq 1 .

This coherence relation, here obtained without considering spin (ordinary polarization) at all, when interpreted in terms of wave/ray or wave/particle duality, occupies a central place in all discussions of duality and complementarity, and has been derived and re-derived by many investigators [6].

The Young scenario can also be discussed for an optical field exhibiting polarization unit vectors explicitly:

E = \hat{h} E_{h} + \hat{v} E_{v} .

Classical polarization optics tells us (see, e.g., Brosseau [7] or Wolf [8]) how to construct

W_{E}

, the polarization coherence matrix for

E

and its degree of polarization

P_{E}

, indexed by spin orientations

\hat{h}

and

\hat{v}

. They are

W_{E} = [\begin{matrix} ⟨ E_{h}^{*} E_{h} ⟩ & ⟨ E_{v}^{*} E_{h} ⟩ \\ ⟨ E_{h}^{*} E_{v} ⟩ & ⟨ E_{v}^{*} E_{v} ⟩ \end{matrix}], P_{E} = \sqrt{1 - \frac{4 D e t W_{E}}{{(T r [W_{E}])}^{2}}} .

Note that polarization also applies to field $F$ , despite the absence of directional unit vectors. We can say that $F$ is strongly “polarized” in the $u_{a}$ “direction” if $| ϕ_{a} (q) |$ has much greater magnitude than $| ϕ_{b} (q) |$ , and the reverse if $| ϕ_{b} (q) | ≫ | ϕ_{a} (q) |$ . There is no intrinsic mathematical difference in the $E$ -polarization and $F$ -“polarization” examples, since $\hat{h}$ and $\hat{v}$ and the $u_{a}$ and $u_{b}$ functions are orthogonal vectors in their own vector spaces of spin and spatial mode. We will designate the $u_{a}$ and $u_{b}$ mode functions as “mode polarization vectors.” The field in Eq. (1) exhibits “mode polarization” in its mode space.

Thus we easily construct $W_{F}$ , the “mode polarization” coherence matrix for $F$ in Eq. (1), and its “degree of mode polarization” $P_{F}$ , indexed by “mode polarization” orientations $u_{a}$ and $u_{b}$ . They are

W_{F} = [\begin{matrix} ⟨ ϕ_{a}^{*} ϕ_{a} ⟩ & ⟨ ϕ_{b}^{*} ϕ_{a} ⟩ \\ ⟨ ϕ_{a}^{*} ϕ_{b} ⟩ & ⟨ ϕ_{b}^{*} ϕ_{b} ⟩ \end{matrix}], P_{F} = \sqrt{1 - \frac{4 D e t W_{F}}{{(T r [W_{F}])}^{2}}} .

Elementary arithmetic then provides

P_{F}^{2} = 1 - 4 \frac{I_{a} I_{b} - | ⟨ ϕ_{a}^{*} ϕ_{b} ⟩ |^{2}}{{(I_{a} + I_{b})}^{2}} = \frac{{(I_{a} - I_{b})}^{2} + 4 | ⟨ ϕ_{a}^{*} ϕ_{b} ⟩ |^{2}}{{(I_{a} + I_{b})}^{2}} = \frac{{(I_{a} - I_{b})}^{2}}{{(I_{a} + I_{b})}^{2}} + 4 \frac{| ⟨ ϕ_{a}^{*} ϕ_{b} ⟩ |^{2}}{{(I_{a} + I_{b})}^{2}},

and this contains elements familiar from Eqs. (3) and (4). In fact, we have just derived

P_{F}^{2} \equiv D_{F}^{2} + V_{F}^{2} .

The

F

degree of polarization satisfies

1 \geq P_{F} \geq 0

, so there is no conflict between the result of Eq. (10) and the multiply derived formula Eq. (5).

It should be obvious that the new identity in Eq. (10) arises equally quickly from Eq. (7) as from Eq. (8). That is, it is independent of the pair of degrees of freedom used to derive it. The identity in Eq. (10) is in fact a very general PCT. It says that the degree of polarization coherence for an optical field, in either the ordinary spin sense or a generalized mode sense, will support the famous duality inequality in Eq. (5). As such, Eq. (10) has unexpected implications for complementarity, which will have to be discussed elsewhere [9], as well as the single-photon version of the present result.

APPENDIX A

We are using analogies that have a firm relationship because they share the exact same vector space basis, and there is a clear connection between the two fields in Eq. (1) and Eq. (6) and thus between our two “polarizations.” Recall that the amplitudes $ϕ (q)$ are dependent on additional degrees of freedom of the light field, unspecified but labeled $q$ above. Similarly, the amplitudes $E_{h}$ and $E_{v}$ depend on spatial and temporal mode amplitudes, not specified, but in the relevant Young context they certainly refer to contributions coming from the spatial modes $u_{a}$ and $u_{b}$ , so we can expand $E_{h}$ and $E_{v}$ in terms of them:

E_{h} = u_{a} H_{a} + u_{b} H_{b}, and E_{v} = u_{a} V_{a} + u_{b} V_{b} .

Then the otherwise arbitrary field $F$ can be interpreted as just the projection of $E$ on an arbitrary intrinsic spin direction $\hat{s}$ :

F = (\hat{s} \cdot E) = (\hat{s} \cdot \hat{h}) (u_{a} H_{a} + u_{b} H_{b}) + (\hat{s} \cdot \hat{v}) (u_{a} V_{a} + u_{b} V_{b}) = u_{a} [(\hat{s} \cdot \hat{h}) H_{a} + (\hat{s} \cdot \hat{v}) V_{a}] + u_{b} [(\hat{s} \cdot \hat{h}) H_{b} + (\hat{s} \cdot \hat{v}) V_{b}],

which is the same as

F = u_{a} ϕ_{a} + u_{b} ϕ_{b}

.

Funding

National Science Foundation (NSF) (PHY-1203931, PHY-1505189, PHY-1539859); Army Research Office (ARO) (W911NF-16-1-0162); Office of Naval Research (ONR) (N00014-14-1-0260).

REFERENCES

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6. An examination by W. K. Wootters and W. H. Zurek, in Phys. Rev. D19, 473 (1979) opened for discussion a quantification of complementarity that is available via the Young experiment, and almost uncountably many derivations and re-derivations related to the result V²+D²≤1 given in Eq. (5) have appeared since 1985: R. J. Glauber, Ann. New York Acad. Sci. 480, 336–372 (1986); D. M. Greenberger and A. Yasin, Phys. Lett. A 128, 391 (1988); L. Mandel, Opt. Lett. 16, 1882 (1991); G. Jaeger, M. A. Horne, and A. Shimony, Phys. Rev. A 48, 1023 (1993); G. Jaeger, A. Shimony, and L. Vaidman, Phys. Rev. A 51, 54 (1993); B.-G. Englert, Phys. Rev. Lett. 77, 2154 (1996); S. Dürr, T. Nonn, and G. Rempe, Phys. Rev. Lett. 81, 5705 (1998); B.-G. Englert, M. O. Scully, and H. Walther, J. Mod. Opt. 47, 2213 (2000); M. Lahiri, Phys. Rev. A 83, 045803 (2011); H.-Y. Liu, J.-H. Huang, J.-R. Gao, M. S. Zubairy, and S.-Y. Zhu, Phys. Rev. A 85, 022106 (2012); F. De Zela, Phys. Rev. A 89, 013845 (2014).

7. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

8. E. Wolf, Phys. Lett. A 312, 263 (2003). [CrossRef]

9. X.-F. Qian, private communication.

Polarization coherence theorem

Abstract

APPENDIX A

Funding

REFERENCES

Cited By

Figures (1)

Equations (12)

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