## All linear optical devices are mode converters |

Optics Express, Vol. 20, Issue 21, pp. 23985-23993 (2012)

http://dx.doi.org/10.1364/OE.20.023985

Acrobat PDF (812 KB)

### Abstract

We show that every linear optical component can be completely described as a device that converts one set of orthogonal input modes, one by one, to a matching set of orthogonal output modes. This result holds for any linear optical structure with any specific variation in space and/or time of its structure. There are therefore preferred orthogonal “mode converter” basis sets of input and output functions for describing any linear optical device, in terms of which the device can be described by a simple diagonal operator. This result should help us understand what linear optical devices we can and cannot make. As illustrations, we use this approach to derive a general expression for the alignment tolerance of an efficient mode coupler and to prove that loss-less combining of orthogonal modes is impossible.

© 2012 OSA

## 1. Introduction

1. Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator,” Opt. Lett. **30**(2), 141–143 (2005). [CrossRef] [PubMed]

7. D. Dai, Y. Tang, and J. E. Bowers, “Mode conversion in tapered submicron silicon ridge optical waveguides,” Opt. Express **20**(12), 13425–13439 (2012). [CrossRef] [PubMed]

1. Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator,” Opt. Lett. **30**(2), 141–143 (2005). [CrossRef] [PubMed]

3. B. Zhu, T. F. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “112-Tb/s space-division multiplexed DWDM transmission with 14-b/s/Hz aggregate spectral efficiency over a 76.8-km seven-core fiber,” Opt. Express **19**(17), 16665–16671 (2011). [CrossRef] [PubMed]

8. Z. Yu and S. Fan, “Integrated nonmagnetic optical isolators based on photonic transitions,” IEEE J. Sel. Top. Quantum Electron. **16**(2), 459–466 (2010). [CrossRef]

3. B. Zhu, T. F. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “112-Tb/s space-division multiplexed DWDM transmission with 14-b/s/Hz aggregate spectral efficiency over a 76.8-km seven-core fiber,” Opt. Express **19**(17), 16665–16671 (2011). [CrossRef] [PubMed]

9. R. N. Mahalati, D. Askarov, J. P. Wilde, and J. M. Kahn, “Adaptive control of input field to achieve desired output intensity profile in multimode fiber with random mode coupling,” Opt. Express **20**(13), 14321–14337 (2012). [CrossRef] [PubMed]

## 2. The mode converter basis set

### 2.1 Device operator

*ϕ*〉 and to generate a corresponding output function |

_{I}*ϕ*〉, that is

_{O}*ϕ*〉 will be in one mathematical function (Hilbert) space – the input space

_{I}*H*– and the possible output functions |

_{I}*ϕ*〉 will in general be in another, which we can think of as the output space

_{O}*H*. (See Appendix A for the notation and general properties of these functions and spaces.) We are generally free to choose these spaces (and hence the inputs and outputs we are interested in) to be whatever we want for our device of interest.

_{O}8. Z. Yu and S. Fan, “Integrated nonmagnetic optical isolators based on photonic transitions,” IEEE J. Sel. Top. Quantum Electron. **16**(2), 459–466 (2010). [CrossRef]

10. B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. **14**(12), 630–632 (1989). [CrossRef] [PubMed]

11. D. H. Broaddus, M. A. Foster, O. Kuzucu, A. C. Turner-Foster, K. W. Koch, M. Lipson, and A. L. Gaeta, “Temporal-imaging system with simple external-clock triggering,” Opt. Express **18**(13), 14262–14269 (2010). [CrossRef] [PubMed]

### 2.2 Example optical systems

### 2.3 Singular value decomposition of the device operator

*ϕ*〉 (|

_{DIm}*ϕ*〉) as its column vectors and D

_{DOm}*is a diagonal matrix with diagonal complex number elements*

_{diag}*s*.

_{Dm}*s*are the singular values and the sets of functions |

_{Dm}*ϕ*〉 and |

_{DIm}*ϕ*〉 each form orthogonal sets in their respective spaces

_{DOm}*H*and

_{I}*H*. These sets of functions are the solutions of the two eigenvalue problems Note that both D

_{O}^{†}D and DD

^{†}are Hermitian (self-adjoint) operators or matrices even if the matrix D is not, and that these two equations have the same eigenvalues |

*s*|

_{Dm}^{2}.

*ϕ*〉 and |

_{DIm}*ϕ*〉 for which the singular value

_{DOm}*s*is non-zero – are complete in the following specific sense: Any function in the output space that can be generated by the device from some function in the input space can be written as a linear combination of the set of functions |

_{Dm}*ϕ*〉 corresponding to non-zero singular values, and any function that can be generated by the device in the output space can be generated by some function in the input space that is a linear combination of the |

_{DOm}*ϕ*〉 corresponding to non-zero singular values. In this sense, we will call the sets |

_{DIm}*ϕ*〉 and |

_{DIm}*ϕ*〉 complete orthogonal sets, and we will understand we are including only those associated with non-zero singular values; they certainly can describe all the functions in the input and output space that are of actual interest for the device operation. We will generally take these functions also to be normalized, giving orthonormal basis sets for the spaces of interest.

_{DOm}*ϕ*〉 that will give rise, one by one, to a set of corresponding orthogonal output functions |

_{DIm}*ϕ*〉, with non-zero coupling coefficients

_{DOm}*s*.

_{Dm}*ϕ*〉 and |

_{DIm}*ϕ*〉 and values

_{DOm}*s*can always be evaluated given D, and are unique (at least within normalization and phase factors and the usual arbitrariness of orthogonal linear combinations of degenerate eigenfunctions). The resulting device matrix or operator is diagonal when expressed in these basis sets. Hence, any linear optical device can be written as a mode converter, from specific orthogonal input modes to specific orthogonal output modes.

_{Dm}## 3. Derivation of example results

*ϕ*〉 and |

_{DIm}*ϕ*〉, the corresponding device operator D and the singular values

_{DOm}*s*to solve two particular problems: First, we derive the alignment tolerance of an efficient mode coupler; second, we prove that the loss-less combination of the power from two orthogonal modes is impossible. We are not aware of prior published derivations either of this general expression for alignment tolerance or of a formal proof in terms of modes of the impossibility of such loss-less combination.

_{Dm}### 3.1 Mathematical preliminaries

4. L. H. Gabrielli and M. Lipson, “Integrated Luneburg lens via ultra-strong index gradient on silicon,” Opt. Express **19**(21), 20122–20127 (2011). [CrossRef] [PubMed]

6. P. Markov, J. G. Valentine, and S. M. Weiss, “Fiber-to-chip coupler designed using an optical transformation,” Opt. Express **20**(13), 14705–14713 (2012). [CrossRef] [PubMed]

12. V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. **28**(15), 1302–1304 (2003). [CrossRef] [PubMed]

*ϕ*〉 that we shine onto the input face of the coupler, with resulting output waves |

_{I}*ϕ*〉 just inside the waveguide, as in Fig. 1(b).

_{O}*ϕ*〉 and |

_{DIm}*ϕ*〉 with corresponding singular values

_{DOm}*s*. As discussed in Appendix A, we can choose the functions to be normalized so that 〈

_{Dm}*ϕ*|

_{DIm}*ϕ*〉 = 1 and 〈

_{DIm}*ϕ*|

_{DOm}*ϕ*〉 = 1 correspond to unit power (in the case of steady beams) or energy (in the case of pulses) in each case.

_{DOm}*s*, there will be at least one with largest possible magnitude, |

_{Dm}*s*|. It is possible that there are several different pairs of functions (and hence values of the index

_{max}*m*) that all have the same magnitude of singular value, all of those magnitudes being equal to |

*s*|; in such a case, we are free to form any set of orthogonal linear combinations of this subset of input functions (and the same linear combination of this subset of output functions) and use those in the sets |

_{max}*ϕ*〉 and |

_{DIm}*ϕ*〉 at our convenience, as is usual in handling the multiple eigenfunctions of degenerate eigenvalues.

_{DOm}*s*, of singular value; obviously we cannot have a larger singular value than this because that would have to correspond to more than 100% efficiency, which is impossible by definition for our loss-less linear optical component. Because of our choice of power or energy normalization of the basis functions in each space, |

_{max}*s*| = |

_{max}*s*|

_{max}^{2}= 1 for such a 100% efficient coupler. Either (i) the specific mode of interest in the guide and its corresponding input mode already are uniquely the only function pair with this magnitude |

*s*| of singular value, or (ii) we are free to construct linear combinations, as discussed above, such that one of the function pairs with this magnitude |

_{max}*s*| of singular value corresponds to our function pair of interest for 100% efficient mode coupling; we can number the singular value functions such that increasing

_{max}*m*corresponds to progressively smaller magnitudes of singular values, so in either case (i) or (ii) we can choose to call this pair |

*ϕ*

_{DI}_{1}〉 and |

*ϕ*

_{DO}_{1}〉.

### 3.2 Limit to the alignment tolerance of high-efficiency mode couplers

*ϕ*

_{DI}_{1}〉 we have some other input function |

*ϕ*〉, corresponding to some misaligned input. Because we are interested in relative efficiencies of coupling, we take this input function also to have unit power or energy, so 〈

_{Imis}*ϕ*|

_{Imis}*ϕ*〉 = 1. Now, we can decompose this misaligned input function |

_{Imis}*ϕ*〉 into a linear combination of the singular value decomposition set |

_{Imis}*ϕ*〉 plus possibly some other function |

_{DIm}*ϕ*〉 that is orthogonal to all the |

_{N}*ϕ*〉 (i.e., 〈

_{DIm}*ϕ*|

_{N}*ϕ*〉 = 0 for all

_{DIm}*m*), givingwhereHere we come to the key point in the argument. Each of the components

*a*|

_{m}*ϕ*〉 in the input wave leads to a corresponding wave proportional to |

_{DIm}*ϕ*〉. Because any function |

_{DOm}*ϕ*〉 orthogonal to all the |

_{N}*ϕ*〉 necessarily leads to no outputs in

_{DIm}*H*, D|

_{O}*ϕ*〉 = 0, and for all components

_{N}*a*|

_{m}*ϕ*〉 other than for

_{DIm}*m*= 1, the resulting generated waves in the waveguide are orthogonal to our output mode of interest. In other words, none of these other components in the input wave leads to any coupling whatsoever into our desired output mode |

*ϕ*

_{DO}_{1}〉. We can show this formally by considering the wave generated in the output space (i.e., the waveguide), which is, by definition,So, the component of |

*ϕ*〉 that is coupled into our mode of interest isso that the field in our output mode is

_{Omis}*a*

_{1}

*s*|

_{max}*ϕ*

_{DO}_{1}〉. Since by choice here a field of

*s*|

_{max}*ϕ*

_{DO}_{1}〉 corresponds to unit power transfer efficiency – i.e., 〈

*ϕ*

_{DO}_{1}|

*ϕ*

_{DO}_{1}〉 = |

*s*|

_{max}^{2}= 1 corresponds to unit power in the output beam – then the power efficiency for coupling into our desired output mode iswhich is our general result here for the coupling efficiency of a misaligned beam into a 100% efficient coupler.

*x*and

*y*,For the specific case where the “misaligned” field |

*ϕ*〉 is merely a displaced version of

_{Imis}*x*and Δ

*y*will beFor any mode coupler that is 100% efficient when it is perfectly aligned,

*η*(Δ

*x*, Δ

*y*) is therefore the alignment tolerance of the power coupling efficiency; at any given displacements Δ

*x*and Δ

*y*, this is also as big as the power coupling can possibly be. Note that we have not only established a bound on the power coupling efficiency from such a misaligned beam; rather we have shown that, for a mode coupler that is 100% efficient when perfectly aligned, this expression Eq. (12), or, more generally, Eq. (10),

*is*the power coupling efficiency when the input beam is misaligned. Note that this expression is now only a function of the input beam shape itself, not of anything else; specifically, it does not depend on the size of the waveguide into which we are coupling.

7. D. Dai, Y. Tang, and J. E. Bowers, “Mode conversion in tapered submicron silicon ridge optical waveguides,” Opt. Express **20**(12), 13425–13439 (2012). [CrossRef] [PubMed]

12. V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. **28**(15), 1302–1304 (2003). [CrossRef] [PubMed]

*ϕ*

_{DO}_{1}〉 in the expressions instead of the input beam |

*ϕ*

_{DI}_{1}〉.

### 3.3 Proof of impossibility of loss-less beam combination of multiple modes

*ϕ*

_{DI}_{1}〉 and |

*ϕ*

_{DO}_{1}〉, as discussed above. Necessarily, these two modes are a pair of the mode-converter basis modes for the device operator D. Suppose, then, that we consider some other input mode |

*ϕ*〉, orthogonal to |

_{Dextra}*ϕ*

_{DI}_{1}〉. Since by choice |

*ϕ*〉 is orthogonal to |

_{Dextra}*ϕ*

_{DI}_{1}〉, then there is no component of |

*ϕ*

_{DI}_{1}〉 in the expansion for |

*ϕ*〉. |

_{Dextra}*ϕ*〉 is a linear combination of the modes |

_{Dextra}*ϕ*〉 for

_{DIm}*m*≥ 2 and/or contains functions orthogonal to all the |

*ϕ*〉. Hence, there is no coupling of the power from |

_{DIm}*ϕ*〉 into the output mode |

_{Dextra}*ϕ*

_{DO}_{1}〉; instead, because of the one-to-one mapping of the mode-converter basis functions, with input mode |

*ϕ*〉 coupling only into output mode |

_{DIm}*ϕ*〉, any such power is coupled into the other orthogonal modes, |

_{DOm}*ϕ*〉 for

_{DOm}*m*≥ 2, or it is not coupled into any of them. Hence, loss-less coupling from two orthogonal modes into one is not possible for any linear optical component (without an amplification mechanism).

## 4. Conclusions

## Appendix A – Notation and Hilbert spaces

*ϕ*〉, because we want to have a general notation that allows us to consider many different possible kinds of fields; examples of the fields could include monochromatic waves varying in space, vector fields (such as electromagnetic fields), pulsed fields, or conceivably other more complicated fields including other attributes such as quantum mechanical spin. Linear operators are written with

*sans serif*upper-case characters, e.g., A. (See, e.g., Ref [14]. for an extended discussion of Dirac notation.)

*ϕ*〉 and |

_{A}*ϕ*〉 in a given space, 〈

_{B}*ϕ*|

_{A}*ϕ*〉 = 0 means that the functions are orthogonal by definition. With the additional property 〈

_{B}*ϕ*|

_{A}*ϕ*〉 = 〈

_{B}*ϕ*|

_{B}*ϕ*〉* (where the superscript * denotes the complex conjugate) – a property we typically expect anyway for the overlap integrals of complex wavefunctions of many different types – then essentially we have the conditions for the

_{A}*H*and

_{I}*H*to be Hilbert spaces. Since the inner product of a function with itself – i.e., 〈

_{O}*ϕ*|

*ϕ*〉 – is necessarily real by the above property, we can choose 〈

*ϕ*|

_{I}*ϕ*〉 and 〈

_{I}*ϕ*|

_{O}*ϕ*〉 so that they each represent the power or energy in the field in the input or output space, respectively (though it is not essential that we make that choice).

_{O}*H*and

_{I}*H*are disjoint – they can overlap, with some or all functions of interest being in both spaces, though we will usually think of them as disjoint spaces.

_{O}## Appendix B – Conditions for and properties of singular value decomposition

*ϕ*〉 and |

_{DIm}*ϕ*〉. The mathematical subtlety here is connected with functions associated with singular values

_{DOm}*s*that are zero. In our actual physical problem, we have little or no interest in such functions, since they correspond to inputs that lead to no device output or outputs that cannot be generated by any input. If we choose to work only with functions corresponding to non-zero singular values, then we can avoid these mathematical subtleties, and we have given in the main text above a suitable statement of that restriction and hence of the specific sense of completeness of sets that we use.

_{Dm}^{†}D and DD

^{†}are also compact, and they are also necessarily Hermitian, because they are products of an operator and its Hermitian adjoint. The arguments on completeness of the sets |

*ϕ*〉 and |

_{DIm}*ϕ*〉 are formally slightly different depending on whether D is of finite rank. For finite rank, because of the completeness of the sets of eigenfunctions of finite rank compact Hermitian operators (see [15], p. 250, Theorem 4.38), we can conclude that, for any finite number of input and output modes, the resulting sets of eigenfunctions |

_{DOm}*ϕ*〉 and |

_{DIm}*ϕ*〉 of the finite rank Hermitian operators D

_{DOm}^{†}D and DD

^{†}are then orthogonal and complete for their respective spaces. Even if we are not restricted to finite numbers of input and/or output modes, since D

^{†}D and DD

^{†}are still compact Hermitian operators, by the Hilbert-Schmidt theorem (see [15], p. 257, Theorem 4.40), the resulting eigenfunction sets |

*ϕ*〉 and |

_{DIm}*ϕ*〉 (corresponding to non-zero eigenvalues

_{DOm}*ϕ*〉 and |

_{DIm}*ϕ*〉 to be complete orthogonal sets in our specific sense for their respective spaces

_{DOm}*H*and

_{I}*H*.

_{O}## Acknowledgments

## References and links

1. | Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator,” Opt. Lett. |

2. | G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. |

3. | B. Zhu, T. F. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “112-Tb/s space-division multiplexed DWDM transmission with 14-b/s/Hz aggregate spectral efficiency over a 76.8-km seven-core fiber,” Opt. Express |

4. | L. H. Gabrielli and M. Lipson, “Integrated Luneburg lens via ultra-strong index gradient on silicon,” Opt. Express |

5. | M.-C. Wu, F.-C. Hsiao, and S.-Y. Tseng, “Adiabatic mode conversion in multimode waveguides using chirped computer-generated planar holograms,” IEEE Photon. Technol. Lett. |

6. | P. Markov, J. G. Valentine, and S. M. Weiss, “Fiber-to-chip coupler designed using an optical transformation,” Opt. Express |

7. | D. Dai, Y. Tang, and J. E. Bowers, “Mode conversion in tapered submicron silicon ridge optical waveguides,” Opt. Express |

8. | Z. Yu and S. Fan, “Integrated nonmagnetic optical isolators based on photonic transitions,” IEEE J. Sel. Top. Quantum Electron. |

9. | R. N. Mahalati, D. Askarov, J. P. Wilde, and J. M. Kahn, “Adaptive control of input field to achieve desired output intensity profile in multimode fiber with random mode coupling,” Opt. Express |

10. | B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. |

11. | D. H. Broaddus, M. A. Foster, O. Kuzucu, A. C. Turner-Foster, K. W. Koch, M. Lipson, and A. L. Gaeta, “Temporal-imaging system with simple external-clock triggering,” Opt. Express |

12. | V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. |

13. | R. W. Boyd, |

14. | D. A. B. Miller, |

15. | G. W. Hanson and A. B. Yakovlev, |

16. | E. Kreysig, |

**OCIS Codes**

(000.6850) General : Thermodynamics

(230.0230) Optical devices : Optical devices

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 25, 2012

Manuscript Accepted: September 26, 2012

Published: October 4, 2012

**Citation**

David A. B. Miller, "All linear optical devices are mode converters," Opt. Express **20**, 23985-23993 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23985

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### References

- Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator,” Opt. Lett.30(2), 141–143 (2005). [CrossRef] [PubMed]
- G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett.105(15), 153601 (2010). [CrossRef] [PubMed]
- B. Zhu, T. F. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “112-Tb/s space-division multiplexed DWDM transmission with 14-b/s/Hz aggregate spectral efficiency over a 76.8-km seven-core fiber,” Opt. Express19(17), 16665–16671 (2011). [CrossRef] [PubMed]
- L. H. Gabrielli and M. Lipson, “Integrated Luneburg lens via ultra-strong index gradient on silicon,” Opt. Express19(21), 20122–20127 (2011). [CrossRef] [PubMed]
- M.-C. Wu, F.-C. Hsiao, and S.-Y. Tseng, “Adiabatic mode conversion in multimode waveguides using chirped computer-generated planar holograms,” IEEE Photon. Technol. Lett.23(12), 807–809 (2011). [CrossRef]
- P. Markov, J. G. Valentine, and S. M. Weiss, “Fiber-to-chip coupler designed using an optical transformation,” Opt. Express20(13), 14705–14713 (2012). [CrossRef] [PubMed]
- D. Dai, Y. Tang, and J. E. Bowers, “Mode conversion in tapered submicron silicon ridge optical waveguides,” Opt. Express20(12), 13425–13439 (2012). [CrossRef] [PubMed]
- Z. Yu and S. Fan, “Integrated nonmagnetic optical isolators based on photonic transitions,” IEEE J. Sel. Top. Quantum Electron.16(2), 459–466 (2010). [CrossRef]
- R. N. Mahalati, D. Askarov, J. P. Wilde, and J. M. Kahn, “Adaptive control of input field to achieve desired output intensity profile in multimode fiber with random mode coupling,” Opt. Express20(13), 14321–14337 (2012). [CrossRef] [PubMed]
- B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett.14(12), 630–632 (1989). [CrossRef] [PubMed]
- D. H. Broaddus, M. A. Foster, O. Kuzucu, A. C. Turner-Foster, K. W. Koch, M. Lipson, and A. L. Gaeta, “Temporal-imaging system with simple external-clock triggering,” Opt. Express18(13), 14262–14269 (2010). [CrossRef] [PubMed]
- V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett.28(15), 1302–1304 (2003). [CrossRef] [PubMed]
- R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).
- D. A. B. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge, 2008).
- G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics (Springer-Verlag, 2002).
- E. Kreysig, Introductory Functional Analysis with Applications (Wiley, 1978).

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