Idriss J Aberkane
email: idriss.aberkane@polytechnique.edu

On the Syracuse conjecture over the binary tree

We investigate recurrent behaviors of the 1 2 (3x + 1) action ("Syr") over the binary tree. The original composition of otherwise simple algebraic behaviors (eg: if a = 4b + 1, then Syr(a) = 4(Syr(b))) allows us to identify the existence of a relation merging orbits over alternate pairs of any odd branch of the tree. We prove that it is possible to compress any Syracuse orbits to critical positions in the tree and give new hints as to how to predict those positions. This allows us to establish a novel research program for the resolution of the Collatz conjecture, of which we also introduce original, and simpler conjectures. This article proves that for any even number e, the orbits of V ert(e) := 4e + 1 and S(V ert(e)) := 2(4e + 1) + 1 merge, but also that for any k that is even, S k (V ert(e)) and S k+1 (V ert(e)) can be proven to merge. For any odd number o, S(V ert(o)) and S 2 (V ert(o)) also merge, and so do any S k (V ert(o)) and S k+1 (V ert(o)) for any odd number k. Another significant result is that for any odd number o, either 8o + 1 and 16o + 1 merge or 8o + 1 merges with 16(V ert(o)) + 1 and 16o + 1 merges with 2o -1. The main result of this paper is that for any odd number o, proving that the orbits of 4o + 1 and 2(4o + 1) + 1 merge will also prove the Collatz conjecture. That such orbits could be systematically proven to merge we call the Golden Gate Conjecture and outline some aspects of a research program attacking this conjecture.

Introduction

In this paper we study the 3x + 1 problem, also known as the Collatz problem, the Syracuse conjecture, Kakutani's problem and several other names.

The 3x + 1 problem concerns the iteration of the following function

T (x) = 3x+1 2 , if x ≡ 1(mod2); x 2 , if x ≡ 0(mod2).
The Syracuse conjecture asserts, that for all x ≥ 1 exist a number k ∈ N such that T k (x) = 1. Iterations of the T function are however known to produce complicated -albeit strictly deterministicorbits when recurrently applied to natural numbers. As these orbits are still very poorly understood, the Syracuse conjecture is currently unsolved, and considered a textbook example of wild deterministic chaos, so much so that Paul Erdos famously declared "mathematics may not be ready for such problems". [START_REF] Lagarias | The Ultimate Challenge: The 3x+1 Problem[END_REF] (p.57). A good start to attack Syracuse then, would consist of making the problem -and the tools -ready for further enquiry. In this paper we attempt to identify, predict and compose some of the behaviors of the Syracuse orbits in the binary tree. The interest of this approach is twofold. Firstly, successful predictions could more thoroughly open the problem to artificial intelligence. Secondly, it could expose otherwise unknown algebraic vulnerabilities, as well as a more general strategy to attack problems in discrete mathematics. As we will prove, our attack of the Syracuse problem already yields the significant result that demonstrating that, for any odd integer o, the orbits of 4o + 1 and 2(4o + 1) + 1 merge proves the conjecture.

We begin by representing all numbers in the form of the following tree, in which all numbers x ∈ N are vertices. The lowest level of the tree consists of numbers 1 and 2 with number 0 (not represented here) at level 0. Every level n > 1 of this tree consists of all numbers from 2 n-1 + 1 up to 2 n .

Any vertex a is connected with two vertices 2a and 2a + 1. In the figure we will show the following connections. If a > 1 and a = 2 k the vertex 2a will be connected with 2 2 a. And for any a > 1 the vertex 2a + 1 will be connected with 2 • (2a + 1) and 2 • (2a + 1) + 1.

We will call a number x red if it is odd, and if x is even we will call it blue.

The Figure 1 thus shows the first five levels of the tree, with red numbers marked with one circle and blue numbers with two circles. We now wish to find regular behaviors of the Syracuse orbits of any numbers within this tree.

Decreasing of the rank

Let us consider an action ρ(x) = (x-1) 2 . The rank of odd number x, denoted by rank(x), is defined to be the number of times we will need apply ρ to the element x until we get an even number. The rank is defined as positive non-zero only for odd numbers ; even numbers are of rank zero by definition. A number x = 2 n+1 • t + (2 n -1) has a rank n. (This is a general formula for a n-rank number.)

We call an element y an image of x if y = Syr(x), this action is defined by the following expression

Syr(x) = (3x + 1) 2 = 3 2 (x + 1) -1.
Denote Syr(. . . Syr n (x)) = Syr n (x).

We will call the sequences generated by 3x+1 and /2 the orbit of element x and denote this sequence by Orb(x).

Theorem 1. Let rank(x) = n, where n > 1. Then rank(Syr(x)) = n -1.

Proof. The number x = 2 n+1 • t + (2 n -1) is odd and rank(x) = n. We first find a successor of x Syr

(x) = 1 2 (3x + 1) = 1 2 3 • 2 n+1 • t + (2 n -1) + 1 = 1 2 2 n+1 • 3 • t + 2 n • 3 -3 + 1 = = 1 2 2 n+1 • 3 • t + 2 n • 3 -2 = 2 n • 3 • t + 2 (n-1) • 3 -1.
Further we perform a few algebraic transformations

Syr(x) = 2 n • (3 • t) + 2 (n-1) • (2 + 1) -1 = 2 n • (3 • t) + 2 (n) + 2 (n-1) -1 = 2 n • (3t + 1) + (2 (n-1) -1)
.

By the formula of a n-rank number we find that rank(Syr(x)) = n -1. This finishes the proof.

Further definitions and statements

For the next two definitions let the number x have a rank(x) = 1.

Definition 1. We call a number x vertical blue if 1 4 (x -1) is even. Definition 2. We call a number x vertical red if 1 4 (x -1) is odd. The set of integers is closed under the action (4x + 1) but not closed under the action 1 4 (x -1).

Let V ert + (x) = V ert(x) = 4x + 1 and V ert -(x) = 1 4 (x -1). Note that V ert + (V ert -(x)) = x and V ert -(V ert + (x)) = x.
Definition 3. The orbits of two numbers are said to merge if they have at least one common number. Definition 4. Let x be odd and y = V ert + (x) then x is v-related to y. Proposition 1. Let x and y be v-related. Then the orbits of x and y merge.

Proof. Since x is v-related to y, we have Definition 5. Let a be an odd number of rank 1. The infinite set of numbers {a, S(a), S(S(a)), . . .} is called a red branch of root a. Definition 6. Let x be an odd number. If there is such a number y that Syr(y) = x, then x is called an odd number of type A and a remainder of dividing x by 3 is 2. If there is such a number y that Syr(y) = 2x, then x is called an odd number of type C, and a remainder of dividing x by 3 is 1. If x is divisible by 3 it is not of type A or type C and we call it of type B. Also note that if x is a type B number then 4x+1 is of type C and therefore there is such a y that Syr(y) = 2(V ert + (x)). In any case then, the orbits of x and y merge. Proposition 2. For any odd number x, there are infinitely many Syracuse orbits either leading to it or to Syr(x).

y = V ert + (x) = 4x + 1. Then 1 4 Syr(y) = 1 8 (3(4x + 1) + 1) = = 1 8 (12x + 4) = 1 2 (3x + 1) = Syr(x). Since
Proof. Let x be an odd number. If x is divisible by 3, then it is of type B and there is a number y so that Syr(y) is the double of V ert + (x). If x + 1 is divisible by 3 then it is of type A and there is a number y so that Syr(y) = x. If x + 2 is divisible by 3 then it is of type C and there is a number y so that Syr(y) = 2x.

As for any y so defined there are infinitely many V ert + (y) numbers merging to its orbit, and for any x so defined, also infinitely many positive numbers k defining V ert k (x) with each of them having at least one z of which the orbit will merge with either itself, its double, or the double of its vertical, there are indeed infinitely many orbits leading to the forward orbit of any odd number.

It is therefore possible, for any odd number x, to define a certain Syr -function.

Definition 7. We call a r-related to S(a) if Syr (rank(a)-1) (a) is vertical blue.

Theorem 2. Let a be vertical blue, then Syr(S(a)) is v-related to 1 2 Syr(a). Proof. Let a be vertical blue,

then b = V ert -(a) is even. Hence b = 2k, then a = V ert + (b) = 8k + 1. Then 1 2 Syr(a) = 1 2 1 2 3(8k + 1) + 1 = 1 4 (24k + 4) = 6k + 1, and
Syr(S(a)) = Syr(2(8k + 1) + 1) = Syr(16k + 3) = 1 2 3(16k + 3) + 1 = 24k + 5.

Since (6k + 1) is odd and V ert + ( 1 2 Syr(a)) = V ert + (6k + 1) = 4(6k + 1) + 1 = 24k + 5, number Syr(S(a)) is v-related to 1 2 Syr(a).

Let us remark that Syr(S(a)) = 24k + 5 is vertical red because V ert -(24k + 5) = 6k + 1 is odd. From the proof we also have that V ert -(Syr(S(a))) = 1 2 Syr(a). Proposition 3. For any number a ∈ N and natural number n ≤ rank(a) we have the following identity S(Syr n (a)) = Syr n (S(a)).

Proof. We will consider only n ≤ rank(a) because for any greater n the Syr n (a) is not defined in a sufficiently monotonous manner (see "Avalanche" later in this article), as Syr rank(a) (a) will be an even number.

We will use a mathematical induction to prove the statement.

Let n = 1 then

S(Syr(a)) = S( 1 2 (3a + 1)) = 2 • 1 2 (3a + 1) + 1 = 3a + 2 = 1 2 (3(2a + 1) + 1) = Syr(S(a)).
Then let us assume that for any number less than k the identity S(Syr k (a)) = Syr k (S(a)) holds. Then we will show that it holds for k + 1 Let a be vertical red, then Syr(S(a)) is vertical blue

S(Syr

Proof. Let a be vertical blue with

V ert + (b) = a then V ert -(Syr(S(a))) = V ert -1 2 3(2a + 1) + 1 = V ert -1 2 6a + 3 + 1 = V ert -(3a + 2) = = 1 4 (3a + 2) -1 = 1 4 (3a + 1) = 1 4 3(4b + 1) + 1 = 1 4 (12b + 4) = 3b + 1.
Then if b is an even number, 3b + 1 has to be odd, and if b was an odd number, 3b + 1 would be even.

Proposition 5. Let a be r-related to S(a). Then a and S(a) merge.

Proof. Let a be r-related to S(a) then Syr (rank(a)-1) (a) is vertical blue by definition and rank Syr (rank(a)-1) (a) = 1.

By Proposition 3

Syr S Syr (rank(a)-1) (a) = Syr Syr rank(a)-1 S(a) = Syr rank(a) (S(a)).

Since Syr (rank(a)-1) (a) is vertical blue, by Theorem 2 number Syr(S(Syr (rank(a)-1) (a))) is v-related to 1 2 Syr(Syr (rank(a)-1) (a)).

Hence by the Proposition 1 numbers Syr rank(a) (S(a)) and 1 2 Syr rank(a) (a) merge. Then we get that a and S(a) merge.

Definition 8. The rank of pair {a, S(a)} is equal to rank(a).

Remark 1. Note that Proposition 3 actually explains the alternative distribution of r-relations along red branches:

(i) let the root a of an red branch be vertical red then for any k that is odd we have S k (a) and S k+1 (a) r-related;

(ii) if for some odd k numbers S k (a) and S k+1 (a) r-related, then the root a of an red branch is vertical red;

(iii) let the root a of an red branch be vertical blue then for any k that is even we have S k (a) and S k+1 (a) r-related;

(iv) if for some even k numbers S k (a) and S k+1 (a) r-related, then the root a of an red branch is vertical blue.

Proof. (i) Let a be the root of an red branch, and vertical red. Then V ert -(a) = 2k + 1 is an red number, then 1 4 (a -1) = 2k + 1 and a = 8k + 5. Then S(a) = 16k + 11 and S 2 (a) = 32k + 1. The pair (S(a), S 2 (a) ) is the first pair of even rank. To check if S(a) is r-related to S 2 (a) we should check if V ert -(Syr rank(S(a))-1 (S(a))) is red or blue. We have that

V ert -(Syr rank(S(a))-1 (S(a))) = V ert -(Syr(16k + 11)) = V ert -(24k + 17) = 6k + 4.
Since 6k + 4 is an even number Syr rank(S(a))-1 (S(a)) is vertical blue and S(a) is r-related to S 2 (a).

Let us then find the rank of pair (S(a) 2t-1 , S 2t (a))

rank( (S 2t-1 (a), S 2t (a)) ) = rank(S 2t-1 (a)) = 2t.
Then we can find that

S 2t-1 (a) = 2 2t-1 a + 2 2t-1 -1 and Syr 2t-1 (S 2t-1 (a)) = 3 2t-1 a + 3 2t-1 -1. Hence V ert -( Syr 2t-1 (S 2t-1 (a)) ) = 1 4 (3 2t-1 a + 3 2t-1 -2) = a = 8k + 5 = 1 4 (3 2t-1 (8k + 5) + 3 2t-1 -2) = = 1 4 (3 2t-1 8k + 5 • 3 2t-1 + 3 2t-1 -2) = 1 4 (3 2t-1 8k + 6 • 3 2t-1 -2) = 1 4 (3 2t-1 • 2 • 4 • k + 2 • 3 • 3 2t-1 -2) = = 3 2t-1 • 2 • k + 1 4 (2(3 • 3 2t-1 -1) = 3 2t-1 • 2 • k + 2 1 4 (3 2t -1) = 3 2t-1 • 2 • k + 1 2 (3 2t -1) = = 3 2t-1 • 2 • k + 1 2 (3 t -1) • (3 t + 1).
Since (3 t -1) and

(3 t + 1) are even, 1 2 • (3 t -1) • (3 t + 1
) is even and the whole expression

V ert -( Syr 2t-1 (S 2t-1 (a)) ) = 3 2t-1 • 2 • k + 1 2 • (3 t -1) • (3 t + 1) is even. Hence Syr 2t-1 (S 2t-1 (a)) is vertical blue then S 2t-1 (a) is r-related to S 2t (a).
(ii) Let k = 2h -1 be an odd number and S k (a) is r-related to S k+1 (a).

Hence

Syr (rank(S k (a))-1) (S k (a)) is vertical blue that is V ert -(Syr (rank(S k (a))-1) (S k (a)
)) ia blue. We already have the next chain of equalities

V ert -(Syr (rank(S k (a))-1) (S k (a))) = V ert -(Syr (rank(S 2h-1 (a))-1) (S 2t-1 (a))) = V ert -( Syr 2h-1 (S 2h-1 (a)) ) = = 1 4 (3 2h-1 (8k + 5) + 3 2h-1 -2). Hence 1 4 (3 2h-1 a + 3 2h-1 -2) is even. Let us transform last expression 3 2h-1 a -3 2h-1 4 + 2 • 3 2h-1 4 - 2 4 = 3 2h-1 • a -1 4 + 2 4 (3 2h-1 -1) = 3 2h-1 • a -1 4 + 1 2 (3 • (3 2h-2 -1) + 2) = = 3 2h-1 • a -1 4 + 3 • (3 h-1 -1)(3 h-1 + 1) 2 + 1.
We know that 3

•(3 h-1 -1)(3 h-1 +1) 2 + 1 is odd. Since 3 2h-1 • a-1 4 + 3•(3 h-1 -1)(3 h-1 +1) 2 + 1 is even we get that 3 2h-1 • a-1
4 is odd. Hence a-1 4 is odd and a is vertical red. The proof of the case (iii) and (iv) can be obtained analogously.

Definition 9. Let a be an odd number. The Glacis of bottom a is an infinite set of odd numbers

{2a + 1, 4a + 1, 8a + 1, . . . , 2 n a + 1, . . .}.
Note that numbers 2 n + 1, where n ∈ N are glacis numbers from the glacis of bottom 1.

Definition 10. For each number 2 n a+1 in a glacis we define its glacis coordinates as the pair (a; n-2), where the first coordinate is the unique glacis bottom and the second coordinate is called the glacis order of the element.

By this definition the glacis coordinates of number 2a + 1 are (a; -1), for number 4a + 1 they are (a; 0) and number 8a + 1 has coordinates (a; 1).

We will call glacis numbers (a; 2k) with an even glacis order Type Vert numbers and the others shall be Type Succ numbers.

We will later explain the logic of this naming.

Main results

We shall now study the behavior of glacis numbers and establish the importance of studying the transformation of glacis under Syracuse.

Lemma 1. Let g1 and g2 be glacis elements of glacis coordinates (x; 1) and (x; 2) respectively, and 3x be either vertical blue or r-related to S(3x). Then g1 and g2 merge.

Proof. Let elements g1 and g2 belong to the glacis of bottom x, then these elements are respectively of the form g1 = 8x + 1 and g2 = 16x + 1.

Let us study their orbits further. Firstly, we consider the elements of Orb(g2). We have that

Syr(g2) = 3 2 (16x + 1 + 1) -1 = 3(8x + 1) -1 = 24x + 2
is an even number, then, the next number in Orb(g2) is

1 2 Syr(g2) = 12x + 1.
The number 1 2 Syr(g2) = 12x + 1 belongs to the glacis of bottom 3x and 3x is odd because x is odd. Then (12x + 1) is vertical red because V ert -(12x + 1) = 1 4 ((12x + 1) -1) = 3x is an odd number. Hence (12x + 1) and 3x are v-related therefore by Proposition 1 they merge. Hence (16x + 1) and 3x merge. Now we consider some consecutive elements of Orb(g1).

1 2 Syr(g1) = 1 2 ( 3 2 (8x + 1 + 1) -1) = 6x + 1.
The number 6x + 1 also belongs to the glacis of bottom 3x. Since 3x is vertical blue or r-related to S(3x), by Remark 1 numbers 3x and (6x + 1) merge. Thus (8x + 1) and 3x merge. Since (16x + 1) and (8x + 1) merge with the same number 3x, these numbers also merge, which finishes the proof.

Definition 11. When Lemma 1 holds for glacis numbers g1 and g2, namely, when Syr(g1) and Syr(g2) belong to a glacis of which the bottom is either vertical blue or r-related to its successor, we will call these numbers vanilla-related.

Figure 2 shows how vanilla can be represented on the binary tree. Let us also define g 2 = Ban -(g2), g 1 = Ban + (g1), and 3x be vertical red or r-related to 1 2 (3x -1). Then g1 and g 1 merge, and so do g2 and g 2 Proof. Let us find how numbers g1, g 1, g2, g 2 depend from the glacis bottom x. We already have g1 = 8x + 1, g2 = 16x + 1. Also g 1 = 8 • (8x + 1) + 9 = 16 (4x + 1) + 1 = 16 • V ert(x) + 1 and has glacis coordinates (V ert + (x); 2). And

g 2 = 1 8 (16x + 1 -9) = 2x -1. Since 1 2 Syr(g2) = 1 2 ( 1 2 (3(16x + 1) + 1)) = 4 • 3x + 1 = V ert(3x)
, it follows that g2 and V ert(3x) merge. Since by Proposition 1 numbers V ert(3x) and 3x merge, we have that g2 and 3x merge. We have 1 2 Syr(g

2) = 1 2 ( 1 2 (3(2x -1) + 1)) = 1 2 (3x -1) from this g 2 and 1 2 (3x -1) merge. Since 1 2 Syr(g1) = 1 2 ( 1 2 (3(8x + 1) + 1)) = 2 • 3x + 1 = S(3x)
we have that g1 and S(3x) merge. The equality 1 2 Syr(g

1) = 1 2 ( 1 2 (3(64x + 17) + 1)) = 1 4 (192x + 52) = 48x + 13 = 4 • (2 • (2 • 3x + 1) + 1) + 1 = V ert(S 2 (3x)) implies that g 1 and S 2 (3x) merge. If 3x is vertical red then V ert -(3x) exists and 1 2 Syr(g 2) = 2 • 1 4 (3x -1) = 2V ert -(3x). Hence 1 4 Syr(g 2) = V ert -(3x) and V ert + (V ert -(3x)) = 3x
then numbers g 2 and 3x merge. Since 1 2 Syr(g2) = V ert(3x) numbers g2 and 3x merge and we have that g2 and g 2 merge.

Let 3x be vertical red then by Remark 1 number S(3x) is r-related to S 2 (3x). Then S(3x) and S 2 (3x) merge. Hence g 1 and g1 merge.

Let 3x be r-related to 1 2 (3x -1) then by Proposition 5 numbers 3x and 1 2 (3x -1) merge. Hence g2 and g 2 merge. Since by the Remark 1 S(3x) and S 2 (3x) merge it follows that g 1 and g1 merge Definition 12. If Lemma 2 holds for glacis numbers g 1, g1 and g 2 and g2 we will say that g 1, g1 are banana-related and that so are g 2 and g2

Figure 3 shows how banana can be represented on the binary tree. The algebraic phenomenon behind the Banana and Vanilla relations is the following: for any two consecutive first glacis numbers g1 and g2 where g1 is of order 1 and g2 of order 2, and the glacis is of bottom a 1 2 Syr(g2) = V ert(3a),

1 2 Syr(g1) = Succ(3a).
Then only two cases are possible: either 3a is r-related to S(3a) or it is not. If 3a is r-related to S(3a) then it means g2 and g1 merge. If 3a is not r-related to S(3a) it means S(3a) is r-related to S 2 (3a) and that either 3a is vertical red or is r-related to S -(3a) = 1 2 (3a -1). Now 3a is necessarily a type B number, therefore V ert -(3a), if it exists, is a type A number, so 3a -1 = 4 • V ert -(3a) has a number y such that Syr(y) = 3a -1 this number y is the number we have called g 2.

When 3a is r-related to 1 2 (3a -1), this latter number is of type C, therefore there exists a number y such that 1 2 Syr(y) = 3a -1 which is number g 2 again. Since S 2 (3x) is r-related to S(3x) and 3x is of type B, S 2 (3x) is of type C, therefore there is a number z such that Syr(z) = 2 • S 2 (3x) and this number is the one we have called g 1.

Therefore we have the following :

Theorem 3. (Banana-Split Theorem) For numbers g2 with glacis coordinates (x; 2) and g1 with glacis coordinates (x; 1), only one of the two following statements holds: (i) g2 and g1 merge;

(ii) g2 and g 2 = Ban -(g2) = 1 8 (g2 -9) merge and g1 and g 1 = Ban + (g1) = 8 • g1 + 9 merge.

For the record, the author, having to find a name for the banana-relation on his handwritten notes, and being used to noting them as three black lines from one number to the other in the binary tree, with the three lines being curved, found it evoked a banana, hence the name.

Any Syracuse orbit is a correspondence mapping red branches to glacis and vice versa. By correspondence we mean that any Syracuse orbit is a sequence of odd numbers of which the rank is either greater than one, and they are therefore red branch numbers, or equal to one, in which case they are red branch roots and therefore glacis numbers.

Since any odd number of rank n greater than one will be reduced to an odd number of rank 1 in a finite number of Syr action, and that any odd number of rank 1 is in a glacis, any element of a red branch will finitely orbit to a glacis.

Glacis numbers, in turn, all finitely orbit to a red branch (get to a red branch in a finite number of steps), with only two possible patterns (ie. cases): either they are of T ypeV ert and their glacis reduction to order 0 under a finite amount of 1 2 Syr actions will be the V ert + of a power of three of the bottom of their initial glacis.

Or they are T ypeSucc and their glacis reduction to order -1 under a finite amount of 1 2 Syr actions will be the Successor of a power of three of the bottom of their initial glacis.

Therefore, an important observation is that the forward orbit of any odd number (i) If it is a red branch number of rank above 1, can only occupy red branches of type A until it reaches a glacis.

(ii) If it is a glacis number of order above 0, Can only occupy glacis of type B bottom (let us call it b) and therefore of type C glacis numbers until it reaches a red branch in either a typeB number

(3 k • b) or a type C number S(3 k • b)).
Definition 13. A bud is any pair (a; S(a)), where a is vertical red.

Definition 14. Let a be a number for which V ert -(a) is not an integer. If a is even, the cob of base a is the infinite set of buds {(V ert + (a); S(V ert + (a)), (V ert +2 (a); S(V ert +2 a)), . . .}.

If a is odd, the cob of base a is the infinite set of buds {(a; S(a)), (V ert + (a); S(V ert + (a)), (V ert +2 (a); S(V ert +2 a)), . . .}

and it has the base a.

Since any number, either odd or even, has a V ert + number, there are indeed two kinds of cobs: those of blue base (blue cob), and those of red base (red cob).

Definition 15. Let a be a vertical red number. The verticality of number a are the coordinates (b, m) such that b is the smallest number and m the highest number satisfying the equation a = V ert m (b).

Definition 16. The verticality of a bud (V ert +n (a); S(V ert +n a)) is equal to the verticality of its smallest element V ert +n (a). Proposition 7. Let (a, S(a)) be a bud, and of verticality (b, m). (Nota Bene: that (a, S(a)) be defined as a bud implies that if b is even, m is greater than 1) Then if b is even, Syr(S(a)) will have the glacis coordinates ( 12 Syr(V ert(b)); 2(m -1)) and will therefore be of type V ert. If b is odd, Syr(S(a)) will have the glacis coordinates ( 12 Syr(b); 2m -1) and will therefore be of type Succ. Therefore note that the image of any cob of base b under one Syr action is either, if the cob is of an even base, all the typeV ert glacis numbers of the glacis of bottom Also note that in the same way that for x a number of rank n > 0 Syr n (a) may be calculated as

(x + 1) • ( 3 2 ) n -1.
For a glacis number of order 2k + 1 or 2k T yr k+1 (a) may be calculated as (a -1) • ( 3 4 ) k+1 + 1. Proof. This is a re-stating of propositions that were already proven in the Banana-Split Theorem Also, the latter two propositions implies that Proposition 9. Let S(a) be the largest bud number of bud (a, S(a)), with vertical coordinates (b, m).

Then if b is odd T yr m (Syr(S(a))) = S(Syr(b) • 3 m ), if b is even then T yr m (Syr(S(a))) = V ert(Syr(b) • 3 m ).
Proof. This is also a re-stating of propositions that we proved earlier.

Therefore, if we consider any red branch, and take a number a of this branch, with rank n > 1 that is r-related to 1 2 (a -1) we know that Syr n (a) will be vertical red of a certain verticality, and therefore Syr n (a) and S(Syr n (a)) will be forming a bud. If we could predict exactly the verticality (b, m) of the pair (Syr Meaning for any red branch number, we could predict the next red branch its orbit will intersect, namely the next time it will have a rank above 1, after it has been reduced to rank 1 for the first time by a finite number of Syr actions.

In a further work, we shall outline a method to achieve such a prediction.

Definition 17. Let (a, S(a)) be a bud. Under one Syr, a is transformed into a power of 4 of the bottom of the glacis of which Syr(S(a)) will be an element. This critical separation of two adjacent red branch numbers of the binary tree into a non-adjacent pair composed of a glacis bottom and a glacis number we call an avalanche.

The avalanche phenomenons accounts for a decisive part of the chaotic behavior of Syracuse orbits. In fact, the only two sources of the wild behavior of all Syracuse orbits are Definition 18. The consequence of the Syr n on the vertical coordinates of buds, namely, for a bud (a, S(a)) of vertical coordinates (b, m) what will be the vertical coordinates of (Syr n (S n (a)), Syr n (S n+1 (a)))?

We call this problem the B2G problem, standing for "Branch to Glacis". Definition 19. The consequence of the 3x action on the red branch coordinates of a number a, namely, given a number a that has a certain rank and of which the root of its branch has certain vertical coordinates, what will be the vertical coordinates and the rank of 3 n a? We call this problem the G2B problem, standing for "Glacis to Branch".

The next conjectures, which are in fact equivalent, would be helpful in searching for the solution of the Syracuse Problem.

Conjecture 1. (Golden Gate Conjecture) Any bud is solvable, namely for any odd number a, the orbits of V ert(a) and S(V ert(a)), either backward or forward, can be proven to have at least one common number.

Conjecture 2. (Golden Avalanche Conjecture) Any glacis number can be proven to merge with its glacis bottom.

Proposition 10. Proving the Golden Gate Conjecture proves the Collatz Conjecture Proof. Along any red branch, we know that if a is not r-related to S(a) it means that Syr rank(a)-1 (a) is vertical red and Syr rank(a)-1 (S(a)) = S(Syr rank(a)-1 (a)). Suppose we had a demonstration that for any odd number b, V ert(b) and S(V ert(b)) can be proven to merge, it would imply that a and S(a) will merge as well. Therefore, such a demonstration would mean that, besides the r-relation which we have exposed in this paper, there is a -evidently much more complex -w-relation between any two r-related pairs along any red branch. This in turn would demonstrate that all the elements of the binary tree will merge.

For the record, the author of this article established the Golden Gate conjecture at the Lange Special Collection Reading Room of the University of California, San Francisco, with a view of the Golden Gate Bridge, a name altogether fitting for the definition of a "bridge" connecting two "red numbers" as they were colored in his personal notes. Obviously, proving the Golden Gate Conjecture will be no trivial work, but we may already make a few simple observations: Proposition 11. The solving of some buds implies the solving of some other buds Proof. Suppose bud (13, 27) is solved. Number 13 belong to the orbit of number 45, and number 91 belong to the orbit of number 27. Indeed ) is a bud. Proposition 12. It is possible for a bud to solve itself by having each two orbits of its elements cross an composition of r-relation demonstrating they merge, and this without having to bruteforce compute the orbits of the two elements until they reach number 1.

Proof. Let us consider the bud (157, 315). and its forward orbit. Syr(315) = 473 and Syr(157) = 236 = 4 • 59. Then Syr(59) = 89 and 1 2 Syr(473) = 355 = S(177) and 177 is vertical blue therefore 355 and 177 are r-related. Numbers 89 and 177 are two consecutive glacis numbers with coordinates (11, 1) and (11, 2) respectively. If 3 • 11 = 33 was r-related to S(33) = 67, then 89 and 177 would be proven to merge by a vanilla relation. It so happens that 33 is vertical blue, and is therefore r-related to 67, implying a vanilla-relation between 89 and 177, because 1 2 Syr(89) = 67 and 1 2 Syr(177) = 133 = V ert(33) Therefore 157 and 315 are proven to merge.

From a purely epistemological perspective, the chaotician that is experienced in theoretical biology may not fail to notice a certain intellectual similarity between the way bud (157, 315) is solved and that in which new covalent bonds are made between different atoms by the active sites of enzymes in biochemistry. With bud (157, 315) we have two unrelated "atoms" so to speak, and their manipulation in a few critical steps, similar to that between the few active sites of enzymes, "catalyzes" a relation, thus "bonding" them.

Conclusion

It seems possible to "make mathematics ready" to expose critical vulnerabilities in the Syracuse problem. In this paper, we have already proven that the simple algebraic relation Syr(4a + 1) = 4(Syr(a)) produces non-trivial emerging behaviors in the Binary Tree. Firstly, it establishes the existence of a relation merging the orbits of alternate pairs of odd numbers along any odd branches of the tree. The existence of this relation implies that whoever can prove that for any odd number a, the orbits of 4a + 1 and S(4a + 1) merge solves Syracuse and that this be feasible we have called the Golden Gate conjecture. Secondly, we have demonstrated that a special, inevitable relation also merging their orbits exists between precisely defined pairs of odd numbers of rank 1, and the proof of this relation we have called the Banana-Split Theorem. Since any odd number of rank greater than 1 will be finitely transformed into an odd number of rank 1 under the Syr action, this result has some significance as it will manifest in the orbit of absolutely any odd number. We have also remarked that the forward orbit of any odd number of rank above 1 may only occupy red branches of type A, and that the forward orbit of any glacis number of order above 0 may only intersect a red branch number of either type B or C, and fly only through glacis of type B bottom.

Moreover, as whoever can successfully attack buds will successfully crack Syracuse, we may now outline a research program to expose vulnerabilities -whether decisive or not -in the buds of the Binary Tree; let us call it the "GoldenP rogram", and split it in two scientific tasks that may be endeavored in parallel of each other, one regarding automatic theorem proving and deep learning, for example, in the vein of [START_REF] Bowling | Heads-up limit hold'em poker is solved[END_REF], and the other regarding a more analytic approach. Let us call the first program that of the "GoldenAutomaton", and the second one, the "GoldenF ormula".

The first approach would consist of coding any family of "Golden Automata", able to solve any buds, namely, to demonstrate that for any two numbers forming a bud in the Binary Tree, their orbit can be proven to merge. Such automata could be assembled by mobilizing the current techniques of artificial evolution and deep learning and -which is an innovation per se -considering the Binary Tree as a self-calculating, infinite dataset.

The second approach may be defined more precisely, and will consist of solving and composing what we have called, in this paper, the B2G and G2B problems. Each of the solutions to these problems will imply the existence of a precise formula mapping branches to glacis, and glacis to branches respectively, the composition of which will provide us with a completely new understanding of the Syracuse orbits.

We shall contribute to the advancement of these two programs in a next paper.

Acknowledgments

The Finally, the author fully acknowledges the remarkably thorough work of a reviewer under the alias of Krystal Désirée Okanda for improvement, correction and cleaner demonstration especially that of theorem 1. The author unambiguously acknowledges that her contribution in sharpness and clarity is so good that, in his opinion, Krystal D. Okanda may be acknowledged as an official contributor of this

Figure 1 .

 1 Figure 1. Binary tree

( 1 4

 1 Syr(y)) ∈ Orb(y) and 1 4 Syr(y) = Syr(x) the orbits of x and y merge. Denote S(a) = 2 • a + 1 and S(. . . S n (a)) = S n (a) = 2 n a + 2 n -1, we call S(a) a successor of a.

  k+1

  (a)) = S(Syr k (Syr(a))) = Syr k (S(Syr(a))) = Syr k (Syr(S(a))) = Syr k+1 (S(a)). Hence S(Syr n (a)) = Syr n (S(a)) holds for any number a ∈ N and natural number n ≤ rank(a).

Proposition 4 .

 4 Let a be vertical blue, then Syr(S(a)) is vertical red.

1 vanillaFigure 2 .Lemma 2 .

 122 Figure 2. Vanilla Let us define an action Ban + (a) = 8a + 9 and Ban -(a) = 1 8 (a -9). By default, just as we use the notation V ert(a) for the exact V ert + (a) we will also use Ban(a) for Ban + (a). Lemma 2. Let g2 have glacis coordinates (x; 2), g1 have glacis coordinates (x; 1)Let us also define g 2 = Ban -(g2), g 1 = Ban + (g1), and 3x be vertical red or r-related to 1 2 (3x -1). Then g1 and g 1 merge, and so do g2 and g 2

  Figure 3. Banana

1 2

 1 Syr(V ert(b)), or if the cob is of odd base, all the typeSucc numbers of the glacis of bottom Syr(b) Proof. If b is odd, then we know that Syr(V ert m (b)) = 4 m Syr(b) and Syr(S(a)) = S(4 m Syr(b)) = 2(4 m Syr(b)+1) By definition of the glacis coordinates, Syr(S(a)) is therefore of coordinates ( 1 2 Syr(b); 2m -1). If b is even, for any m greater than 1, Syr(V ert m (b)) = 4 m-Syr(V ert(b)) And Syr(S(a)) = S(4 m-Syr(V ert(b))) = 2(4 m-1 Syr(V ert(b)) + 1) By definition of the glacis coordinates Syr(S(a)) is therefore of coordinates ( 1 2 Syr(V ert(b)); 2m -1) Proposition 8. Let a be a glacis number of coordinates (b, 2k + 1) where k is greater or equal to 0 Then defining T yr(a) := 1 2 Syr(a) T yr k+1 (a) = S(3 k+1 Syr(b)) if a is a glacis number of coordinates (b, 2k) then T yr k+1 (a) = V ert(3 k+1 Syr(b))

  n (a), S(Syr n (a))), we could know that either T yr m (Syr(S(Syr n (a)))) = S(Syr(b) • 3 m ) or T yr m (Syr(S(Syr n (a)))) = V ert(Syr(b) • 3 m )

  author acknowledges this work could not have been accomplished without the decisive support of the late Stanford University Professors Alan Tower Waterman Junior and Solomon Feferman who respectively hosted and invited him to conduct preliminary research in concept theory and metamathematics as a Visiting Scholar of Stanford University in 2006-2007.This work could not have existed without the latter invitation, and also without the Summer School of the Kozmetsky Global Collaboratory of Stanford University of 2008. The author wishes to thank Prof. Yakov Eliashberg who was head of the Department of Mathematics at the time of his first invitation, and Dr. Syed Shariq -most especiallyco-Chair of the Kozmetsky Global Collaboratory. The author also wishes to thank Prof. Paul Bourgine, founding President of the Unesco-Unitwin Complex Systems Digital Campus, and Prof. Pierre Collet, co-chair. The institutional merit of this research is equally shared by the CS-DC, Stanford University and 42 Silicon Valley.

work and even a second author, a title to which nobody else may pretend.

In Memoriam

Solomon Feferman Alan Tower Waterman Jr.

Proposition 6. Let the glacis coordinates of x be (a; n), where n > 0. Then the glacis coordinates of 1 2 Syr(x) are (3a; n -2). Proof. By the statement of the proposition x = 2 n+2 a + 1. Then 1 2 (Syr(x)) = 1 4 (3(2 n+2 a + 1) + 1) = = 2 n • (3a) + 1. Hence glacis the coordinates of 1 2 Syr(x) are (3a; n -2).

and number y is of glacis order m = 2 in the same glacis then 1 2 Syr(x) = S(V ert -( 1 2 Syr(y))) This explains why we have called glacis numbers with an odd order "type Vert" and the ones with an even order "type Succ".

We may now generalize the formula to calculate the progression of glacis numbers. Let a be any odd number. All type Succ numbers of its glacis are written

and all type Vert numbers are written

Any glacis number g of order 2k or 2k-1 may be reduced to rank 0 or -1 by the following transformation

then on we have (i) for type Succ numbers :

(ii) for type Vert numbers :

As these equalities will fit any number a, we have indeed that any glacis number of order 2k will be finitely mapped to 4a • 3 k + 1 = V ert(a • 3 k ) and that any glacis number of order 2k -1 will be mapped to 2a • 3 k + 1 = S(a • 3 k ).


Sonia Boscolo

Frédéric Audo

Julien Fatome

Christophe Finot
email: christophe.finot@u-bourgogne.fr

Nonlinear spectral compression in optical fiber: A new tool for processing degraded signals

Keywords: Nonlinear fiber optics, spectral compression, signal processing

We propose two new applications of the spectral focusing by self-phase modulation that occurs in a nonlinear optical fiber. We numerically show the possibility of using nonlinear spectral compression to improve the optical signal to noise ratio and mitigate the amplitude jitter of the signal pulses. We also demonstrate experimentally that use of spectral focusing in a combination with an external sinusoidal phase modulation achieves efficient suppression of coherent spectral background.

INTRODUCTION

The effect of self-phase modulation (SPM) on an optical pulse propagating in a nonlinear optical fiber is ordinarily associated with broadening of the pulse spectrum [START_REF] Stolen | Self-phase modulation in silica optical fibers[END_REF]. However, this is not always the case with chirped pulses, where pulses with an initial negative chirp (such as that imparted by an anomalously dispersive element) can experience spectral narrowing [START_REF] Zohrabian | Compression of the spectrum of picosecond ultrashort pulses[END_REF][START_REF] Oberthaler | Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers[END_REF]. Indeed, as a result of SPM, the propagating pulse will have the frequencies of the leading edge downshifted and the frequencies of the trailing edge upshifted. Therefore, in the case of an initially negatively chirped pulse, both the trailing and leading edges of the pulse will experience a spectral shift in the direction of the central frequency. This method of spectral compression (SC) has been implemented using various types of fibers and wavelengths [START_REF] Andresen | Spectral Analog of the Gouy Phase Shift[END_REF][START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF]. The process can also sustain simultaneous amplification of the pulse [START_REF] Limpert | SPM-induced spectral compression of picosecond pulses in a single-mode Yb-doped fiber amplifier[END_REF], thereby providing an attractive solution to convert ultra-short pulses delivered by femtosecond oscillators into high-power, near-transform-limited picosecond pulses, and to counteract the spectrum expansion that usually occurs with the direct amplification of picosecond structures. The concept can be extended to fiber laser cavities [START_REF] Boscolo | Amplifier similariton fiber laser with nonlinear spectral compression[END_REF].

In this paper, we propose two new applications of the nonlinear SC process. We design and numerically assess the performance of an optical power limiter combining nonlinear SC with centered optical band-pass filtering for the regeneration of intensity-modulated signals. We also experimentally demonstrate that the combination of SC with an external sinusoidal phase modulation applied to the pulses can significantly increase the extinction ratio of a signal degraded by a coherent background.

OPTICAL POWER LIMITING FUNCTION

Impact of input signal-to-noise ratio

Several recent theoretical and experimental studies of SC by SPM [START_REF] Andresen | Spectral Analog of the Gouy Phase Shift[END_REF][START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF][START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF][START_REF] Boscolo | Enhanced nonlinear spectral compression in fibre by external sinusoidal phase modulation[END_REF] have been concerned with the reduction of the intensity level of the residual side lobes in the compressed spectrum that, in the nonlinearitydominant regime of propagation, result from the mismatch between the initial linear chirp of the pulse and the SPM-induced nonlinear chirp. However, to date, the problem of the stability of the SC process against various forms of quality degradation of the initial pulses, such as amplitude fluctuations or broadband noise, has not been yet considered. Here, we analyze the impact of amplitude jitter and degraded optical signal-to-noise ratio (OSNR) of the seed pulses on the compression process by means of extensive numerical simulations of the nonlinear Schrödinger equation. Remarkably, our results show that the spectral compression is rather stable against these pulse degradation factors. An example of this stability is illustrated in Fig. 1(a), which shows the averaged spectra of 1000 pulses after experiencing SC in a highly nonlinear fiber obtained for different input OSNR values. The pulses at the fiber input are initially 2-ps Gaussian pulses after temporal stretching by a factor of 20. It is remarkable that the OSNR of the spectrally focused pulses is nearly an order of magnitude higher than that of the seed pulses for input OSNR values above 10 dB [Fig. 1(b)]. The fall-off in OSNR improvement for highly degraded input pulses can be partly ascribed to the inherent sensitivity of the SPM effect driving SC to fluctuations in initial pulse duration. Even though improvement of the extinction ratio is still observed, the statistical distribution of the one-bit level at the regenerator output maintains a Gaussian shape but is narrower than the input distribution. The improvement in signal quality is therefore marginal. We attribute this reduction of the device's performance to the influence of the jitter of the temporal width incurred by the input pulses because of the added amplified spontaneous emission noise. Indeed, as already demonstrated in the Mamyshev's regenerator [START_REF] Finot | Numerical study of an optical regenerator exploiting self-phase modulation and spectral offset filtering at 40 Gbit/s[END_REF], contrary to other nonlinear regenerators such as nonlinear optical loop mirrors whose TFs only depend on the instantaneous pulse power, a variation in the input pulse duration affects the TF of our device by shifting the optimum operational power, thus impacting its beneficial effect.

Figure 2. (a) TF of the power limiter. (b-c) Eye diagrams of a 4096 pseudorandom binary pulse train and distribution of the one-bit pulse level. Panels 1: input pulse train impaired by an rms amplitude jitter of 8% of ideal

power, filtered by a 20-GHz OBPF; panels 2: pulse train at the regenerator output; panels 3: pulse train at the regenerator output when the initial signal has an OSNR of 17 dB.

EXTINCTION RATIO IMPROVEMENT

Principle of operation

Here we describe how SC can help enhance a pulsed signal degraded by a coherent background such as, for instance, the pulses generated using a lithium niobate modulator. Because of the finite extinction ratio of the modulator (typically around 23 dB), at low pulse repetition rates the energy contained in the unwanted residual background is far from negligible and may impair the practical applications of the generated pulses. The method that we propose to eliminate such spurious background is illustrated in Fig. 3 and relies on three successive stages. We first impose on the initial signal (a 120-ps Gaussian pulse with an extinction ratio of 23 dB in the example of Fig. 3) a sinusoidal modulation of the temporal phase with amplitude A and frequency mod. The effect of this modulation is to create spectral replica located at frequencies n mod and with amplitude Jn(A),

where Jn is the Bessel function of the first kind of order n. By choosing A = 2.405 the replica at the central frequency vanishes and all of the energy is transferred to the higher-order replicas. The second stage involves the SC of the modulated signal in a Kerr medium such as an optical fiber operating in the highly nonlinear regime of propagation. The effect of the SPM experienced by the propagating signal is to focus the energy back to the central frequency. It is noteworthy that the use of an initial sinusoidal temporal phase modulation is particularly well-suited to the enhancement of the performance characteristics of the SC process [START_REF] Boscolo | Enhanced nonlinear spectral compression in fibre by external sinusoidal phase modulation[END_REF]. As the nonlinear process intrinsically discriminates the pulsed part of the signal from the coherent residual background, the energy of the background remains spread across several frequency-shifted coherent lines. Therefore, filtering the signal with an OBPF provides a simple way to remove the spurious spectral peaks and improve the extinction ratio by several tens of dB. The resulting intensity pulse profile is slightly broadened but remains close to a Fouriertransform limited waveform.

Proof-of-principle experiment

We experimentally validated the proposed concept using the setup sketched in Fig. 4(a), which relies on commercially available components dedicated to optical telecommunications. Gaussian-like pulses with a fullwidth at half maximum of 120 ps are generated at a repetition rate of 78.13 MHz. The extinction ratio of the lithium niobate modulator being used is 23 dB, which causes a significant portion of the energy to be contained in the coherent background. The signal is first temporally phase-modulated with a sinusoidal wave of frequency 4.7 GHz and amplitude A = 2.405. This leads to the formation of equally spaced spectral replicas, as seen in

CONLUSION

We have proposed and demonstrated two new applications of the process of SC arising from nonlinear pulse propagation in an optical fiber. We have numerically assessed the performance of an optical power limiter that combines nonlinear SC with optical band-pass filtering for the regeneration of intensity-modulated signals.

Further, we have presented a proof-of-concept of how SC can help improve the extinction ratio of a signal deteriorated by a coherent background.

Figure 1 .

 1 Figure 1. Impact of OSNR of the initial pulses on the SC process: (a) The averaged output spectrum of 1000 pulses at the point of optimum compression (solid) is compared with the input spectrum (dotted) for different input OSNR levels: 10 dB (blue), 20 dB (green) and 30 dB (red). (b) OSNR improvement versus input OSNR.

Figure 3 .

 3 Figure 3. Operating principle of the coherent background cancellation. (a) Temporal and (b) spectral characteristics of the signal at different stages: the initial degraded signal (blue), after external phase modulation (red), after spectral focusing (green), and after band-pass filtering (black, and dotted grey in subplot b3). The circles represent the intensity profile of an equivalent transform-limited pulse.

Fig. 4 (

 4 c1). The spectral focusing occurs in a 500-m long section of highly non-linear fiber with a nonlinear coefficient of 10 /W/km, and efficiently brings back the energy of the pulsed part of the signal to the central part of the spectrum [Fig.4(c2), yellow curve]. The energy associated the coherent background is then efficiently suppressed by a narrow-band optical filter. The resulting temporal waveform is slightly broadened compared with the initial signal but remains close to the Fourier transform limit. The estimated improvement in extinction ratio of the pulse train is more than 20 dB.

Figure 4 .

 4 Figure 4. (a) Experimental setup.(b) Temporal intensity profile of the pulses at different stages: input profile (black), after phase modulation and spectral compression (red), and after filtering (blue). The blue open circlesrepresent the transform-limited profile that can be derived from the spectrum after processing. (c) Spectral intensity profile of the pulses at different stages: input spectrum (black dotted line), after temporal phase modulation (red), after nonlinear propagation (yellow) and after spectral filtering (blue).

ACKNOWLEDGEMENTS

This work was supported by the European Union within the framework of the operational Program FEDER-FSE Bourgogne 2014/2020, by the Région Bourgogne (Pari Photcom) and by the Agence Nationale de la Recherche (Labex Action ANR-11-LABX-01-01).


Manon Lamy

Kamal Hammani
email: kamal.hammani@u-bourgogne.fr

Juan Arocas

Julien Fatome

Jean-Claude Weeber

Christophe Finot

Titanium Dioxide Waveguides for Data Transmissions at 1.55 µm and 1.98 µm

Keywords: Titanium Dioxide waveguides, mid-infrared telecommunications, metal grating couplers

We demonstrate error free transmissions of 10 Gbps signals in titanium dioxide waveguides at wavelengths of 1.55 or 2 µm. An efficient coupling of light is achieved thanks to metal grating couplers and we have checked that the component could be used with standard CWDM SFP+ devices.

INTRODUCTION

Exploring new spectral bands of optical transmission is one of the solutions to overcome the inexorable increase of data traffic. The emergence of thulium doped fiber amplifiers (TDFA) has focused the attention on the wavelengths around 2 µm [START_REF] Li | Thulium-doped fiber amplifier for optical communications at 2 µm[END_REF]. This trend has stimulated studies of dedicated photonic components such as InPbased modulators [START_REF] Ye | InP-Based Active and Passive Components for Communication Systems at 2 μm[END_REF][START_REF] Sadiq | 10 Gb/s InP-based Mach-Zehnder modulator for operation at 2 μm wavelengths[END_REF] or arrayed waveguide grating [START_REF] Zhang | Dense WDM transmission at 2 µm enabled by an arrayed waveguide grating[END_REF]. High bit rate communications over distances exceeding a hundred meters have already been successfully demonstrated in low-loss hollow core bandgap photonic fibers [START_REF] Petrovich | Demonstration of amplified data transmission at 2 µm in a low-loss wide bandwidth hollow core photonic bandgap fiber[END_REF][START_REF] Zhang | 100 Gbit/s WDM transmission at 2 µm: transmission studies in both low-loss hollow core photonic bandgap fiber and solid core fiber[END_REF].

Regarding integrated photonics, optical transmissions also deserve interest in the context of on-board connections as well as photonic routing. Therefore it becomes of a great interest to find a suitable platform operating both in the 2-µm spectral region as well as in the others more conventional spectral bands (ranging from 800 nm to 1550 nm). Titanium dioxide (TiO 2 ) is a good candidate for such platforms. This cost-efficient material is transparent from the visible to the mid-infrared wavelengths [START_REF] Kischkat | Mid-infrared optical properties of thin films of aluminum oxide, titanium dioxide, silicon dioxide, aluminum nitride, and silicon nitride[END_REF], has a negligible two photons absorption in the C-band and handles high power intensities. Moreover, it can be considered as a complementary metal-oxide semiconductor (CMOS) compatible material with an easier deposition process possible at lower temperature. Here, we present titanium dioxide waveguides operating at 1.55 µm and 1.98 µm wavelengths. After having described the design of the TiO 2 devices, we detail experimental set-ups and validation of the components for error-free transmission of a 10 Gbps On-Off keying signal in two different spectral regions.

DESIGN AND TEST OF THE PHOTONIC STRUCTURES

Design and fabrication of the couplers and the waveguides

Whatever the platform used, a crucial step is always to efficiently inject the light into the device. Basically, there are two main ways to couple light from an optical fiber to a photonic waveguide: the end-fire coupling (also known as butt-coupling) or grating coupling. Here, we have chosen metal grating couplers, because in addition to not requiring an accurate alignment, the fabrication appears easier thanks to an etching free process. Recently, such metal grating couplers have been used to demonstrate high coupling efficiencies [START_REF] Taillaert | Compact efficient broadband grating coupler for silicon-oninsulator waveguides[END_REF]. Unlike previous studies, we considered a metal (Au) grating buried between two TiO 2 layers on a glass substrate. This configuration offers an additional degree of freedom and a better coupling efficiency. We consider two different waveguides: first, a 1.65 µm-wide waveguide of a width around 1.65 µmincluding a taper at each end and secondly an 8.0 µm-wide waveguide without taper. The grating is depicted in Fig. 1a with the SEM images of the gratings (Fig. 1b) for each considered waveguide (Fig. 1c).

We optimized the geometric parameters with 2D numerical simulations using a commercial finite element based software, namely COMSOL Multiphysics, to target the best performance, around 1.55 µm wavelength on one side and around 2.0 µm on the other side. Note that an incident angle of 30° was fixed by the experimental setup constraints. The simulated source has a TM polarization for the considered embedded gratings. For the two considered spectral bands, the best set of parameters is obviously different.

Then the devices have been fabricated combining widely-spread techniques such as reactive DC magnetron sputtering to deposit titanium dioxide layers, electron beam lithography, thermal evaporation and reactive-ion etching. In this paper, we focus our attention on three fabricated devices: one waveguide with taper (Fig. 1c 1 ) for each studied wavelength and a wider waveguide without taper (Fig. 1c 2 ) for 2.0 µm spectral region. The total length of the devices is 575 µm (including the 85 µm long tapers if present). The fabricated devices have the following measured parameters, respectively at 1.55 µm and at 2 µm: a top layer thickness h top of 189 nm and 235 nm, a grating period Λ of 1375 nm and 1900 nm and a grating line width w 1 of about 610 nm and 630 nm. For the whole devices, the bottom layer thickness h bottom is 69 nm and the gold thickness h Au is 56 nm. At 1550 nm, the light source used is a supercontinuum with a spectrum ranging from 1300 nm to 1900 nm, generated a high peak power picoseconds laser, centred at 1550 nm and injected into a highly nonlinear fiber. We have measured losses as low as -7 dB per facet with a -3-dB transmission bandwidth covering the whole C-band (Fig. 2a) for the waveguide with taper, which is in fair agreement with numerical simulations.

To test waveguides at 2.0 µm wavelengths, an amplified spontaneous emission (ASE) source spanning from 1900 nm to 2050 nm is used. The total coupling losses for the single-mode waveguide (around 1.65 µm width) and the multimode slab (8.0 µm wide waveguide without taper) are represented in Fig. 2b. Note that oscillations of high amplitude and short period are particularly marked for the multimode waveguide. We can connect these spectral oscillations to a Fabry-Perot effect. The difference between the numerical simulations and the experimental results can be in a large extend assigned to discrepancies between the targeted values and the values reached at the end of the various fabrication processes. Future improvements may involve metal mirror under the gratings [START_REF] Van Laere | Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides[END_REF] or parabolic gratings [START_REF] Wang | Embedded metallic focus grating for silicon nitride waveguide with enhanced coupling and directive radiation[END_REF]. Let us however note that, as we will see in the next section, the obtained coupling efficiencies are already sufficient for the purpose of our study, i.e. to transmit an optical coded signal.

VALIDATION OF A HIGH SPEED OPTICAL TRANSMISSIONS

Transmission at 1.55 µm with commercial components

To demonstrate the availability of high speed optical transmission at conventional telecommunication wavelengths, we used enhanced small form-factor pluggable (SFP+) to transmit and receive light. These transceivers are commonly used as part of wavelength-division multiplexing (WDM) optical networks. The 10 Gbps signal is coded through a 2 31 pseudo random bit sequence (PRBS) generator. A bit error tester and a high speed sampling oscilloscope to image eye diagrams are used as diagnostic tools of the transmission quality.

Since the photonic devices are polarization-sensitive, polarization controllers are inserted before the focusers. Firstly, four SFP+ emitting at different wavelengths are tested in a single channel configuration. For SFP+ emitting at 1530 nm, 1550 nm and 1570 nm, error-free transmission are achieved with bit error rates (BER) lower than 10 -12 . For the SFP+ emitting at 1510 nm, the BER decreases to 1.6.10 -11 due to extra losses in the transmission line. Then, in order to go beyond, WDM experiments with 1550 nm and 1570 nm wavelengths have been performed in co-and in counter-propagation configurations. The experimental set-up for co-propagating configuration is illustrated in Fig. 3a. At 1550 nm, the transmission is still error-free for both configurations whereas at 1570 nm, the BER is about 4.10 -8 and 7.5.10 -7 , respectively. The difference with the single channel configuration can be explained by the presence of two optical couplers at the input and output adding 2 dB losses approximately.

Transmission at 1.98 µm

For the 2.0 µm band, SFP+ do not exist commercially. Then, we have established a more complex experimental set-up in order to demonstrate the suitability of our devices for 10-Gbps data transmissions around 2 µm. The experimental set-up is detailed in Fig. 4a. The transmitter is based on a laser diode emitting at 1.98 µm and intensity modulated by a lithium niobate modulator. The resulting coded signal is based on Non-Return-to-Zero On-Off-Keying modulation. Fig. 4b summarizes the experimental results obtained at 1.98 µm for the two considered waveguides (1.5 µm wide and 8.0 µm wide). Error-free optical transmissions have been be achieved with the TiO 2 waveguides. The eye diagram on the graph illustrates one of these transmissions. Finally, a more quantitative study of the transmission quality was made by means of BER measurements as a function of the OSNR incoming on the receiver for both devices. The results obtained in the back-to-back configuration and in presence of the waveguides are very similar. For the considered waveguide length (575 µm), the waveguide width and thus its multimode character seems to have no impact on the transmission quality.

CONCLUSIONS

We have shown for the first time to our knowledge an efficient coupling with embedded metal gratings in titanium dioxide waveguides which allowed us to transmit 10 Gbps optical signal in titanium waveguides at two different spectral regions. At 1.55µm, we have demonstrated that it was possible to use commercial integrated transceivers (SFP+) to transmit such coded signals without any error even in WDM configuration [START_REF] Lamy | Broadband etching-free metal grating couplers embedded in titanium dioxide waveguide[END_REF]. At 2 µm, we have demonstrated an error free high speed transmission for both 1.65 and 8.0 µm wide waveguides [START_REF] Lamy | Demonstration of high speed optical transmission at 2 µm in titanium dioxide waveguides[END_REF]. No significant power penalty has been observed. This study paves the way to integrated photonics at 2 µm and introduces titanium dioxide as a serious candidate for nonlinear applications in near-infrared [START_REF] Evans | Spectral broadening in anatase titanium dioxide waveguides at telecommunication and near-visible wavelengths[END_REF] and for photonics from the visible to the near-infrared.

Figure 1 .

 1 Figure 1. (a) Embedded metal grating in titanium dioxide layout (side view). (b) Electronic scanning microscope images of (b 1 ) metal grating at the end of waveguide with taper and (b 2 ) metal grating at the end of waveguide without taper (slab). (c) Optical images of (c 1 ) a waveguide with taper (around 1.65 µm-width) and (c 2 ) a waveguide without taper (8.0 µm-width) 2.2 Test of the coupling efficiency of the photonic component Before transmitting an optical signal in the titanium dioxide waveguides, we have measured their total losses. The structures under-test should have a good total coupling efficiency to enable high speed optical transmissions at 1.55 µm and 1.98 µm. Their losses are measured with a set-up composed of two optical focusers based on lensed fibers. IR and visible cameras allow us to adjust the focusers onto the grating couplers to optimize the injection and collection of the light. A powermeter and optical spectrum analyzer allow us to optimize the focusers adjustment and record the output light.At 1550 nm, the light source used is a supercontinuum with a spectrum ranging from 1300 nm to 1900 nm, generated a high peak power picoseconds laser, centred at 1550 nm and injected into a highly nonlinear fiber. We have measured losses as low as -7 dB per facet with a -3-dB transmission bandwidth covering the whole C-band (Fig.2a) for the waveguide with taper, which is in fair agreement with numerical simulations.To test waveguides at 2.0 µm wavelengths, an amplified spontaneous emission (ASE) source spanning from 1900 nm to 2050 nm is used. The total coupling losses for the single-mode waveguide (around 1.65 µm width) and the multimode slab (8.0 µm wide waveguide without taper) are represented in Fig.2b. Note that oscillations of high amplitude and short period are particularly marked for the multimode waveguide. We can connect these spectral oscillations to a Fabry-Perot effect. The difference between the numerical simulations and the experimental results can be in a large extend assigned to discrepancies between the targeted values and the values reached at the end of the various fabrication processes. Future improvements may involve metal mirror under the gratings[START_REF] Van Laere | Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides[END_REF] or parabolic gratings[START_REF] Wang | Embedded metallic focus grating for silicon nitride waveguide with enhanced coupling and directive radiation[END_REF]. Let us however note that, as we will see in the next section, the obtained coupling efficiencies are already sufficient for the purpose of our study, i.e. to transmit an optical coded signal.

Figure 2 .

 2 Figure 2. (a) Total coupling losses as a function of the injected wavelength for a waveguide optimised in the 1.55 µm spectral region. The black dashed curve is the simulation for the best set of parameters. The grey dashed curve corresponds to the simulation with the measured parameters of our fabricated waveguide. (b) Total coupling losses for a waveguide and a slab in the 2.0 µm spectral region. The curves in dashed line correspond to the total coupling losses obtained in simulations. The grey dashed line is the simulation for the best set of parameters we found. The black dashed line is the simulation with the measured parameters of our fabricated slab.

Figure 3 .

 3 Figure 3. (a) Experimental set-up for a 10 Gbps transmission in co-propagating configuration at 1550 nm and 1570 nm µm. Each SFP+ is composed of a transmitter Tx and a receiver Rx. WDM : wavelength division multiplexer. PC : polarisation controller. Eye diagrams (b) at 1550 nm and (c) at 1570 nm in counterpropagation configuration.

Figure 4 .

 4 Figure 4. (a) Experimental set-up for a 10 Gbps optical transmission at 1.98 µm. CW : continuous wave ; IM: intensity modulator ; PC : polarization controller ; TDFA : thulium doped fiber amplifier ; OVA : optical variable attenuator ; PM : powermeter ; PRBS : pseudo-random binary sequence ; OSA : optical spectrum analyzer ; BERT : bit error rate tester ; PD : photodiode. (b) BER measurements as a function of the OSNR for a waveguide and a slab at 1.98 µm. The associated eye diagram has been recorded for error-free measurement and after the transmission in the 1.5 µm wide waveguide.

ACKNOWLEDGEMENTS

This work was supported by the European Union within the framework of the operational Program FEDER-FSE Bourgogne 2014/2020, by the Région Bourgogne (Pari Photcom), by the Agence Nationale de la Recherche (Labex Action ANR-11-LABX-01-01) and by Carnot Arts Institute (PICASSO 2.0 project).


Manon Lamy

Christophe Finot

Julien Fatome

Mickael Brun

Pierre Labeye

Sergio Nicolleti

Adonis Bogris

Dimitris Syvridis

Mohammed A Ettabib

David J Richardson

Periklis Petropoulos

Kamal Hammani
email: kamal.hammani@u-bourgogne.fr

Data Transmissions at 1.98 µm in cm-long SiGe Waveguides

Keywords:

We demonstrate an error-free transmission of 10-Gbit/s optical signals along a SiGe waveguide at a wavelength of 1.98 m. Bit error rate measurements confirm the absence of penalty during the transmission through a 2.5cm long waveguide having a width of 2.2 m.

INTRODUCTION

So far, silicon photonics has attracted much attention for applications in the telecommunication C-band. However, more recently, a shift towards middle infrared (Mid-IR) wavelengths has also been proposed [START_REF] Soref | Group IV Photonics: Enabling 2 µm communications[END_REF]. Indeed, further in the Mid-IR, the nonlinear losses decrease drastically and even vanish beyond 2.2 µm, paving the way to a significant improvement of performance.

At the same time, it is now established that the usual spectral band exploited for optical telecommunications is close to reaching saturation, approaching the so-called 'capacity crunch' [START_REF] Richardson | Filling the Light Pipe[END_REF]. Therefore, the recent emergence of thulium doped fiber amplifiers (TDFA) has mainly focused the attention on the wavelengths around 2 µm [START_REF] Li | Thulium-doped fiber amplifier for optical communications at 2 µm[END_REF]. This trend has stimulated studies of dedicated photonic components such as InP-based modulators [START_REF] Sadiq | 10 Gb/s InP-based Mach-Zehnder modulator for operation at 2 μm wavelengths[END_REF]- [START_REF] Ye | InP-Based Active and Passive Components for Communication Systems at 2 μm[END_REF] or arrayed waveguide gratings [START_REF] Zhang | Dense WDM transmission at 2 μm enabled by an arrayed waveguide grating[END_REF]. High bit rate communications over short distances [START_REF] Xu | Transmission of IM/DD Signals at 2 µm Wavelength Using PAM and CAP[END_REF] but also over optical links exceeding a hundred meters have already been successfully demonstrated mainly in low-loss hollow core bandgap photonic crystal fibers [START_REF] Petrovich | First Demonstration of 2 µm Data Transmission in a Low-Loss Hollow Core Photonic Bandgap Fiber[END_REF]- [START_REF] Petrovich | Demonstration of amplified data transmission at 2 µm in a low-loss wide bandwidth hollow core photonic bandgap fiber[END_REF].

However, high bit-rate transmission at 2 µm into silicon-based waveguides still call for experiments. In this paper, we focus our attention on Silicon Germanium (SiGe) waveguides. Such components have been thoroughly studied in the C-band [START_REF] Hammani | Optical properties of silicon germanium waveguides at telecommunication wavelengths[END_REF] where their nonlinear properties have been beneficial for ultrafast optical processing [START_REF] Ettabib | PSA-based phase regeneration of DPSK signals in a silicon germanium waveguide[END_REF]- [START_REF] Ettabib | Polarization Insensitive Wavelength Conversion in a Low-Birefringence SiGe Waveguide[END_REF]. However, such waveguides are expected to reach even better performance at mid-IR wavelengths [START_REF] Brun | Low loss SiGe graded index waveguides for mid-IR applications[END_REF]- [START_REF] Barritault | Mlines characterization of the refractive index profile of SiGe gradient waveguides at 2.15 µm[END_REF]. Even though they have been used to convert a signal from mid-IR to C-band [START_REF] Hammani | Towards nonlinear conversion from mid-to near-infrared wavelengths using Silicon Germanium waveguides[END_REF], no high bit-rate transmission has been reported so far. Here, we demonstrate, for the first time, an error-free transmission of a 10-Gbit/s on-off keying signal at 1.98 µm in a 2.5-cm long SiGe waveguide.

DESIGN AND PROPERTIES OF THE SI-GE WAVEGUIDE

The device consists of several epi-layers of Si 1-x Ge x grown on a Si substrate. First, a 1.4-µm thick SiGe layer with 20 % germanium concentration (x = 0.2) was grown by reduced pressure chemical vapor deposition (RP-CVD) to control precisely the Germanium concentration and thus preserve uniformity. Then, standard photolithography and deep reactive ion etching were used to form the strips (see Fig. 1a). Finally, the waveguides were encapsulated with a 12-µm Si cladding layer epitaxially grown with the same RP-CVD technique.

In the following, we focus our attention on a 2.2-µm wide strip waveguide with 5-µm wide input and output tapers. The total length of the strip is 2.5 cm given each taper is 0.5 mm long. The refractive index of the SiGe layer is about 3.54 at 1.98 µm [START_REF] Hon | The third-order nonlinear optical coefficients of Si, Ge, and Si1-xGex in the midwave and longwave infrared[END_REF]- [START_REF] Humlicek | Infrared refractive-index of germanium-silicon alloy crystals[END_REF] whereas the silicon has an index close to 3.45. This leads to a predominant mode shown in Fig. 1b and being TE 0,0 with an effective index of 3.50.

Butt-coupling is implemented to inject and collect the light into the 2.2 µm SiGe waveguide by means of lensed fibers. First-of-all, the experimental spectral transmission of the component has been measured thanks to an amplified spontaneous source (ASE) and is reported in Fig. 1c. The average total fiber-to-fiber losses are close to 26 dB. We can also observe the oscillatory nature of the transmission function. Such a wavelength dependence is attributed to the multimode nature of the waveguide. More precisely, the waveguide is not single-mode at these wavelengths and can handle two TE modes and two TM modes with almost identical effective indices.

EXPERIMENTAL SETUP AT 2 m

In order to demonstrate the suitability of SiGe waveguides for high speed optical transmissions around 2.0 µm, we have implemented the experimental set-up depicted in Fig. 2. The transmitter is based on a laser diode (CW) centered at a wavelength of 1.98 µm and intensity-modulated by means of a lithium-niobate modulator. The transmitted signal corresponds to a 2 31 -1 pseudo-random bit sequence (PRBS) at 10 Gbit/s encoded by a Non-Return-to-Zero On-Off-Keying modulation format. A polarization controller (PC) is inserted after the intensity modulator so as to mitigate the polarization sensitivity of the SiGe waveguide. Two optical variable attenuators (OVA) are inserted into the set-up: the first one is used to control the power injected into the device and thus adjust the optical signal-to-noise-ratio (OSNR) at the receiver, whereas the second one preserves a constant power on the photodiode (PD). A power meter and an optical spectrum analyzer (OSA) measure the power before and after the transmission, respectively. Fiber-to-fiber transmission losses of 30 dB are reached at 1.98 µm. The OSA is also used to evaluate the OSNR of the received signal. The PD can be connected either to a high speed sampling oscilloscope or to a bit error rate tester (BERT) in such a way to monitor the output eye diagrams or perform the bit error rate measurements, respectively. For the back-to-back reference configuration, the sample and the lensed fibers are replaced by an equivalent amount of losses induced by the OVA. In order to further assess the quality of the transmission, we have performed bit error rate measurements as a function of the OSNR, as summarized in Fig. 4. The results obtained for the back-to-back configuration and for the SiGe waveguide appear very similar and confirm that the transmission through the waveguide does not induce any detrimental distortions. Complementary measurements have demonstrated BER lower than 10 -12 (error free).

CONCLUSIONS

We have shown for the first time to our knowledge, the error free transmission of a 10 Gbit/s optical signal in a 2.5-cm long Silicon Germanium waveguide at a wavelength of 1.98 µm. Despite the fact that the waveguide appears slightly multimode, no significant power penalty has been recorded on the BER measurements. This experimental study confirms that silicon-based integrated photonics is a serious candidate for enabling transmission in the Mid-IR around 2 µm. Moreover, it should be noticed that such SiGe waveguides are more nonlinear in this spectral region than in the C-band [START_REF] Hon | The third-order nonlinear optical coefficients of Si, Ge, and Si1-xGex in the midwave and longwave infrared[END_REF] therefore, our results call for further studies on nonlinear processing at 2 µm.

Figure 1 .

 1 Figure 1. (a) Sketch of the waveguide before encapsulation (b) Field of the TE fundamental mode for a waveguide design similar to the experiments: the width is 2.2 µm and height 1.4 µm (c) Waveguide coupling efficiency according to the input wavelength. The dashed vertical line indicates the wavelength of our continuous wave source used in the following setup.

Figure 2 .

 2 Figure 2. Experimental set-up for the 10 Gbit/s optical transmission at 1.98 µm. CW : continuous wave ; IM: intensity modulator ; PC : polarization controller ; TDFA : thulium doped fiber amplifier ; OVA : optical variable attenuator ; PM : power meter ; PRBS : pseudo-random bit sequence ; OSA : optical spectrum analyzer ; BERT : bit error rate tester ; PD : photodiode.

4 .

 4 EXPERIMENTAL VALIDATION OF THE 10 GBIT/S TRANSMISSION AT 2 mError-free transmissions have been achieved through the 2.5-cm long SiGe waveguide. Fig.3illustrates the typical eye diagrams monitored during these experiments. Both eye diagrams recorded in the back-to-back configuration and after transmission into the waveguide appear well open and do not exhibit any visible degradation.

Figure 3 .

 3 Figure 3. Eye diagrams for (a) back-to-back configuration and (b) SiGe waveguide output.

Figure 4 .

 4 Figure 4. BER as a function of OSNR for a 2.5-cm long 2.2-m wide SiGe waveguide. The black circles are associated with the back-to-back configuration while red crosses indicate BER measurements at the output of the waveguide. The red line corresponds to the associated linear regression.

ACKNOWLEDGEMENTS

This work was supported by the European Union within the Seventh Framework Programme FP7/2007-2013 under Grant 288304 (STREP CLARITY) and within the framework of the operational Program FEDER-FSE Bourgogne 2014/2020. It was also supported by the Région Bourgogne (Pari Photcom), by the Agence Nationale de la Recherche (Labex Action ANR-11-LABX-0001-01) and by Carnot Arts Institute (PICASSO 2.0 project).